The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity 3-3-35 Yamate, Suita Osaka 564-8680 Japa Abstract I the behavior aalysis of a compay we propose a method cosiderig subjectivity of each compay. We itroduce some motives for strategy selectio by a compay ad costruct his subjective game by usig his motive distributio. It is desirable that each compay follows his Nash equilibrium strategy for his ow subjective game. We apply this method to the Courot oligopoly market model ad obtai the subjective Nash equilibrium strategy for each player. Moreover by some umerical examples we examie the ifluece of the motive distributio o the equilibrium strategy. Keywords : The Courot oligopoly model, strategy selectio, motive distributio, subjective game. 1. Itroductio The cocept of the Nash equilibrium (Nash [5]) i o-cooperative game has bee utilized broadly i the area of micro-ecoomics. The Nash equilibrium is a equilibrium which all players reach by aimig at maximizig persoal payoff respectively. Although all players should take the Nash equilibrium strategies i order to realize this Nash equilibrium, may game experimets by huma beigs report that a player does ot E-mail: gh3d901@edu.kasai-u.ac.jp E-mail: akai@iecs.kasai-u.ac.jp Joural of Iformatio & Optimizatio Scieces Vol. 27 (2006), No. 1, pp. 55 68 c Taru Publicatios 0252-2667/06 $2.00 + 0.25
56 S. FURUYAMA AND T. NAKAI ecessarily take his Nash equilibrium strategy. These results show that a player does ot aim at oly maximizatio of his ow payoff. The it is ecessary to explai ratioally why a player takes a strategy except his Nash equilibrium strategy. Nakai [4] otices motives for the strategy selectio by a player, deotes the ucertaity for the oppoet s motives as a motive distributio ad costructs a subjective game for each player. He proposes a subjective Nash equilibrium strategy as oe solutio of the above problem, ad explais the diversity of player s actios ratioally. Furuyama ad Nakai [2] develops the cocept of subjective game i a -perso o-cooperative game. The above-metioed argumet ca be applied eve if a player is a compay. We cosider a oligopoly market. A compay i the iterdepedece situatio geerally iteds to esure his ow profit by elaboratig all sorts of strategic thought i which he predicts others actios ad uses a artifice i order to forestall oppoets i decidig his ow actio. Game theory is ow idispesable to the aalysis i the case that the oppoet s actio iflueces this side. The traditioal game theory shows the resultat equilibrium whe all players aim at maximizig their ow payoffs. But subjective thoughts of players, for example, differeces i the stadpoits of players or the methods of approaches, etc. are ot take ito cosideratio at all. Therefore, for example i the ew etry problem, the etry compay caot discrimiate betwee the case that the existig compay has the cooperative motive to aim at the syergistic effect with the etry compay, ad the case that it has the offesive motive to stop the ew etry. This paper proposes a aalysis method for the case that each compay has oe of various motives ad that the motive is ukow for other players. That is to say, a method of behavior aalysis of a compay i cosideratio of the subjective motive uder strategy selectio is show, applyig the cocept of subjective game to the Courot model (Courot [1]) i a oligopoly market. Moreover we calculate the subjective Nash equilibrium ad examie its chage by the chage of motives i some umerical examples i order to aalyze the ifluece of motive o the equilibrium. 2. Courot oligopoly model I this sectio we cosider the traditioal Courot oligopoly model ad give its Nash equilibrium solutio. We cosider compaies
COURNOT OLIGOPOLY MODEL 57 (players) which produce the same goods. Let q i (q i 0, i = 1,..., ) be the quatity of productio by player i ad therefore Q = i=1 q i is the aggregate supply i the market. We assume that the market price is a liear fuctio of the aggregate supply, that is, a Q if Q a P(Q) = (1) 0 if Q a where a ( 0) is the potetial demad. Let c i be the productio cost per uit quatity of player i, π i (q 1,..., q ) be the payoff of player i whe players 1,..., produce q 1,..., q respectively, that is, π i (q 1,..., q ) = {P(Q) c i }q i (a Q c i )q i if Q a = c i q i if Q a. (2) The, we calculate the Nash equilibrium i case that compaies determie their quatities of products simultaeously. Lettig (q 1,..., q ) be the Nash equilibrium poit, q i is the solutio of max q i >0 π i(q 1,..., q i 1, q i, q i+1,..., q ). The solutio is give by q i = a c i 1 + 1 (a c j ). (3) Therefore the aggregate supply Q, market price P(Q ) ad expected payoff π i (q 1,..., q ) of player i i the equilibrium are as follows: ( ) Q = 1 a + 1 c j (4) P(Q ) = 1 + 1 π i (q 1,..., q ) = { ( a + ) c j a c i 1 + 1 (5) (a c j )} 2. (6) If c i = c (c < a) (i = 1,..., ), we obtai q i = a c + 1 (7)
58 S. FURUYAMA AND T. NAKAI Q (a c) = + 1 (8) P(Q ) = a + c + 1 (9) π i (q 1,..., q ) = ( ) a c 2. + 1 (10) If becomes large, the aggregate supply Q icreases, the market price P(Q ) ad the expected payoff π i of compay i decreases. Namely as the umber of compaies icreases, the competitio itesifies; as a result, the price is reduced. The, the market power of each compay becomes weak ad makes the social surplus icrease. The the icrease i the umber of compaies is a importat factor which makes a oligopolistic market trasform structurally ito a more competitive market. Therefore the chage of regulatios with respect to a ew etry has a importat meaig i market ecoomy. The literatures [3, 6, 7, 8, 9] are good guides o oligopoly market models. 3. Itroductio of subjective motives i a strategy selectio I this sectio, we costruct a model which itroduces some subjective motives for a strategy selectio by each player i the Courot oligopoly model. Nakai [4], ad Furuyama ad Nakai [2] advocate subjective games which take ito cosideratio motives for a strategy selectio by a player i the field of o-cooperative games. The subjective game is costructed by motives of all players which he expects i the competitio situatio from his subjective viewpoit. Each player will play his ow subjective game based o his thought. By this method we ca explai the behavior of a player with the altruistic character who hopes the icrease of the oppoet s profit with his self-sacrifice. There is o ivestigatio with respect to the behavior aalysis of a compay by usig a subjective game. That is to say, i the traditioal micro-ecoomics a market competitio model is costructed uder objective coditios (for example, the legal framework ad the relatio betwee supply ad demad etc.) i which the oppoet s subjective thought has ot bee cosidered explicitly. But i the behavior aalysis of a compay the maagemet attitude of the compay ad the subjective aticipatio for the attitude of the oppoet compay have importat effects. For example, sice a major compay is rich i iformatio, fuds ad techology, it ca afford to have log-term prospects, as a result, it
COURNOT OLIGOPOLY MODEL 59 may behave by the differet thought from a smaller compay. Therefore it has a importat sigificace that we apply the subjective game to the behavior aalysis of a compay. I this paper we propose a method for the behavior aalysis of a compay (player) by usig the above-metioed subjective game. First we itroduce some motives for a behavior of a compay. Though various motives ca be cosidered, we cosider four motives: the selfish motive, the cooperative motive, the competitive motive, the offesive motive. We cosider a motive distributio o these motives which deotes a aticipatio with respect to a motive of each player by a certai player P. Usig the origial payoff fuctio ad the motive distributio by player P, we defie a subjective game for player P. It is desirable that player P follows his Nash equilibrium strategy for his ow subjective game. We shall obtai actually the subjective game for the Courot model. As motives i selectig a strategy, we cosider four motives m 1,...,, m 4, as follows: m 1 : The selfish motive (to aim at maximizatio of his ow persoal payoff). max π i. q i >0 m 2 : The cooperative motive (to aim at maximizatio of the average of all members payoff). max q i >0 1 π j. m 3 : The competitive motive (to aim at maximizatio of the differece betwee his ow payoff ad the average of the others payoff). ( max π i 1 ) q i >0 1 π j. m 4 : The offesive motive (to aim at miimizatio of the average of the others payoff). ( max 1 ) q i >0 1 π j. Now, we cosider oe uspecified player P (ayoe of all players).
60 S. FURUYAMA AND T. NAKAI Let θ i = θ i 1,θi 2,θi 3,θi 4 (11) be a player i s motive distributio which player P estimates, where θ i k is the probability that player P thiks that player i follows the motive m k, θ i k 0 (k = 1,..., 4), 4 θk i k=1 = 1 (i = 1,..., ). The, whe player 1,..., produce quatities q 1,..., q respectively, the expected payoff fuctio with respect to player i s motive distributio θ i cosidered by player P is give by πi P(q 1,..., q θ i ) = θ1 i π i + θ2 i 1 +θ i 4 = θ i 1 π i + θi 2 = θi 4 ( π j + θ3 i π i 1 ) 1 π j ( 1 ) 1 π j 1 ( ( π j + θ3 i π i θi 3 ) π 1 j π i π j π i ) ( θ i 1 + θi 3 + θi 3 + θi 4 1 ) θ i π i +( 2 θi 3 + ) θi 4 π 1 j = A i π i + B i π j (12) where A i = θ i 1 + θi 3 + θi 3 + θi 4 1 (13) B i = θi 2 θi 3 + θi 4 1. (14) By the equatio (2), we obtai πi P(q 1,..., q θ i ) (A i +B i )qi 2 + {(A i +B i )(a c i ) (A i +2B i )Q i }q ) i = +B i (aq i Q 2 i c j q j if Q a (A i + B i )c i q i B i c j q j if Q a (15)
COURNOT OLIGOPOLY MODEL 61 where Q i = q j. We call the above-metioed o-cooperative game (π1 P,..., π P ) the subjective Courot game for player P. 4. Subjective Nash equilibrium I this sectio, we shall obtai a Nash equilibrium of the subjective Courot game for player P defied i the previous sectio. The Nash equilibrium i this game is called the subjective Nash equilibrium for player P. Let q = (q 1,..., q ) be the subjective Nash equilibrium poit. Furthermore we put Q = q j, Q i = q j. The q i is the solutio of the followig maximizig problem π P i (q 1,..., q i 1, q i, q i+1,..., q θ i ) max q i ( 0). (16) From the equatio (15), the fuctio πi P is a cocave quadratic fuctio with respect to q i i regio q i a Q i ad a decreasig liear fuctio i regio q i a Q i. Furthermore the fuctio π i P is cotiuous at q i = a Q i. The the optimal solutio q i exists i the iterval [0, a Q i ]. Whe 0 q i a Q i, from π i P/ q i = 0, we obtai q i = (A i + B i )(a c i ) (A i + 2B i )Q i 2(A i + B i ) (17) if A i + B i = 0, that is, θ i 4 < 1. Substitutig Q i = Q qi i the equatio (17) ad rewritig it, we obtai q i = (A i + B i )(a c i ) (A i + 2B i )Q A i (18) if A i = 0, that is, θ i 2 < 1. Calculatig the sum Q = K j (a c j ) 1 + (2K j 1) of (18), we obtai i=1, (19)
62 S. FURUYAMA AND T. NAKAI where K i = 1 + B i = ( 1)θi 1 + ( 1)θi 2 + ( 1)θi 3 A i ( 1)θ1 i + 2 θ3 i +. (20) θi 4 Substitutig equatio (19) i equatio (18), q i = K i (a c i ) 2K i 1 1 + K j (a c j ) q i. (21) (2K j 1) Let q i be the value of the right had side of the equatio (21). I order that q i is optimal, it must satisfy the coditio 0 q i a Q i. The to be exact, q i = mi{max( q i, 0), a Q i }. (22) The the followig theorem is acquired. Theorem 1. Suppose that θ2 i = 1, θi 4 = 1 for ay i, amely, player P cosiders that o player is cooperative ad offesive with probability 1. The subjective Nash equilibrium poit for player P is give by q = (q 1,..., q ) where 0 if q i < 0 qi = q i if 0 q i a Q (23) i a Q i if a Q i < q i. The value q i is give by (21). The coditio q i a Q i is equivalet to the followig relatio: K j (a c j ) 1 + (2K j 1) a. (24) The subjective Nash equilibrium strategy of player P is q P. The equilibrium price is P(Q ) = a Q = a K j (a c j ) 1 + (2K j 1). (25)
COURNOT OLIGOPOLY MODEL 63 The expected payoff of player i at the subjective Nash equilibrium poit is π P i (q 1,..., q θ i ) = (a Q c i )q i 5. Numerical examples = K i (a c i ) 2 (a c i )(3K i 1) +(2K i 1) { K j (a c j ) 1 + (2K j 1) K j (a c j ) 1 + (2K j 1) } 2. (26) I this sectio we give some umerical examples showig the equilibrium productio quatity ad the equilibrium payoff for each player uder various motive distributios. I all examples we suppose that = 3, a = 1000, c i = c = 2 ad that θ2 i = 1, θi 4 = 1 (i = 1, 2, 3). Moreover we restrict our attetio to the case that the costrait (24) is satisfied, that is, where 499(K 1 + K 2 + K 3 ) K 1 + K 2 + K 3 1 1000 (27) K i = 2(3θi 1 + θi 2 + 3θi 3 ) 3(2θ i 1 + 3θi 3 + θi 4 ). (28) The i the followig figures, it may occur that the graph breaks halfway. Example 1. We cosider the case that all players have oly two motives m 1 ad m 2 ad have the same motive distributio θ i 1,θi 2,θi 3,θi 4 = 1 θ,θ, 0, 0 (i = 1, 2, 3) where θ is the probability that each player follows the cooperative motive m 2. The commo equilibrium productio quatity q i is show by the curve m 2 i Figure 1. The curve m 2 is decreasig getly. Namely if all players become more cooperative, the each player aims to maximize the social payoff (the average of all player s payoffs) ad refrais from high productio. Similarly we cosider the case that all players have the same motive distributio 1 θ, 0, θ, 0 ( 1 θ, 0, 0, θ ). The commo equilibrium productio quatity is show by the curve m 3 (m 4 ) i Figure 1. The curve m 3 is icreasig getly. This meas that if all players become more
64 S. FURUYAMA AND T. NAKAI competitive, the each player icreases his productio quatity, as a result, the society falls i overproductio. The curve m 4 is icreasig rapidly. Namely if all players become more offesive, the each player icreases his productio quatity explosively, as a result, the price falls heavily. Figure 1 The equilibrium productio quatity i Example 1 Example 2. We cosider the case that player II ad III have the same motive distributio 0.7, 0.1, 0.1, 0.1, that is, they are, if aythig, selfish ad that player I follows the motive distributio 1 θ, 0,θ, 0 ( 1 θ, 0, 0,θ ). The equilibrium productio quatity qi ad the equilibrium payoff πi P for each player are show i Figure 2, 3 (Figure 4, 5). From Figure 2, 3 we kow the followigs: If player I becomes more competitive, the he ca expect the icrease of his ow payoff by icreasig his productio quatity. O the other had player II ad III must decrease their productio quatities, as a result, their expected payoffs decrease. Namely player I is active ad player II ad III are passive. From Figure 4, 5, we kow the followigs: If player I becomes more offesive, the the above-metioed tedecy becomes large, but the equilibrium payoff of player I decreases rapidly after passig the peak sice the decrease of the other players payoffs reach the limit. Example 3. We suppose that player II ad III have motive distributio 0.7, 0.1, 0.1, 0.1 ad 0.1, 0.1, 0.7, 0.1 respectively, that is, player II (III) is, if aythig, selfish (competitive). Whe the motive distributio of player I is 1 θ, 0,θ, 0) ( 1 θ, 0, 0,θ ), the equilibrium productio
COURNOT OLIGOPOLY MODEL 65 Figure 2 The equilibrium productio quatity i the case of 1 θ, 0,θ, 0 i Example 2 Figure 3 The expected payoff i the case of 1 θ, 0,θ, 0 i Example 2 Figure 4 The equilibrium productio quatity i the case of 1 θ, 0, 0,θ i Example 2
66 S. FURUYAMA AND T. NAKAI Figure 5 The expected payoff i the case of 1 θ, 0, 0,θ i Example 2 quatity ad the equilibrium payoff of each player are show i Figure 6, 7 (Figure 8, 9). From Figure 6, 7 we kow the followigs: If player I becomes more competitive, the his productio quatity icreases ad the productio quatities of player II ad III decrease slightly. O the other had the equilibrium payoff of player I chages from the slight icrease to the slight decrease ad the equilibrium payoffs of player II ad III decrease. From Figure 8, 9 we kow the followigs: If player I becomes more offesive, the the above-metioed tedecy becomes large. The equilibrium payoff of player I decreases rapidly as the expected payoffs of other players decrease. Figure 6 The equilibrium productio quatity i the case of 1 θ, 0,θ, 0 i Example 3
COURNOT OLIGOPOLY MODEL 67 Figure 7 The expected payoff i the case of 1 θ, 0,θ, 0 i Example 3 Figure 8 The equilibrium productio quatity i the case of 1 θ, 0, 0,θ i Example 3 Figure 9 The expected payoff i the case of 1 θ, 0, 0,θ i Example 3
68 S. FURUYAMA AND T. NAKAI 6. Cocludig commets I this paper, we itroduced the behavior aalysis model of the compay cosiderig the subjective motive i the strategy selectio. The fudametal model is the Courot oligopoly model. By applyig the subjective game to it, we reflect the subjective thoughts of compaies with differet stadpoits ad attitudes, etc. i decisio-makig. We cosider four types of motives for the costructio of the model, ad give three umerical examples. Of course, more motives ca be cosidered actually. With the icrease of the umber of motives, the equilibrium aalysis seems to become more complicated. From umerical examples we ca kow that i the Courot oligopoly market the o-cooperative motive (competitive motive ad offesive motive) brigs a advatageous result. I the traditioal ecoomics may discussios have bee made o the assumptio of the perfect ratioality ad objective coditios of the market. But there are a few ivestigatios treatig subjectivities of players. The our model with subjective game is hoped for oe of market aalysis models pursuig subjectivities of compaies. Refereces [1] A. Courot, Researches ito the Mathematical Priciples of the Theory of Wealth, N. Baco (ed.), Macmilla, New York, 1838. [2] S. Furuyama ad T. Nakai, The costructio of subjective games by motive distributios i -perso o-cooperative game, Joural of Iformatio ad Optimizatio Scieces, Vol. 25 (3) (2004), pp. 533 541. [3] R. Gibbos, Game Theory for Applied Ecoomists, Priceto Uiversity Press, 1992. [4] T. Nakai, Subjective games i a o-cooperative game, Joural of Iformatio ad Optimizatio Scieces, Vol. 21 (1) (2000), pp. 129 147. [5] J. F. Nash, No-cooperative games, Aals of Mathematics, Vol. 54 (1951), pp. 286 295. [6] L. Phlips, Applied Idustrial Ecoomics, Cambridge Uiversity Press, 1998. [7] W. G. Shepherd ad J. M. Shepherd (2004), The Ecoomics of Idustrial Orgaizatio, Wavelad Press, Ic. [8] O. Shy, Idustrial Orgaizatio, The MIT Press, 1996. [9] J. Tirole, The Theory of Idustrial Orgaizatio, The MIT Press, 2001. Received March, 2005