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2013 129 Eirini Spiliotopoulou Information Reliability in supply chains: the case of multiple retailers Departamento Director/es Zaragoza Logistics Center Gurbuz, Mustafa Cagri Donohue, Karen Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Eirini Spiliotopoulou INFORMATION RELIABILITY IN SUPPLY CHAINS: THE CASE OF MULTIPLE RETAILERS Director/es Zaragoza Logistics Center Gurbuz, Mustafa Cagri Donohue, Karen Tesis Doctoral Autor 2013 Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA UNIVERSIDAD DE ZARAGOZA TESIS DOCTORAL FIABILIDAD DE LA INFORMACION EN CADENAS DE SUMINISTRO: EL CASO DE LOS DISTRIBUIDORES MINORISTAS Eirini Spiliotopoulou Máster de Ingeniería en Logística y Gestión de la Cadena de Suministro, Programa Internacional de Logística MIT-Zaragoza Zaragoza Logistics Center (ZLC), Universidad de Zaragoza (España) Máster en Administración y Dirección de Empresas (MBA), Universidad de Económicas y Administración de Empresas de Atenas (Grecia) Licenciada en Investigación Operativa y Marketing, Universidad de Económicas y Administración de Empresas de Atenas (Grecia) 20 de Junio de 2013 © Eirini Spiliotopoulou. Reservados todos los derechos Autor: Dña. Eirini Spiliotopoulou, Doctorando Director de tesis: Dr. Mustafa Çagri Gürbüz Profesor de Gestión de la Cadena de Suministro, Programa Internacional de Logística MIT-Zaragoza Zaragoza Logistics Center (ZLC) Director del Zaragoza Logistics Center (ZLC): Dr. David Gonsalvez Fiabilidad de la información en cadenas de suministro: el caso de los distribuidores minoristas por Eirini Spiliotopoulou en relación con el cumplimiento parcial de los requisitos para la obtención del título de Doctor en Logística y Gestión de las Cadenas de Suministro Resumen En esta tesis doctoral abordamos el estudio relativo al intercambio de información sobre la demanda dentro de una cadena de suministro cuando las partes interactúan de una forma estratégica. Los distribuidores minoristas forman una agrupación y delegan la gestión del inventario (los pedidos y la asignación) a un planificador central benévolo (CP, por sus siglas en inglés). Cada uno de los minoristas debe enfrentarse a una demanda incierta y dispone de información privada sobre ella como consecuencia de su proximidad al mercado; nos centramos en determinar si entre los minoristas y el CP se produce un intercambio fiable de información sobre la demanda. En primer lugar estudiamos el impacto de diversos mecanismos de asignación sobre el comportamiento en materia de pedidos de los minoristas, cuando la cantidad de inventario total en el almacén central es fija. Los minoristas efectúan los pedidos después conocer de manera privada su demanda. Demostramos analíticamente que los minoristas comunicarán sus necesidades reales, es decir, sus demandas realizadas, de acuerdo a una norma de asignación uniforme pero no de acuerdo a otras normas comunes como, por ejemplo, la noma proporcional o lineal; posteriormente, estudiamos una configuración donde la cantidad de inventario agrupado no es fija, sino más bien una variable de decisión, determinada por el CP después de haber solicitado información de demanda prevista de los minoristas. Las asignación del inventario total, en este caso también, se efectúa después de conocerse las realizaciones de demanda final, pero las demandas finales son de conocimiento común. Entonces, los minoristas pueden influir su asignación solo a través de la cantidad de inventario total. Mediante modelos teóricos asociados a tácticas podemos ver que el reconocimiento de la verdad y la confianza no se encuentran en una situación de equilibrio. A continuación, en un entorno de laboratorio controlado que simula la configuración de la cadena de suministro objeto de consideración, estudiamos el impacto de a) la competencia por el inventario común y b) la incertidumbre del mercado sobre la distorsión de la información, la confianza y la eficacia de la cadena de suministro. Nuestros resultados sugieren que existe una confianza continua cuando los incentivos pecuniarios están alineados y cuando no lo están, lo que viene a desmentir los casos teóricos extremos de minoristas completamente dignos de confianza o que no son fiables en absoluto; incluso aunque la información no sea totalmente fiable, el valor de la comunicación es importante. En última instancia, estudiamos el impacto de la propiedad del inventario sobre las motivaciones de las partes implicadas de cara a compartir de manera honrada sus previsiones de demanda; los inventarios específicos tampoco inducen a decir la verdad. Comparamos los inventarios resultantes y los beneficios de acuerdo con la toma de decisiones a nivel local con información más precisa con respecto a la toma de decisiones centralizada, mediante la cual se logra la coordinación de los pedidos. Director de tesis: doctor Mustafa Cagri Gurbuz Cargo: Profesor de Gestión de la Cadena de Suministro, Zaragoza Logistics Center Lista de figuras 3-1 El calendario de eventos en la t´actica de asignaci´on del inventario . . 44 4-1 El calendario de eventos en la t´actica de intercambio de informaci´on coninventariocom´un........................... 59 4-2 (Q1 −E[QCP ]) como funci´on de la se˜nal recibida para alta variabilidad de seal, peque˜na incertidumbre de mercado y dimensiones asim´etricas dedichomercado............................. 73 4-3 Registros de los minoristas frente a las se˜nales reales de la demanda . 83 4-4 Selecci´on de inventario de los CP frente a las previsiones registradas (Q(ˆ θ)) ................................... 84 5-1 El calendario de los eventos para LMI y CPMI . . . . . . . . . . . . . 97 5-2 Beneficio total esperado como funci´on de q1yq2............ 107 5-3 qLMI i(θH)−qLMI i(θL) como funci´on de la incertidumbre del mercado . 116 13 Lista de tablas 1.1 1.1. Resumen de los captulos principales . . . . . . . . . . . . . . . . 24 3.1 Comparaci´on de diversos mecanismos de asignaci´on . . . . . . . . . . 52 4.1 Resumen de hallazgos analticos del cap´ıtulo 4 . . . . . . . . . . . . . 69 4.2 Inventario ´optimo para el minorista 1 como funci´on de θ1?θ2cuando cr=0,5................................... 70 4.3 Q1 f−QCP f,comofunci´ondeθ1yθ2, cuando cr=0,75 and cr=0,25 . . 71 4.4 Comparaci´on del inventario ´optimo para el minorista 1 y el inventario esperado en el sistema si se dice la verdad, como funci´on de θ1.... 73 4.5 Dise˜no experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.6 Resumen de estadsticas en el valor |ˆ θ−θ|............... 82 4.7 Resumen de estadsticas en el valor Q.................. 82 4.8 Definici’on de variables en ecuaciones (4.6)—(4.8) . . . . . . . . . . . 87 4.9 Impacto de la competencia y de la incertidumbre del mercado sobre el hecho de decir la verdad, la confianza y la eficacia . . . . . . . . . . . 88 5.1 Elecciones de inventario optimas de acuerdo con LMI: (qLMI i(θi)= (qiL,q iH )) ................................. 113 5.2 Elecciones de inventario optimas de acuerdo con LMI y conocimiento com´un................................... 115 5.3 Comparaci´on del inventario total y de los beneficios esperados de acuerdo con el LMI, el CPMI y el CPMI con toda la informaci´on . . . . . . . 117 15 5.4 Comparaci´on del inventario local de acuerdo con el LMI, el CPMI y el CPMI con toda la informaci´on . . . . . . . . . . . . . . . . . . . . . . 117 A.1 Resumendenotas............................. 133 C.1 Comparaci´on del inventario ´optimo para el minorista 1 y el inventario esperado en el sistema si se dice la verdad, como funci´on de θ1, cuando cr=0,75ycr=0,25 ............................ 153 C.2 El inventario que maximiza el beneficio medio del minorista, para diversos valores de theta1,frentealinventario´optimoparaelsistema, cuando la demanda es uniforme y discreta . . . . . . . . . . . . . . . 154 Conclusiones y debate En esta tesis hemos estudiado los aspectos relacionados con el intercambio de informaci´on sobre la demanda – previsiones y demanda realizada – dentro de una coalici´on asociada con la puesta en com´un de inventarios. Incluso a pesar de que el valor de la informaci´on es un tema bien estudiado en la bibliografa especializada relativa a la gesti´on de operaciones y de cadenas de suministro, normalmente se parte de la base de que el intercambio de informaci´on, cuando se produce, se realiza de una manera cre´ıble. Por otro lado, muchos fallos bien documentados en las actividades comerciales se deben a la desinformaci´on en materia de datos privados, como por ejemplo la exageraci´on de pedidos ante la expectativa de escasez del inventario o pedidos imprecisos demasiado optimistas que nunca se materializan. Centramos nuestro trabajo en determinar si el intercambio fiable de informaci´on sobre la demanda se produce entre minoristas y un planificador central ben´evolo (CP) que coordina la asignaci´on de pedidos y de inventario dentro de una coalici´on asociada con la puesta en com´un de inventarios. Los minoristas no compiten por la demanda, pero pueden competir por el inventario; cada uno de ellos cuenta con informaci´on privada sobre la demanda en su regi´on como consecuencia de la proximidad al mercado, la cual puede ser transmitida de forma fidedigna, o no, al planificador central. En primer lugar, estudiamos anal´ıticamente el impacto de diversos mecanismos de asignaci´on en lo que respecta al comportamiento de los minoristas al efectuar pedidos, una vez que se ha establecido la cantidad total de inventario en el almac´en central. Para hacer esto, en primer lugar mostramos que, cuando la asignaci´on se basa en la demanda realizada (es decir, la demanda realizada pasa a ser conocida para todas las partes interesadas), todas las normas de asignaci´on consideradas, es decir, proporcional, lineal y uniforme, resultan eficaces (excluyen p´erdidas) y con un resultado Pareto ´optimo. Sin embargo, cuando las demandas realizadas en cada regi´on permanecen en el ´ambito privado con respecto a los minoristas, la norma de asignaci´on empleada desempe˜na un importante papel; solo a trav´es de la norma de asignaci´on uniforme los minoristas podr´an informar de sus necesidades reales mediante la presentaci´on de un pedido definitivo equivalente a su demanda realizada. Este resultado es an´alogo al caso t´ıpico de racionamiento de la capacidad, incluso si en la configuraci´on que est´e consider´andose (a) la asignaci´on se determina despu´es de resolver la incertidumbre de la demanda y (b) cada minorista puede recibir por encima o por debajo de su pedido final. La diferencia fundamental es que la asignaci´on uniforme basada en los pedidos definitivos, en nuestro caso, induce a decir la verdad y resulta ´optima desde el punto de vista de Pareto. En el caso del racionamiento de la capacidad, la asignaci´on uniforme no resulta ´optima desde el mencionado punto de vista de Pareto, puesto que no es receptiva desde la perspectiva individual, lo cual constituye una condici´on necesaria para que resulte ´optima desde dicho punto de vista. Adem´as, proponemos una norma modificada de asignaci´on uniforme que no solo sea ´optima desde el punto de vista de Pareto e induzca a decir la verdad, sino que tambi´en garantice a cada minorista un beneficio superior al que habr´ıamos obtenido en un sistema puramente descentralizado de acuerdo con cualquier realizaci´on de la demanda. A continuaci`on, procedemos a efectuar un estudio anal´ıtico y experimental del intercambio de la previsi´on de la demanda entre minoristas y el CP que solicita esta informaci´on para establecer el inventario total. Las demandas realizadas pasan a ser de dominio com´un cuando se produce la asignaci´on, y nos centramos en la t´actica de registrar se˜nales de la demanda (previsi´on) para influir sobre el inventario contenido en el sistema. Valoramos la t´actica en los casos en los que se utiliza un mecanismo de asignaci´on proporcional (para la demanda realizada a nivel local): un mecanismo ampliamente utilizado en la pr´actica y con multitud de propiedades atractivas. Mediante el uso de modelos basados en la teor´ıa de juegos determinamos que, cuando existe una incertidumbre no resuelta en materia de demanda antes de la comunicaci´on, el hecho de decir la verdad y la confianza no forman un equilibrio Bayesiano Perfecto; adem´as, en un sistema de inventario automatizado que toma como datos las previsiones registradas por los minoristas y solicita el nivel de inventario ´optimo para toda la coalici´on, no existe un equilibrio Bayesiano de Nash puro entre los diversos minoristas. Posteriormente, en un entorno de laboratorio controlado que simula la configuraci´on de la cadena de suministro objeto de consideraci´on, estudiamos el impacto de a) la competencia por el inventario com´un y b) la incertidumbre del mercado sobre la distorsi´on de la informaci´on, la confianza y la eficacia de la cadena de suministro. Nuestros resultados sugieren que existe una confianza continua cuando los incentivos pecuniarios est`an alineados y cuando no lo est´an, lo que viene a desmentir los casos te´oricos extremos de minoristas completamente dignos de confianza o que no son fiables en absoluto. Adem´as, determinamos que tanto la competencia por el inventario com´un como la incertidumbre en materia de previsiones da˜nan notablemente tanto el hecho de decir la verdad como la cooperaci´on entre las diversas partes de la cadena de suministro. Ese es el motivo por el que la acumulaci`on de inventario con arreglo a una asimetr´ıa de la informaci´on puede tener resultados negativos, a pesar de la agregaci´on de riesgos de la demanda. Incluso aunque la informaci´on no fuera ´ıntegramente fiable, el valor de la comunicaci´on era significativo en todos nuestros experimentos. En ´ultimo lugar, estudiamos el impacto de la propiedad del inventario sobre los alicientes de las partes interesadas para compartir sus previsiones de forma fidedigna. Cuando valoramos dos ubicaciones independientes que deciden a nivel local sobre su nivel de inventarios y toman en consideraci´on la posibilidad de transferir la propiedad del inventario despu´es de realizar las demandas, existe un equilibrio Bayesiano de Nash con caracter´ısticas ´unicas. La elecci´on ´optima de un inventario en un emplazamiento se incrementa en su se˜nal de demanda recibida; cuando el CP adopta la decisi´on de efectuar el pedido, lo que importa es ´unicamente la cantidad total de inventario. A menos que concurran algunas condiciones especiales de conservaci´on, ´este no podr´a ser separado entre los dos emplazamientos antes de que se realice la demanda, de forma que los incentivos locales y del sistema queden alineados. Procedemos a comparar en t´erminos num´ericos los inventarios y los beneficios resultantes en condiciones de toma de decisiones a nivel local con informaci´on m´as precisa frente a la toma de decisiones centrales en las que se consigue la coordinaci´on de pedidos. Vemos que, cuando el fractil fundamental es alto, la toma de decisiones a nivel central desemboca en inventarios m´as elevados, mientras que cuando el fractil fundamental es bajo se cumple la circunstancia contraria. Las comparaciones direccionales de beneficios esperados dependen del valor de la informaci´on local que se pierde cuando pasamos a la toma de decisiones a nivel central frente al valor adicional de la coordinaci´on del inventario (en el equilibrio babbling o equilibrio no informativo). Este trabajo tiene varias limitaciones, como consecuencia de la complejidad analtica del problema. Estudiamos por separado la estrategia de asignaci´on de inventarios cuando el planificador central no conoce las demandas finales y aquella relacionada con el intercambio de informaci´on sobre la previsi´on de la demanda para influir sobre el nivel de inventario de la coalici´on. El hecho de saber cu´ales ser´ıan las interacciones cuando ambos temas se valoraran de manera conjunta sigue siendo una pregunta abierta. Por ejemplo, si la asignaci´on final estuviera vinculada a la previsi´on registrada, ¿c´omo cambiar´ıan las din`amicas del intercambio de informaci´on sobre previsiones? ¿Cu´al ser´ıa el impacto sobre la actitud de los minoristas a la hora de efectuar pedidos? ¿Ser´ıa eficaz la asignaci´on final? Hay muchas ampliaciones interesantes de esta tesis doctoral; para empezar, podramos analizar con mayor detalle el impacto de los factores del comportamiento en la estrategia de compartir informaci´on relativa a las previsiones de la demanda. Cuando el inventario es com´un, es interesante investigar c´omo el tama˜no de la coalici´on y el de la demanda media los minoristas afecta relaciones de confianza. Adem´as, quisi´eramos estudiar si el nivel de confiabilidad de los minoristas cambia cuando hay una garant´ıa de c´omo se utilizan sus previsiones de la demanda para ajustar el nivel de inventario com´un. Consideramos esto una pregunta de investigaci´on interesante con implicaciones gerenciales potencialmente muy relevantes. Como investigaci´on futura, estamos planeando ejecutar experimentos adicionales cuando el n`umero de minoristas aumenta a 3 y 4, el CP est´a automatizado y los minoristas no son id´enticos. Ser´ıa interesante estudiar, desde un punto de vista experimental, c´omo la propiedad sobre el inventario tiene un impacto sobre la estrategia de registro de las previsiones por parte de los minoristas. Incluso aunque los inventarios espec´ıficos no alinearan los alicientes individuales y los pecuniarios del sistema, ¿la incertidumbre reducida en relaci´on con la asignaci`on final incrementar´ıa la confianza de los minoristas y mejorara la cooperaci´on? Un segundo tema de inter´es es estudiar, mediante experimentos asociados al comportamiento, el impacto de los mecanismos de asignaci`on en las actitudes de formulaci´on de pedidos de los minoristas, para as´ı arrojar luz sobre los componentes que, potencialmente, pudieran “faltar” en esta interacci´on. Los conceptos de equilibrio, que parten de la base de que las partes interesadas son perfectamente racionales, ¿exageran notablemente la tendencia de los minoristas a pedir m´as / menos de que lo que necesitan? ¿En virtud de qu´e mecanismos de asignaci´on es m´as pronunciada la distorsi´on en los pedidos? Otra v´ıa de investigaci´on futura es estudiar el intercambio de informaciones sobre la demanda en una coalici´on para la puesta en com´un de inventarios y centrarse en las implicaciones conductuales a la hora de presentar pedidos imprecisos (no vinculantes) frente al intercambio de predicciones sobre la demanda. Una vez m´as, partimos de la base de que todos los minoristas tienen una mejor informaci´on sobre la demanda gracias a su proximidad al mercado, y que la comparten (quiz´a de manera falsa) con el CP, ya sea en forma de intercambio de previsiones (enviando su se˜nal de demanda) o de una orden no vinculante antes de que la demanda se realizara. ¿C´omo se comparan los niveles de inventario y los beneficios con arreglo a estas dos formas diferentes de intercambio de informaci´on? Una configuraci´on de cadena de suministro distinta donde el intercambio de las previsiones de la demanda desempe˜na un papel fundamental es la de proveedor – fabricantes / minoristas. Cuando el proveedor es una unidad de negocio independiente con su propio margen de beneficio, ¿c´omo cambia la din`amica de la comunicaci´on? El hecho de saber cu´al ser´ıa el impacto de confianza y confiabilidad sobre el intercambio de informaci´on estrat´egica, el nivel inventario (o capacidad) y / o la asignaci´on, en una configuraci´on de proveedor ´unico – m´ultiples minoristas, sigue siendo una pregunta abierta. Otra ampliaci´on interesante para trabajos futuros es estudiar, tanto a nivel anal´ıtico como experimental, las din´amicas de comunicaci´on y de intercambio de informaci´on (demanda final o previsiones) en problemas multiper´ıodo. Cuando se toman en consideraci´on las interacciones repetidas, ¿en virtud de qu´e condiciones pueden formar un equilibrio sostenible la confianza y el hecho de decir la verdad? Los contratos de relaci´on (p. ej., basados en estrategias de desencadenamiento), ¿inducen a la colaboracin? En estrategias multiperodo, muchos aspectos adicionales pueden desempe˜nar tambi`en un papel importante, como por ejemplo las opiniones y el aprendizaje, las consideraciones relativas a la reputaci´on, la posibilidad de sancionar un comportamiento enga˜noso o la confianza en las relaciones a largo plazo. Author.............................................................. Eirini Spiliotopoulou, PhD Candidate MIT-Zaragoza International Logistics Program Thesis Supervisor.................................................... Dr. Mustafa Cagri Gurbuz Professor of Supply Chain Management, Zaragoza Logistics Center Thesis Supervisor.................................................... Dr. Karen Donohue Associate Professor of Supply Chain and Operations, Carlson School of Management, University of Minnesota Director, Ph.D. Program ............................................ Dr. Maria Jesus Saenz Professor of Supply Chain Management, Zaragoza Logistics Center 8 Contents 1 Introduction 17 1.1 Motivation................................. 17 1.2 Commonsetting.............................. 18 1.3 Researchquestions ............................ 21 2 Literature Review 25 2.1 Inventory pooling decisions . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Capacity choice and allocation . . . . . . . . . . . . . . . . . . . . . . 30 2.3 Randomyield............................... 32 2.4 Central versus local decision making . . . . . . . . . . . . . . . . . . 34 2.5 Demand forecast information sharing . . . . . . . . . . . . . . . . . . 35 3 Impact of Inventory Allocation Mechanisms on Demand Information Sharing 39 3.1 Introduction................................ 39 3.2 Setting and timing of events . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Properties of Pareto optimal allocation mechanisms . . . . . . . . . . 44 3.4 Allocation with symmetric information . . . . . . . . . . . . . . . . . 46 3.5 Allocation under asymmetric information . . . . . . . . . . . . . . . . 50 3.6 Concludingremarks............................ 53 4 Forecast Information Sharing and the Order Quantity Decision: Impact of Inventory Competition and Market Uncertainty 55 9 4.1 Introduction................................ 55 4.2 The analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.1 Setting and timing of events . . . . . . . . . . . . . . . . . . . 58 4.2.2 The information sharing game for n=1 ............ 60 4.2.3 The information sharing and allocation game for n>1 .... 61 4.3 Numericalstudy.............................. 69 4.3.1 Common knowledge . . . . . . . . . . . . . . . . . . . . . . . 70 4.3.2 Asymmetric information . . . . . . . . . . . . . . . . . . . . . 71 4.4 Hypotheses ................................ 74 4.5 Experiments................................ 76 4.5.1 Experimental design and procedures . . . . . . . . . . . . . . 76 4.5.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 81 4.6 Concludingremarks............................ 90 5 Forecast Information Sharing and the Order Quantity Decision: Impact of Inventory Ownership 93 5.1 Introduction................................ 93 5.2 Thesetting ................................ 95 5.3 Locally Managed Inventory (LMI) . . . . . . . . . . . . . . . . . . . . 97 5.4 Central Planner Managed Inventory (CPMI) . . . . . . . . . . . . . . 104 5.4.1 Is truth-telling an equilibrium? . . . . . . . . . . . . . . . . . 108 5.5 Numericalanalysis ............................ 111 5.6 Coordinating transfer prices . . . . . . . . . . . . . . . . . . . . . . . 119 5.7 Concludingremarks............................ 122 6 Conclusions and Discussion 125 A Notation 131 B Proofs 135 B.1 Proofsofchapter3 ............................ 135 B.2 Proofsofchapter4 ............................ 139 10 B.3 Proofsofchapter5 ............................ 147 C Additional Numerical Analysis 153 D Experiment Instructions 155 D.1 Handout to regional managers: Case R2DU............... 155 D.2 Handout to central planners: Case R2DU................ 159 D.3 Summarysheets.............................. 163 E Snapshots from the Experiment Software 165 E.1 Snapshots of a retailer’s computer screen: Case R2DU......... 165 E.2 Snapshots of a central planner’s computer screen: Case R2DU.... 168 11 12 List of Figures 3-1 The timing of events in the inventory allocation game . . . . . . . . . 43 4-1 The timing of events in the information sharing game with common inventory.................................. 59 4-2 (Q1−E[QCP ]) as a function of the signal received, for high signal variability, small market uncertainty and asymmetric market sizes . . 73 4-3 Retailers’ reported versus actual demand signal . . . . . . . . . . . . 83 4-4 CP’s inventory choice versus reported forecasts (Q(ˆ θ))......... 84 5-1 The timing of events in the information sharing game with common inventory.................................. 97 5-2 Total Expected Profit as a function of q1and q2............ 107 5-3 qLMI i(θH)−qLMI i(θL)asafunctionofmarketuncertainty . . . . . . . 116 13 14 List of Tables 1.1 Summary of main chapters . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 Comparison of various allocation mechanisms . . . . . . . . . . . . . 52 4.1 Summary of analytical findings of Chapter 4 . . . . . . . . . . . . . . 69 4.2 Optimal inventory for retailer 1 as a function of θ1and θ2when cr=0.5 70 4.3 Q1 f−QCP f,asafunctionofθ1and θ2, for cr=0.75 and cr=0.25 . . . . 71 4.4 Comparison of the optimal inventory for retailer 1 and the expected inventory in the system under truth-telling, as a function of θ1.... 73 4.5 ExperimentalDesign ........................... 78 4.6 Summary Statistics on Reported |ˆ θ−θ|................. 82 4.7 Summary Statistics on Total Order Quantity Q............ 82 4.8 Variable Definition in Equations (4.6)—(4.8) . . . . . . . . . . . . . . 87 4.9 Impact of competition and market uncertainty on truth-telling, trusting and efficiency ............................. 88 5.1 Optimal inventory choices under LMI: (qLMI i(θi)=(qiL,q iH ))..... 113 5.2 Optimal inventory choices under LMI and common knowledge . . . . 115 5.3 Comparison of total inventory and expected profits under LMI, CPMI and CPMI with full information . . . . . . . . . . . . . . . . . . . . . 116 5.4 Comparison of local inventory under LMI, CPMI and CPMI with full information ................................ 117 A.1 Summaryofnotation........................... 133 15 C.1 Comparison of the optimal inventory for retailer 1 and the expected inventory in the system under truth-telling, as a function of θ1,for cr=0.75andcr=0.25 ........................... 153 C.2 The inventory that maximizes average profit for retailer 1, for various values of θ1,andthesystemoptimalinventorylevelfordiscreteuniform demand .................................. 154 16 impact of behavioral factors, such as trust and trustworthiness on the communication. In this work we define trust as the CP’s willingness to rely on retailers’ forecasts to determine total inventory. Trustworthiness is measured by the difference between a retailer’s real forecast and the one he reports to the CP. We also study, both analytically and experimentally, the role of supply chain environment (i.e., number of retailers, market uncertainty) on the result of the communication. The main research questions are: (a) Do retailers truthfully transmit their demand forecasts (private information) to the central planner when they compete for inventory? (b) Does the CP incorporate retailers’ transmitted forecast information in her inventory decision (i.e., trust the shared demand information)? (c) How are communication dynamics affected by the supply chain environment, i.e., number of retailers, level of forecast uncertainty, retailers’ trust in the CPs inventory decision process? In chapter 5, we continue studying the demand forecast sharing game between locations and the central planner, keeping the same information structure. In contrast to chapter 4 where the inventory held centrally is common, in this chapter we examine the impact of inventory ownership on individual players’ incentives. We study the quality of communication when each region has a dedicated inventory quantity and unilateral inventory reallocations are allowed after demands are realized. Unilateral transshipments (or reallocations of centrally held inventories) is a common practice in many industries such as OEMs [46] and pharmaceuticals. We first study the impact of inventory decision rights placement (locally versus centrally) on the resulting inventory levels and profits. We then examine whether inventory ownership may align retailers’ and system’s incentives so that, in the case of central decision making, truthful forecast information sharing can be expected. The main research questions are: (a) What is the impact of inventory ownership on the reliability of information transmitted? To be more specific, do the dynamics of information sharing change when each retailer has a dedicated inventory quantity held centrally? (b) What are the resulting inventory levels when inventory decision rights are placed locally (i.e., to retailers) versus centrally (i.e., the right to decide on inventories is transferred to 23 the central planner)? Table 1.1 summarizes the settings we consider in the following chapters, comparing them along the dimensions of decisions studied and information transmitted. The pooling coalition (retailers and the central planner) is the system we are considering. For example, when we refer to “endogenous” inventory quantity, we mean that it is a decision variable for either retailers or the central planner. Before presenting the main chapters of the thesis, in the next chapter we review the related literature, mainly within the field of operations and supply chain management but not exclusively, and we position our research within each research stream. Inventory Quantity Allocation Information Asymmetry Chapter 3 Exogenous Various rules Final demands Chapter 4 Endogenous Proportional Demand forecasts (signals) Chapter 5 Endogenous Dedicated Inventories Demand forecasts (signals) w/ reallocation possibility Table 1.1: Summary of main chapters 24 Chapter 2 Literature Review In this chapter, we review the streams of literature this thesis is mainly related to, considering both analytical and experimental papers. We first review the literature related to inventory pooling decisions and associated behavioral issues. We next examine related literature on capacity choice and allocation, random yield, central versus local decision making and demand forecast information sharing, focusing on the issue of information credibility. We comment on how our work fits into each of these fields and why it differs from previous research. 2.1 Inventory pooling decisions Benefits of inventory pooling Pooling, the strategy of aggregating demand streams, has long been studied under two main contexts: reducing product variety (i.e., through SKU rationalization) or geographical variety (i.e. inventory centralization). A third dimension of aggregation is that of demand aggregation across time (i.e., order consolidation). Reducing variety will generally allow a company to provide the same customer service level at a lower cost, or to improve it without incurring extra costs. Eppen (1979) [18] in his seminal work showed that total holding and stockout costs are lower when demand is aggregated, under the assumption of independent and normal demand distributions 25 and identical cost parameters (one-period setting). Eppen and Schrage (1981) [19] extend the result to the multiple period problem while Corbett and Rajaram (2006) [15] generalize Eppen’s model to (almost) arbitrary multivariate dependent demand distributions. In this thesis we study the benefits of pooling when there is information asymmetry. We assume that a central planner has less information about the demand distributions than that of the decision makers in the decentralized case (due to her distance from the markets). To the best of our knowledge this is the first attempt to incorporate information asymmetry in the problem of inventory pooling. Sharing the benefits of inventory pooling Another stream of papers studies how the benefits of inventory pooling should be shared among retailers, so that the pooling coalition is stable, in the sense that all participating retailers have higher expected profits. Hartman et al. (2000) [26] employ cooperative game theory to prove the existence of a cost allocation scheme so that all retailers are better off(incur lower costs) by pooling their inventories together, under the assumption of symmetric and independent demand distributions, identical overage and underage costs, or a multivariate normal demand distribution. In other words, they show that, under these assumptions, the core of the inventory centralization game is non-empty, implying stability of the system. They do not study which cost allocation schemes or mechanisms of inventory rationing are on the core of the game. Muller et al. (2002) [40] extend this result for all possible joint distributions of random demands and give sufficient conditions under which the core is a singleton and conditions under which at a core allocation every newsvendor shares a nonnegative cost. Gerchak and Gupta (1991) [22] study whether “popular” cost allocation methods, namely cost allocation by demand volume, by individual safety stock requirements, by incremental contribution to joint costs, and proportional to stand-alone costs, are potentially unfair (in the sense that a retailer may be charged more under consolidation than his cost would be under a dedicated inventory system). 26 Only apportioning costs according to each retailer’s stand-alone costs is found to be “fair” in centralized continuous review inventory systems. Robinson (1993) [45] uses cooperative game theory and, through a counterexample, proves that this basis of allocation is not always in the core of the game; some customers may be worse off when new customers join up. He uses the Shapley value allocation rule to determine “fair” cost allocations in which no one is worse offafter consolidation [48]. This allocation ensures that each customer is charged somewhere between their incremental and stand-alone costs. Kemahlioglu-Ziya and Bartholdi (2011) [31] study a two-stage supply chain consisting of a supplier and multiple retailers and find that if savings of centralized inventory are allocated among supply chain members by Shapley value, the pooling coalition is farsightedly stable. This allocation may not always belong to the core of the centralization game if retailers are not identical but it always coordinates the supply chain and distributes profits in a “fair” way. They move one step further by identifying a quantity allocation rule (a modified linear allocation) of shared inventory where players’ expected after-pooling profits are equal to their Shapley value allocations plus their before-pooling profits and the supplier carries the supply-chain-optimal level of inventory. In this stream of research, cost or profit allocation is based on full information. In our work we abstract from the issue of excess profit allocation by assuming that all retailers join the inventory pooling mechanism. This is representative of situations where retail stores belong to the same parent company or are otherwise constraint to accept a pooling policy. In other words, we study the information sharing dynamics within a coalition, after its synthesis has been set. In chapter 3, where different allocation mechanisms are compared, we identify one that guarantees each individual retailer profit’s is at least as high as in the decentralized system. 27 Behavioral issues in inventory pooling decisions It has long been acknowledged in the decision making (i.e. Tversky and Kahneman, 1974 [56]; Thaler, 1980 [55]) and economics literature (Mullainathan and Thaler, 2000) [39] that decision makers are not perfectly rational expected profit maximizers. Their decisions may systematically deviate from optimality for a variety of reasons. For example, people exhibit systematic biases in their judgments and rely on a limited number of heuristic principles to assess uncertain situations [56], they have limited cognitive abilities and bounded self-interest [39], they may have other social considerations, e.g., fairness, reputation, social norms [14]. In the field of operations and supply chain management, there has been a growing interest in behavioral research. Behavioral research studies the effects of human behavior in processes and performance by incorporating social and cognitive psychology considerations [17]. For a comprehensive review of this stream, please see Bendoly et al. 2010 [3] and Donohue and Siemsen 2010 [17]. We continue with reviewing behavioral literature that is specifically related to inventory and pooling decisions. There is a vast literature on how people make newsvendor problem decisions [4, 5, 7, 47]. We focus on multi-player newsvendor settings. Su (2008) [53] builds a decision model based on the quantal choice model (best decision need not always be made but better decisions are made more often) and applies it to the newsvendor setting. He predicts that when inventory for multiple locations is held centrally, apart from the benefits associated with variance reductions, behavioral benefits also exist as inventory centralization helps by pooling decision errors across locations (“supply uncertainty” pooling). Lavaro and Corbett (2003) [33] study analytically and through simulations the pooling effect in the context of SKU rationalization when inventory policies are suboptimal and demand is non-normal (single planning period). They find that the value of pooling may be negative when the inventory policy in use is suboptimal while it varies little across the distributions they studied. More recently, Ho et al. (2010) [28] study experimentally the ordering behavior in multilocation 28 inventory systems and they show that systematic biases eliminate the risk-pooling benefit when the demand across stores is strongly correlated. They propose a behavioral theory based on reference dependence (psychological aversion to leftovers is greater than the disutility of stockouts) to explain / predict ordering behavior in a multi-location newsvendor framework, under both a centralized and a decentralized inventory structure. Kremer (2007) [32] studies experimentally the impact of secondary markets (opportunity of inventory rebalancing after demands are realized) on newsvendor’s ordering decisions. Inventory reallocations are possible, at exogenous or endogenous market prices determined through a market-clearing auction mechanism, allowing for demand risk pooling. He finds that the option to trade units in the secondary market increases supply chain profitability in all cases. More interestingly, it also has a beneficial impact on inventory decisions ahead of the selling season as it induces average order quantities to regress towards system-optimal levels. Behavioral issues related to transshipment as a pooling strategy are studied as well by Bostian et al. (2012) [6]. Unlike Kremer’s work, transshipment decisions are considered automatic (both quantities and prices) and the main focus is on whether behavioral bias in ordering nullify the risk-pooling benefit of transshipments. They find that transshipment is a behaviorally robust risk-pooling technique in the sense that it gives an even greater benefit in the presence of behavioral biases than in the absence of these biases. Our work considers behavioral factors in information sharing within an inventory pooling coalition. On the one hand, we study the role of trust and trustworthiness in a pooling coalition where information asymmetry among players is present. We compare the predictions of analytical models based solely on pecuniary payoffs to the results of controlled laboratory experiments with human subjects. On the other hand, we show that under information asymmetry and when behavioral factors are taken into consideration, the benefit of inventory pooling may, in practice, be inexistent or even negative, even when demands across locations are independent. 29 2.2 Capacity choice and allocation The framework employed to study the dynamics of a centralized inventory system when retailers strategically interact closely parallels that of capacity choice and allocation in a single supplier, multiple retailer (or manufacturer) context. One main characteristic of the capacity choice and allocation literature is that it employs a two-stage supply chain in contrast to the classical inventory pooling literature where retailers decide to collude among themselves (i.e. forming an aggregate retailer) to increase total profits. Cachon and Lariviere (1999, 1999 b) [8,10] study how the capacity allocation scheme affects retailers’ orders and supply chain performance when the supplier has limited capacity and the retailers have private information about their optimal stocking levels. They find that under any Pareto allocation mechanism (e.g., proportional or linear), all retailers truthfully reporting their optimal allocations is not a dominant Bayesian equilibrium and that a manipulable mechanism (not truth revealing) may lead to higher capacity and higher profits for everyone. Our work differs in that “the central planner”, who decides about the inventory (rather than capacity) to be held centrally, is a total system profit maximizer. Cachon and Lariviere (1999 a) [9], in a two period setting and with two retailers, find that linking a retailer’s current allocation to his previous sales rate (“turn and earn” allocation) does not generally coordinate the system. Retailers sell more but in equilibrium no one gains an advantage. Lu et al. (2010) [37] extend this work to an infinite horizon game with multiple retailers and find a richer set of equilibria. “Turn and earn” allocation may reduce demand variability placed on the supplier as retailers absorb local demand fluctuations. Compared to this stream of literature, the timing of the allocation in our work, and therefore the information it is based on, differs. In the capacity rationing literature, manufacturers usually order from a capacitated supplier a quantity at the start of the selling season, before their demands are revealed. In the pooling coalition case considered in this thesis, total quantity is procured from an uncapacitated external supplier at the beginning of the selling season and it is held centrally until regional 30 demands are realized. Hence, the final allocation happens after demand uncertainty is resolved because of the different supply chain tier considered. In our work, allocation by the central planner is based on realized demands if they are known (chapters 5 and 6) or on final orders that are placed by the retailers after they see their final demands (chapter 3). In chapter 3, we consider inventory allocation mechanisms that are similar to the capacity allocation mechanisms studied in these papers (modified accordingly to represent our setting), while in chapters 4 and 5 we take the allocation rule as given and we focus instead on the issue of optimal central inventory choice for the coalition. Li et al. (2011) [35] consider situations where firms buy options to use the capacity of a supplier. They explore whether the supplier benefits from providing transfer rights with these options such that a firm that cannot use all its purchased capacity can sell it to another firm that may need it. They examine what happens when a buyer requires more or less than her reserved capacity under three policies: a) individual reservations are final and excess demand at any buyer is lost, b) any unused capacity by each buyer returns to the supplier and he has the right to sell it to another buyer if there is demand for it and c) each buyer owns her reserved capacity and can resell any unused capacity to another buyer after demands are realized. They find that providing transfer rights to the buyers can be better for the supplier than reserving the transfer rights to itself in a wide variety of situations because the buyers value their reserved capacity more which in turn allows the supplier to charge higher reservation prices. Their setting parallels ours, mainly the one considered in Chapter 5, in the sense that they are comparing final “individual capacity reservations” to “pooled” capacity with minimum guarantees (allowing “capacity transshipments”) and they are focusing on the issue of who has the right to transfer unused capacity. In the case where the supplier owns the property rights to reserved capacity, and both buyers need capacity above their reserved quantities, the uniform capacity allocation rule is employed in order to exclude retailers’ strategic ordering and to keep the model tractable. We are considering a central total system profit maximizer instead and we are focusing on 31 the gaming between the players in the presence of information asymmetry. The only work, to the best of our knowledge, that explicitly models and experimentally estimates behavioral factors in an allocation game, is that of Chen et al. (2012) [13]. They consider a setting of complete information, capacity shortage, known demands and proportional allocation. They find that retailers do not order that much more than what they need, as game theory would predict as a result of their strategic interaction. They propose a model of bounded rationality, based on the quantal response equilibrium, to explain the observed ordering behavior in the lab. For our experimental work, we base the allocation of total inventory on realized demands (instead of orders placed) and we focus on the information sharing game between the retailers and the central planner who sets the inventory quantity. 2.3 Random yield In our setting, the final allocation that a retailer or a region gets is random because it is based either on realized demands at two or more locations (chapters 4 and 5) or on retailers’ final orders that, in turn, depend on random demand realizations (chapter 3). The allocation that each retailer receives can be thought of as a random yield of the total inventory quantity and hence our work is also related to the random yield literature. The yield factor will depend on the allocation rule employed. Analytical models of determining lot sizes when production or procurement yields are random can be classified in two main categories: single-stage continuous time review models where demand is constant or random and periodic review (discrete time) models for single or multiple production stages and single or multiple periods (known or random demand). Yano and Lee (1995) [59] provide a comprehensive review on quantitatively-oriented approaches for determining lot sizing with random yields. Our setting is closer to the discrete time, single period, single run model studied first by Shih (1980) [49]. He focuses on the case where yield uncertainty is caused by defective units and he shows that when the distribution of the % of defectives is 32 Chapter 3 Impact of Inventory Allocation Mechanisms on Demand Information Sharing 3.1 Introduction When a number of retailers form an inventory pooling coalition, the issue of how inventory is allocated after demands are realized becomes very important. In the absence of a pricing mechanism that would balance supply and demand, inventory rationing through quantity limits (upper or lower) in case of shortage or surplus are necessary. In such cases though, retailers may have an incentive to misreport their needs in order to gain a more favorable allocation. Allocation, or rationing of limited common inventory, is an ongoing issue in many industries where there is only one opportunity for production or procurement, before the start of the selling season. For example, allocation mechanisms have been employed in the fashion apparel, consumer electronics and, automotive industries [8,9,38]. The issue of capacity rationing in a supply chain with one supplier selling to multiple retailers is a phenomenon well-studied in the literature [8–11]. When several retailers compete for limited capacity, a broad class of allocation mechanisms are prone to 39 manipulation. When inventory is limited, prior analytical research has shown that retailers may order more than what they need to gain a higher allocation [10]. Our setting differs from this prior research in three ways: (a) the manager of the pooling coalition (CP) is interested in maximizing total supply chain profits (she is not a seperate business entity trying to maximize her own profits), (b) when allocation takes place, retailers already know their realized demands (not their optimal stocking levels under demand uncertainty), and (c) retailers are responsible for both understocking and over-stocking costs of the coalition. In contrast to the case of capacity rationing, total inventory of the coalition is held in the central warehouse until local demands are realized. After demands are realized and final orders are placed to the central warehouse, retailers may receive less or more than their final order. Retailers collectively assume all demand uncertainty risk, as the central planner is not a separate unit with its own financial objectives. In this chapter, we abstract from the issue of how the quantity in the central warehouse is set (i.e. the level of inventory held at the CP is for now assumed to be given) and study how retailers pass orders under different allocation mechanisms. We study three allocation mechanisms that are commonly used in practice and analyzed in the literature. These include proportional, linear and uniform. Perhaps the most intuitive mechanism is the proportional allocation where each retailer receives a proportional amount of his announced demand (or order when demands are not common knowledge). Linear allocation gives each retailer his final demand (or order) plus / minus a common quantity when there is inventory surplus / shortage. Under uniform allocation, each retailer gets the same quantity, under some conditions. No retailer gets more than what he asks for when total orders exceed total inventory and no retailer gets less than what he asks when the reverse is true. Under common knowledge, these conditions guarantee that there are no unsold units when there is inventory shortage and that there is no unmet demand when there is inventory surplus. Furthermore, we introduce a new allocation rule, the modified uniform allocation rule, which has some attractive properties in this setting. This allocation mechanism connects retail40 ers’ initial (soft) orders to the final allocation they receive after demands have been realized. The research questions we address in this chapter include: (a) Which allocation mechanisms are Pareto optimal under common knowledge of realized demands? (b) When final demands remain retailers’ private knowledge, under which allocation rules, if any, will retailers truthfully report these to the central planner (who allocates inventory)? (i.e. place a final order equal to their realized demand) (c) Do Pareto optimal allocations that induce truthful demand telling exist? The rest of the chapter is organized as follows: we start by further defining the supply chain setting we are studying and the timing of events. Then, we identify properties that Pareto optimal allocation mechanisms have in this setting. Next, we introduce the various allocation mechanisms we are considering in this chapter. We first study the case of symmetric information (the CP knows the realized demands when she allocates inventory) which serves as a benchmark. Then, we analyze the asymmetric information case. Each retailer knows his realized demand (and only his) while the CP does not see actual demands but only the orders retailers place. 3.2 Setting and timing of events Anumberofretailersnhave decided to form a pooling coalition to better satisfy their uncertain demand. At the beginning of the season, each retailer receives a signal θiabout the demand at his region (i=1...n). Demand at each region is given by di=µi+θi+�i,whereµiis a positive constant representing the average demand at location iand θiis retailer’s iprivate information about demand, a zeromean random variable with cumulative distribution function Fi(·), probability density function fi(·)andsupport[θi,θi] (adopted from Ozer, 2011 [42]). Market uncertainty is represented by �i, a zero-mean random variable with cdf Gi(·), probability density function gi(·)andsupport[�i,�i]. Both retailer iand the central planner know µiand Gi(·). Retailer ialso knows at the beginning of each selling season the realization of θi, 41 while the central planner knows only its distribution Fi(·). We assume that signal and market uncertainty are independent across regions and that within a region market uncertainty is independent of the signal received (i.e., Cov(�i,� j)=0,Cov(θi,θ j)=0 ∀i, j =1,2...n,i�=jand Cov(�i,θ j)=0∀i, j =1,2...n). Cost parameters are also common knowledge to all players. Each retailer receives pfor each unit he sells and pays cfor each unit he receives from the central warehouse. The salvage value of the product at the end of the selling season is normalized to zero. In this chapter, we focus on the information transmission and retailers’ incentives at the inventory allocation stage. Hence, the level of inventory held by the CP is for now assumed to be a given quantity (i.e., it is a parameter). Central inventory can be determined in many ways, e.g., as the sum of local inventory level calculations, or set optimally by the CP given his full knowledge of the demand distributions at the time that the inventory level decision is made. In all cases though, we assume that the level of common inventory is independent of the allocation mechanism employed. For the rest of the chapter we consider the case where inventory level is determined as the sum of retailers’ initial (soft) orders qa i. It is important to point out that these initial orders could be decided in a number of ways (e.g., optimal newsvendor quantity for each retailer) but retailers do not have freedom in choosing the way qa iis calculated. Hence allocation does not change the way retailers behave. In this sense, we ignore at this stage the interaction between the allocation mechanism and the total inventory set. The same assumption would hold if the CP announces the allocation mechanism along with the inventory level set for the coalition at the same time. The central planner (CP) is responsible for managing total inventory at the central warehouse and for distributing it to retailers after their local demands are realized, according to a publicly announced mechanism. After he observes his realized demand, each retailer places a second, final, order (qb i≥0) to the central warehouse. We denote by αthe vector of quantities sent to retailers. The timing of events, depicted graphically in Figure 3-1, is as follows: 42 0. Before the period begins, the central planner announces the allocation mechanism she will use once demands are realized (or after retailers have placed their final orders in the asymmetric information case). 1. At the beginning of the period, each retailer iobserves a private signal θi(only his) about his demand at location iand places an order qa ito the central planner. 2. The central planner calculates total inventory to be held centrally, Q=�n iqa i. This quantity becomes common knowledge to all players. 3. Demand is realized. In the symmetric information case after local demands are realized they become common knowledge to all players. In the asymmetric information case, where only retailers know their own di’s, retailers simultaneously place a second order qb ito the central planner. 4. The allocation of inventory to retailers (α) is calculated according to the posted allocation mechanism based on Qand the available information. The central warehouse fulfills the orders submitted by sending to each retailer αi.Each retailer is charged with c·αi. Figure 3-1: The timing of events in the inventory allocation game Before we analyze the symmetric and asymmetric information case, we begin by formally defining a Pareto optimal allocation mechanism and identifying some properties of such mechanisms in this context. 43 3.3 Properties of Pareto optimal allocation mechanisms An allocation mechanism in this case, where demand uncertainty is resolved before allocations are determined, is efficient when it excludes wastage. System efficiency implies that there are no unsold units when total demand is equal to or higher than Qand there is no locally unsatisfied demand when total demand is lower than Q.In other words, no retailer faces a shortage while at the same time another retailer faces a surplus (i.e., it cannot be that αi>d iand αj<d jfor some i, j =1...n). We denote by dand qbthe vector of local demand realizations and the vector of final orders placed by retailers, respectively. An allocation mechanism is Pareto optimal if it maximizes the sum of retailers profits and thus system profits, assuming in the asymmetric information case that all retailers truthfully report their realized demand by ordering qb=d. Please note that initial orders qa iare soft orders used to determine total inventory and are not sent to retailers (qb iis not on the top of qa i). Proposition 3.1: (a) A Pareto optimal allocation mechanism satisfies: ∂πi(αi,di) ∂αi=∂πj(αj,dj) ∂αj∀i, j. (b) An allocation mechanism is Pareto optimal if and only if it is efficient. All proofs are provided in Appendix B.1. Next we show that in our setting, a Pareto optimal allocation mechanism is not necessarily individually responsive. An individually responsive mechanism ensures that if a retailer is receiving not zero but positive allocation (αi>0), his allocation quantity increases when his demand (or his final order quantity in the asymmetric information case) increases, unless he has already been allocated the total quantity. Similarly, under such a mechanism, allocation quantity decreases when demand (or order quantity) decreases. 44 Following the definition of Cachon and Lariviere (1999) [10], an allocation mechanism is individually responsive if, for all i,0<α i(di)<Qimplies αi(ˆ di,d −i)>α i(di,d −i),ˆ di>d i(3.1) Proposition 3.2 states that being individually responsive is not a necessary condition for Pareto optimality in our setting. Proposition 3.2: A Pareto optimal allocation mechanism is not necessarily individually responsive. This is an important result because the (modified) uniform allocation mechanism, which is not individually responsive, is not excluded from the set of potentially Pareto optimal allocations. This result is different from the result of Cachon and Lariviere (1999) [10] who show that when allocation of scarce capacity to retailers is done before demand uncertainty is resolved if an allocation mechanism is not individually responsive it cannot be Pareto optimal. The intuition behind their result is that a Pareto mechanism “must recognize the smallest change in every retailer’s marginal valuation of stock and thus must be individually responsive”. If a retailer has a higher optimal stocking level, it is optimal to receive a higher allocation. In the setting under consideration, because allocation is decided after demand uncertainty is resolved at each location (and retailers have identical cost parameters), the incremental value of an additional unit of stock is constant; it is p−cwhen there is inventory shortage and −cin case of total inventory surplus. Hence, responsiveness is not a necessary condition for allocation optimality (unlike the capacity allocation setting considered in papers [8–10]). 45 3.4 Allocation with symmetric information We start by introducing the structure of the three allocation functions of interest and checking whether they are Pareto optimal. We build on the allocation rules for capacity rationing proposed by Cachon and Lariviere (1999, 199b) [8,10] and we modify them so that �n i=1 αi=Q. Please note that in the context under consideration a retailer may be allocated a quantity higher than his final demand, if total demand is lower than the total quantity held centrally. Proportional allocation αi(d)= di �n i=1 di Q(3.2) Under symmetric information, it is trivial to show that proportional allocation is efficient and therefore Pareto optimal. Linear allocation Index retailers in decreasing order of their demand, i.e. {d1≥ d2≥... ≥dn}. Case 1: �n i=1 di≥Q, retailer iis allocated αi(d,˜n), where αi(d,˜n)=   di−1 ˜n(�˜n jdj−Q)fori≤˜n 0fori>˜n (3.3) and ˜nis the largest integer such that α˜n(d,˜n)>0. Case 2: �n i=1 di<Q, retailer iis allocated αi(d), where αi(d)=di+1 n(Q− n � i=1 di) (3.4) Linear is also an efficient allocation because if �n i=1 di≥Q,thenαi≤di∀i. Similarly, when �n i=1 di<Q, it is guaranteed that αi≥di∀i.Thereforewastageis excluded in both cases. Although this is an allocation mechanism that maximizes the 46 sum of retailer profits, when there is inventory shortage it may assign zero inventory to retailers with low demand [8]. Consequently, implementing this allocation rule by itself may not satisfy the individual rationality constraints of the retailers and hence may not encourage inventory centralization [31]. Uniform allocation Index retailers in decreasing order of their demand, i.e. {d1≥ d2≥... ≥dn}. Case 1: �n i=1 di≥Q, retailer iis allocated αi(d,ˆn), where αi(d,ˆn)=   1 ˆn(Q−�n j=ˆn+1 dj)fori≤ˆn difor i>ˆn (3.5) and ˆnis the largest integer such that αˆn(d,ˆn)<d ˆn. Case 2: �n i=1 di<Q, retailer iis allocated αi(d,ˆn), where αi(d,ˆn)=   difor i<ˆn 1 n−ˆn+1 (Q−�ˆn−1 j=1 dj)fori≥ˆn (3.6) and ˆnis the smallest integer such that αˆn(d,ˆn)>d ˆn. Again, as in linear allocation, the uniform allocation rule guarantees that when �n i=1 di≥Q,αi≤di∀iand when �n i=1 di<Q,αi≥di∀i.Thereforeitisefficient and Pareto optimal. Uniform allocation favors retailers with low demand in the case of inventory shortage, and retailers with high demand in the case of inventory surplus. These allocation mechanisms are based solely on realized demands and not on retailers’ initial orders qa i.Hence,theydonotallowforcomparisonsbetweenthe profit of a retailer as part of the pooling coalition and what he could have earned in a decentralized system, assuming that he would have ordered qa i.Themechanismsalso do not provide any minimum inventory or profit guarantee to individual retailer. 47 We are interested in identifying an allocation mechanism that provides a guaranteed and a maximum allocation. A guaranteed allocation is the amount of inventory that the retailer is assured to receive if he wants it. Maximum allocation is the retailer’s largest possible allocation given his demand. A mechanism that satisfies condition (3.7) provides such guarantees and gives a retailer a greater control over his own destiny. min(di,qa i)≤αi(di,qa i,Q)≤max(di,qa i)∀i& n � i=1 αi=Q(3.7) An allocation mechanism that satisfies condition (3.7) guarantees retailer iaquantity at least equal to his initial order and up to his demand, when his realized demand is higher than his initial order. When his local demand is lower than his order quantity, retailer’s demand is guaranteed to be satisfied but he may be allocated and charged a quantity less than his initial order. An allocation mechanism satisfying these properties is Pareto efficient because it excludes wastage and eliminates all profitable trade among retailers (total profit maximizing). An allocation mechanism that satisfies (3.7) also guarantees a retailer a profit larger or equal to what he would have earned under a decentralized system. Proposition 3.3: Each retailer’s profit when his is part of a pooling coalition that sets Q=�qa iand the CP allocates inventory according to a mechanism that satisfies condition (3.7) is larger or equal to his profit under a pure decentralized system. In Proposition 3.3 we assume that in a decentralized system (separate newsvendors) each retailer will set inventory qa i. Please note that we put no restrictions on how qa i is calculated. We assume though that it is the same and it does not depend on the allocation function. It follows immediately from Proposition 3.3. that total profits as well will be larger or equal to sum of individual profits in a decentralized setting. This is due to the benefit of excess demand and stock rebalancing opportunity after demands are realized (unilateral change of inventory ownership). 48 Chapter 4 Forecast Information Sharing and the Order Quantity Decision: Impact of Inventory Competition and Market Uncertainty 4.1 Introduction When we consider the case of multiple retailers that form a pooling coalition and delegate inventory management to a central planner (CP), reliability of demand forecast sharing becomes crucial. The present chapter analyzes the forecast-sharing game between multiple retailers and a central planner who sets common inventory under information asymmetry. In particular, early demand signals are known by retailers but the CP needs this information to make appropriate inventory decisions for the coalition. Reliable demand information sharing in a supply chain depends on parties’ incentives as well as on behavioral factors such as trust and trustworthiness between supply chain parties [42]. For example, in a typical supplier-retailer setting, where a supplier solicits private forecast information from a retailer to set his capacity, standard game 55 theory predicts that parties do not cooperate and the only equilibrium is uninformative. However, recent research reveals that in controlled laboratory experiments parties cooperate even in the absence of reputation-building mechanisms and complex contracts (e.g., [42]). The underlying reason for cooperation seems to be supplier’s trust in retailer and the retailer’s associated trustworthiness. In the case of multiple retailers trading the same product, an additional level of complexity is added due to their strategic interactions. Retailers compete for common inventory and this may affect their incentives when sharing demand information. In such a case, the free-rider problem may arise. The alignment of system and individual incentives becomes important as well as the ability of the CP to induce or detect truthfulness in forecast sharing. For example, observing multiple forecasts may increase the CP’s ability to detect truthful information sharing1but on the other hand the increased competition among retailers may distort incentives or harm trust. The same dynamics arise in the case of a single company that owns multiple stores and decides to hold inventory centrally, under the assumption that each store is a separate business unit that strives to maximize local profit. Each branch has private information regarding local market factors that may or may not transmit truthfully to the central decision maker who sets and controls the common inventory. The research focus of this chapter is on whether reliable forecast information sharing occurs when multiple retailers form an inventory pooling coalition managed by a benevolent CP who (a) sets common inventory level before demand is realized and (b) allocates inventory to the retailers after demand uncertainty is resolved in proportion to their realized demands. As in the previous chapter, we consider the case of a benevolent CP in the sense that she is interested in maximizing total system profits (sum of retailer profits). 1In our work we do not model this situation, the CP cannot make inferences whether retailers are lying or not. 56 Furthermore, we want to study what the impact of the size of the pooling coalition and the market uncertainty is on the level of trust and trustworthiness of supply chain parties. We define trust as the CP’s willingness to rely on retailers’ forecasts to determine total inventory. Trustworthiness, similarly to Ozer et al. (2011) [42], is measured by the difference between a retailer’s real forecast and the one he reports to the CP. To be more specific, we are interested in answering the following questions: (a) Do retailers transmit truthfully their demand forecasts to the CP in such a setting where they strategically interact and compete for common inventory? (b) Does the CP trust the retailers’ shared demand information? (c) Does retailers’ truthfulness depend on their forecast accuracy? (d) Does the number of retailers forming the coalition have an impact on their trustworthiness and the CP’s level of trust? To answer these research questions, we first study players’ incentives and the result of their strategic interaction using game theoretical models. After developing a series of analytical results, we develop and execute a series of laboratory experiments to test the theories implied by these results. The rest of this chapter is organized as follows: first we analyze the forecast communication game with one-time interaction to obtain the standard model prediction. We proceed by doing an extensive numerical analysis to gain some insights on the analytical results. Next, we present four hypotheses in forecast information sharing and cooperation, established based on the analytical results of the game theoretic model and on the existing literature about trust, trustworthiness and human behavior biases. We test these hypotheses in a controlled laboratory environment. The last section describes the experimental procedure, analysis and findings. 57 4.2 The analytical model 4.2.1 Setting and timing of events We consider, again, a setting where a group of nretailers each face uncertain demand (newsvendor type problem) and have private information (receive a signal) about it due to their proximity to the market. We denote by ˜ dithe demand that retailer i faces, after he receives his signal θi, a random variable with mean µi+θi, cumulative distribution function Li(·), probability density function li(·)andsupport[µi+θi+ �i,µ i+θi+�i], and by Dithe random variable representing total demand from the point of view of retailer iafter he receives θi. Each retailer reports his signal (maybe untruthfully) to the CP (ˆ θi). The CP makes a single ordering decision (Q) for the whole coalition after she solicits demand information from each retailer and before demand is realized. She allocates the total inventory quantity to retailers after demand uncertainty is resolved, in proportion to their realized demands. In contrast to the previous chapter, demand realizations at the end of the period are common knowledge to all players. Let dibe the realized demand at location iand αidenote the allocation retailer igets. Then, we define proportional allocation as follows: αi(d)= di �n i=1 diQ. The proportional allocation rule is widely used in practice and easy to enforce, as it is perhaps the most intuitive scheme [8]. Furthermore, it has several attractive properties. First, as we showed in chapter 3, when realized demands become common knowledge before allocation, proportional rule is a Pareto optimal mechanism from the system’s perspective. It is an efficient mechanism, in the sense that it excludes wastage (there are no unsold units when total demand is equal to or higher than Qand there is no locally unsatisfied demand when total demand is lower than Q). Hence, it maximizes total system profit. Second, because total inventory is assigned to retailers (even when inventory exceeds total demand), the profit function for the retailer is not monotonically increasing in Q. Third, the proportional rule provides 58 the same service level to all retailers, a property expected in a system with identical retailers. Last, the proportional allocation rule is chosen for analytical tractability. The timing of the events, described schematically in Figure 4-1, is as follows: 0. Before the period begins, the size of the pooling coalition and the allocation rule are announced. 1. At the beginning of the period each retailer observes his (and only his) private demand signal (θi). 2. Each retailer sends his forecast (reports a demand signal ˆ θi)totheCP. 3. The CP sets the inventory level of the system (Q) and this quantity becomes common knowledge to all players. 4. Local demands (di’s) are realized and revealed to all players, the allocation of inventory to retailers is done according to the proportional rule and profits are calculated. Figure 4-1: The timing of events in the information sharing game with common inventory Compared to Figure 3-1, the timing of events is very similar. The main differences are: (a) retailers send their forecasts to the CP and not their soft orders and (b) retailers do not place a final order to the central warehouse before allocation is calculated. In terms of decision rights, in the case described in Figure 3-1, the CP just 59 aggregates soft orders placed by the retailers to set the quantity (she has no decision rights on the inventory level). We first study the information sharing game when there is a single retailer and a CP (n= 1) and, second, when multiple retailers form a coalition (n>1). In the case of multiple retailers, we present both the case where the coalition has an automated ordering system in place and the case where there exists a CP with decision rights on setting inventory. We also study the special cases of infinitely many retailers and the case where demand uncertainty is resolved for each retailer before information transmission. 4.2.2 The information sharing game for n=1 We start our analysis with the simplest setting of one retailer to gain intuition on the dynamics between a retailer and the CP when there is no competition for inventory. In this case, there is only one retailer who sends his signal to the CP who in turn decides the quantity to be held for the retailer. For given Qand known θi,the retailer’s and the CP’s expected profits are given by: πi(Q, θi)=E�i[pmin[(µi+θi+�i),α i(Q)] −cαi(Q)] (4.1) Π(Q, θi)=E�i[pmin[(µi+θi+�i),Q]] −cQ (4.2) The allocation of retailer iafter demand is realized reduces to αi(Q)=Qand hence πi(Q, θi) = Π(Q, θi). If the CP knew θi,shewouldmaximizeherexpectedprofitby setting inventory Q(θi)=µi+θi+G−1 i(p−c p). This is the quantity that maximizes retailer’s expected profit, when he has received θi. Thus, in this interaction, the retailer has no incentive to distort the report of his forecast, and the CP has no reason not to consider the reported forecast as credible. Observation 4.1: In the case of a single retailer, agents’ interests coincide and a truthful information sharing equilibrium exists. 60 This setting is different from the supplier-manufacturer (or retailer) case considered in Ozer et al. (2011) [42] because the retailer’s profit function is not monotonically increasing in the CP’s inventory choice Q.Evenifheincursnodirectcostbyreporting θi, in the case of just one retailer his forecast indirectly works as an enforceable order. The CP allocates the retailer the total quantity she orders no matter if it is higher than retailer’s demand. The retailer assumes all demand uncertainty risk while the CP has no individual profit margin; she is benevolent instead, trying to maximize retailer’s profit. Please note that this equilibrium is not unique; other equilibria exist in such a setting which vary based on the CP’s belief concerning how ˆ θiand θiare related. For example, for any δi, the retailer reporting ˆ θi=θi+δiand the CP setting Q(ˆ θi)= µi+ˆ θi−δi+G−1 i(p−c p) constitute a BNE. In such a case the retailer is partially trustworthy and the CP is partially trusting. The resulting equilibrium though is totally informative in the sense that the CP can infer from the forecast reported the true value of θi. 4.2.3 The information sharing and allocation game for n>1 In the case of multiple retailers, the issue of allocation becomes important. The ultimate allocation that each retailer gets is a function of the realized demands and the total inventory set by the CP: αi(d,Q)= di �n i=1 diQ. We note that there is no dedicated inventory to each retailer and no quantity commitment based on his forecast. A retailer’s allocation is a (random)2fraction of the inventory held centrally. Hence, a retailer may influence the allocation he expects to get after the demands are realized only through influencing the total quantity that is held centrally. In other words, when retailer ireports a signal ˆ θi, his allocation depends on ˆ θionly through Qthat may depend on the signal sent by retailer i.ForgivenQand θi’s, retailer’s iand 2At the time of forecast information transmission allocation is a random fraction of the total quantity because final demands have not been yet realized. 61 CP’s expected profits are: πi(Q, θi)=E�,θj,j�=i[pmin[(µi+θi+�i),¯αi(d,Q)] −c¯αi(d,Q)] (4.3) Π(Q, θ)=pE�[min[ n � i=1 (µi+θi+�i),Q]] −cQ (4.4) where ¯αi(d,Q)= ˜ di DiQand θ=[θ1,θ 2, ..., θ n]. Common knowledge We begin by studying the common knowledge case where all players have the same information about local and total demand (i.e. θis common knowledge to all players). This case serves as a benchmark and a way to better understand the incentives of the players who have private information in the general setting. When the CP knows the demand signals at each retail location, she faces a newsvendor problem with total demand equal to the convolution of all individual retailers’ demand distributions updated according to the realized demand signals. She maximizes Equation (4.4) by setting inventory as: QCP f(θ)= n � i=1 (µi+θi)+(G1◦G2... ◦Gn)−1�p−c p�(4.5) We then study how the quantity given by equation (4.5) compares to the quantity that maximizes each retailer’s profit, when he has the same information about total demand. We denote by Qi fthe optimal quantity to be held centrally from the point of view of retailer i(the quantity that maximizes retailer’s iexpected profit) when he has the same information about total demand like the CP who sets Q(retailer knows the realized demand signals in all retail locations). Lemma 4.1: When demand signals are common knowledge, the CP will set QCP f= Qi fiffE[˜ di �n i=1 ˜ di|�n i=1 ˜ di>Q i f]=E[˜ di �n i=1 ˜ di|�n i=1 ˜ di<Q i f]. All proofs are in Appendix B.2. 62 Lemma 4.1 shows that with the same information about total demand, individual retailers’ and CP’s incentives may not coincide; the CP sets an inventory level that is different than the optimal inventory from the point of view of a retailer iunless the expected ratio of own demand to total demand at Qi fis the same in case of shortage and in case of surplus. Corollary 4.1: When E[˜ di �n i=1 ˜ di|�n i=1 ˜ di>Q i f]>E[˜ di �n i=1 ˜ di|�n i=1 ˜ di<Q i f]then Qi f>Q CP f. When the reverse is true, Qi f<Q CP f. Corollary 4.1 states that when the expected ration of own demand to total demand in case of shortage is higher than the expected ratio in case of surplus, the optimal total inventory from retailer’s ipoint of view is higher than the system optimal inventory. If the expected ration of own demand to total demand is higher when there is surplus compared to when there is shortage, retailer iprefers a lower total inventory than the system optimal quantity. Automated inventory ordering system Under common knowledge, even if the desired quantity by retailer imay be different than the quantity set by the CP, the retailer does not have the chance to influence CP’s decision by misreporting demand information. But, under asymmetric information, due to this discrepancy in incentives, retailers, anticipating how the CP will set common inventory, may have an incentive to misreport to her their private information. In the context of strategic communication and information sharing, in which a better informed sender sends a possibly noisy signal to a receiver, who then takes an action that determines the welfare of both, the amount of information that the sender shares is related to the similarity of agents’ interests [16]. We begin with the case of an automated central inventory ordering system that takes as input the forecasts reported by the retailers and automatically calculates and orders the optimal inventory level for the whole coalition, based on the demand information provided. The total quantity ordered is given by the equation QCP(ˆ θ)= 63 �n i=1(µi+ˆ θi)+(G1◦G2...◦Gn)−1(p−c p) and satisfies the sufficient optimality condition (p−c)Pr[ˆ D>Q CP]=cPr[ ˆ D<Q CP], where ˆ Ddenotes the total demand, updated according to the reported signals. In this case, retailer ihas an incentive to distort his report of θi,if and only if QCP is different than Qithat maximizes Equation (4.3). As the next theorem formally states, when such an automated inventory ordering system is in place, we expect QCP �=Qi∀i. Theorem 4.1: Under an automated inventory ordering system, the optimal quantity from the point of view of retailer isolves the equation (p−c)Pr[Di>Q i]E[˜ di Di |˜ di>¯αi(d,Q i)] = cPr[Di<Q i]E[˜ di Di |˜ di<¯αi(d,Q i)] and consequently QCP �=Qi∀i, unless ��˜ di>¯αi¯riΓ(¯ri,˜ di)d¯rid˜ di=Pr[ ˆ D>Q i]E[¯ri], where ¯ri=˜ di Diand Γis the joint distribution of ¯riand ˜ di. Hence, retailer imay have an incentive to distort his reported signal. Theorem 4.1 shows that the optimal total quantity for retailer iwill differ from the one that the automated system sets, for two reasons. First, each retailer has partial information about total demand (knows only his demand signal) while the automated system uses reported signals about all local markets (which in turn may be distorted). To be more specific, total demand for retailer iis characterized by the random variable Di=[ �n i=1 µi+θi]+[ �n j=1,j�=iθj+�n i=1 �i] where the random part is �n j=1,j�=iθj+�n i=1 �i,whiletheautomatedsystemsetsinventorybasedon ˆ D=�n i=1(µi+ˆ θi)+�n i=1 �i, where the random part is �n i=1 �i.Thus,forany given Q, retailer ihas different belief about the probability of inventory shortage (or surplus) for the coalition than the one the automated system calculates (Pr[Di> Q]�=Pr[ˆ D>Q]). This is also the case when all the reported signals to the system are true (when ˆ θi=θi∀i,ˆ D=�n i=1(µi+ˆ θi)+�n i=1 �i,adifferent random variable than Di). 64 system optimal quantity. When the demand signals received by the two retailers are identical, then Q1 f=QCP f, suggesting that the expected ratio of own demand to total demand is the same in case of shortage and in case of surplus (Lemma 4.1). We perform the same analysis, when p= 4 and c= 1, resulting in a critical ratio of 0.75, and for p=2andc=1.5, resulting in a critical ratio of 0.25. The difference between the optimal quantity for retailer 1 and the system optimal one are summarized in Table 4.3. We observe that the comparison gives the same directional results, independent of the critical ratio considered. We therefore continue our numerical analysis with the asymmetric information case, using cr=0.5 while the calculations for the base case under asymmetric information and critical ratios 0.75 and 0.25 are delegated into Appendix C. Q1 f −QCP fwhen cr=0.75 Q1 f −QCP fwhen cr=0.25 θ2θ2 θ1-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 0.000 0.009 0.017 0.024 0.030 0.036 0.041 0.000 0.010 0.019 0.027 0.034 0.040 0.046 -2 -0.008 8.000 0.007 0.013 0.019 0.024 0.029 -0.009 0.000 0.008 0.015 0.021 0.027 0.032 -1 -0.013 -0.006 0.000 0.006 0.011 0.015 0.019 -0.015 -0.007 0.000 0.006 0.012 0.017 0.021 0 -0.017 -0.011 -0.005 0.000 0.005 0.009 0.013 -0.019 -0.012 -0.006 0.000 0.005 0.010 0.014 1 -0.019 -0.014 -0.009 -0.004 0.000 0.004 0.007 -0.022 -0.015 -0.010 -0.005 0.000 0.004 0.008 2 -0.021 -0.016 -0.011 -0.007 -0.003 0.000 0.003 -0.023 -0.018 -0.013 -0.008 - 0.004 0.000 0.003 3 -0.025 -0.020 -0.016 -0.013 -0.009 -0.006 0.000 -0.024 -0.019 -0.015 -0.011 - 0.007 - 0.003 0.000 Table 4.3: Q1 f−QCP f,asafunctionofθ1and θ2, for cr=0.75 and cr=0.25 4.3.2 Asymmetric information We calculate the optimal quantity for retailer 1 (Q1), as a function of the market signal he receives, when he does not observe the signal received by the other retailer, but he knows only its distribution. In order to study retailer’s incentives in the information 71 sharing game, we compute the expected quantity an automated inventory system would set, assuming that ˆ θi=θi,fori=1,2, and we compare it to Q1. We note that QCP(ˆ θ1,ˆ θ2)=K+ˆ θ1+ˆ θ2,whereK=µ1+µ2+(G1◦G2)−1(p−c p). Retailer 1, does not know the reporting rule of retailer 2 and therefore we cannot calculate Eˆ θ2[QCP] as a function only of ˆ θ1. But it is reasonable to assume that retailer’s 1 belief about retailer’s 2 reporting rule does not depend on θ1and therefore E[ˆ θ2]=λ,whereλis a constant. Hence, E[QCP (ˆ θ1)] = K+λ+ˆ θ1. For our calculations of Q1−E[QCP], we assume λ=0(orˆ θ2=θ2)andˆ θ1=θ1, but it is straightforward to compute how the difference would change for different beliefs about λor different reporting rules of retailer 1. In the base case, the average market size at both locations (before demand signals are realized) is the same (µ1=µ2= 10), the demand signals are independent, normally distributed, with mean 0 and variance 1 (θi∼N(0,1)) and market uncertainty at each location follows as well the standard normal distribution (�i∼N(0,1)). For the base case, the optimal quantity for retailer 1 and the expected system quantity had the two retailers reported the true signals are presented in Table 4.4. It is interesting to note that in all instances Q1<E[QCP] and the absolute difference is increasing in the signal received by retailer 1. Retailer’s 1 perception about total demand uncertainty is higher because he does not know the demand signal of retailer 2. The CP instead, in these numerical experiments, takes the received signals as given. But since the critical fractile is 0.5, this difference in total demand uncertainty does not influence the optimal total inventory level from the point of view of each player. The fact that retailer 1 always prefers a smaller quantity is in line with the result that “for linear holding/shortage costs the optimal probability of shortage under random yield is no smaller than the probability of shortage under certain yield” [27]. We continue by studying the effect of private information variability, the effect of market uncertainty and that of the magnitude of local demands. Figure 4-2a) shows 72 θ1Q1E[QCP]Q1−E[QCP ] -3 16.966 17 −0.034 -2 17.958 18 −0.043 -1 18.952 19 −0.048 019.949 20 −0.051 120.947 21 −0.053 221.946 22 −0.054 322.946 23 −0.054 Table 4.4: Comparison of the optimal inventory for retailer 1 and the expected inventory in the system under truth-telling, as a function of θ1 (Q1−E[QCP]) for the base case, for high signal variability (θ2∼N(0,3)) and for very small market uncertainty (�i∼N(0,0.5)) while Figure 4-2b) shows the results when the average market size of retailer 1 doubles (µ1= 20) and when instead the average market size of retailer 2 doubles (µ2= 20). Figure 4-2: (Q1−E[QCP]) as a function of the signal received, for high signal variability, small market uncertainty and asymmetric market sizes Results suggest that when market signal variability increases, the difference between retailer’s and system’s preferred inventory level increases in magnitude. On 73 the contrary, market uncertainty does not have a strong effect on the difference of preferred quantities. This is because both the retailer and the central system have the same information about total market uncertainty, while the automated inventory system treats the reported signals as true constants. In addition, results suggest that the relative size of the retailer plays an important role in his incentives. In our example, we observe that when µ1= 20, retailer 1 always prefers for total inventory a quantity smaller than the system optimal inventory (independent of θ1). On the other hand, when his market is relatively small (µ2= 20), he may have an incentive to inflate his forecast when his market signal is low and under-report his signal when it is high. To summarize, numerical analysis suggests that the market size of a retailer compared to the other member of the coalition plays an important role in his preferred total inventory level compared to the system optimal one. A retailer that expects lower demand than the other retailer has a higher expected allocation ratio in case of shortage than in case of surplus and this drives up his preferred inventory level. On the other hand, uncertainty in the allocated quantity, which increases with signal variability, drives the locally optimal inventory level down compared to the system optimal level. 4.4 Hypotheses When there is only one retailer sharing information with a benevolent CP, game theory provides a definitive prediction of the result of the information sharing game: any equilibrium is totally informative. This result is the opposite to the case of one supplier-one manufacturer, where the only theoretical equilibrium in demand forecast sharing for capacity investment is uninformative [42]. In that case, players’ incentives are opposed as the manufacturer always prefers a higher capacity. In the setting we consider, a single retailer’s and CP’s incentives coincide (Observation 4.1). Thus, truth-telling, in this “cheap talk” setting, is an equilibrium even if it may not be 74 unique. The retailer truthfully communicates his demand forecast to the CP who, in turn, updates accordingly her belief about demand when setting the inventory. Based on this equilibrium, we formulate the following hypothesis: Hypothesis 1: In the information sharing game between a retailer and a CP, (a) the retailer is fully trustworthy (ˆ θ=θ) and (b) the CP is fully trusting (Q(ˆ θ)= µ+ˆ θ+G−1(p−c p)). In the case of two or more retailers, they compete for common inventory. Theorem 4.3 predicts that in the forecast communication and inventory allocation game, retailers being fully trustworthy when reporting their demand forecasts and the CP fully trusting them do not constitute a Bayesian Equilibrium. Retailers that want to maximize their pecuniary payoffs have an incentive to distort the information they send and thus a rational CP will not consider their forecasts as credible. But in this strategic information transmission setting, even though there are no exogenous signaling costs when reporting a forecast, there are endogenous signaling costs resulting from the fact that the total inventory quantity held centrally by the CP will be allocated to the retailers after local demands are realized. Retailers are held responsible for both system understocking and overstocking costs. If the central inventory quantity is very high, individual expected overstocking costs increase while if the quantity is very small, individual expected under-stocking cost increase. Motivated by these results and the work of Ozer et al. (2011) [42] that suggest that in reality a continuum of trust exists when supply chain parties share forecast information (in contrary to the all-or-nothing view adopted by the extant literature), we formulate the following two hypotheses: Hypothesis 2: In the information sharing and allocation game, retailers’ reports ˆ θi’s are informative about their private forecasts θi’s. More specifically, ˆ θiis positively correlated with θi(retailers are are partially trustworthy). 75 Hypothesis 3:TheCPreliesonˆ θi’s to determine inventory that is held centrally. More specifically, Qis positively correlated with �n i=1 ˆ θi(the CP is partially trusting). The second factor in the supply chain environment that we study is the impact of market uncertainty. When demand uncertainty is resolved before information transmission, i.e. there is no market uncertainty after retailers receive their demand signals, Theorem 4.5 predicts that the resulting equilibrium is totally informative. Motivated by this prediction, we examine the following hypothesis. Hypothesis 4: When demand uncertainty is resolved before information sharing (i.e. retailers’ private forecasts are perfectly accurate), (a) retailers are fully trustworthy (ˆ θi=θi) and (b) the CP is fully trusting (Q(ˆ θ)=�n i=1[µi+ˆ θi]). To summarize, theory predicts reliable information transmission only in the case of a single retailer and in the case of known demand by retailers when communication takes place. On the other hand, when more than one retailers compete for common inventory and demand is uncertain, truth-telling and trust do not form an equilibrium. We therefore hypothesize, that competition for common inventory and demand uncertainty harm reliable information sharing. To be specific, compared to the other two cases, we expect that the signal sent is less informative (lower correlation between ˆ θiand θi)andtheCPrelieslessontheinformationreceivedtosetinventory(lower correlation between �n i=1 ˆ θiand Q). 4.5 Experiments 4.5.1 Experimental design and procedures We conducted a series of human-subject controlled laboratory experiments3to investigate the aforementioned hypotheses. We conducted three treatments / experiments 3All experiments were conducted at the Social and Behavioral Sciences Laboratory, at the University of Minnesota. 76 as summarized in Table 4.5. Each treatment is labeled as RiDj,wherei=1,2denotes the number of retailers and j∈{U, K}stands for “uncertain” and “known” demand, respectively. In case R1DU,onlyoneretailerinteractswithaCP.Incases R2DUand R2DK,thenumberofretailersincreasestotwoandtheissueofinventory allocation becomes relevant. Under case R2DK, demand uncertainty is resolved before demand information transmission. Under two treatments retailers compete for common inventory (cases R2DUand R2DK) and under two treatments there is demand uncertainty when information is transmitted (cases R1DUand R2DU). All other supply chain parameters are kept constant across the different treatments. We fix average demand, signal variability and market uncertainty (for the cases that is relevant).4For the demand parameters used in our experiments, we calculate a retailer’s optimal inventory level, for various demand signal realizations, and we compare it to the system optimal had the CP known the real signals (i.e., the expected inventory level under the assumption that the other retailer reports his true signal and the CP trusts the reported information). The results, analogous to Table 4.4 but for discrete uniform demand distributions, are presented in Appendix C. We note that in all cases, the difference between the quantity that retailer 1 prefers and the system optimal inventory level is less than 1%. We use revenue and cost parameters that result in critical ratio of 0.5. In that case, the optimal order quantity for the CP is equal to the mean of the demand. By doing so, we avoid experimental results to be influenced by the “pull-to-center” effect when setting the inventory quantity; the phenomenon of systematically ordering too little when the cost of underage is high and ordering too much when the cost of overage is high. This phenomenon may be explained by some well-known decision biases, e.g., anchoring and insufficient adjustment, observation bias, reference-dependent preferences and is well documented in the literature in single newsvendor [5, 7, 47] and 4For average demand and signal distribution we adopted the values used for the information sharing experiments reported in Ozer et al (2011). Regarding market uncertainty, we use the average of the aforementioned experiments. 77 multilocation newsvendor models [28]. The number of participants in each treatment was decided based on the supply chain group synthesis in each case (one vs. two retailers), taking into account the probability that a specific group will play more than once and the average number of rounds a given group is expected to play. Treatment No of retailers (n) Demand Uncertainty No. of participants No. of rounds Case R1DUn=1 Yes 10 30 Case R2DUn=2 Yes 15 30 Case R2DKn=2 No 12 30 Notes. In all treatments, µi= 250, θi∼U[−150,150], �i∼U[−50,50] (all discrete), p=2andc=1. Table 4.5: Experimental Design We used a between-subjects design; i.e., each treatment involves an independent group of participants. We recruited students of the University of Minnesota for the experiments, through the Carlson School of Management Subject Pool 5. Participants in each treatment were randomly assigned the role of retailer or central planner; this role assignment remained unchanged for all rounds. At the beginning of each round, all participants were randomly and anonymously assigned to a supply chain group. Players were informed that they would not be assigned to the same supply chain group in consecutive rounds. The experimenter also stressed out that rounds are independent, both in terms of group assignment and demand realizations, to avoid reputation effects and demand chasing. Each subject received a detailed sheet of experimental instructions and a short summary sheet with the important info about supply chain parameters, the sequence 5Carlson School of Management (CSOM) Subject Pool was launched on September 26, 2008 and consists of a database of individuals who have previously agreed to be part of the pool. CSOM Subject Pool members are able to go online, view studies available for their participation, and sign up to participate in any study they are interested in, provided they meet the filtering criteria set by the researcher. 78 of events and a reminder about how profits are calculated. The detailed instructions to the CP and to the retailers for case R2DUcan be found in Appendix D. Each participant was informed about the task performed by the other role, including the information available and profit objective. After the instructions were read, subjects were allowed to ask questions and then directed to the computers’ room where the experiment was implemented. The experiment was programmed and conducted with the software z-Tree [20]. Participants were not allowed to talk to each other from the time they entered the laboratory until the time they left. During the experiment, participants interacted with each other only through computer terminals and did not know the identity of the person with whom they were playing. Each treatment consisted of 35 rounds. The first 5 rounds served as a trial period and did not affect the final profit; they served as training for the participants to better understand the dynamics of the game. Participants were not informed how many rounds they would play after this trial period6to avoid end-of-treatment effects. Participants played the information sharing inventory game specified in section 4.2.1. Briefly, each round consisted of 3 periods. In period 1, each retailer iobserved his private information (the exact value of µ+θi)andwasaskedtosubmitareportto the CP. In period 2, the CP, after observing the report(s) sent by the retailer(s), decided on the (total) order quantity to place. The software indicated to the CP the quantity that maximized her expected profit, if the reports sent are considered to be true. After the decisions were made, in period 3, demands were revealed and profits were calculated. At the end of each round, participants observed realized demand(s) (and total demand in case of two retailers), (total) order quantity, own inventory allocation and own profits from the round. We provide sample snapshots of the CP and retailer screens in Appendix E. In the experiment, the demand values generated (θiand �i)variedbetweenthetwo retailers in the same supply chain group (for case R2DUand case R2DK)andacross 6Participants were told they are going to make decisions for up to 40 independent rounds (5 trial and up to 35 game rounds). 79 rounds. However, they remained the same across treatments whenever that was applicable (e.g. θ1in round 1 was the same across all cases, for all groups, while �1in round 1 was the same between case R1DUand case R2DU, for all groups). This design feature is particularly useful because it allows to compare decisions across treatments while controlling for individual specific effects and to compare profits across treatments (in some cases), while controlling for the impact of demand realizations on actual profits. At the beginning of each treatment, participants were asked for basic demographic information, including major, class year, specific courses and level of experience in supply chain management. At the end of each treatment, and after all rounds were played, participants were required to complete a post-game survey. The survey contained questions asking to comment on the choices they made during the study (report of private information or order quantity placement), on how their strategy changed throughout the session and on their degree of trust towards the other players in their supply chain. The level of trust they placed in the other supply chain group members was measured, for each role, on a Likert-type scale from 1 to 5, where 1 denotes “No trust at all” and 5 represents “Absolute trust”. Finally, every participant received payment proportional to the total experimental dollars he or she earned, with a minimum participation fee of $10 and maximum potential earnings of $20. 7 7Experimental earnings that a player would earn under full information, common knowledge and optimal decisions (i.e., had all supply chain members known the actual demand and ordered inventory quantity equal to demand) were calculated. Given the demand stream each player faced, a corresponding rate from experimental to USD dollars was calculated so that the maximum profit a player could make was $10. Experimental profits of each player were then divided by his/her corresponding rate to calculate his/her earnings in USD. These dollar earnings were rounded up to the highest integer and added to the player’s $10 participation fee. 80 Variable Definition Dependent Variables ˆ θit −θit Difference between the average reported signal and the signal observed by retailer type iin round t(i=1,2) Qjt −Q(ˆ θ)jt Difference between the (Total) Order Quantity placed by CP jin round tand the quantity suggested as optimal, if the CP believed the reported signals. Ekt System efficiency under Case kin round t,k∈{R1DU,R 2DU,R 2DK} Treatment dummies CIndicator variable for competition for common inventory; C=1 if the data are from Case R2DUor Case R2DKand 0 otherwise UIndicator variable for demand uncertainty when information is transmitted; U=1 if the data are from Case R1DUor Case R2DUand 0 otherwise Other Independent variables tRound, t=1,2...30 Dkt Total Demand in round tunder Case k,k∈{R1DU,R 2DU,R 2DK} Error terms �it Independent error across retailer “types” and periods ωjIndividual specific error for CPs ejt Independent error across (Total) Order Quantity decisions vkt Independent error across cases (treatments) and periods Table 4.8: Variable Definition in Equations (4.6)—(4.8) 87 We note from the responses of the subjects to the post-experiment survey that the average CP’s level of trust to the reported information was similar in all cases (4, 3.6 and 3.8 for cases R1DU,R2DUand R2DKrespectively, on a scale from 1 to 5). Estimate (Standard error) Variable ˆ θit −θit Qjt −Q(ˆ θ)jt Efficiency (%) Intercept −8.226** (2.547) 14.470 (13.197) 0.908 ** (0.026) C15.920** (1.920) −17.633 (10.25 ) −0.089 ** (0.020) U9.801** (1.567) −9.307 (10.872) −0.035 * (0.014) t−0.199* (0.081) −0.340** ( 0.119) 0.002 * (0.001) D- - 0.0003** (0.00006) *p<0.05 and **p<0.01. Table 4.9: Impact of competition and market uncertainty on truth-telling, trusting and efficiency Next, we examine the impact of treatment effects on cooperation and system efficiency. We find statistically significant evidence that system efficiency decreases with competition (almost by 1%) and market uncertainty (less than 0.5%) on average. This is expected as both these two factors lead to an average increase in forecast distortion, while the CP fails to account adequately for it, but keeps his level of trust almost unchanged (according to post-experiment survey answers). System profits decrease because the decision to set the common inventory is based on less accurate information. It is worth mentioning though, that average efficiency in all cases is very high (above 93.5%). Last, we note the significant coefficients for tin all regressions indicate that retailers tend to inflate less their forecasts over time, the CP discounts more the received signals and sets a quantity closer to the system optimal and system efficiency increases over time. 88 What is the value of information transmission and does the benefit of risk pooling exist under information asymmetry? In this section we first investigate what is the value of communication among the supply chain parties, even if information transmitted is not fully reliable. For this reason, we compare system profits under each case to profits had the CP ignored the signal(s) sent by the retailer(s). We compute the system profits had the CP set the inventory optimally based only on her knowledge about the mean demand in each region and the distribution of signals and market uncertainty, if any. The realized average experimental profits under each treatment were 15.6%, 27.0% and 23.1% higher than the computed profits with no information transmission, for case R1DU,R2DU and R2DK,respectively. Wehaveevidencethatthereissignificantvalueinsupply chain communication, even when parties do not fully cooperate. This observation is consistent with the finding that the demand signal transmitted is partially informative and the CP that sets inventory partially incorporates it in her decision to set common inventory. We also examine if the risk pooling benefits of centralization survive information asymmetry when information transmission is not fully reliable. For this reason, we compare the average profits of retailer 1 under case R1DU(n= 1) and under case R2DU(n= 2). Under both these cases, by design, retailer 1 sees the same demand streams while market uncertainty after information transmission suggests that there may be benefits from demand risk pooling when n= 2 and common inventory is held centrally. We observe, instead, that the average, over all rounds, experimental profits of retailer 1 are lower under case R2DUthan that under case R1DU($6,353 versus $6,362). Additionally, we compare the average profits per round of retailer 1 under case R1DUand case R2DUand we cannot reject that they are not different (twosided Wilcoxon rank test). This is consistent with the finding that system efficiency decreases when there is competition for common inventory. Even if inventory pooling increases expected profits under common knowledge (e.g., from $6,539 to $6,684 for retailer 1) the actual benefits under information asymmetry may be inexistent or even 89 negative due to a decrease in cooperation and quality of information the inventory decision is based on. In short, our experimental results suggest that there is value in information transmission, in all cases studied, even when theory predicts completely uninformative communication between the parties (case R2DU). Other things being equal (i.e., number of retailers, market uncertainty) it is always beneficial to allow communication between the more informed retailer(s) and the CP who sets inventory. On the other hand, we observe that the increase in information distortion, when we move from a single retailer to two retailers setting, out-weights the benefit of demand uncertainty pooling. Our results suggest that when it comes to deciding on inventory pooling in practice, supply chain parties should be aware that pooling may not always be a beneficial strategy under information asymmetry among players. 4.6 Concluding remarks In this chapter, we study whether demand forecast sharing between retailers, who are better informed about local demand, and the central planner, who sets common system inventory, is reliable both in theory and in practice. We want to further investigate the influence of supply chain environmental factors on trust and performance. These factors include the number of retailers, market uncertainty, and level of automation. We find that in the communication game between a single retailer and a benevolent CP, truthful information sharing and full trust is an equilibrium. However, when we consider multiple retailers, players’ incentives do not coincide and therefore a truthful information sharing equilibrium is not sustainable, unless market uncertainty is resolved before information transmission (i.e., demand uncertainty is zero after local market signal is received by the retailer). The difference between individually and system-wide optimal inventory quantity is attributed to two factors: a) a retailer has partial information about total demand and therefore different belief about its 90 distribution and b) the expected ratio of a retailer’s own demand to total demand in case of shortage and in case of surplus may not be equal. Furthermore, we find that as the size of the coalition approaches infinity, truth-telling becomes sustainable again. Most importantly though, the results of our extensive numerical analysis show that the difference between the optimal inventory quantity from the point of view of a retailer and that of the system is extremely small. This implies that players’ incentives are not far apart and information distortion could be in practice minimal. Our experimental results suggest that a continuum of trust exists both when pecuniary incentives are aligned and misaligned. Experimental data refutes the extreme theoretical cases of fully trustworthy or fully not trustworthy retailers, but it suggests a directional shift of information reliability consistent with theory. To be more specific, both competition for common inventory and forecast uncertainty harm truthtelling, trust and cooperation (measured by the resulting system efficiency). Despite the fact that information is not fully reliable, in all our cases, the value of communication was significant. On the other hand, we observed that actual inventory pooling benefits may be inexistent or even negative due to a decrease in cooperation and quality of information the inventory decision is based on. 91 92 Chapter 5 Forecast Information Sharing and the Order Quantity Decision: Impact of Inventory Ownership 5.1 Introduction In the previous chapter, we show that when retailers compete for common inventory they will have, in general, an incentive to misreport private information about their local demands to the decision maker who sets total inventory, as long as demand forecasts are not perfectly accurate (part of demand uncertainty remains unresolved when the communication takes place). In this chapter, adopting a similar information structure setting, we study the role of inventory ownership on individual players’ incentives. We compare the resulting inventory levels in the system under local and central decision making, when each unit of inventory in the system, even if held centrally, belongs to a specific retailer. We also study, if in such a setting of allocation “guarantees”, through dedicated inventories, credible information sharing between players forms an equilibrium. To do so, we model a profit-maximizing firm that sells its product in two horizontal markets (e.g., geographical regions) that are subject to demand uncertainty. The firm 93 has to decide on its inventory structure that determines who makes the inventory quantity decisions for each market. In all cases, inventory for both regions is held centrally. But for each region, a separate dedicated inventory quantity is held. In other words, each unit of inventory in the system belongs to a specific market, even before demand is known. After local demands are realized and before inventory is sent to each market, we allow for change of inventory ownership between regions. In such a setting, a minimum quantity of inventory allocation is guaranteed for each market in case of shortage, and a maximum quantity is guaranteed in case of surplus. Compared to the proportional rule studied in chapter 4, this allocation rule reduces uncertainty for locations in regards to their final allocation. With the proportional to realized demands allocation there is neither maximum nor minimum (other than zero) quantity that a location can be allocated. This will depend not only on final demand realizations but also on the total inventory set by the CP based on her beliefs about demand (that in turn may depend on overoptimistic or very pessimistic forecasts from other locations). As in the previous chapter, our model differentiates between demand information that is available to all (e.g., past sales data) and local knowledge that is available only to the regional managers (e.g., “feel” about the market, knowledge about trends in colors, styles, sizes, etc.). Local knowledge can be communicated efficiently to the central decision maker (without cost) but maybe untruthfully if the regional managers have an incentive to do so. We assume that each region is a separate business unit (local profit maximizer). We focus on the impact of who holds the right to manage inventory (to determine the inventories to be held centrally) on supply chain players’ incentives and strategic interaction. We employ a game theoretical model of information sharing to explore (a) how the placement of inventory decision rights (central versus regional) influences the total inventory level, and (b) whether a transfer pricing mechanism can be designed 94 that incentivizes regional managers to truthfully share their local knowledge about demand in their regions. 5.2 The setting We consider two distinct locations (geographical regions), indexed i, j =1,2, that face stochastic demands and hold stocks at a central location (newsvendor type problem). Each regional manager gains more knowledge about his local demand as time unfolds, which is captured through a demand signal. The demand at each region is given by di=µi+θi+�i(as described in section 3.2). We consider two inventory arrangements: locally managed inventory (LMI) where inventory decision making is done locally and central planer managed inventory (CPMI), where the right to decide the level of inventory is transferred to the central planer. Under both arrangements, inventory is held at a central location until regional demands are realized; we separate the ownership (with decision rights) and the location of inventories in the system. We use the notion of claims (see Anupindi et al, 2001) [2] that establish ownership for each unit of inventory in the system, regardless of its location. The claims give the units’ owner ex ante decision rights regarding its level and ex post decision rights regarding its usage. Under LMI, each regional manager (RM) determines and owns the quantity to be held centrally for him (dedicated inventory). In other words, the regional manager has a claim to the units of inventory held for him in the central warehouse. After demands are realized, a region’s local demand that exceeds its available stock at the central location is satisfied using excess stocks, if any, belonging to the other region. Under CPMI inventory arrangement, the central planner (CP) solicits local demand information from the regional managers (RMs) and sets the inventory quantity that maximizes total system profit. We consider the case where the CP holds for each region a dedicated inventory quantity. The sum of these quantities maximizes total expected profit for the system, given the belief of the CP about regional demands. 95 Regional managers know the way their inventory is calculated as a function of the CP’s beliefs. To be more specific, to study the information sharing game between RMs and the CP under CPMI, we assume a certain partition of the expected profit maximizing total inventory quantity into dedicated regional inventories. This allows us to study how the behavior of RMs changes with inventory “guarantees”. Inventory is shipped to each region after local demands are realized. We allow for “change of ownership” of regional stocks. In other words, each retailer is guaranteed its dedicated inventory quantity if needed, but change of inventory ownership is allowed, after demand uncertainty is resolved, to balance supply and demand in different regions (exactly as in RM case). We note that the inventory arrangement and the inventory claims regime are all agreed ex ante, before RMs receive their private information. It is a “contract” signed under symmetric information. However, the decision on total inventory is taken under asymmetric information. This is the interim stage of the game. Under LMI, when RMs determine their inventories they have incomplete information. Each RM has received his demand signal but he does not know the precise characteristics (demand signal received) of the other player. Market uncertainty is still unresolved for both players (�i’s are random). Similarly, under CPMI, the CP has incomplete information when setting inventories, as she does not observe local demand signals. The allocation of inventory (final shipments to regional markets) is done ex post, under perfect information; market uncertainty is realized and local demands become common knowledge to all players. Figure 5-1 presents the events and their timing under the two inventory decision arrangements. For comparison purposes, we use the two extreme cases (benchmarks): pure decentralized inventory structure where regional managers do not cooperate in managing their inventories and central decision making with complete information (demand signals are common knowledge to all players). In all cases, the CP is benevolent in the sense of total system profit maximizer. 96 Theorem 5.1:For0<c<p, a pure strategy Bayesian Nash Equilibrium exists. Any pair of functions (qLMI i(θi),qLMI j(θj)) that satisfy (5.14) for i=1,2isaBayesian Nash equilibrium: ηi(qLMI i(θi),θ i)+p−c pEθj[γi(qLMI i(θi),qLMI j(θj),θ i,θ j)] −c pEθj[βi(qLMI i(θi),qLMI j(θj),θ i,θ j)] = p−c p(5.14) According to Theorem 5.1, finding an equilibrium in this game requires solving 2 integral equations simultaneously with variable limits. Solving the system of equations to obtain a closed form solution is not possible. Using Proposition 5.1, we know that the optimal inventory choice is monotonic as well. We continue by characterizing the functional form of the optimal inventory choice, as a function of the received signal. Proposition 5.2: In all equilibria, the inventory choice function of location iis linear in its own received signal and more specifically of the form qLMI i(θi)=µi+θi+δi. It is interesting to note that if the received signal increases by xunits, the optimal inventory choice at location iwill increase by the same amount. The received signal determines the mean demand at location iand it is incorporated in location’s optimal inventory choice as it is (one-to-one relationship), no matter its magnitude (e.g., regardless of whether it is low or high). In addition, multiplying equilibrium conditions by fi(θi) and integrating both sides over θiwe get that E[ηi(qi,θ i)] + p−c pE[γi(qi,q j,θ i,θ j)] −c pE[βi(qi,q j,θ i,θ j)] = p−c p(5.15) From (5.15) we see that under information asymmetry, the optimality conditions for a Bayesian Nash equilibrium is the same as the one under full information where the corresponding probabilities of inventory shortage and surplus are calculated over all possible demand signal realizations. 103 Note that the equilibrium functions described in Theorem 5.1 may result in multiple equilibria that involve different inventory choice functions for each player. In our setting, the two locations have the same revenue and price parameters. If we additionally assume that demand at each location, before the signal transmission, follows the same distribution (i.e., the average market size at each location is the same (µi=µj) and signal and market uncertainty at each location follow the same distribution (fi(·)=fj(·), gi(·)=gj(·)), then we can focus on symmetric equilibria, defined as follows: Corollary 5.1: The symmetric Bayesian Nash equilibrium (qLMI (θ1),qLMI (θ2)) satisfies condition (5.16) for all θi. ηi(qLMI (θi),θ i)+p−c pEθj[γi(qLMI (θi),qLMI (θj),θ i,θ j)]− c pEθj[βi(qLMI (θi),qLMI (θj),θ i,θ j)] = p−c p(5.16) 5.4 Central Planner Managed Inventory (CPMI) Under CPMI, inventory decisions are made centrally to maximize total profits. RMs receive their demand signal and send a report to the CP (communicate a demand forecast). The demand signal each RM receives for his region at the beginning of the selling season is private knowledge. Therefore, RMs can choose to communicate their signal truthfully or not. The CP in turn may use this information to determine inventory levels (q1,q 2) to maximize the sum of profits across locations, based on her beliefs about local demands. This is a case of centralized decision making with demand information asymmetry and information sharing (maybe non-credible) in the form of demand forecasts. The allocation dynamics are similar to the LMI case. Inventory held in the system (Q=q1+q2)isintheformofdedicatedinventoryforeachregionbutheldcentrally. After the CP makes the inventory decision, demands are realized (di’s) and inventory 104 is allocated according to the established policy. Each RM iis guaranteed qiif di≥qi and cannot be allocated more than qiif di<q i.Tobemorespecific,thefinal allocation that region igets (αi) is given by equation (5.2), where τij is defined as in (5.1). Hence, it satisfies the following relationships; αi≤qiif di<q i,αi≥qiif di>q i,αi=qiif di=qiand αi+αj=qi+qj. The expected value of total profits across the two locations, denoted as ΠCPMI,for given θi’s, is given by: ΠCPMI(qi,q j,θ i,θ j)=pE�i,�jmin[( ˜ di+˜ dj),(qi+qj)] −c(qi+qj) =E�i,�j[pmin(di,α i)−cαi+pmin(dj,α j)−cαj] =E�i,�j[psi−c(τji −τij)] −cqi+E�i,�j[psj−c(τij −τji)] −cqj =pE�i,�j(si+sj)−c(qi+qj) (5.17) By taking the derivative of the CP’s profit function and collecting the terms we have. ∂ΠCPMI(qi,q j,θ i,θ j) ∂qi =p(1 −Pr[ ˜ di<q i]−Pr[qi<˜ di<q i+qj−˜ dj] +Pr[qi+qj−˜ dj<˜ di<q i]) −c (5.18) The intuition parallels that of the Locally Managed Inventory with the main difference that the marginal unit generates revenue pwhen it is used to cover either excess demand at location i(that happens with probability (1 −Pr[ ˜ di<q i]−Pr[qi<˜ di< qi+qj−˜ dj]) as in the LMI case) or excess demand at j(which happens when there is excess stock in iand shortage in j,i.e.,whenqi+qj−˜ dj<˜ di<q i). Furthermore, the marginal cost is cwith probability 1 (no matter whether or where it is sold). 105 By setting (5.18) to zero we get the first order necessary optimality condition for qi,i=1,2. Therefore, the profit-maximizing inventory choices needs to satisfy, for i=1,2, the condition (we later show that this condition is also sufficient) ηi(qi,θ i)+γi(qi,q j,θ i,θ j)−βi(qi,q j,θ i,θ j)=p−c p(5.19) We note that of the left side condition (5.19) can be re-written as: ηi+γi−βi=Pr[˜ di<q i]+Pr[qi<˜ di∩qi+qj>˜ di+˜ dj]−Pr[ ˜ di<q i∩qi+qj<˜ di+˜ dj] =Pr[˜ di<q i∩qi+qj>˜ di+˜ dj]+Pr[qi<˜ di∩qi+qj>˜ di+˜ dj] =Pr[˜ di+˜ dj<q i+qj] (5.20) In other words, condition (5.19) for iand j, boils down to the single optimality condition Pr[ ˜ di+˜ dj<Q]=p−c p(5.21) where Q=q1+q2. This is the well known newsvendor critical fractile optimality condition, with demand being the sum of local demands. Because the expected profit is concave in Q,itisalsoconcaveinqi(composition of a concave function with an affine function). Therefore, first-order conditions (5.21) and (5.19) are sufficient for optimality. In the case of CPMI, what matters is the total inventory held centrally because (a) retailers are identical in their cost/revenue parameters and (b) inventory is held centrally and sent to retailers after demands are realized (there are no transshipment costs). QCPMI that maximizes system profits satisfies the optimality condition (5.21); QCPMI =µi+µj+θi+θj+(Gi◦Gj)−1(p−c p). Therefore, the inventory choices (qCPMI i,qCPMI j) that the CP can make to maximize total profits, are infinite. Any combination of qi,qjthat satisfies qi+qj=QCPMI is an optimal solution from the system’s perspective. 106 For expositional purposes, we plot total profits as a function of q1and q2,forthe case where p=2,c= 1 (i.e., critical fractile is 0.5), the average demand in each market is 10 units, θ1+θ2= 0 and market uncertainty in each region is independent and distributed normally, with mean 0 and variance 2 (Figure 5-2). In this case it is optimal from a total profit perspective to set QCPMI = 20. As we observe from the figure, any combination qiand qjthat adds up to 20 units maximizes the expected profit function. Figure 5-2: Total Expected Profit as a function of q1and q2 For given θi’s, qCPMI ican take any value in the interval [0,Q CPMI]. This implies, that before inventory is decided by the CP, and most importantly when information transmission takes place, there are no minimum or maximum inventory guarantees for retailer i. Since there is no unique optimal solution (qCPMI i,qCPMI j), retailers cannot infer the inventory choices of the CP, given her beliefs about local demands. 107 To overcome this issue, we need to specify how the CP decides inventory ownership of QCPMI units between the two locations. For the rest of the chapter, where we study the information sharing game between locations and the CP, we assume that qCPMI i=µi+b(θi)+s,∀i,whereb(θi)denotesthebeliefoftheCPaboutθiand s=(Gi◦Gj)−1(p−c p) 2. Please note that scan be either positive or negative, depending on the critical ratio. 5.4.1 Is truth-telling an equilibrium? When all parties have the same information about demand (common knowledge setting), in the case of CPMI where inventory decisions are centrally coordinated, by definition the aggregate profits across locations are maximized. But under information asymmetry, is this still the case? When the CP solicits private demand information from the RMs and then she sets the inventory quantities (qCPMI i,qCPMI j) to maximize system profits, would RMs have an incentive to truthfully communicate their demand information? To answer this question, we first study whether the incentives of a RM and the CP coincide under common knowledge (all players know the demand signal realization in both regions). In other words, given θiand θj,wecompareqLMI i(qj)toqCPMI i(qj). For notational convenience, for the rest of the section we denote βi(qi,q j,θ i,θ j)and γi(qi,q j,θ i,θ j)byβiand γi,respectively. Proposition 5.3:Forgivenθi,θj,qCPMI i�=qLMI i,∀qjand i=1,2, unless γi βi= p−c cfor some qj, where the probabilities βiand γiare evaluated at (qCPMI i,q j)and qCPMI i=QCPMI −qj. Proposition 5.3 states that locally and centrally optimal inventory choice for region i,giventheinventoryofregionj,willbedifferent, unless the probability that excess demand at region ican be covered by excess inventory at region jover the probability that excess supply at region ican be used to satisfy excess demand at region jequals 108 the markup (profit margin as a percentage of the cost). Both these probabilities are evaluated at the inventory choice qithat maximizes aggregate profits, given inventory for region j. It is therefore implied that if each regional manager had the right to set his own inventory, knowing both his demand signal and that of the other location, he would have chosen a different quantity than the one the CP will set for him. Keeping qjfixed, qCPMI idoes not maximize RMi’s profit as by definition that quantity is qLMI igiven by equation (5.8). This result is similar to the case of proportional allocation under common knowledge (Lemma 4.1). In both cases, when demand signals are common knowledge, the quantity that a retailer / regional manager prefers is different to the one that maximizes system profits, unless special conditions happen to hold. Corollary 5.3: Under common knowledge, if RMs had the right to decide their inventories, none of the centrally chosen inventory choices (qCPMI i,qCPMI j) that maximize system profits would form an equilibrium, unless γi βi=γj βj=p−c cevaluated at some (qCPMI i,qCPMI j). To see this, let us assume that qCPMI j(qi)=qLMI j(qi) for some qi. That would mean that the CP chooses an inventory quantity for region jso that its regional expected profit is maximized, given the inventory choice for location i.Theinventorychoiceof the CP for region i, in turn, needs to be such that the aggregate profits are maximized (qCPMI i=QCPMI −qLMI j). But according to proposition 5.3, for any given qj(and thus also qj=qLMI j), the quantity that maximizes the expected profit of region i is different than the quantity that maximizes system profits (qLMI i�=qCPMI i)unless γi βi=p−c c. The same argument holds for the second region. In short, central optimal inventory choices will coincide with local choice equilibrium quantities under common knowledge if and only if cost and demand parameters are such that equate left-hand sites of Equations (5.8) and (5.19) for i=1,2. 109 But when regional manager ireports his local demand information (sends a signal ˆ θi), he knows θibut not θj. Therefore he cannot directly compare qLMI ito qCPMI i,had both players had the same information. Regional manager idoes not know qCPMI i,no matter the reporting strategy of the other regional manager and the CP’s beliefs given the reporting strategy. To study the information sharing game between a regional manager and the central planner, we assume that among the infinitely many optimal inventory choices given her demand beliefs, the CP sets qCPMI i=µi+b(θi)+s,for i=1,2. We next study whether truth-telling and trusting can form an equilibrium. Suppose that location isends a signal ˆ θi, which may or may not be the same as his true signal θi, but location jtransmits his true demand signal and the central planner believes that both locations transmit their true demand signals. As the next proposition shows, retailer iwill have, in general, an incentive to report his demand signal falsely. Proposition 5.4 holds for any qCPMI ithat depends on b(θi). Proposition 5.4: When regional manager jtruthfully reports his demand signal and the central planner trusts the received information, regional manager ihas an incentive to falsely report his demand signal, unless γi(qi,θi) β(qi,θi)=p−c pevaluated at qCPMI i(θi). Having analyzed regional managers’ inventory choices given their information versus inventories that maximize aggregate profits, we show that truthful information sharing of private demand information will not maximize, in general, the expected profits of regional managers (unless special conditions happen to hold). Therefore, reliable information sharing will not be a sustainable equilibrium in this setting as the next theorem formally states. Theorem 5.2:Letφ(ˆ θi|θi) denote regional manager’s ireporting strategy given θi, (qi(ˆ θi),q j(ˆ θj)) the inventory choices (to be held centrally) of the CP for regions iand jrespectively and b(θ|ˆ θ) the central planner’s posterior belief about θafter observing ˆ θ.Then, 110 -φ(ˆ θ|θ)=θ -(qi(ˆ θi,ˆ θj),q j(ˆ θj,ˆ θj)) = (qCPMI i(ˆ θi),qCPMI j(ˆ θi)) -b(θ|ˆ θ)=ˆ θ do not constitute a perfect Bayesian equilibrium. A perfectly informative Bayesian equilibrium of forecast sharing where regional managers share truthfully their private demand information and the CP trusts the information received and exactly incorporates it in her inventories choices does not exist. On the other extreme, it is easy to show that an uninformative (babbling) equilibrium exists (as in all cheap talk games). In this equilibrium, each regional manager’s reported forecast ˆ θiis independent of θi. The CP does not update her beliefs about local demands after receiving the forecasts. She determines the system optimal quantities based on her initial knowledge about the distribution of θi’s. 5.5 Numerical analysis To complement our analytical findings, we continue by conducting numerical analysis to compare optimal inventory choices, and the corresponding expected profits, under LMI and CPMI when there is information asymmetry and unreliable demand forecast sharing. We then compare the results to the case where the CP has the same knowledge with RMs about local demands (she knows local demand signals) and to the pure decentralized case (two separate newsvendors that keep their inventories separately and no transshipment policy is in place). Numerically solving for the equilibria defined by the integral equations (5.15) requires discretizing the distributions of θi’s to npoints and solving 2 ·nequations simultaneously. If we restrict ourselves to symmetric equilibria, the system of equations reduces to nbut computing the equilibria remains computationally challenging. Furthermore, when we move to the discrete case, as the number of types increases, the problem of estimating the probability of their realization also increases. For these reasons, in inventory management games with incomplete information it is advisable 111 to limit the number of types to a small number in order to successfully solve the problem and obtain useful insights [58]. Thus, for the numerical analysis, we just consider two types of possible demand signals, “Low” and “High”. We assume that θican take only two values, θHwith probability λand θLwith probability 1 −λand that they are independent across locations. Because of independence Pr[θj=θL|θi=θL]=Pr[θj=θL|θi=θH]=1−λ and Pr[θj=θH|θi=θL]=Pr[θj=θH|θi=θH]=λ.Wedenotetheinventorychoice of player ithat receives θLby qi(θL)=qiL,fori=1,2. Similarly, qi(θH)=qiH .The expected payofffor each player as a function of his type is: πLMI 1L(q1L,q 2(θ2)) = (1 −λ)πLMI 1L(q1L,q 2L;θ2L)+λπLMI 1L(q1L,q 2H;θ2H) πLMI 1H(q1H,q 2(θ2)) = (1 −λ)πLMI 1H(q1H,q 2L;θ2L)+λπLMI 1H(q1H,q 2H;θ2H) πLMI 2L(q1(θ1),q 2L)=(1−λ)πLMI 2L(q1L,q 2L;θ1L)+λπLMI 2L(q1H,q 2L;θ1H) πLMI 2H(q1(θ1),q 2H)=(1−λ)πLMI 2H(q1L,q 2H;θ1L)+λπLMI 2H(q1H,q 2H;θ1H) (5.22) To determine the Bayesian Nash equilibrium under LMI in this case, we need to solve the following system of four nonlinear equations with four unknowns: ∂ ∂q1LπLMI 1L(q1L,q 2(θ2)) = (1 −λ)∂ ∂q1LπLMI 1L(q1L,q 2L;θ2L)+λ∂ ∂q1LπLMI 1L(q1L,q 2H;θ2H) ∂ ∂q1HπLMI 1H(q1H,q 2(θ2)) = (1 −λ)∂ ∂q1HπLMI 1H(q1H,q 2L;θ2L)+λ∂ ∂q1HπLMI 1H(q1H,q 2H;θ2H) ∂ ∂q2LπLMI 2L(q1(θ1),q 2L)=(1−λ)∂ ∂q2LπLMI 2L(q1L,q 2L;θ1L)+λ∂ ∂q2LπLMI 2L(q1H,q 2L;θ1H) ∂ ∂q2HπLMI 2H(q1(θ1),q 2H)=(1−λ)∂ ∂q2HπLMI 2H(q1L,q 2H;θ1L)+λ∂ ∂q2HπLMI 2H(q1H,q 2H;θ1H) (5.23) where ∂ ∂qiL πLMI iL (qiL,q jL;θjL)isgivenby(5.7)whereθi=θL,θj=θL,qi=qiL and qj=qjL and ∂ ∂qiL πLMI iL (qiL,q jH;θjH)isgivenby(5.7)whereθi=θL,θj=θH,qi=qiL and qj=qjH. Similarly, for ∂ ∂qiH πLMI iH (qiH ,q jL;θjL)and ∂ ∂qiH πLMI iH (qiH ,q jH;θjH), with the difference that θi=θH,fori=1,2. We compute explicit solutions for the case where demand at the two locations is distributed independently or Cov(θi,θ j)=0,Cov(�i,� j)=0,fori, j =1,2and i�=jand Cov(θi,� j)=0,fori, j =1,2. It is also assumed that retailers are 112 Hence, their optimal inventory choices coincide. This result sheds light to when there is misalignment of incentives between regional managers and the CP; when the critical fractile is different than 0.5 and RMs have private information about their regional demands, the difference between locally and centrally optimal inventory quantities may result in unreliable information sharing. The next question that we would like to answer is whether a transfer pricing mechanism exists that leads to an alignment of incentives and therefore induces reliable information sharing between the players. That is the topic of the next section. 5.6 Coordinating transfer prices In the previous section, change of inventory ownership, after demands are realized, costs the region that gets the additional unit cand generates revenue crespectively to the region that gives the unit. It is therefore implicitly assumed that a profit margin (p−c) is earned by the market where the sale took place, no matter if the product unit sold was initially owned by the other location. In addition, the location that gives out one unit of excess inventory to the market that is stocked out recovers the full procurement cost c. In this section we study the case where when a unit of excess inventory owned by location iis used to cover excess demand at location j, the additional revenue from the sale pis split arbitrarily between the two locations. Is there a way to split the additional revenue so that truthful demand information sharing is induced? To model this situation, we denote by cij the price location icharges location jfor a unit sold at jthat was owned initially by location i. Location jfinds it profitable to use an excess unit initially owned by ito generate a sale at market jwhen the latter is stocked out, only when cij <p. After demands are realized, region iis willing to ”sell” to region jany excess units as long as cij >0. In other words, change of ownership from market ito market jis mutually profitable only whenever there is 119 excess demand at location jand excess supply at location i.Inthiscase,locationi gains from the additional sale cij and location jgains p−cij. In such a setting, under LMI, the expected profit of region i,asafunctionof inventory quantities, for given θiand θj,isgivenbytheexpression: πLMI i(qi,q j,θ i,θ j)=E�i,�j[psi+cijτij −cjiτji]−cqi(5.24) By taking the derivative with respect to qiwe get: ∂πLMI i(qi,q j,θ i,θ j) ∂qi =p(1 −Pr[ ˜ di<q i]−Pr[qi<˜ di<q i+qj−˜ dj]) +cji Pr[qi<˜ di<q i+qj−˜ dj] +cij Pr[qi+qj−˜ dj<˜ di<q i] −c (5.25) The first term is the expected additional revenue from having one more inventory unit available at site i, as in expression (5.7). The difference with (5.7) is that now an additional cost cis incurred with probability 1, while the additional second and third term denote the savings from not transferring one unit from market j(that happens if there is excess demand at iand inventory surplus at j)andtherevenuecij from selling one unit to region j(that happens when there is inventory surplus at iwhile shortage at j). Following the same notation, the first-order sufficient optimality condition, for region i, after demand signals are realized, becomes: ηi(qi,θ i)+p−cji pγi(qi,q j,θ i,θ j)−cij pβi(qi,q j,θ i,θ j)=p−c p(5.26) Under CPMI, optimality conditions (5.24) do not change. From a central planner’s perspective, it does not matter how profit is split between the two regions. 120 Does it exist a pair of transfer prices between the two markets that could induce truthful sharing of private demand information between RMs and the central planner that sets inventories? To answer this, we are looking for transfer prices that align individual and system incentives; which make the optimal quantities set by the CP to simultaneously satisfy the individual optimality conditions of the RMs. In other words, we are looking for transfer prices that under LMI would result in inventory equilibrium quantities equal to the quantities chosen by the CP as total profit maximizing. Theorem 5.3:Foragivenpairofreceivedsignals(θi,θ j), there exist transfer prices cij,fori, j =1,2 that align individual locations’ and company’s incentives. These are given by cij =( βjγi−βiβj γiγj−βiβj )p(5.27) where these probabilities are evaluated at a chosen (qCPMI i,qCPMI j). Theorem 5.3 implies that there are infinitely many pairs of transfer prices that achieve alignment of incentives. For a any pair (qCPMI i,qCPMI j)chosenbytheCP,such that qCPMI i+qCPMI j=QCPM,auniquesetoftransferpricescanbecalculated.These transfer prices make system optimal inventory choices also optimal for individual locations. However, the CP does not know ex-ante QCPM,becauseshedoesnot know the demand signal realizations. She can only credibly commit to inventory choices that maximize individual and system profits, given the transfer prices set. In other words, we model the case where individuals locations report their demand signals to the CP considering the direct effect on inventories and not the indirect one on transfer prices between locations. Inventory choices of the CP are system wide and locally optimal, given the transfer prices set after information transmission. But would locations have an incentive to misreport reported signals in order to influence the transfer prices set by the CP? That would depend on the mechanism used to split QCPMI ownership between the two locations (as a function of reported demand 121 signals) which in turn would determine the corresponding transfer prices. We consider it an interesting extension for future work. 5.7 Concluding remarks In this chapter we compare local to central inventory decision making, when the latter is based on less accurate demand information. We also study whether under central inventory coordination, truthful information sharing is expected between regional managers that have better information about their local demands and the central planner that sets inventory. Compared to chapter 4, we consider the case where a scheme with inventory guarantees is in place through dedicated inventories for each region. Under LMI, each region determines its inventory that is held centrally before the beginning of the selling season. When we consider two inventory locations, a pure strategy Bayesian equilibrium exists. In all equilibria, the inventory choice of a location is increasing monotonically in its received signal (i.e., in its average demand). Under CPMI, what matters for the CP is the total inventory held for both locations and not how this is split between them. Hence, when each regional manager sends his demand information to the CP, he cannot anticipate her inventory choice for his region. We show though, that under common knowledge, the system optimal quantity cannot be split between the two regions (dedicated inventories) so that local and system incentives are aligned, unless special conditions happen to hold. Furthermore, we show that truth-telling and trusting do not, in general, constitute a Perfect Bayesian equilibrium under information asymmetry. We proceed by doing numerical analysis to compare the resulting inventories and the expected profits under LMI and CPMI (considering the babbling equilibrium). In our numerical examples, when the critical fractile is below 0.5, LMI results in higher inventories, on average, than CPMI (and than system optimal inventory under common knowledge). When the critical fractile is above 0.5, the directional results 122 are reverse. Also, we show that, unlike CPMI case, regional demand information asymmetry does not create any additional system inefficiency under LMI. Regarding expected profits, no inferences can be made; which decision making arrangement is preferred will depend on the relative value of local information compared to that of central coordination of inventory choices. Last, even if it is easy to show that a pair of prices, for unit ownership transfers between locations, exists that aligns central and local incentives under common knowledge, it remains an open question if or how such a system could be implemented by the CP, under CPMI and asymmetric information, to incentivize local managers to report truthfully their demand information. 123 124 Chapter 6 Conclusions and Discussion In this thesis, we studied the issue of demand information sharing – forecasts and realized demands – within an inventory pooling coalition. Even though the value of information is a topic well-studied in operations and supply chain management literature, it is usually assumed that information sharing, when it happens, is done in a credible way. On the other hand, many well-documented failures in businesses are due to misreporting of private information, such as order exaggeration in anticipation of inventory shortage or overoptimistic soft orders that never materialize. We focus our work on whether reliable demand information sharing occurs between retailers and a benevolent central planner (CP) who coordinates ordering and inventory allocation within an inventory pooling coalition. Retailers do not compete for demand but they may compete for inventory. Each retailer has some private information about demand in his region due to his proximity to the market, that may transmit truthfully or not to the central planner. First, we study analytically the impact of various allocation mechanisms on the ordering behavior of retailers, after total inventory quantity in the central warehouse is set. To do so, we first show that when allocation is based on realized demands (i.e., realized demands become common knowledge to all players), all allocation rules considered, i.e., proportional, linear and uniform are efficient (they exclude wastage) and Pareto optimal. But when realized demands in each region remain private knowledge 125 to the retailers, the allocation rule employed plays an important role. Only under a uniform allocation rule, retailers will report their true needs by placing a final order equal to their realized demand. This result is analogous to the typical capacity rationing case, even if in the setting under consideration (a) the allocation is determined after demand uncertainty is resolved and (b) each retailer may receive both below or above his final order. The main difference is that uniform allocation based on final orders is both truth inducing and Pareto optimal in our case. In the capacity rationing case, uniform allocation is not Pareto optimal, because it is not individually responsive, a necessary condition for Pareto optimality. We further propose a modified uniform allocation rule that not only is Pareto optimal and truth inducing, but also it guarantees each retailer a profit higher than what he would have earned in a pure decentralized system, under any demand realization. Next, we proceed by studying analytically and experimentally demand forecast sharing between retailers and the CP who solicits this information to set total inventory. Realized demands become common knowledge when allocation takes place and we focus on the game of demand signal (forecast) reporting to influence the inventory held in the system. We consider the game where a proportional (to local realized demand) allocation mechanism is employed: a mechanism widely used in practice with several attractive properties. Using game theoretical models, we find that when there is unresolved demand uncertainty when communication takes place, truth-telling and trusting do not form a Perfect Bayesian equilibrium. In addition, under an automated inventory system that takes as input the forecasts reported by the retailers and orders the optimal inventory level for the whole coalition, a pure strategy Bayesian Nash equilibrium among retailers does not exist. We then study, in a controlled laboratory environment that simulates the supply chain setting into consideration, the impact of a) competition for common inventory and b) market uncertainty on information distortion, trust and supply chain efficiency. Our results suggest that a continuum of trust exists both when pecuniary incentives are aligned or misaligned, refuting the extreme theoretical cases of fully trustworthy or fully non126 trustworthy retailers. Further, we find that both competition for common inventory and forecast uncertainty harm significantly truth-telling and cooperation among supply chain parties. That’s why, inventory pooling under information asymmetry may have negative results despite demand risk aggregation. Even if information was not fully reliable, the value of communication was significant in all our experiments. Last, we study the impact of inventory ownership on the incentives of the players to truthfully share their forecasts. We find that dedicated inventories do not induce truth-telling either. When we consider two separate locations that locally decide on their level of inventories and take into account the possibility of inventory ownership transfers after demands are realized, there is a unique Bayesian Nash equilibrium. The optimal inventory choice of a location is increasing in its received demand signal. When the CP makes the ordering decision, what matters is only the total inventory held. Unless special conditions happen to hold, it cannot be split between the two locations, before demand is realized, in a way that local and system incentives are aligned. We numerically compare resulting inventories and profits under local decision making with more accurate information versus central decision making where coordination of orders is achieved. We find that when the critical fractile is high, central decision making results in higher inventories while the reverse is true when the critical fractile is low. Expected profit directional comparisons depend on the value of local information that is lost when we move to the central decision making versus the additional value of inventory coordination (in the the babbling equilibrium). This work has several limitations given the analytical complexity of the problem. We study separately the inventory allocation game when final demands are not known to the central planner and that of demand forecast information sharing to influence the inventory level of the coalition. It remains an open question what the interactions would be when both issues are considered together. For example, if the final allocation is tied to the forecast reported, how would the dynamics of forecast information 127 sharing change? What would be the impact on retailers’ ordering behavior and would the final allocation be efficient? There are several interesting extensions of this thesis. To begin with, we could further explore the impact of behavioral factors in the demand forecast sharing game. When the inventory is common, it is interesting to investigate how the size of the pooling coalition and that of retailers affects trusting relationships. Furthermore, we would like to study whether the level of trustworthiness of retailers changes when there is a guarantee about how their forecasts are used for setting the common inventory level. We consider this an interesting question with potentially very relevant managerial implications. As future research, we are planning to run additional treatments where the number of retailers increases to 3 and 4, the CP is automated and retailers are non-identical. It would be also interesting to experimentally study how inventory ownership impacts retailers’ forecast reporting strategy. Even if dedicated inventories do not align individual and system pecuniary incentives, would reduced uncertainty with regards to the final allocation increase retailers’ trustworthiness and enhance cooperation? A second topic of interest is to study through behavioral experiments the impact of allocation mechanisms on retailers’ ordering behavior to shed light on potentially “missing” components in this interaction. Do equilibrium concepts, which assume that players are perfectly rational, substantially exaggerate retailers’ tendency to strategically order more / less than what they need? Under which allocation mechanisms is order distortion more pronounced? Another path of future research is to study demand information sharing in an inventory pooling coalition and focus on the behavioral implications when placing soft orders versus sharing demand forecasts. We consider again that each retailer has better demand information due to his proximity to the market and he shares it (maybe untruthfully) with the CP either with the form of forecast sharing (sending his demand signal) or with the form of a non-binding order before demand is realized. How 128