Performance-based pay and limited information access: An agent-based model of the hidden action problem

Reinwald, Patrick; Leitner, Stephan; Wall, Friederike Article Performance-based pay and limited information access: An agent-based model of the hidden action problem Journal of Economics and Statistics Provided in Cooperation with: De Gruyter Brill Suggested Citation: Reinwald, Patrick; Leitner, Stephan; Wall, Friederike (2024) : Performance-based pay and limited information access: An agent-based model of the hidden action problem, Journal of Economics and Statistics, ISSN 2366-049X, De Gruyter Oldenbourg, Berlin, Vol. 244, Iss. 4, pp. 381-423, https://doi.org/10.1515/jbnst-2023-0101 This Version is available at: https://hdl.handle.net/10419/333289 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Patrick Reinwald, Stephan Leitner* and Friederike Wall Performance-Based Pay and Limited Information Access. An Agent-Based Model of the Hidden Action Problem https://doi.org/10.1515/jbnst-2023-0101 Received November 30, 2023; accepted July 26, 2024 Abstract: Models involving human decision-makers often include idealized assumptions, such as rationality, perfect foresight, and access to relevant information. These assumptions usually assure the models’internal validity but, at the same time, might limit the models’power to explain empirical phenomena. This paper addresses the well-known model of the hidden action problem, which proposes an optimal performance-based sharing rule for situations in which a principal assigns a task to an agent and the task outcome is shared between the two parties. The principal cannot observe the action taken by the agent to carry out this task. We introduce an agent-based version of this problem in which we relax some of the idealized assumptions. In the proposed model, the principal and the agent only have limited information access and are endowed with the ability to gain, store and retrieve information from their (finite) memory. We follow an evolutionary approach and analyze how the principal’s and the agent’s decisions affect their respective utilities, the sharing rule, and task performance over time. The results suggest that the optimal (or a close-to-optimal) sharing rule does not necessarily emerge in all cases. The results indicate that the principal’s utility is relatively robust to variations in memory. On the contrary, the agent’s utility is significantly affected by limitations in the principal’s memory, whereas the agent’s memory appears to only have a minor effect. Keywords: robustness; replication; limited rationality; agent-based modeling; simulation JEL Classification: M20; C63; D00 *Corresponding author: Stephan Leitner, University of Klagenfurt, Universitätsstraße 65–67, 9020 Klagenfurt, Austria, E-mail: [email protected]. https://orcid.org/0000-0001-6790-4651 Patrick Reinwald and Friederike Wall, University of Klagenfurt, Universitätsstraße 65–67, 9020 Klagenfurt, Austria. https://orcid.org/0000-0002-2907-7939 (P. Reinwald). https://orcid.org/0000-00018001-8558 (F. Wall) Journal of Economics and Statistics 2024; 244(4): 381–423 Open Access. © 2024 the author(s), published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 1 Introduction In the history of research on behavioral control, the concept of rational expectations emerged as the dominant paradigm (Muth 1961), which went hand in hand with the extensive development of sophisticated mathematical methods and models with sometimes idealized assumptions about individuals. Up to today, such approaches have played an essential role in many fields, such as management and economics. Most of these models assume agents who are rational in their behavior, employ sophisticated optimization methods, usually possess all (or at least most) pieces of information to come up with optimal solutions immediately, and mostly make no errors in doing so (Farmer and Foley 2009; Thaler 2000). However, it has already been recognized that technically correct and valid models often lack the power to explain empirical phenomena (Franco et al. 2020) and calls to include behavioral insights from other disciplines –such as cognitive psychology –have emerged (Hämäläinen, Luoma, and Saarinen 2013; Leitner and Behrens 2015a; Royston 2013; Wall, Chen, and Leitner 2024; Wall and Leitner 2021). Franco and Hämäläinen (2016) point out that increased attention to behavioral aspects becomes prominent whenever scientificfields reach maturity and argue that, amongst others, this is the case in economics, accounting, and strategic management. We place our research in the stream of research that analyzes the robustness of economic models to relaxations of the included and often restrictive assumptions. In particular, in our research we focus on the assumptions of information availability (and limitations thereof) in the well-known model of the hidden action problem introduced by Holmström (1979). This model is based on a principal who assigns a task to an agent. The agent acts on behalf of the principal by makingan effort to carry out the task assigned to him, while the principal’s role is to provide capital and incentives. The modeled situation is further characterized by information asymmetry in the form of hidden actions. The principal can only observe the task outcome but not the action taken by the agent, which is why the principal can only employ a performance-based compensation mechanism. The model provides this mechanism by creating a rule to optimally share the task outcome between the principal and the agent (Caillaud and Hermalin 2000; Eisenhardt 1988; Lambert 2001). The assumptions considered in this model include what Axtell (2007) refers to as the economic sweet spot: Rationality, homogeneity, and equilibrium solutions. We are particularly interested in the conditions of rationality that also include issues related to information availability. Economic models often implicitly follow the idea that decision-makers possess the necessary capabilities to access all relevant pieces of information. Keeping in mind that information can take different forms, such as complex texts, multimedia 382 P. Reinwald et al. materials, or information that is gathered by observing the environment, these required capabilities are truly manifold and rich (McCreadie and Rice 1999). This renders the assumptions implicitly included in economic models nontrivial. In a microeconomic context, Leitner, Rausch, and Behrens (2017) and Leitner, Brauneis, and Rausch (2015) analyze the effects of limited information in capital budgeting by modeling overconfident decision-makers, and principals and agents with limited foresight, respectively. In the context of the hidden action problem, Leitner and Wall (2021, 2022) limit the principal’s and the agent’s respective capabilities to search for the optimal sharing rule. In the vein of this related research, we transfer the hidden action model into an agent-based model (Guerrero and Axtell 2011; Leitner and Behrens 2015a). By doing so, we propose an agent-based model of the hidden action problem with principals and agents who suffer from limitations in information access. We focus on the following questions: How do limitations in the principal’s and the agent’s information availability in the context of the hidden action problem affect (i) the rule to share the task outcome between the principal and the agent, (ii) the agent’seffort, and (iii) the principal’s and the agent’s respective utilities. To demonstrate that our agent-based model effectively integrates the hidden action model, we first establish that the solutions derived from our model align with those suggested by Holmström (1979) under conditions of unlimited information for both the principal and the agent. When information is restricted, our findings suggest that the allocation of task outcomes between the principal and the agent depends solely on the principal’s information. Specifically, greater informational access for the principal leads to an increased share of outcomes for the agent. Moreover, the principal’s information predominantly influences the agent’seffort; consistent with many economic models, the agent’s behavior is driven by the incentives provided. We further observe that the agent’s information does not impact the effort exerted. Regarding utilities, our analysis indicates that the principal’s utility remains largely unaffected by his information state, as he can adjust the incentive mechanisms to consistently maximize utility. In contrast, the agent’s utility is significantly influenced by environmental volatility and the principal’s level of information. This suggests potential scenarios where a risk-neutral principal might withhold a risk premium from a risk-averse agent. Furthermore, since the principal consistently achieves maximum utility, there is minimal incentive to enhance her information state. This leads to scenarios where the agent’s dependence on the principal is disproportionately high. The remainder of this paper is structured as follows: We discuss the relevant background in Section 2. Section 3 discusses our approach to limit the principal’s and the agent’s information availability, formalizes the proposed agent-based model, and An ABM of the Hidden Action Model 383 introduces the simulation setup. The results are presented in Section 4, and Section 5 discusses the results. Finally, Section 6 concludes the paper. 2 Related Work 2.1 Holmström’s Hidden Action Model The hidden action model introduced in Holmström (1979) describes the relationship between one principal who assigns a task to one agent. In particular, the principal designs a contract that includes the task to be carried out and a rule to share the outcome, and offers this contract to the agent. If the agent accepts the contract, he makes an effort (often also referred to as action) to carry out the task assigned to him. Together with an exogenous factor, this action generates the task outcome. At the same time, acting leads to disutility for the agent. The sharing rule included in the contract defines –before the action is taken –how the outcome is to be shared between the principal and the agent. The model presented by Holmström (1979) is a non-repeated model, indicating that it encapsulates a single execution of the specified sequence outlined above. As a result, it does not account for temporal dynamics, including the distribution ofeffort overseveral periods, contract re-negotiations, and the implications of present actions on future payoffs (as, for example, done in Ma 1991). The hidden action model introduced in Holmström (1979) is capable of describing relations between one principal and one agent in a wide range of areas, including the relationship between employer and employee, buyer and supplier, and client and contractor (Caillaud and Hermalin 2000; Eisenhardt 1989; Leitner and Wall 2021; Reinwald, Leitner, and Wall 2020). The sequence of events within the hidden action model is included in Figure 1, whereby the main features can be summarized as follows: In τ= 0, the principal designs the contract and offers it to the agent who makes his decision about whether or not to accept it in τ=1.Inτ= 2, the agent selects an effort level a∈A⊆Rfrom a set of effort levels Athat are feasible to carry out the task. The agent’s selected effort level ais hidden to the principal, i.e. the principal cannot observe it because the costs for observing it are prohibitively high or this information is not accessible to her. Consequently, there is an information asymmetry regarding the effort level in favor of the agent. The exogenous factor θ∼N(μ,θ) takes effect in τ= 3, and it is a random variable that describes the state of nature, which includes, for example, the behavior of suppliers or customers. The outcome xmaterializes in τ= 4; it is a function of the agent’seffort level aand an exogenous factor θand follows the production function x=x(a,θ). Both the principal and the agent can observe the outcome. There is information asymmetry regarding the exogenous factor: Unlike the principal, the 384 P. Reinwald et al. agent can observe the exogenous factor or deduce it from the outcome. If the principal knows of the exogenous factor, she can deduce the agent’seffort level from the outcome, which would render the hidden action problem trivial since all information asymmetry would be resolved, and theprincipal couldpaythe agent basedon his effort. As a consequence of the information asymmetry, the principal can only base the sharing rule on the outcome. The principal is risk-neutral. Her utility comes from the generated outcome x minus the agent’s compensation s(x) so that UP(x,s(x)) = x−s(x).(1) The agent is risk-averse and characterized by the utility function UA(s(x),a)=V(s(x)) − G(a),(2) where V(s(x)) stands for the utility from his share of the outcome and G(a) indicates the disutility of the effort he makes to carry out the task, with V′> 0 and x a ≥0. 1 Given these characteristics of the hidden action model, the principal’s optimization problem to generate the optimal sharing rule is formalized as follows: max s(⋅),aEU Px,s(x) ()() (3a) s.tE U A(s(x),a) () ≥U (3b) a∈arg max a′∈A EU A(s(x),a′) () (3c) Equations (3b) and (3c) are constraints that have to be considered by the principal. In particular, Eq. (3b) represents the participation constraint that ensures that the agent gets a minimum utility U and therefore accepts the contract. This minimum utility is also called reservation utility and represents the agent’s best outside option. Equation (3c) is the incentive compatibility constraint and aligns the agent’s objective (to Figure 1: Sequence of events within Holmström’s hidden action model. 1The subscript adenotes the partial derivative concerning a. An ABM of the Hidden Action Model 385 maximize his utility) with the principal’s objective. This constraint affects the agent’s choice of effort level a. The notation ‘arg max’represents the set of all arguments that maximizes the objective function that follows. The solution to the problem in Eqs. (3a)–(3c) is provided in Appendix A. 2.2 Related Work on Extensions to Repeated Models of the Hidden Action Problem The hidden action model introduced in Holmström (1979) has been extended in several directions, whereby we are particularly interested in works that extend the model towards a multi-period or repeated model, and this section aims to give an illustrative overview of these extensions. Initial steps towards a model that spans multiple periods are introduced by Rubinstein and Yaari (1983) and Radner (1985), focusing on problems repeated indefinitely with no discounting from both the agent and the principal. They explore how the agent’s rewards adapt based on their historical performance, demonstrating that a contract can be formulated to eliminate inefficiencies associated with moral hazard. Conversely, Lambert (1983) proposes a model with a finite horizon, incorporating dynamic production functions and discounting of the principal’s and agent’s utilities. This model highlights the significance of long-term contracts for addressing moral hazard, by offering the agent a long-term commitment and utilizing their performance history to mitigate uncertainty in their actions. Rogerson (1985) also takes into account discounting from both the principal’s and the agent’s perspectives, demonstrating that the history of performance is crucial in crafting an optimal contract. Building upon this foundation, Spear and Srivastava (1987) further develop the concept by also introducing a repeated moral hazard model that incorporates discounting. Utilizing this model, they explore how contracts can feasibly incorporate dependency on historical actions and examine the evolution of these contracts over time. Broadening the analysis, Holmström and Milgrom (1987) propose a continuous time model and investigate instances in which agents are compensated at theend of a finite duration, with Holmström and Milgrom (1991) acknowledging that their findings accurately reflect short-term dynamics. Holmström and Milgrom (1987) demonstrate that contracts adopt a linear format with respect to total output under certain conditions, such as agents with exhibiting exponential utility and the effort entailing monetary costs. This finding has been further explored by researchers like Schättler and Sung (1993), who devise a more general mathematical framework, and Hellwig and Schmidt (2002), who examine the prerequisites for discrete-time models to align with the findings of Holmström and Milgrom (1987). Williams (2015) also extends the model introduced in Holmström and Milgrom (1987). In particular, 386 P. Reinwald et al. he is concerned with situations in which the agent has hidden saving, which is why his consumption and wealth cannot be monitored. Sannikov (2008) contributes to this line of research by showing that optimal long-term contracts exhibit complex non-linear wage and effort patterns. They detail the circumstances under which an agent can retire with an optimal contract, where the decision to continue or not at each step is influenced by prospective wages and effort levels. Edmans and Gabaix (2011) are concerned with fixed contracts in the hidden action context. They reference the work of Grossman and Hart (1992), who demonstrated the complexities arising from fixed contracts in discrete-time scenarios. In response, Edmans and Gabaix (2011) introduce a model featuring contracts that are easier to manage by revising a crucial premise: the agent has the ability to perceive environmental noise prior to exerting effort. They contend that without this modification, agents would behave in a manner aimed at fulfilling expected incentives. Research has also focused on the issue of an agent’s decision to stay indefinitely with the principal. While many studies assume such indefinite commitment, there are models that account for limited commitment. A significant amount of this work is aimed at creating contracts that, in any situation, give agents no reason to exit, ensuring the contract’sinfinite validity (e.g. Kocherlakota 1996; Phelan 1995; Thomas and Worrall 1988). Another approach is followed by Wang and Yang (2019); they address this interaction between the principal and the agent, introducing a model that features stochastic alternatives, dynamic contracts, and a mechanism for endogenous self-enforcement. There are a number of previous works that focus on uncertainty in information about performance metrics. For instance, Chaigneau, Edmans, and Gottlieb (2014) examine information’s role and its constraints concerning performance metrics within hidden action scenarios. They draw on the informativeness principle (Holmström 1979; Shavell 1979), which posits that principals should seek out highly precise performance indicators. Emphasizing this principle, Chaigneau, Edmans, and Gottlieb (2014) highlight the necessity of weighing the informational benefits against the associated costs, a task complicated by the indirect relationship between additional information and contracting problem parameters when deriving optimal contracts proves challenging. Utilizing the moral hazard framework, they introduce a model to scrutinize the benefits principals gain from enhanced information accuracy. MacLeod (2003) takes up the argument provided in Prendergast (1993) who argues that jobs in which objective performance measures are available are very rare, and rather bonuses are often based on subjective performance evaluations. In this vein, MacLeod (2003) explores (one-period) models where performance measures are not always fully accessible or may even be entirely unavailable. In these models, agent evaluations are based more on subjective assessments by the principal. Similarly, Fuchs (2007) argues that certain labor market phenomena, like wage An ABM of the Hidden Action Model 387 compression and periodic reviews, can serve as effective strategies to mitigate moral hazard on the agent’s part, especially when performance evaluations are subjective and held privately. Some of previous works focused on uncertainty regarding the agent’s characteristics. For example, Cohen, Deligkas, and Koren (2022) examine scenarios involving hidden actions, where the principal is unaware of not only the agent’s effort but also their utility function and action space. They present a model wherein the principal iteratively discerns the characteristics of an optimal contract by extending offers to identical agents and monitoring the results. Additionally, Cohen, Deligkas, and Koren (2022) introduce a learning algorithm tailored for this purpose. Prat and Jovanovic (2014) focus into hidden action scenarios with long-term contracts, especially when an agent’s capabilities of perform tasks remain unknown. They assume the persistence of an agent’s capabilities and base their argumentation and the accumulation of information over time to facilitate incentive provision. In a similar vein, Lai, Liu, and Li (2021) address scenarios with unknown agent capabilities, arguing that as the contractual period extends towards infinity, uncertainty diminishes because the agent’s capabilities become fully revealed. Conversely, He et al. (2017) and DeMarzo and Sannikov (2016) tackle stochastic uncertainty through learning, proposing models that equate stochastic future profits from output with uncertainty about agent abilities. Mekonnen (2017) builds upon these works, addressing the limited knowledge regarding the production function, which introduces uncertainty about outcome distributions. In his framework, both the principal and the agent form individual beliefs about this distribution, updating their beliefs based on past outcomes while the agent can influence the principal’s perceptions. To address these information limitations, Mekonnen (2017) incorporates an additional state variable in computing optimal contracts. In addition, there is a line of research that employs simulation-based research approaches to assess the robustness of the hidden action model to its included assumptions. Leitner and Wall (2021) modify the hidden action model’s assumptions, introducing information uncertainty regarding the environmental variable and the action space. In this adaptation, the principal and the agent are equipped with distinct information systems for data retrieval. This study observes the parameters for incentive mechanisms that emerge at the organizational level. Following this, Leitner and Wall (2022) conduct a detailed examination of the micro-level behavioral dynamics in scenarios characterized by limited information about the action space and environment. Furthering this research, Reinwald, Leitner, and Wall (2022) focus solely on the parties’knowledge of the environmental variable and, extending beyond previous research, consider the memory capacities of both the principal and the agent. 388 P. Reinwald et al. where ηdenotes the Arrow-Pratt measure of risk-aversion (Arrow 1973). The notation used in the agent-based model is summarized in Table 2. 3.2.2 Simultaneous and Sequential Learning Model The principal and the agent dispose of an individual memory m P and m A , respectively. The higher m P and m A , the more estimations and observations of the exogenous factor the principal and the agent can store in their memory. Due to different information states, the agent’s actual effort a t might deviate from the incited effort a  t. However, the principal has no information about the actual effort a t and, therefore, Table :Notation for the agent-based model. Description Notation Endogenous variables: Principal’s utility U P Agent’s utility U A Timesteps t Outcome in tx t =a t +θ t Agent’seffort in ta t Incited effort in ta  t Exogenous variable in tθ t Principal’s expected outcome for effort level a′in tx  Ptða′Þ Agent’s expected outcome for effort level a′in tx  Atða′Þ Agent’s share of the outcome in ts(x t )=x t ⋅ρ t Premium parameter included in the contract in tρ t Premium parameter for effort level a′in tρ t (a′) Principal’s set of all feasible actions in tA Pt Principal’s candidates for incited effort in tA  Pt Agent’s set of all feasible actions in tA At Estimations of the exogenous factors available to the principal in tΘ Pt Observations of the exogenous factors available to the agent in tΘ At Principal’slearned expectation of the exogenous factor in tθ  Pt Agent’slearned expectation of the exogenous factor in tθ  At Exogenous variables: Maximum timesteps T Agent’s Arrow-Pratt measure of risk-aversion η Principal’s memory m P Agent’s memory m A An ABM of the Hidden Action Model 395 bases her estimation of the exogenous factor in ton the incited effort. She computes her estimation according to θ  t=xt−a  t.(8) The agent, in contrast, knows the actual effort a t he made and, therefore, can compute the actual exogenous factor in taccording to θt=xt−at.(9) The principal and the agent store their estimations and observations in their memories. Once their capacities, m P for the principal and m A for the agent, are reached, the oldest information is replaced by the latest estimation or observation. Once the simulation moves on to the next timestep, the principal and the agent update the learned expected value for the exogenous factor by averaging all privately stored estimations/observations. To do so, they retrieve the information from their memories. Let us denote the estimations of the exogenous factor available to the principal in tby ΘPt = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ [θ  1,…,θ  t−1]if mP=∞, [θ  t−mP,…,θ  t−1]if mP<∞andt≥mP, [θ  1,…,θ  t−1]if mP<∞andt<mP. (10) For the agent, we denote the observations of the exogenous factor available in tby ΘAt =⎧ ⎪ ⎨ ⎪ ⎩ [θ1,…,θt−1]if mA=∞ [θt−mA,…,θt−1]if mA<∞andt≥mA [θ1,…,θt−1]if mA<∞andt<mA (11) The principal and the agent compute their learned expected value of the exogenous factor in tas the mean ∅(⋅) of the information available to them: For the principal, the learned expected value of the exogenous factor is θ  Pt =∅(ΘPt), while for the agent it is computed according to θ  At =∅(ΘAt). Note that the learned expected exogenous factor can also be interpreted as the principal’s and the agent’s information about (and perception of) the environment since it represents how they regard the environment in that timestep. 3.2.3 The Principal’s and Agent’s Decisions The principal’s and the agent’s decisions revolve around the selection of actions and the computation of the corresponding premium parameters. We denote the set of feasible actions from the perspective of the principal and the agent by A Pt and A At , respectively. The participation constraint defines the lower boundary of these 396 P. Reinwald et al. spaces, and the upper limit is given by the incentive compatibility constraint (Holmström 1979; Leitner and Wall 2021). Recall that the computation of the two constraints includes the expectation about the exogenous factor (see Eqs. (3b) and (3c)). Thus, if the principal and the agent have the same (different) expectations about the environment, A Pt and A At perfectly coincide (are different). 3.2.3.1 The Principal’s Decision In every timestep, the principal can adapt the premium parameter in the contract. To do so, she randomly discovers two alternative effort levels in the action space A Pt ,a  1 and a  2, which together with the incited effort of the previous period a  t−1serve as candidates for the effort she wants the agent to make in period t. Let us denote the set of candidates for the incited effort in tby A  Pt =[a  1,a  2,a  t−1]. Then, the principal undertakes a step-by-step local search, adhering to a hill climbing strategy. This implies that, based on the information available to her at any given moment, she opts for the most promising choice (Cormen et al. 2022). In line with the production function in Eq. (5), the principal computes the expected outcome for all alternatives a′∈A  Pt according to x  Pt (a′)=a′+θ  Pt .(12) The principal also computes the premium parameters for all candidate effort levels a′∈A  Pt according to ρt(a′)=arg maxρ∈[0,1]UP(x  Pt (a′),s(x  Pt (a′))),(13) and finally selects the candidate which maximizes her utility as the incited effort for the period taccording to a  t=arg maxa′∈A  PtUP(x  Pt (a′),s(x  Pt (a′))) (14) Together with the task that is to be carried out (which is the same throughout all time steps), the corresponding premium parameter ρt≔ρt(a  t)is the main element of the contract that is offered to the agent. 3.2.3.2 The Agent’s Decision In every timestep, the agent makes two decisions. First, he decides whether to accept or reject the contract offered by the principal. In particular, if the utility of the offered contract exceeds the reservation utility the agent accepts the contract. To make this decision, the agent computes the effort that maximizes his utility given the offered contract according to a* t=arg maxa′∈AAtUA(s(x  At (a′)),a′),(15) An ABM of the Hidden Action Model 397 where x  At (a′)=a′+θ  At. If the utility of this effort level exceeds or is equal to the agent’s reservation utility, UA(s(xt),a* t)≥U , the agent accepts the contract and makes this effort in t. In consequence, at≔a* t. 3 3.3 Parameter Settings and Observations 3.3.1 Parameter Settings 3.3.1.1 Parameters for Scenarios with Unlimited Information Access The solution proposed in Holmström (1979) aids as the benchmark solution in our study. To show that the proposed model is capable of replicating the benchmark solution, we first run simulations on scenarios with unlimited information access for the principal and the agent, i.e. we set the variables m A and m P equal to ∞.In addition, we take into account the turbulence of the environment: Recall the exogenous variable follows a normal distribution, which allows us to control the turbulence via the standard deviation. In particular, we set the mean of the normal distribution to zero and define its standard deviation relative to the optimal outcome x* of Holmström’s hidden action model (see and Appendix A), so that σ= {0.05x*, 0.25x*, 0.45x*}. All other parameters are kept constant during the simulation runs. For scenarios with unlimited information access, our analysis focuses memory on the first T= 200 timesteps in every simulation round. Every scenario is repeated R= 700 times. 4 We set the Arrow-Pratt measure of risk aversion at 0.5 to represent agents with moderate risk aversion, aligning with the findings of Graham, Harvey, and Puri (2013) and Brenner (2015). These studies indicate that CEOs exhibit lower risk aversion compared to the broader population and that the behavior of executives is consistent with a moderate level of risk aversion. 3.3.1.2 Parameters for Scenarios with Limited Information Access To capture the effects of limited information access in hidden action setups, we analyze scenarios with three different values for the principal’s memory 3Please note that, without loss of generality, we normalize the reservation utility to zero during the simulation experiments, and make sure that the agent accepts the contract in all cases. 4We follow the approach proposed in Lee et al. (2015) and select the number of simulation rounds based on the coefficient of variation. For our settings, this measure stabilizes at ϵ≤0.01 at around 700 repetitions. 398 P. Reinwald et al. (m P = {1, 3, 5}) and the agent’s memory (m A = {1, 3, 5}), whereby this parameter can be interpreted so that the poorer the memory, the lower the information availability. 5 The analysis of scenarios with limited memory focuses on the first T=20 timesteps in every simulation round; we do so because most of the dynamics can be observed within this period. All other parameters are kept constant during the simulation runs. An overview of the parameters considered in this simulation study is provided in Table 3. 3.3.2 Observations In our simulation experiments, we observe four measures: (i) the premium parameter determined by the principal, (ii) the level of effort exerted by the agent in completing the given task, (iii) the utility experienced by the agent throughout the experiments, and (iv) the utility of the principal. As introduced in Eq. (6), the premium parameter defines the agent’s share of the task outcome. We denote the premium parameter that is effective in period tand simulation run rby ρ tr and the premium parameter effective in the optimal solution by ρ* (see Appendix A). We compute the average normalized premium parameter in every timestep according to Table :Simulation parameters for scenarios with unlimited and limited memory. Parameter Notation Limited memory Unlimited memory Subject to variation: Principal’s memory m P ,,∞ Agent’s memory m A ,,∞ Exogenous factor: standard deviation σ.x*, .x*, .x*.x*, .x*, .x* Constants: Exogenous factor: mean μ Agent’s Arrow-Pratt measure η.. Observation period T  Simulation rounds R  5We limit the parameter space because of the computational complexity of the simulation model. The computation time Tcomp for a simulation model with pparameters and llevels per parameter is lp, scaling polynomially with the number of levels lper parameter. For example, with p= 3 and l= 3, the parameter space is 33= 27, and the initial computation time is Tcomp 0=27 ⋅Tsim, with Tsim ≈640 minfor T= 20 time steps per simulation run. Adding nlevels per parameter changes the parameter space to (l+n)pand the computation time to Tcomp n=(l+n)p⋅Tsim, resulting in a computation time ratio of Tcomp n/Tcomp 0=(l+n)/ (l)p. Thus, for n= 1 the computation time approximately doubles (≈2.37), with n= 2 it quadruples (≈4.63), and with n= 3 it octuples. An ABM of the Hidden Action Model 399 ρ  t=1 R∑ R r=1 ρtr ρ*.(16) To capture the effort the agent makes to carry out his task, we report the average normalized actual effort level in every timestep t. We compute this metric as follows: a  t=1 R∑ r=R r=1 atr a*,(17) where a tr indicates the effort made by the agent in timestep tand simulation run r, and a* stands for the optimal level of effort suggested in Holmström (1979) (see Appendix A). Please note that the average normalized effort is also a proxy of the task outcome at the macro level, as the outcome is a function of the agent’seffort and the exogenous variable (see Eq. (5)). Lastly, we document the utility experienced by both the principal and the agent throughout the simulation runs. For the principal, we denote the utility experienced in time step tand simulation run rby U Prt (see Eq. (4)) and normalize it by the utility that can be achieved with the solution of Holmström’s hidden action model, which we denote by U* P(for its computation see Appendix A). Then, the principal’s average normalized utility at time tis computed according to U  Pt =1 R∑ R r=1 UPrt U* P .(18) Similarly, we denote the agent’s utility (see Eq. (7)) in timestep tand simulation run r and the utility following Holmström’s model by U Art and U* A, respectively. We compute the agent’s average normalized utility at time tby U  At =1 R∑ R r=1 UArt U* A .(19) 4 Results The results are organized into four sections. In Section 4.1, we show that the solution emerging from the agent-based model converges to the optimal solution suggested by Holmström (1979). Then, Section 4.2 analyzes the scenarios with limited information access. Specifically, Section 4.2.1 analyzes the effects of limited information on the premium parameter, Section 4.2.2 provides an analysis of the effort made by the agent, and Section 4.2.3 focuses on the dynamics within the agent-based model and particularly emphasizes the principal’s and the agent’s respective utilities that result from the choices related to the premium parameter and the effort. 400 P. Reinwald et al. 4.1 Scenarios with Unlimited Information Access: Replicating the Benchmark Solution Our first simulations aim at demonstrating that the observations from our agent-based model align with the optimal solution proposed by Holmström (1979) (given the utility functions specified in Section 3.2.1). In these simulations, both the principal and the agent are endowed with unlimited memory concerning the environmental variable. It can be expected that with a sufficiently long observation period, the principal’s and the agent’s estimations of the environmental variable will converge to a value close to its expected value, and following this, we expect the emergence of solutions that are close to the ones proposed in Holmström (1979) from the agent-based model. To observe the model’s actual behavior, we run simulations for an observation period of 200 timesteps and track all observations as detailed in Section 3.3.2. For a summary of all parameters, see Table 3. To account for the requirement of long (practically infinite) observation periods, we have fitted power models of the form x (t)=a⋅tb+cto extrapolate the observed time series over longer periods. The parameters for these models are listed in Table 4. Generally, we can see from the results that the exponent bis negative for all models. When tincreases, the term tb converges to zero because of negative values of b; in consequence, the term a⋅tbalso approaches zero, and the result of the function x(t) will converge to the value of the constant c. The values of c(considering the confidence intervals with α= 0.05) are sufficiently close to 1. Therefore (and as we normalize our observations), weconclude that, for a sufficiently extended observation period, our model predictions align closely with the theoretical solutions proposed by Holmström (1979), given the utility functions specified in Section 3.2.1. To assess the convergence rate of x(t) toward the constant c,wedefine the coefficient-exponent influence ratio (CEIR), R=|a/b|. This ratio is calculated for each pair (a,b) and presented in Table 4. The CEIR facilitates a comparative analysis; a lower CEIR indicates faster convergence of x(t)toc, whereas a higher CEIR suggests slower convergence. 6 The results reveal that environmental turbulence has a negligible impact on the convergence rates of both the agent’seffort and the 6The convergence rate of x(t) towards cis dictated by its derivative’s magnitude, expressed as |x′ (t)|=|a|⋅|b|⋅tb−1. Please note that the term a⋅tbin the power function x(t)=a⋅tb+cis dominated by b, due to the power law dynamics; bdetermines the rate of change and aonly scales this effect. Consequently, a higher |b|results in a more responsive model to variations in t. High CEIRs, where |a| exceeds |b|, suggest that the scaling effect of apredominates, leading to a relatively slower convergence of x(t) towards c. Conversely, low CEIRs imply that b’sinfluence is greater relative to a, facilitating a quicker convergence of x(t)toc. An ABM of the Hidden Action Model 401 Table :Estimated parameters for fitted power models for scenarios with unlimited information access. Observation Estimated power model parameters CEIR Goodness aCI bCI cCI |a/b|RRMSE Relatively stable environment: Premium parameter −. ±. −. ±. . ±. . . . Effort level −. ±. −. ±. . ±. . . . Principal’s utility −. ±. −. ±. . ±. . . . Agent’s utility −. ±. −. ±. . ±. . . . Mid-turbulent environment: Premium parameter −. ±. −. ±. . ±. . . . Effort level −. ±. −. ±. . ±. . . . Principal’s utility −. ±. −. ±. . ±. . . . Agent’s utility −. ±. −. ±. . ±. . . . Turbulent environment: Premium parameter −. ±. −. ±. . ±. . . . Effort level −. ±. −. ±. . ±. . . . Principal’s utility −. ±. −. ±. . ±. . . . Agent’s utility −. ±. −. ±. . ±. . . . Estimated power models have the form of x(t)=a⋅tb+c. CI columns report the confidence intervals for α=.. CEIR column reports the coefficient-exponent influence ratio, |a/b|. Goodness columns report the coefficient of determination (R) and the root mean squared error (RMSE) as two measures for the models’fit. 402 P. Reinwald et al. principal’s utility. In contrast, the convergence of the premium parameter and the agent’s utility to cis notably slower in turbulent conditions. Moreover, the analysis of the CEIRs indicates that the agent’seffort and the principal’s utility consistently converge more rapidly to cthan the premium parameter and the agent’s utility. 4.2 Scenarios with Limited Information Access For every scenario, we monitor Rtime series of the measurements described in Section 3.3.2, each with a duration of Tperiods. This includes observing 700 time series of length 20 for scenarios with limited memory, covering all four metrics. To verify if a series length of 20 is adequate, we define two time windows within the series and compare their central tendencies with a Mann-Whitney Utest (Mann and Whitney 1947), and their variances with a Levene test (Levene 1960). Our focus is on the series’last quarter, identifying segments from periods 15 to 17 and 18 to 20, utilizing samples from 700∗3 = 2, 100 observations. We consider a time series length of 20 adequate if we can accept the null hypotheses of the Mann-Whitney U test (indicating equal central tendencies) and the Levene test (suggesting equal variances) at significance levels of either p≤0.01 or 0.05. The corresponding results are included in Tables 5 through 8. To evaluate if altering the principal’s or the agent’s memory impacts the observed time series, we conduct a Mann-Whitney Utest comparing two related time series (Lee et al. 2015). If we cannot accept the null hypothesis of equal central tendencies at p≥0.05 or 0.01, we infer that the memory capacity influences the central tendency. Furthermore, to determine the magnitude of this impact, we calculate the rank biserial correlation as an effect size indicator. This is done using the approach outlined in Wendt (1972) who computes the rank biserial correlation as follows: r rb =1−(2 ⋅U)/(n 1 ⋅n 2 ), where Uis the Mann-Whitney Utest statistic, and n 1 and n 2 are the sizes of the two compared samples. The results are detailed in Tables 9 and 10. 4.2.1 Premium Parameter This section examines the impact of information constraints on the premium parameter. We document our findings by first presenting the mean of the average normalized premium parameters throughout the entire observation period and, at the end of this period, alongside their respective confidence intervals in Table 5. Further, the analysis of the time series behavior is also included in Table 5. All time series for the premium parameter demonstrate stability in both their central tendencies and variances. An ABM of the Hidden Action Model 403 The results indicate that, in general, the premium parameter decreases with higher environmental turbulence. Please note that the turbulence perceived by the principal and the agent is also affected by their memories. Recall the principal’s and Table :Expected premium parameters and stationarity of time series. Memory Expected premium parameter Stationarity Principal Agent Periods : CI Period  CI Central tendency Variance Relatively stable environment: . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ Mid-turbulent environment: . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗ . ±. . ±. ∗∗∗ Turbulent environment: . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ . ±. . ±. ∗∗ ∗∗ CI columns report the confidence intervals for α=.. Stationarity columns report results of a Mann-Whitney Utest (central tendency) and a Levene test (variance). ∗null hypothesis can be accepted with p≤..∗∗ null hypothesis can be accepted with p≤.. 404 P. Reinwald et al. rises, highlighting that the most substantial positive effects are seen in more stable conditions. 5 Discussion The results presented in Section 4 allow to put the effects of limitations in the principal’s and agent’s respective information and environmental turbulence in the following framework (see Figure 3): Recall, the principal and the agent estimate and observe the exogenous variable in every timestep, respectively. Then, they store their estimations/observations in their memories. Limitations in their respective information take effect in the form of constraints in their memories. The more informed the principal and the agent are, the more information stored in their memories are considered when computing the learned expectation of the exogenous variable. Thus, their respective information moderates the principal’s and agent’s perceptions of the environment. The principal’s perception affects her choice of the premium parameter. One might expect that the agent’s perception of the environment impacts his effort choice, but we could not observe significant influences for this relationship. Together with the actual environment, the principal’s and the agent’s decisions about the premium parameter and the effort, respectively, affect their utilities. The analysis in this paper focuses mainly on the effects of turbulence in the environment and limitations in the principal’s and agent’s respective information in this framework (which is indicated by the gray boxes in Figure 3). Figure 3: Framework for limited information in hidden action situations. An ABM of the Hidden Action Model 411 5.1 Results Related to the Premium Parameter The main findings presented in Section 4.2.1 are that (i) the value of the premium parameter decreases with environmental turbulence and (ii) increases with the principal’s information. This means that the agent’sshareoftheoutcomeis relatively small (large) if the environment is rather turbulent (stable). However, iftheturbulenceishigh,theagentcannoteasilycontroltheoutcome,which puts his compensation at risk (Miceli and Heneman 2000). Accordingly, the finding that the (perceived) turbulence decreases the premium parameter is (in part) contrary to the risk premium hypothesis.Thishypothesisstatesthatthe principal will have to increase the risk-averse agent’s total compensation in turbulent environments to protect him from risk (Burns and Stalker 2001; Eisenhardt 1988; Umanath, Ray, and Campbell 1993). Counter-intuitively, the results presented in Section 4.2.1 indicate the contrary. Even though the riskbearing is initially within the principal’s role (Fama and Jensen 1983), she shifts a part of the risk to the agent. This finding not only contradicts the risk premium hypothesis but is also in contrast to previous research on task programmability and the predictions of classical organization theory. Stroh et al. (1996) and Sung and Choi (2012) link environmental turbulence with task programmability by arguing that more (less) turbulence can be interpreted in terms of less (more) programmable tasks since task situations and problems change more frequently in turbulent environments. Furthermore, it is often suggested that task programmability is negatively correlated with the magnitude of variable compensation components (Gomez-Mejia and Balkin 1992). This translates into the expectation that more environmental turbulence (and less task programmability) should be linked to higher proportions of variable compensation and vice versa (Gerhart and Milkovich 1990; Stroh et al. 1996). This expectation is also in line with classical organization theory (Thompson 1967), which predicts that organizations that operate in turbulent environments rely more heavily on variable compensation to ensure the appropriate behaviors of their agents. Our findings do not support this expectation but can be linked to the argumentation provided in Smith (1984). There are not just drawbacks but also advantages of turbulent environments (Eisenhardt 1989), and an analogy of agents who are hired in risky environments and ‘unfair lotteries’can be established: The average pay is not necessarily high, but there is a chance to receive a high compensation, even though perhaps this chance is relatively slight (see also Miceli and Heneman 2000). 412 P. Reinwald et al. 5.2 Results Related to the Agent’sEffort The results presented in Section 4.2.2 indicate that the agent’seffort follows the patterns observed for the premium parameters. The effort decreases with environmental turbulence so that the agent makes more (less) effort in relatively stable (turbulent) environments. If a more informed principal sets the premium parameter, the agent makes significantly more effort. This relation is a fundamental assumption in economic contexts and is supported by experimental research (Dickinson 1999; Takahashi, Shen, and Ogawa 2016) and field research (Banker, Lee, and Potter 1996; Lazear 2000). Since the agent’s average normalized effort is also a proxy for how well the organization performs, these observations can be related to environmental turbulence and firm performance research. Environmental uncertainty and dynamism (Aldrich 2008; Chen, Reilly, and Lynn 2005; Milliken 1987) increase the difficulty of organizational decision-making and significantly affect organizational performance (Reinwald, Leitner, and Wall 2022). This is in line with the results presented in Yu, Wang, and Brouthers (2016), who find that (perceived) environmental uncertainty affects the identification of competitors so that firms identify more competitors if the environment is certain. This directly translates into a more competitive advantage and, as a consequence, a better (worse) performance in certain (uncertain) environments (Dutta and King 1980; Porter 1997). For the hidden action context, it is also shown in Reinwald, Leitner, and Wall (2020) that increases in (perceived) environmental turbulence lead to a drop in firm performance. This finding is also supported by the contingency management accounting literature, which is, amongst others, concerned with the fit between organizations and their environment (Otley 1999, 2016). If the environment is turbulent and/or the principal is not very well informed about the environment, she cannot design the incentive scheme so that it fits the actual environment (Chenhall and Morris 1986; Ezzamel 1990; Ghosh and Olsen 2009; Hoque 2005). Since the agent respondsto the incentives set by the principal, suboptimal incentive parameters directly translate into adverse effects on performance. 5.3 Results Related to the Principal’s and the Agent’s Utilities Above, it was established that the principal’s choice of the premium parameter might indicate that she transfers some of the risks to the agent by reducing the premium parameter as environmental turbulence increases. Now, as we take the results related to the agent’s utility presented in Section 4.2.3 into account, this conjecture becomes even more evident. One would expect that the agent compensates for the An ABM of the Hidden Action Model 413 additional risk and the missing risk premium by making less effort, which –in turn – eventually increases his utility. However, the results indicate the opposite: The agent’s utility decreases with increases in environmental turbulence. If a more informed principal sets the premium parameter, the agent’s utility appears to increase. For the principal, the results presented in Section 4.2.3 indicate that the volatility of his utility is relatively high. However, the principal’s utility appears to be less sensitive to limitations in her own and the agent’s information. As a consequence we can conclude that the risk-neutral principal withholds a risk premium from the risk-averse agent, which, in turn, assures the robustness of the principal’s utility to environmental turbulence. The results indicate that the agent is over-dependent on the principal, which results in a dilemma for the agent: First, if the agent were able to increase his memory, doing so would not allow him to escape the situation, since his memory has no significant effect on his utility. Second, the principal appears to set the premium parameter to punish the agent for the risky environment. However, this would still be the best option for the agent since otherwise, he would have rejected the contract and followed the outside option. Shirking (i.e. putting in less effort) would not be an option either, since it would decrease the agent’s utility (Nilakant and Rao 1994). Third, even if the agent were more informed, he would have no incentive to disclose his private information about the environment. If the principal was informed about the actual exogenous variable (instead of estimating it), she could deduce the agent’seffort from the outcome (Caillaud and Hermalin 2000). As soon as the principal realizes that the agent discloses this information, she could switch from performance-based pay to effort-based compensation. As a consequence, the principal could further increase her utility at the cost of the agent. Previous research merely addresses the issue of overly dependent agents. It tends to focus on overdependence on the principal’s side: Huang, Raimo, and Humfrey (2016), for example, state that the principal’s power to exert control decreases with the dependence of the agent. Willcocks and Choi (1995) also focus on the agent’s perspective and argue that in vendor–client relationships, the client might be overly dependent on the vendor (see also Hancox and Hackney 2000). Our results indicate that limited information –of both the principal and the agent –appears to empower the principal to (unintentionally) siphon-offutility from the agent by capitalizing on her control over the compensation (David, Kochhar, and Levitas 1998). Recall that one fundamental foundationof the principal-agent theory is that both the principal and the agent are driven by self-interest (Huang, Raimo, and Humfrey 2016). Now that we know that the principal experiences (almost) the same utility in all cases, she has no incentive to stop transferring risk to the agent or to gather information about the environment on her own (i.e. to increase her information). The principal, thus –perhaps unintentionally –behaves in a way that might be 414 P. Reinwald et al. interpreted as opportunism, i.e. self-interest-seeking with guile (Williamson 1975), whereby limitations in the principal’s information appear to reinforce behavioral patterns that appear as guile. 5.4 Methodological Contributions From a methodological point of view, we have introduced an agent-based model of the hidden action problem (Holmström 1979) by employing the research approach put forward by Guerrero and Axtell (2011) and Leitner and Behrens (2015a), and we show that the solution of the agent-based model converges to the solution proposed by the original model. Our approach is different from the classical principal-agent theory at a conceptual level. In particular, agent-based modeling and simulation allow systematically analyzing the robustness of the solutions derived from closedform models to deviations from the assumptions included in these models (Guerrero and Axtell 2011; Leitner 2024; Wall and Leitner 2021). The principal-agent theory often includes some idealized assumptions. Some researchers are concerned that (over-)simplified behavioral assumptions might come at the cost of the validity and explanatory power of the findings (Eden 1989; Mingers 2011; Mingers and Rosenhead 2004). Usually, it is assumed that individual behavior is driven by optimization and rationality, the modeled individuals’choices are representative for the entire population, and equilibrium solutions can be achieved. Our approach, however, allows for the explicit consideration of emergence, limitations in information, and heterogeneity (Chang and Harrington Jr 2006; Chen 2017; Mealy, Farmer, and Teytelboym 2019). These features of our approach allow to overcome some of the limitations of the formal approaches in analytical research: While solutions derived from analytical models are related to narrow behavioral assumptions, the model presented here allows to take rich environmental contexts and relaxed–and perhaps even more realistic –behavioral assumptions into account (Wall 2024; Wall and Leitner 2021). However, the use of agent-based modeling and simulation in microeconomic contexts is rather scarce. Therefore, the approach presented here can be regarded as a step towards a more open approach in microeconomic research that allows for relaxing (some of) the well-established assumptions. 6 Conclusions In this paper, we proposed an agent-based model of the hidden action problem. In particular, we transferred the closed-form model introduced in Holmström (1979) into an agent-based model (Guerrero and Axtell 2011; Leitner and Behrens 2015a). An ABM of the Hidden Action Model 415 Doing so allowed us to relax some of the idealized assumptions in Holmström’s model related to the principal’s and the agent’s respective information. We focus on the memory of information about the environment. Our results indicate that the principal’s information is the key to good performance, whereas the agent’s information does not significantly affect performance. Surprisingly, the principal appears to behave very selfishly by siphoning offutility from the agent to maintain nearoptimal personal utility by exerting her control over the agent’s compensation. We model (more realistic) human behavior by employing operational research methods and by considering (and bringing together) findings from the disciplines of cognitive psychology, economics, management, and operational research. Of course, our research is not without limitations. Several further incentive mechanisms –also nonformal ones –might be employed in the modeled situation. Granting the principal degrees of freedom in her choices related to the control mechanism might be a fruitful avenue for future research. We limit the principal’s and the agent’s respective information concerning memory only. Further research might focus on extending the limitations in information by also taking into account, for example, calculation errors, limitations in other types of information, and biases in information processing. Additionally, the agent makes decisions based on the learned mean value of the environmental variable. Future research could enhance the agent’s utility function by incorporating the variance of the environmental variable, thereby more accurately reflecting the agent’s risk aversion. Currently, we model scenarios where there is a one-to-one delegation relationship between a single principal and a single agent. Future studies could broaden this framework to include delegation relationships involving multiple agents. Although the focus of this paper is on a theoretical analysis of how the hidden action model withstands limitations in information access, subsequent research might include laboratory experiments to empirically validate the model. Research funding: This work was supported by funds of the Österreichische Nationalbank [Austrian Central Bank Anniversary Fund, project number: 17930]. Data and code availability: Simulation data and code are available via the following link: https://github.com/sforstephan/JBNST24. Appendix A: Solution to Holmström’s Hidden Action Problem There are two different approaches that can be used to solve the program formalized in Eqs. (3a)–(3c). Important for us is the approach of Mirrlees (1976), who suppresses 416 P. Reinwald et al. θand views xas a random variable with distribution F(x,a). In this approach, it is assumed that ∀a∈A∃x∈IR:F a (x,a) < 0 so that a change in ahas nontrivial effect on the distribution of x. For a given distribution of θ,F(x,a) is the distribution induced on x=x(a,θ) (Holmström 1979). In the following program, f(x,a) is the density function of Fwith f a and f aa well defined for all (x,a) and Eq. (3c) is replaced with a first-order constraint. Furthermore, s(x) is restricted to lie in the interval [c,d+x] to guarantee an existing solution to Eqs. (3a)–(3c) for the class of functions in Eq. (20), where Vb′ bis the total variation of sin the interval [b,b′] (Holmström 1979; Kolmogorov and Fomin 1970). SK=s(x)∈[c,d+x]|Vb′ b(s)≤K⋅(b′−b) {} ,(20) max s(x)∈[c,d+x],a∫G(x−s(x))f(x,a)dx(21a) subject to ∫[U(s(x)−V(a)]f(x,a)dx≥H  ,(21b) ∫U(s(x))fa(x,a)dx=V′(a).(21c) We denote the multipliers for Eqs. (21b) and (21c) by λand μ, respectively. After a pointwise Lagrangian optimisation, the optimal sharing rule yields the following characterization: G′(x−s(x)) U′(s(x)) =λ+μ⋅fa(x,a) f(x,a),(22) for almost every xfor which Eq. (22) has a solution s(x)∈[c,d+x]. 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