Company valuation under interaction in discrete time (real game options model)

© 2019 Published by VŠB-TU Os a a. All igh s ese ed. ER-CEREI, Volume 22: 37–44 (2019). ISSN 1212-3951 (P in ), 1805-9481 (Online) doi: 10.7327/ce ei.2019.06.01 Company alua ion unde in e ac ion in disc e e ime ( eal game op ions model) Dana DLUHOŠOVÁa, Zdeněk ZMEŠKALa * a Depa men o Finance, Facul y o Economics, VŠB – Technical Uni e si y o Os a a, Sokolská řída 33, 702 00 Os a a, Czech Republic Abs ac Company alua ion is a c ucial opic in inancial decision making. The ad anced alua ion me hod is he eal op ions app oach ealised unde isk and lexibili y. I e lec s a s ochas ic ea u e o he unde lying asse and dynamic manage ial decision making. Ano he aspec o alua ion, which is o en neglec ed, is in e ac ion, meaning he mu- ual impac o o he companies on he calcula ed alue. Game heo y models his aspec . The pape ’s objec i e is o desc ibe and apply company wo-phase eal game op ions alua ion in disc e e ime. A gene alised eal game op- ions alua ion model based on he wo-phase me hod, disc e e ime, isk-neu al p obabili y, and swi ching cos is o mula ed. The game ca ego isa ion is in oduced, especially ma ke s uc u e games, including equilib ium calcu- la ions ollowing pu e and mixed s a egies, and he eal game op ions model is o mula ed. A company wo-phase alua ion me hod in he Cou no p oduc ion duopoly ma ke s uc u e unde andom demand is de eloped, and an illus a i e example is p esen ed. The pape con i ms he possibili y o modelling company wo-phase alue h ough eal game op ions alua ion models. Neglec ing an in e ac ion unde a non-pe ec ma ke s uc u e can unde alue a company, so his aspec is essen ial. Keywo ds Game heo y, eal game op ion, eal op ion, alua ion JEL Classi ica ion: C7, D4, G13, G32, L1 * [email p o ec ed] (co esponding au ho ) The esea ch was unded by he VSB-Technical Uni e si y o Os a a, SGS P ojec s SP2021/57 and SP2020/124. Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 22, 2019 38 Company alua ion unde in e ac ion in disc e e ime ( eal game op ions model) Dana DLUHOŠOVÁ, Zdeněk ZMEŠKAL In oduc ion Company alua ion is an impo an p oblem in inan- cial decision making and managemen . The choice o alua ion me hod depends on a ious aspec s. The p ac ical alua ion app oach is a wo-phase discoun ed cash low me hod. The complexi y o alua ion leads o he equen applica ion o disc e e binomial models. Due o he alua ion en i onmen , isk, lexibili y, and in e ac ion a e subs an ial aspec s. Te m isk cha ac e - ises andomness (s ochas ic p ocess), lexibili y ep e- sen s a dynamic decision (op ion alua ion), and in e - ac i i y means ha one company's decision is in lu- enced by o he companies' decisions (game heo y). The eal op ions me hod is applied unde isk and lex- ibili y. Howe e , in e ac i i y is o en neglec ed, e en hough i ep esen s a signi ican aspec o a company's alua ion, supposing a non-pe ec ma ke . Resea che s and p ac i ione s deal wi h game eal op ions. Many aspec s conce ning goals, o mula ions, me hodological concep ions, and applica ion possibili- ies a e in es iga ed (see e.g. Aze edo and Paxson, 2014; Che alie -Roignan and T igeo gis, 2011; G en- adie , 2000a, 2000b, 2002; Huisman, 2001; Huisman e al., 2004; Smi and T igeo gis, 2004, 2017). The p ob- lem can be o mula ed in disc e e o con inuous ime using Bellman's dynamic p og amming p inciple. Va - ious op imisa ion echniques can be applied, game ypes in es iga ed, he p ojec NPV o alue o he company compu ed, and p oblems o a ious sec o s in es iga ed and analysed. The pape ocuses on he eal game op ions alua- ion o a company using he wo-phase me hod in dis- c e e ime. This app oach has no ye been desc ibed in dep h in he li e a u e. The pape 's objec i e is o de- sc ibe and apply company wo-phase eal game op ions alua ion in disc e e ime. P ima ily, an in e ac ion ea u e is in es iga ed. The wo-phase discoun ed cash low is applied, and he bi- nomial model in disc e e ime is assumed. The i s sec- ion is de o ed o he de elopmen o alua ion me h- ods and a desc ip ion o he eal op ions alua ion p in- ciples. The chosen aspec s o game heo y a e hen de- sc ibed and analysed. Subsequen ly, he me hods o game eal op ions a e ou lined, especially ma ke s uc- u e games. The applica ion o he alua ion model wi h andom demand and a duopoly ma ke is e i ied. 1. Valua ion me hods unde isk and lexibili y ( eal op ions) The alue o company V can be s a ed h ough he dis- coun ed cash low me hod as he p esen alue o cash low ( ) 1 1 V FCF R − = =  +  . In he case o he wo-phase me hod ( ) ( ) 1 1 1 2 1 1 T T T V V V FCF R V R −−− = = + =  + +  +  , (1) 1V is he alue o he i s phase, 2V is he alue o he second phase, FCF is he ee cash low, R is he isk- ee a e, and T V is he e minal (con inual) alue, being a alue a he beginning o he second phase. Thus, FCF EAT DEP NWC INV= + − − and, o ex- ample, cons an pe pe ui y 1/ TT V FCF R + = o g owing pe pe ui y ( ) 1/ TT V FCF R g + =− . Then, he p esen alue can be o mula ed as ollows: ( ) ( ) ( ) ( ) ( ) 1 1 1 1 2 1 1 1 1 1 1 ...... TT T V R FCF VR R FCF R FCF − − − − − −    + +      =  +   + +    + + +  . This implies ha he ecu en o mula o one s ep is ( ) 1 11 V FCF V R − + = +  + , (2) and he alue equals he ee cash low plus he p esen alue o he one-s ep u u e alue. The alue can be cal- cula ed h ough he backwa d induc ion and dynamic p og amming (Bellman's equa ion) me hod. Valua ion unde isk means ha he unde lying as- se ( ac o ) is a andom p ocess. In a disc e e binomial model, he alue can be s a ed using isk-neu al alua- ion and de i ed om he eplica ion s a egy. The basic idea is o c ea e a po olio alue  om he unde lying ee cash low (asse ) FCF and isk- ee asse B so a de i a i e alue V can be epli- ca ed. Fo he po olio alue a ime , s a e i is ,, i i a FCF B V   + = , D. Dluhošo á, Z. Zmeškal – Company alua ion unde in e ac ion in disc e e ime ( eal game op ions model) 39 he po olio alue a ime 1 + in upwa d mo emen 1i+ is ( ) 1, 1 1, 1 1, 1 1 i i i a FCF B R V + + + + + +    +  + = , he po olio alue a ime 1 + in downwa d mo e- men 1i− is ( ) 1, 1 1, 1 1, 1 1 i i i a FCF B R V − + − + − +    +  + = , and he e symbol a is he unde lying asse quan i y. Sol ing he h ee equa ions wi h a iables a, B, and ,i V , he alue o mula is he ollowing: ( ) ( ) 1 , , 1, 1 1, 1 11 i i i i V FCF R p V p V − + + − + = + +   + −    , (3) whe e he e m p is he isk-neu al (no ma ke ) p ob- abili y such ha eplica ion is eached. ( ) , 1, 1 1, 1 1, 1 1i i i i R FCF FCF pFCF FCF −+ + + − + +  − =− . I he ee cash low depends on ano he unde lying asse S , which is a unc ion o S , ( ) FCF S= , hen he isk-neu al p obabili y is ( ) , 1, 1 1, 1 1, 1 1i i i i R S S pSS −+ + + − + +  − =− . The gene alised eal op ions alua ion model is a mul i-mode (mul i-swi ching model) allowing swi ch- ing be ween mo e han wo modes. The ecu en equa- ion is he ollowing: ( ) ( ) ,, ,1 1, 1 1, 1 max 11 q m q i i qQ i i C FCF VR p V p V − + + − +  ++  =  +   + −     . (4) He e, q is a pa icula mode, Q is a mode se , and m is an ini ial mode. ,mq C is he swi ching cos be ween modes, o which a nega i e alue means a cos and a posi i e alue means e enue. Fo mo e in o ma ion abou he eal op ions opic, see, o example, T igeo gis (1998). 2. Game heo y appa a us The impo an aspec o alua ion is in e ac ion. I means ha a pa icula subjec ’s decision depends on he decisions o o he in elligen subjec s and ice e sa. This opic is he subjec o game heo y, which could be conside ed o be a gene alised decision-mak- ing heo y. Basic e e ences a e, o example, Dlouhý and Fiala (2009), Maňas (1974), Pe e s (2015), and Tadelis (2013). Games a e ca ego ised acco ding o a ious c i e ia: he numbe o playe s ( wo, mo e han wo), he numbe o s a egies ( ini e – disc e e, in ini e – con inual), he coope a ion ype (coope a i e, non-coope a i e), syn- ch onisa ion (simul aneous, sequen ial), ime (s a ic, dynamic), solu ion esul s a egies (pu e, mixed), in- o ma ion ( ull, pa ), and he game o mula ion (s a e- gic – no mal, ex ensi e). The c ucial e m o game heo y is equilib ium, ha is, playe s a egies sea ching o equilib ium. The basic p inciple is he Nash equilib ium; in o he wo ds, equilib ium playe s a egies ep esen he bes e- sponses o pa icula playe s o o he playe s' s a egies. Al e na i ely, i equilib ium exis s, i any playe s di e om he equilib ium s a egy, hey a e damaged (achie e less u ili y). Usually, a pe ec ma ke is supposed in a alua ion; all he pa icipan s a e p ice ake s. This does no o en su icien ly e lec eali y, and he ma ke s uc u e mus espond o i . I is necessa y o conside and pe - o m a alua ion me hod in coincidence wi h a ma ke s uc u e; o he wise, he alua ion canno be co ec and habi ual asse s a e unde alued. When aluing companies and p ojec s, games ha e o be ca ego ised due o ma ke s uc u es: a pe ec ma ke (conside able playe numbe s), oligopoly ( ini e playe numbe s), du- opoly ( wo playe s), and monopoly (one playe ). Fo he Cou no model, he c ucial a iable is he p oduc ion quan i y and simul aneous decisions o playe s. Ano he possibili y is a compe i ion by p ice, as desc ibed by he Be and model. In he case o se- quen ial decisions, he ma ke consis s o leade s and ollowe s. The S ackelbe g model ep esen s his si ua- ion. Duopoly is he ma ke s uc u e o wo companies. He e, he s a egy choice o he i s company in luence he s a egy selec ion o he second company and ice e sa, so mu ual in e ac ions a e espec ed. In he Cou no p oduc ion duopoly, he goal is p o i maximi- sa ion ( 12 ,zz ) and he s a egic decision abou p oduc- ion quan i y ( 12 ,QQ ) conce ns gi en posi i e in e - als. The in e se demand cu e p o ides he p ice. Sales ( 12 ,TT ) and cos s ( 12 ,NN ) a e exp essed h ough a linea unc ion; 12 , a e uni a iable cos s. The p ice is o mula ed as an in e se linea demand cu e unc ion, ( ) 12 P a b Q Q= −  + . The p o i is s a ed as ollows: 1 1 1 1 1 1 z T N P Q Q= − =  −  , subs i u ing o a p ice,   1 1 2 1 1 1 2 1 1 1 1 2 () () z a b Q Q Q Q a Q b Q b Q Q = −  +  −  = −  −  −   . Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 22, 2019 40 The maximal p o i is an ex emal alue as ollows: 11 1 2 1 ( ) 2 0 za b Q b Q Q = − −  −  =  . I implies 12 1 () 2 a b Q Qb − −  = ; analogically, 21 2 () 2 a b Q Qb − −  = . Bo h unc ions a e so-called e- sponse unc ions s a ing equilib ium p oduc ion in coun e pa y p oduc ion. By mu ual subs i u ion, he equilib ium p oduc ion is 12 1 2 3 a Qb −+ = and 21 2 2 3 a Qb −+ = . The o al p oduc ion is he ollow- ing: 12 12 2 ( ) 3 a Q Q Q b −+ = + = . Subs i u ing he o al p oduc ion in o he equilib ium p ice equa ion, he p ice is 12 () 3 a P++ = . The las s ep is o s a e he p o i wi h knowledge o p oduc ion, ( ) 2 12 1 2 9 a zb −+ = , ( ) 2 21 2 2 9 a zb −+ = . The a io a Qb  has o be posi i e o gi e he p oblem an eco- nomic a ionale. Simila ly, equa ions o o he ma ke s uc u es (ol- igopoly, monopoly, and pe ec ma ke ) can be ob- ained. The esul s a e p esen ed in Table 3–1. The uni a iable cos and many companies a e supposed o he pe ec ma ke . Table 3–1 P oduc ion, p ice, and p o i due o he ma ke s uc u e Ma ke s uc u e P oduc ion o he company i i Q To al p oduc ion Q P ice P P o i o he company i i z Monopoly 2 i a b − 2 i a b − 2 i a + ( ) 2 4 i a b − Cou no duo- poly 2 3 ij a b −+ ( ) 2 3 ij a b −+ 12 () 3 a ++ ( ) 2 2 9 ij a b −+ Cou no oli- gopoly ( ) 1 1 ii a n n n nb − −  + − + 1 n a nb −   + 1 a n n + + ( ) 2 1 1 1 ii a n n bn −  −  + −   +  Pe ec ma - ke 1 1 a nb − + 1 n a nb −   + 1 a n n + + 2 1 1 a bn −   +  Legend: 1n ij ji − −  = , n j j = A bima ix game signi ies a wo-playe game wi h dis- c e e s a egies. The ma ix o each playe p esen s pay- o s o all combina ions o s a egies. Commonly, he game is a non-ze o-sum one. The p ima y objec i e o he playe s' choice o s a egy is o ind an equilib ium s a egy. The solu ion can in ol e pu e s a egies o mixed s a egies. Fi s ly, e e y playe selec s only one s a egy. The me hods o bes esponses and i e a i e elimina ion non-domina ed s a egies can be applied. Secondly, he playe s selec s a egies wi h p obabili y. The op imisa ion mixed p oblem is used. The playe sea ches o he bes esponses (maximal alue) o coun e -playe s a egies by applying he bes esponse me hod. The bes esponses a e shaded in Fig- u e 3–1. Fo playe A, i playe B selec s s a egy B1, he bes eac ion is A1; i he playe selec s B2, he bes esponse is A1. The eac ion o playe B is he ollow- ing: i playe A chooses A1, B selec s B2; o choice A1, playe B selec s B1. The equilib ium poin p esen s he combina ion o A1 and B2 wi h a payo o 500 o A and 700 o B. Payo ma ix Playe A Playe B S a egy B1 S a egy A1 Playe A S a egy A2 600 300 S a egy B2 500 400 Payo ma ix Playe B Playe B S a egy B1 S a egy A1 Playe A S a egy A2 500 300 S a egy B2 700 200 Figu e 3–1 Bes esponse equilib ia in a bima ix game D. Dluhošo á, Z. Zmeškal – Company alua ion unde in e ac ion in disc e e ime ( eal game op ions model) 41 In pu e s a egies, excep o one unique equilib- ium, he e can be mo e o no solu ions. In he case o mo e solu ions, he mos s able solu ion is sough , and he concep o he so-called minimisa ion o shaken hands is used. The common solu ion is in mixed s a e- gies, encompassing pu e s a egies as a subse . The o - mula ion o he non-linea op imisa ion p oblem is e- po ed, o example, by Dlouhý and Fiala (2009), Maňas (1974), and Pe e s (2015) as he ollowing: P oblem I (bima ix game) , , , max ij M N M N i ij j i ij j p q w i j i j p a q pb q w+ − −   , N ij j j a q i    , M ij i i b p w j    He e, i p and j q a e he s a egy p obabili ies o playe A and, espec i ely, playe B, ij a and ij b a e he payo s o playe A and, espec i ely, playe B, and w a e a iables, i and j a e he indexes o he s a egies o playe A and, espec i ely, playe B, and M and N a e he numbe o s a egies o playe A and, espec- i ely, playe B. P oblem I can be modi ied in o a mo e sui able compu a ion o mula ion. A e di iding he equa ions by and w and subs i u ing / ii x p = and / ii x p w= , P oblem II is as ollows: P oblem II (modi ied bima ix game) ,,, max 1 1 M N M N i ij j i ij j p q a b i i i i x a y xb y+ − −   1, N ij j j a y i    1, M ij i i b x j    3. Real game op ions alua ion model The payo unc ion depends on he unde lying asse andom p ocess and independen ly on in e ac ions du - ing he alua ion in he eal op ions me hod. In his sec- ion, he app oach is gene alised. I is supposed ha a payo unc ion is de e mined, excep o he unde ly- ing andom asse mo emen , by mu ual in e ac ions, so i depends on o he playe s’ decision (choice o s a e- gies). The alua ion o he game eal op ions is simila o he eal op ions alua ion, excep he payo unc ion is only gi en by game heo y wi h in e ac ions. The opic in disc e e ime is wo ked ou o example by Che alie -Roignan (2011) and Smi and T igeo gis (2004, 2017). Valua ion conside ing he ac ions o o he compa- nies is a gene alised app oach including isk, lexibil- i y, and in e ac ion. Game heo y ins umen s se e o model in e ac i i y. The c ucial e m o game heo y is equilib ium. The e m mode is subs i u ed by he e m s a egy in compa ing he eal op ions model. The gen- e alised mul i-mode eal op ions wi h in e ac ion model is modi ied o a gene alised mul i-s a egy model as ol- lows: ( ) ( ) , ,, ,1 1, 1 1, 1 11 kk k ww n w i i i i C FCF VR p V p V − − + + − +  ++  = +   + −      , (5) whe e, o he k h playe , k k wW (scala o ec o ) is he equilib ium s a egy and k k wW − − ( ec o o ma- ix) is he s a egy se , o o he playe s, k w− ( ec o o ma ix) is he equilib ium s a egy and k W− (ma ix) is he s a egy se , and ,k nw C (scala o ec o ) is he swi ching cos be ween s a egies. The binomial model wi h isk-neu al alua ion, he wo-phase me hod, and he game payo unc ion is p e- sen ed. Valua ion p ocedu e o he eal game op ions model (i) De e mina ion o he unde lying asse ( ac- o ) andom p ocess An app oach based on an expe 's es ima ion o andom p ocess calib a ion (e.g., B owns, CIR, Ho-Lee). (ii) Equilib ium game payo de e mina ion Equilib ium payo s a e calcula ed due o he game model. (iii) S a ing e minal (con inuum) alue Te minal alue calcula ion (e.g. pe pe ui y and g owing pe pe ui y) due o s a es ,iT V . (i ) De e mina ion alue o pa icula s a es Backwa d induc ion p ocedu e due o (5): ( ) ( ) ,, ,1 1, 1 1, 1 11 k k w n w i i i i C FCF VR p V p V − + + − +  ++  =  +   + −     ( ) De e mina ion alue o an op ion The alue o he eal game op ion a he begin- ning 0 V . Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 22, 2019 42 4. Company alua ion in a duopoly ma ke s uc- u e unde andom demand (simpli ied example) The objec i e is o s a e he alue o he companies (company A and company B) ope a ing in a p oduc ion duopoly. We simplis ically assume ha dep ecia ion equals he in es men , o he ixed cos s a e no consid- e ed, he ne wo king capi al change is ze o, and he ansac ion cos and axes a e neglec ed. The e o e, he ee cash low is iden ical o he p o i . A wo-phase discoun ed cash low alua ion me hod is applied, and a non-coope a i e Cou no p oduc ion duopoly de e - mines he equilib ium p o i . An in e se linea demand cu e gi es he p oduc ion p ice. Random de elopmen is gi en by pa ame e a o he demand cu e, which obeys a geome ic B ownian p ocess. Fo he calcula- ion, a binomial model is used. Wi h inpu pa ame e alue a = 10, he companies’ uni a iable cos s a e A = 2, and B = 3. Hence, com- pany A is mo e e ec i e han company B, he cos o capi al o bo h companies AB R = 20%, and he isk- ee a e R= 10%. The de elopmen o pa ame e a is appa en om he binomial model (Figu e 5–1), and he inpu pa ame e s, including he calcula ed up-index and down-index, along wi h he isk-neu al p obabili- ies, a e shown in Table 5–1. 210 ime s a e 15,63 10,00 6,40 12,50 8,00 10,00 2 1 0 -1 -2 u uu ud = du d dd Figu e 5–1 De elopmen o pa ame e a (binomial model) Table 5–1 Inpu and calcula ed pa ame e s I em Pa ame e Value Pa ame e Value P ice a 10 b 0,5 Uni a iable cos A 2 B 3 Ra es R 0,10 AB R 0,2 Indices U 1,25 D 0,8 P obabili- ies p 0,67 1p− 0,33 Valua ions applying he wo-phase me hod, bino- mial model, eplica ion app oach, Ame ican op ions, and wo payo ma ices a e implemen ed, and a pu e s a egy is supposed. The ecu en alua ion equa ion o mula is as ollows: ( ) ( ) , , , 1 1, 1 1, 1 11 AB i ww i A AA i i z VR p V p V − + + − +  +  =  +   + −     , (6) ( ) ( ) , , , 1 1, 1 1, 1 11 BA i ww i B BB i i z VR p V p V − + + − +  +  =  +   + −     , (7) whe e ,i A V and ,i B V a e alues, , A i z and , B i z a e p o i , A A wW and B B wW (scala ) a e he equilib ium s a egy, A W and B W a e ec o s o s a egies, p is he isk-neu al p obabili y, and R is he isk- ee a e. Duopoly companies' equilib ium p o i o conc e e nodes, shown in Table 5–1, a e ( ) 2 2 9 AB A a zb −+ = and ( ) 2 2 9 BA B a zb −+ = . The backwa d induc ion p ocedu e, wi h he i s alue a a e minal ime using pe pe ui y, is calcula ed as AB z VR = , hen he node alues a e cal- cula ed using (6) and (7). Table 5–1 p esen s he esul s o nodes, including p ice () 3 AB a P++ = and p o- duc ion 2 3 AB A a Qb −+ = , 2 3 BA B a Qb −+ = . The de elopmen equilib ium p o i and company alue a e shown in Figu e 5–1. D. Dluhošo á, Z. Zmeškal – Company alua ion unde in e ac ion in disc e e ime ( eal game op ions model) 43 Table 5–2 Calcula ed alues, p ocess, p oduc ion, and p o i o companies A and B S a e beginning u d uu ud = du dd company A B A B A B A B A B A B coe . a 10,00 12,50 8,00 15,63 10,00 6,40 p ice P 5,00 5,83 4,33 6,88 5,00 3,80 p oduc ion Q 1,50 1,00 1,92 1,42 1,17 0,67 2,44 1,94 1,50 1,00 0,90 0,40 p o i z 18,00 8,00 29,39 16,06 10,89 3,56 47,53 30,03 18,00 8,00 6,48 1,28 237,66 90,00 32,40 200,70 75,25 162,44 210 2 1 0 -1 -2 ime s a e Company A 47,53 18,00 6,48 29,39 10,89 18,00 2 1 0 -1 -2 P o i Value 150,16 40,00 6,40 119,18 29,74 89,24 210 Company B 30,03 8,00 1,28 16,06 3,56 8,00 u uu ud = du d dd u uu ud = du d dd Figu e 5–2 Equilib ium p o i and alue de elopmen o company A and company B I is e iden ha company A’s alue is 162,44 m. u., company B’s alue is 89.24 m. u., and he impac o he companies' e ec i eness is appa en . The compu ed alues e lec he p oduc ion duopoly condi ions and he economic le el o he pa icula companies. Ac- co ding o s a es and ime, he p oblem allows he anal- ysis o he alua ion ci cums ances and equilib ium pa- ame e s; see Table 5–1, which p esen s he p ices P , companies’ p oduc ion A Q and B Q , and p o i A z and A z . I is easy o show ha company alues in a duopoly ma ke could be compu ed compa ably unde an oli- gopoly ma ke s uc u e by applying he o mulas o Table 3–1. 5. Conclusion The eal op ions app oach could be conside ed as a company alua ion concep ha e lec s unce ain y and lexibili y. An essen ial elemen o he alua ion en i onmen is in e ac ion. This phenomenon, embod- ying he mu ual ela ionships among companies, is deal wi h using game heo y. Howe e , he in oduced aspec is o en neglec ed e en i i subs an ially in lu- ences he company alue unde speci ic non-pe ec ma ke s uc u es. Hence, he eal game op ions me hod encompasses his phenomenon. The me hodology o he game eal op ions alua ion model, based on a wo-phase me hod in disc e e ime, was de eloped and o mula ed and an illus a i e ex- ample was p esen ed in he pape . The compu a ion p ocedu e o eal game op ions was desc ibed. Games Ekonomická e ue – Cen al Eu opean Re iew o Economic Issues 22, 2019 44 wi h non-pe ec ma ke s uc u es we e o mula ed, speci ically duopoly and oligopoly Cou no p oduc ion games. The duopoly ma ke s uc u e was implemen ed and calcula ed in he illus a i e example. I was ound ha wo-phase eal game op ions al- ua ion in disc e e ime is a sui able alua ion app oach o companies e lec ing non-pe ec ma ke s uc u es. Re e ences AZEVEDO A., PAXSON, D. (2014). De eloping Real Op ion Game Models. Eu opean Jou nal o Ope a io- nal Resea ch 237 (3): 909–920. h ps://doi.o g/10.1016/j.ejo .2014.02.002 DLOUHÝ, M., FIALA, P. (2009). Ú od do eo ie he . 2. up a ené ydání, P aha: Oeconomica. DLUHOŠOVÁ, D. (2004). Přís upy k analýze inanční ýkonnos i i em a od ě í na bázi me ody EVA – Eco- nomic Value Added. Finance a ú ě – Czech Jou nal o Economics and Finance 11–12, 54:541–559. DLUHOŠOVÁ, D. e al. (2014). Financial Ma- nagemen and Decision-Making o a Company. VSB- Technical Uni e si y o Os a a, Facul y o Econo- mics. CHEVALIER-ROIGNANT, B., TRIGEORGIS, L. (2011). Compe i i e S a egy: Op ions and Games. USA: Massachuse s Ins i u e o Technology. h ps://doi.o g/10.7551/mp ess/9780262015998.001.0 001 GRENADIER, S. R. (2000a). Op ion Exe cise Games: he In e sec ion o Real Op ions and Game. Theo . Jou nal o Applied Co po a e Finance. 13: 99–107. h ps://doi.o g/10.1111/j.1745-6622.2000. b00057.x GRENADIER, S. R. (2002). Op ion Exe cise Games: An Applica ion o he Equilib ium In es men S a e- gies o Fi ms. Re iew o Financial S udies. 15: 691– 721. h ps://doi.o g/10.1093/ s/15.3.691 GRENADIER, S. R. (2000b). Game Choices: The In- e sec ion o Real Op ions and Game Theo y. Risk Bo- oks. h ps://doi.o g/10.1111/j.1745-6622.2000. b00057.x HUISMAN, K. (2001). Technology In es men : A Game Theo e ic Real Op ions App oach. Sp inge h ps://doi.o g/10.1007/978-1-4757-3423-2 HUISMAN, K. J. M., KORT, P. M., PAWLINA, G., THIJSSEN, J. J. J. (2004). S a egic in es men unde unce ain y: Me ging eal op ions wi h game heo y. Zei sch i ü Be iebswi scha . 3: 97–123. h ps://doi.o g/10.1007/978-3-663-12338-5_4 GUTHRIE, G. (2009). Real Op ions in Theo y and P ac ice. Ox o d Uni e si y P ess. MAŇAS, M. (1974). Teo ie he a op imální ozhodo- ání. P aha: SNTL. PETERS, H. (2015). Game Theo y: A Mul i-Le eled App oach. Sp inge . h ps://doi.o g/10.1007/978-4- 431-55339-7, h ps://doi.o g/10.1007/978-4-431- 55336-6 SMIT, H. T. J., TRIGEORGIS, L. (2004). S a egic in- es men : eal op ions and games. P ince on: P ince on Uni e si y P ess. h ps://doi.o g/10.1515/9781400829392 SMIT, H. T. J., TRIGEORGIS, L. (2017). S a egic NPV: Real Op ions and S a egic Games unde Di e- en In o ma ion S uc u e. S a egic Managemen Jou nal, 38: 2555–2578. h ps://doi.o g/10.1002/smj.2665 TADELIS, S. (2013). Game Theo y-In oduc ion. P in- ce on Uni e si y P ess. TRIGEORGIS, L. (1998). Real Op ions – Manage ial Flexibili y and S a egy in Resou ce Alloca ion. Ha- a d Uni e si y. ZMEŠKAL, Z. (2008). Applica ion o he Ame ican Real Flexible Swi ch Op ions Me hodology A Gene a- lized App oach. Finance a ú ě -Czech Jou nal o Eco- nomics and Finance. 5–6, 58:261-275. ZMEŠKAL, Z. (2010). Gene alised so binomial Ame ican eal op ion p icing model ( uzzy-s ochas ic app oach). Eu opean Jou nal o Ope a ional Resea ch. 207(2):1096–1103. h ps://doi.o g/10.1016/j.ejo .2010.05.045 ZMEŠKAL, Z. (2013). Game heo y and eal op ions decision-making hyb id models unde andom demand. In: 31 h In e na ional Con e ence on Ma hema ical Me hods in Economics. P ague: 1057–1062.