Research on dynamic adjustment of cooperation in price duopoly game

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1 3rd Internatonal Conferene on Mehatrons and Informaton Tehnology (ICMIT 06 Researh on dynam adjustment of ooeraton n re duooly game Gung ShaDehang XabYujng gao3bentu L4dDehua Wang5e 345 Shandong Voatonal College of Sene and TeonologyNo.6388 Xhuan RoadWefangShandong ProvneChna a sg.98@63.om bxad985@63.om gaoyjng@63.omdlbentu@63.omewangdehua000@63.om Keywords: Betrand game nomlete nformaton ooeraton Pareto Otmalty. Abstrat: The aer onsders a Betrand model wth nomlete nformaton. Two strateges wth re adjustment are manly dsussed: the tt-for-tat strategy and the tt-for-tat strategy wth ooeratve ntenton and ther dynam systems resetvely. Introduton Duooly s refer to a market stuaton n whh the atons of several frms affet suly and re of same or homogeneous roduts. It s ths haraterst of nterdeendene that makes olgoolst onsder reatons of the other omettors.under erfet ometton all frm are outut-takers or re-takers n olgoolst ometton []. Two of the most famous and mortant olgooly models are Cournot model and Bertrand model. In Cournot model eah olgoolst frm assumes that other frms hold ther oututs onstant. Whle Bertrand model s based uon the remse that eah olgoolst frm assumes other omettors hold ther re onstant. For maxmzng rofts all frms selet a quantty to rodue by other s oututs n Cournot model [][3]. Whereas n Bertrand model eah frm maxmzes rofts by settng a re that underuts omettors res when omettors res exeed ost. In the above models eah frm s moton s ndeendent. However the frms s very lmted n the duooly market and they an realze the nterdeendene between them and that an easy to ause the ooeraton between them. In game theory Nash equlbrum s the bas onet whh refers to omettve equlbrum and t reflets ndvdual ratonalty but t volates olletve ratonalty Nash equlbrum of the duooly game s not Pareto otmal. The rsoners dlemma shows that there s a ontradton between ndvdual ratonalty and olletve ratonalty and the orret hoe based on ndvdual ratonalty wll redue everybody s welfare. The man queston whh the rsoners dlemma oses s whether a ooeratve behavour an emerge among ratonal and self-nterested layers whenever there s no formal agreement [4]. Theoretal and exermental studes have ndated several ways by whh the ooeratve soluton an emerge [5][6]. For examle admttng an nfnte number of nteratons the so-alled folk theorem shows that ooeraton an be establshed through a system of unsh ments and rewards although not ayoff-maxmzng n any sngle stage. Axelorod has also demonstrated that the best behavour allowng the ahevement of ooeraton n reeated games s the tt-for-tat ondut onsstng n dong what the oonent dd n the revous move. In ths aer we study that how frms get bgger rofts by adjustng ther own re wthout the nformaton of the omettor s outut and roft and onsder the ooeratve behavour n duooly ometton wth the tt-for-tat ondut. 06. The authors - Publshed by Atlants Press 308

2 The model In Betrand game frms hoose the res of ther roduts nstead of the quanttes they wll rodue as n Cournot game. We onsder two frms rodung smlar roduts n a olgooly market. Let t reresent the re of th frm at dsrete tme erods t The quantty eah frm sells equatons: q a lnear nverse demand funton s determned by the followng q a b + q a b + ( a and b are ostve onstants. The ost funton has the lnear form C q. ( the ostve arameters are margnal osts of the frm resetvely. Wth above assumtons the roft of the frm at tme t s gven by π π ( q ( ( a b ( q ( ( a b + t t t t t t + t t t t t t Ths aer s about ooeraton under the nomlete nformaton and the followng models are based on whh the frms omare ther own rofts wth the ooeratve roft. The solvng of the ooeratve roft has been ntrodued n duooly game theory. The ooeratve roft means the roft whh s solved by maxmzng the sum of all frms roft. We onsder the symmetral: and get the ooeratve roft ( π 4( b ( ( b a+ b. a b ( 3 and the ooeratve re The tt-for-tat dynam strategy The tt-for-tat strategy s the best behavour allowng the ahevement of ooeraton n reeated games [8]. Its haraterst s that every layer onssts n dong what the oonent dd n revous move. In ths aer the Betrand model s studed wth the tt-for-tat ondutng and the dynam equatons are based on the nomlete nformaton. Although eah roduer annot obtan the omettor s omlete nformaton he omletely knows about hs own re and roft. The frm an omare hs roft π t at tme t wth the ooeratve roft π whh s Pareto otmal. Whle hs own roft s more than the ooeratve roft ( π π > 0 he extraolates that the omettor s ooeratve then he wll roerly rases hs re n order to ontnue the ooeraton as a reward ; Otherwse f π π < 0 the frm annot realze the ooeratve roft and t t Under the ondton that the market demand s onstant rasng the re an redues the sales volume and the roft dereases. 309

3 extraolates that the omettor s not ooeratve then he wll redues hs re as enalty. Based on these thoughts a dynam equaton wth the re s adjustment s bult as follows: + + u t t t ( π π ( ( t + u t a bt jt π + u ( s a adjustng arameter and u > 0. j and j ( 4 Wth above assumtons the duooly game wth heterogeneous layers s desrbed by a ' ' two-dmensonal nonlnear ma T( ( defned as ( ( ( ( ' u a b+ π T : ' u a b + π ' denotes the unt-tme advanement that s f the rght-hand sde varables are re of erod t then the left-hand ones reresent re of erod t +. In ths aer we are onsderng a eonom model only non-negatve equlbrum onts are meanngful. So that we only ay attenton to the nonnegatve fxed onts of the ma ( 5.e. ( 5 the soluton of the nonlnear algebra system as ( ( a b ( ( a b + π 0 + π 0 ( 6 whh s obtaned by settng ' n system ( 5. We have an only fxed ont of system ( 6 E ( ( a+ b ( 7 The study of the loal stablty of the fxed ont of the two-dmensonal system ( 5 deends on the egenvalues of the Jaoban matrx of ( 5. The Jaoban matrx J at the ont ( has the form ( J We estmate the Jaoban matrx J at E whh s ( ( u ( + u ( a b + + b + u a b + + b u ( J The haraterst equaton s ( ( ( ( u a b u a b. u a b u a b ( 8 Under the ondton that the market demand s onstant redung the re an nrease the sales volume and the roft nreases. 30

4 ( λ λ Trλ+ Det 0 Tr s the trae and Det s the determnant of the Jaoban matrx J(. ( ( u a b u a b Tr ( ( u a b u a b Det Then we have two egenvalues of matrx J( λ and λ ( ( u a b u a b. From the ondton that u ( s very small we have that λ <. Sne λ s seal we an t know the stablty of the system ( 5. But the followng numeral exerments show that ts stablty s senstve to the arameters. The tt-for-tat dynam strategy wth ooeratve ntenton For further studes wth the ooeratve behavour we mrove the equaton ( 4. Through addng feedbak ontrol we have the form ( π π ( ( ( π ( u + v t t t t u a b + v t t t jt t u ( s an adjustment arameter and u > 0 0 ontrol of the system. In the equaton ( 9 the frst tem ( ( ( tt-for-tat strategy and the seond tem ( v( t v > v( t ( 9 s the feedbak u t a bt + jt π orresonds wth the shows that the frm wll redue the re of ths erod whle the revous one surasses the ooeratve re ( > 0 otherwse ( < 0 the frm wll roerly nrease the re of ths erod. Ths shows that the frms t have ertan ooeratve onsousness. Beause of the feedbak ontrol the frms have the sontanety to the ooeratve behavour. That s to say the frms both have the ooeratve ntenton from ther own angles. In ths system ( matrx J of ( ( + ( 9 E ( 9 at the ont ( has the form a b s also ts fxed ont. The Jaoban t ( J We an get the Jaoban matrx J at E whh s ( ( ( ( + u a b + + b v u u + u a b + + b v ( 0 3

5 ( J E ( ( ( ( u a b u a b v u a b u a b v and we have two egenvalues of matrx J( E λ v and λ ( ( u a b u a b v. So long as 0 v < < and ( v 0< u+ u < a absolute value of λ and λ s smaller than and hene the equlbrum s loal stable. the Conluson The aer onsders a Betrand model wth nomlete nformaton. Two strateges wth re adjustment are manly dsussed: the tt-for-tat strategy and the tt-for-tat strategy wth ooeratve ntenton and ther dynam systems resetvely.analyss results ndate that the frms both have the ooeratve ntenton from ther own angles n the tt - for - tat strategy.but the followng numeral exerments show that ts stablty s senstve to the arameters.beause of the feedbak ontrol the frms have the sontanety to the ooeratve behavour. the system stablze n a ertan range. Referenes [] Jxang Zhang Qngl Da Yanhua Wang. The dynams of Betrang model wth bounded ratonalty. Chaos Solutons and Fratals (007. [] H.N. Agza A.A. Elsadany. Chaot dynams n nonlnear duooly game wth heterogeneous layers. Aled Mathemats and Comutaton 49 ( [3] M.T. Yassen H.N. Agza. Analyss of a duooly game wth delayed bounded ratonalty. Aled Mathemats and Comutaton 38 ( [4] Vttoro Cafagna Paolo Coorese. Dynamal systems and the arsng of ooeraton n a Cournot duooly. Chaos Soltons and Fratal 5 ( [5] Davd K. Levne Wolfgang Pesendorfer. The evoluton of ooeraton through mtaton. Games and Eonom Behavor 58 ( [6] Gary Charness Gullaume R. Fréhette Cheng-Zhong Qn. Endogenous transfers n the Prsoner s Dlemma game:an exermental test of ooeraton and oordnaton. Games and Eonom Behavor 60 ( [7] Axelrod R. The evoluton of ooeraton. New York: Bas Books;