Management Science Letters

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1 Maageme Scece Leers 6 (216) Coes lss avalable a GrowgScece Maageme Scece Leers homepage: The combao of sysem dyamcs ad game heory aalyzg olgopoly markes Al Mohammad a, Alagh Mosleh Shraz b, Ahmad Talebezhad c, Ahmad Sadraee Javaher b ad Ehsa Javamard d a Professor, School of Ecoomcs, Maageme ad Socal Sceces, Uversy of Shraz, Shraz, Ira b Assocae Professor, School of Ecoomcs, Maageme ad Socal Sceces, Uversy of Shraz, Shraz, Ira c Asssa Professor, School of Ecoomcs, Maageme ad Socal Sceces, Uversy of Shraz, Shraz, Ira d Ph.D. Sude, School of Ecoomcs, Maageme ad Socal Sceces, Uversy of Shraz, Shraz, Ira C H R O N I C L E A B S T R A C T Arcle hsory: Receved Ocober 28, 215 Receved revsed forma November 28, 215 Acceped Jauary 28, 216 Avalable ole February 2, 216 eywords: Sysems dyamc Game heory Olgopoly marke I hs paper, we prese a hybrd mehod of game heory ad dyamc sysems o sudy he behavor of frms a olgopoly marke. The am of hs sudy s o buld a model for a olgopoly game o he bass of feedback loops ad sysem dyamcs approach ad o solve he resuled problems uder some specal codos where radoal game heory mehods are uable o hadle. The mehod cludes a combao of qualave mehods cludg ervews wh dusry expers o prepare he model ad quaave mehods of sysem dyamcs, smulao mehodologes ad game heory. The resuls dcae ha compeve behavor ad he mpora parameers such as volume of demad, eres raes ad prce flucuao wll be sablzed afer a raso perod. 216 Growg Scece Ld. All rghs reserved. 1. Iroduco Whe a ey's prof does o ecessarly deped o hs/her behavor, bu could be flueced by he behavor of oe or more oher ees, ad he decsos of ohers, boh could have posve ad egave mpacs o hs/her profs, a game bewee wo or more ees are formed (Ahmed & Hegaz, 26). I maagg games, dffere sraeges could be cosdered. Feedback Sackelberg sraeges, for sace, are cosdered for woperso lear mulsage games wh quadrac performace crera ad osy measuremes (Casao & Ahas, 1976). I he world of commerce ad busess, here s always a ogog game ad wheever here s oly lmed umber of supplers for a parcular produc, a olgopoly game s formed. I hs srucure, he acvy of each seller wll affec he behavor of oher vedors. Aoher po a olgopoly marke srucure s ha here s a erdepedecy bewee he frms compared wh he oher markes ad hs s a aural cosequece of he lmed umber of supplers. The prmary obecve of hs sudy s o evaluae he compeve behavor of he frms a olgopolsc marke based o a hybrd of sysem dyamcs ad game heory Correspodg auhor. Emal address: avamard.ehsa@gmal.com (E. Javamard) 216 Growg Scece Ld. All rghs reserved. do: /.msl

2 266 (Akyama & aeko, 22). Baard e al. (215) proposed a evoluoary model of olgopoly compeo where ages could choose bewee varous behavoral rules o make decsos o producos. Merloe ad Szdarovszky (215) vesgaed dyamc olgopoles wh coge workforce ad vesme coss. Zhag e al. (215) preseed a gameheorec ecoomc operao of resdeal dsrbuo sysem wh hgh parcpao of dsrbued elecrcy prosumers. They deermed a ew roles of ules ad dsrbued elecrcy prosumers he fuure real elecrcy marke. The gameheorec algorhms were mplemeed o deec he real elecrcy marke prce by cosderg he group coalo scearos of mulple elecrcy prosumers. Lamber ad Maova (214) proposed a feedback equlbra a dyamc reewable resource olgopoly. They examed a rece leraure o producve asse exploao descrbg a dffereal olgopoly game of resource exraco uder sac, lear feedback ad olear feedback sraeges, where hey permed for he possbly of resource exhauso. They repored ha, frs, feedback rules could eal resource exhauso for a fe umber of frms. I addo, feedback sraeges were more aggressve ha sac oes as log as he resource sock was bg eough, accordace wh he acqured vew based o he radoal preempo argume assocaed wh feedback formao. Akyama ad aeko (2) preseed a heorecal framework called dyamcal sysems game, whch he game self could be chaged due o he effec of players behavors ad saes. Asker (27) formulaed a dyamcal muleam Couro game for a reewable resource. 2. The proposed sudy The ma problem wh he maory of curre models s o predc he fuure, fac, mos games he evrome s assumed o be cosa. However, realworld, mos players chage her sraeges based o dffere eves. I fac, each player mus cosder dffere crcumsaces ad make hs/her decso accordg o he cosequeces, whch could happe fuure. A dyamc game ca be defed as follows (Nash, 195, 1951; Mgers, 24), G : E( ), S( ) E( 1), S( 1) where G represes a game, E deoes he saus of evrome ad S s assocaed wh saus of each player. Fally, dcaes he saus of he game over me. Therefore, we have u : E(), S() E(), S() v : E( ), S( ), O( ) E( 1), S( 1) G : u v Here u represes aural laws, v deoes he effecs of players acos ad o s assocaed wh acos ha players accomplsh. Each player seup hs aco as follows, Z : E(), S() O () Here represes he player, Z demosraes he mechasm of decso makg ad O represes aco plas wh Z = {Z 1, Z 2,, Z }. Thus, Z : E (), S () O () Le N 1,2,..., be he se of he umber of players, E represes he saus of player, ,,..., ad O o, o,..., o S s s s represe he sae ad aco of each player, respecvely. Fg. 1 demosraes he srucure of game sysem dyamcs. (1) (2) (3) (4)

3 A. Mohammad e al. / Maageme Scece Leers 6 (216) 267 Fg. 1. The srucure of sysem dyamcs game Here, each player res o maxmze hs/her prof as follows, D. P C (5) Here P, D ad C are prce, demad ad cos, respecvely. However, each player s behavor chages over he me so we have he followg, E Le D S R,, 1,2,..., e where S ad R represe he saus ad marke share of player a me, ad (6) (7) e represe oal ad expeced prof of player a me, respecvely. I geeral, we may expec o have a reverse relaoshp bewee demad ad prce as follows, dd dp wh 1 dd, dp dd dp dd (II) 1 dp Therefore, he obecve fuco of he proposed sudy ca be summarzed as follows, D f ( P, D ) C f ( D,, ) DP. CD. PC ( P) ( D,, ) Based o he Eqs. (513) we may show he relaoshps bewee dffere players Fg. 2 as follows (Wes & Lebere, 21), (I) (8) (9) (11) (12) (13)

4 268 S R,, 1,2,..., e E D D. P C S, S 2.2. Modelg a olgopoly mehod Fg. 2. The relaoshps bewee dffere players As we have already saed, he reveue of each frm a olgopoly marke cosss of he prese prof plus he accumulaed profably over me, whch ca be calculaed as follows, Max D. P C (14) d B( O ) O (15) do where O ad B ( O ) represe he opmum aco ad decso of player agas player, respecvely. I lear form of demad, he relaoshp ca be summarzed as follows, D D. P. P,,, D C. D,, where α ad β represe varable ad fxed coss of each player, respecvely. Therefore, he Nash (195, 1951) equlbrum equao s as follows, (16) d D. P C B( P ) P D 2. P. P. dp Solvg Eq. (18) yelds, (17) D. P. ( ) 2 B P P. I case we have oly wo compeors, we have, D1 12. P D D B1( P2 ) P1 P D2 21. P D D B2( P1 ) P P D D D D N ( P1, P2 ) (, ) (19) (2)

5 A. Mohammad e al. / Maageme Scece Leers 6 (216) 269 Accordg o Eq. (19) ad Eq. (2), he profably of each player depeds o each player s varable cos ad prce elascy o demad of he oher player. I real world, he relaoshps are o lear ad here are oher olear relaoshps, whch could be used such as Mulplcave Compeve Ieraco (MCI) ad Mulomal Log (ML) as follows (Elereby & Masour, 212), D e M k k 1.. ( X ). k MCI Model : D D k M e. ( X ). M D. e MNL Model : D D M e 1 1 k ( k. X k ) k 1 ( k. X k ) k 1 Sce he relaoshps are olear, akg he dervaves ad solvg he olear equaos are o easy. Usg logceerg, oe may learzes he frs equao Eq. (23) as follows (Harsay, 24), k (23) D D e M k 1 D 1 1 k 1.. ( X ). k M e. ( X ). log( D ) log( R D ) log( D ) log( R ) k k M log( D ) log( D ) log( ) log( D ) log( M ) log( M ) k 1 M 1 k k k k k 1 1 k 1 log( D ) log( D ).log( X ) log( ) log( e. X. ) Usg a sraghforward mah yelds, (24) (25) k k k k k 1 1 k 1 log( D ) log( D ).log( X ) log( ) log( e. X. ) where D, X k ad represe he arhmec mea ad represes he geomerc mea. Dffereag Eq. (26) from Eq. (25) yelds, (26) D X log( ).log( ) D where k k k 1 X k (27). log( ) Smlarly, we may smplfy he MNL model as follows,

6 27 D log( ).( ) D k X k X k k 1 We ow demosrae he proposed sudy of he paper usg a graphcal sysem dyamcs represeao Fg. 3 as follows (m & m, 1997), (28) Delay Compeors Prce Marke Prce Average Sale Prce Prce adusme Prof Coverage Cumulave Prof Toal Prof Frm Prce Rao Delay ρ Frm s Marke Share Frm Demad Toal Marke Demad Naural Effecs Fg. 3. The cause ad effecs of prce o marke share a olgopoly marke Accordg o Fg. 3, prce of each frm s compared wh he average prce of he marke ad ulmaely deermes he marke share of each frm. I addo, he role of aural facors oal demad of he marke, as well as he mpac of he demad o frm s profs ad eargs mpac o prces, whch s he saus of he marke s cosdered. The cause ad effec relaoshps cosder he effecs of player s prcg sraegy o marke equlbrum. Noe ha hs model, we assume all players are lookg for far reur cosdered for each secor of dusry. Whe a frm cosders a dscou, he frm wll expec gag hgher marke share ad compeors may o reac o such dscou decsos. I addo, Fg. 4 shows he cause ad effecs of demad of each frm o prof (loss) a olgopoly marke. Idusry Prof Marg Expeced Prof Prof Coverage Produc o Rae Toal Prof Toal Cos Frm Demad Toal Marke Demad Iveory U Cos Fg. 4. The cause ad effecs of demad of each frm o prof (loss) a olgopoly marke Accordg o Fg. 4, oal marke demad flueces posvely o frm s demad ad flueces o oal prof, oal cos ad u cos, accordgly. I addo, a crease o frm demad reduces veory, whch flueces posvely o oal cos. Moreover, a crease o produco rae flueces posvely o veory ad expeced prof. Based o he descrpo gve Fg. 3 ad Fg. 4 ad deals of he formao provded we prese he proposed model Fg. 5 as follows,

7 A. Mohammad e al. / Maageme Scece Leers 6 (216) 271 Fg. 5. The proposed cause ad effec relaoshps olgopoly marke 3. The resuls For he proposed sudy of hs paper, we have cosdered a marke wh he followg daase, Prce coeffce γ =.975, he effec of frm compared wh mea marke s.65, whch meas he proposed frm has beer performace compared wh oher frms o he marke, he marke s cosdered o weekly bass ad he model has bee smulaed for 14 weeks or wo years. I addo, varable ad fxed coss are equal o 1 ad 5,, respecvely. Moreover, weekly holdg cos s 2 ad dusry average prof s also 2%. We assume wll ake wo weeks ul oher players recogze a player s chagg sraegy. There are 8 players o he marke ad each player produces 41, us. Fally, he al prce a he begg of he plag s se o 2. The smulao s execued Vesm sofware ad Fg. 6 shows he resuls of veory crculao. Idusry prof margs Expeced prof Rae of prof Prof coverage Toal Reveues Cumulave Prof Toal Coss Fxed Cos U varable cos <Sale> <Prce> Dscou Rae Holdg cos Ial Prce Prce Adusme Prce Produco Toal Iveory Sale <Marke Prce Average> Prce Icresg rae Udersadg of prce compeo Comperors Prce Adusme Ial Comperors Prce Compeors Prce Average <Toal Marke Demad> Marke Prce Average Compeors o Frm Prce Rao Produco rae Frm's marke share Toal Marke Demad <Gamma> Comparave Markeg Effecveess Base marke Demad usual facors mpac Demad Effec udersadg of Demad <Prce> Gamma Fg. 6. The oupu of veory flow

8 272 As explaed earler, we have bee lookg o he effec of chage o prce o oher players behavors, he effecs of prce chage o oher players decsos. Our resuls have dcaed ha all chages wll be sablzed over a log perod of me. Fg. 7 shows he chages o prce of he frm, he chage o volume of produco over me. 1, Dollar/To 2, To/Week 1, Dollar/To 2, Dollar/To 5 Dollar/To 1, To/Week 5 Dollar/To 1, Dollar/To Dollar/To To/Week Tme (Week) Dollar/To Dollar/To Tme (Week) Prce : Curre Sale : Curre Dollar/To To/Week Prce : Curre Compeors Prce Average : Curre Prce versus sales volume over me Prce versus compeors prces Fg. 7. The resuls of he chage o prce o oher players produco ad prce Dollar/To Dollar/To As we ca observe from he resuls of Fg. 7, afer approxmaely oe year, he sysem becomes sable. Fg. 8 also preses he red o prce ad sales fgures. Oce more me, prce ad sales become sable afer oe year. 2 Frm Prce Rao 2, Sale , 1 1,.5 5, Tme (Week) Tme (Week) Frm Prce Rao : Curre Dml Sale : Curre Frm prce rao Fg. 8. The red o prce ad sales fgures Sales To/Week 4, Toal Iveory 2, Toal Marke Demad 3, 15, 2, 1, 1, 5, Tme (Week) Tme (Week) Toal Iveory : Curre To Toal Marke Demad : Curre To/Week Tred of veory Tred of oal marke demad Fg. 9. The red o veory ad marke demad

9 A. Mohammad e al. / Maageme Scece Leers 6 (216) 273 The resuls of Fg. 9 also shows ha veory ad marke demad become sable afer oe year. Fally, Fg. 1 shows he chages o prof ad as we ca see, alhough here some flucuaos o profably bu afer almos oe year, here s seady red o profably. 4 Prof coverage 4 M Rae of prof 2 2 M 2 2 M Tme (Week) 4 M Tme (Week) Prof coverage : Curre 1/Week Rae of prof : Curre The rae of prof coverage The rae of prof Fg. 1. The red o prof coverage ad prof Dollar/Week 4. Cocluso I hs paper, we have preseed a hybrd of game heory ad dyamc sysems o sudy he behavor of frms a olgopoly marke. The am of he sudy was o model a complex sraegy for olgopoly game o he bass of feedback loops ad sysem dyamcs, explored he dyamcs prevalg a game he real world. The resuled model has bee solved uder some specal codos where radoal game heory mehods were uable o hadle. The mehod corporaed a combao of qualave mehods cludg ervews wh dusry expers o prepare he model ad quaave mehods of sysem dyamcs, smulao mehodologes ad game heory. The resuls have dcaed ha compeve behavor ad he mpora parameers such as volume of demad, eres raes ad prce flucuao could be sablzed afer a raso perod. Refereces Ahmed, E., & Hegaz, A. S. (26). O dyamcal muleam ad sgalg games. Appled Mahemacs ad Compuao, 172(1), Akyama, E., & aeko,. (2). Dyamcal sysems game heory ad dyamcs of games. Physca D: Nolear Pheomea, 147(3), Akyama, E., & aeko,. (22). Dyamcal sysems game heory II: A ew approach o he problem of he socal dlemma. Physca D: Nolear Pheomea, 167(1), Asker, S. S. (27). O dyamcal muleam Couro game exploao of a reewable resource. Chaos, Solos & Fracals, 32(1), Baard, L. C., Lamaa, F., & Rad, D. (215). Evoluoary compeo bewee boudedly raoal behavoral rules olgopoly games. Chaos, Solos & Fracals, 79, Casao, D., & Ahas, M. (1976). O sochasc dyamc Sackelberg sraeges. Auomaca, 12(2), Elereby, M. F., & Masour, M. (212). O Couro dyamc muleam game usg complee formao dyamcal sysem. Appled Mahemacs ad Compuao, 218(21), Harsay, J. C. (24). Games wh complee formao played by Bayesa players, : par. he basc model&. Maageme scece,5(12_suppleme), m, D. H., & m, D. H. (1997). Sysem dyamcs model for a mxed sraegy game bewee polce ad drver. Sysem Dyamcs Revew, 13(1), 3352.

10 274 Lamber, L., & Maova, A. (214). Feedback equlbra a dyamc reewable resource olgopoly: preempo, voracy ad exhauso. Joural of Ecoomc Dyamcs ad Corol, 47, Merloe, U., & Szdarovszky, F. (215). Dyamc olgopoles wh coge workforce ad vesme coss. Mahemacs ad Compuers Smulao,18, Mgers, J. (24). Realzg formao sysems: crcal realsm as a uderpg phlosophy for formao sysems. Iformao ad orgazao, 14(2), Nash, J. F. (195). Equlbrum pos perso games. Proc. Na. Acad. Sc. USA, 36(1), Nash, J. (1951). Nocooperave games. Aals of mahemacs, 51, Wes, R. L., & Lebere, C. (21). Smple games as dyamc, coupled sysems: Radomess ad oher emerge properes. Cogve Sysems Research, 1(4), Zhag, N., Ya, Y., & Su, W. (215). A gameheorec ecoomc operao of resdeal dsrbuo sysem wh hgh parcpao of dsrbued elecrcy prosumers. Appled Eergy, 154, by he auhors; lcesee Growg Scece, Caada. Ths arcle s a ope access arcle dsrbued uder he erms ad codos of he Creave Commos Arbuo (CCBY) lcese (hp://creavecommos.org/lceses/by/4./).