ON THE EXISTENCE OF EQUILIBRIUM IN GAMES AND ECONOMIES MURAT ATLAMAZ

Transcription

1 To Betul

2 ON THE EXISTENCE OF EQUILIBRIUM IN GAMES AND ECONOMIES The Insttute of Economcs and Socal Scences of Blkent Unversty by MURAT ATLAMAZ In Partal Fullfllment of the Requrements for the Degree of MASTER OF ECONOMICS n THE DEPARTMENT OF ECONOMICS BILENT UNIVERSITY ANARA July 2001

3 I certfy that I have read ths thess and have found that t s fully adequate, n scope and n qualty, as a thess for the degree of Master of Economcs Assocate Professor Dr. Farhad Hussenov I certfy that I have read ths thess and have found that t s fully adequate, n scope and n qualty, as a thess for the degree of Master of Economcs Professor Dr. Bulent Ozguler Examnng Commttee Member I certfy that I have read ths thess and have found that t s fully adequate, n scope and n qualty, as a thess for the degree of Master of Economcs Assstant Professor Dr. Erdem Basc Examnng Commttee Member Approval of the Insttute of Economcs and Socal Scences Prof. Dr. ürşat Aydoğan Drector

4 ABSTRACT ON THE EXISTENCE OF EQUILIBRIUM IN GAMES AND ECONOMIES Atlamaz, Murat M.A., Department of Economcs Supervsor: Assocate Professor Farhad Hussenov July 2001 There are three man contrbutons of ths thess n equlbrum theory. The frst s about the exstence of equlbrum n dscontnuous games. We fnd suffcent condtons for the exstence of ε-nash equlbrum n games wth dscontnuous payoff functons. In the second one, under tme-varyng dscount factors we restate the Folk Theorems on the exstence of equlbra of nfntely repeated games. The thrd one s about the exstence of equlbrum n economes wth ndvsble goods. We re-formulate the model and the exstence result of Danlov et al(2001) wth more realstc cost functons, whch depend on prces as well as the output level. eywords: Exstence of equlbrum, ε-nash equlbrum, dscontnuous games, tme-varyng dscount factors.

5 ÖZET OYUNLARDA VE EONOMİLERDE DENGENİN VARLIĞI ÜZERİNE Atlamaz, Murat Yüksek Lsans, İktsat Bölümü Tez Yönetcs: Doç.Dr. Farhad Hüssenov Temmuz 2001 Bu çalõşma le denge kuramõna üç ana katkõda bulunmak amaçlanmõştõr. Brncs süreksz oyunlarda dengenn varlõğõ le lgldr. Bu kõsõmda süreksz kazanç fonksyonu olan oyunlarda ε-nash dengesnn varolablmes çn yeter şartlar gösterlmektedr. İknc olarak, zamanla değşen ndrm katsayõlarõ kullanõlarak, sonsuz tekrarlõ oyunlarda dengenn varlõğõna lşkn Folk teoremler şekllendrlmektedr. Üçüncü olarak se bölünemeyen mallardan oluşan ekonomlerde dengenn varlõğõ ele alõnmaktadõr. Danlov(2001) n çalõşmasõnda kurulan model ve denge varlõk sonuçlarõ, daha gerçekç olan, üretm mktarõ le brlkte fyatlara da bağlõ malyet fonksyonlarõ kullanõlarak tekrar formüle edlmektedr. Anahtar sözcükler: Denge varlõğõ, ε-nash denges, süreksz oyunlar, zamanla değşen ndrm katsayõlarõ. v

6 ACNOWLEDGEMENTS I wsh to express my deepest gradtute to Assocate Professor Farhad Hussenov for hs gudance and helpful comments throughout my graduate study. I am very thankful to Assstant Professor Erdem Basc for hs comments whch I benefted a lot. I also wsh to thank Assstant Professor Serdar Sayan because he beleved n me when I promsed to complete my graduate study at Blkent. I am ndebted to Betul for beng so patent wth me and supportng me. I am also thankful to Bars Yaslan and Ozcan oc for ther supports and frendshps. v

7 TABLE OF CONTENTS ABSTRACT ÖZET v ACNOWLEDGEMENTS......v TABLE OF CONTENTS...v TABLE OF FIGURES...v CHAPTER I: INTRODUCTION CHAPTER II: NASH EQUILIBRIUM IN STRATEGIC GAMES Strategc Games Nash Equlbrum Mxed Nash Equlbrum Exstence of a Nash Equlbrum ε-nash Equlbrum...16 CHAPTER III: EXISTENCE OF EQUILIBRIUM IN DISCONTINUOUS GAMES...21 CHAPTER IV: EQUILIBRIUM IN EXTENSIVE GAMES WITH PERFECT INFORMATION...27 v

8 4.1 Extensve Games Extensve Games wth Perfect Informaton Subgame Perfect Nash Equlbrum...34 CHAPTER V: EQUILIBRIUM IN REPEATED GAMES Infntely Repeated Games Folk Theorems...46 CHAPTER VI: EXISTENCE OF EQUILIBRIUM IN PRODUCTION ECONOMY WITH INDIVISIBLE GOODS The Model of Producton Economy Convexfcaton of an Economy Exstence of an Equlbrum...57 CHAPTER VII: CONCLUSION...63 REFERENCES...64 v

9 LIST OF FIGURES 1. Prssoner s Dlemma An example for a strategc game Another example for a strategc game Another example for a strategc game Quas upper sem-contnuty Two examples for extensve games Another example for extensve games Strategc form of an extensve game A stage game for an nfntely repeated game...43 v

10 CHAPTER I: INTRODUCTION The theory of exstence of equlbrum has played a core role n the feld of game theory. In developng ths theory, the man tool was topologcal analyss. The topologcal propertes of the strategy and/or the commodty sets and the behavours of payoff and/or producton functons have determned ths relatonshp between the topologcal ssues and Game Theory. The concept of equlbrum n Game Theory nvolves the dea of stablty. The payoff of any agent depends on the actons of all agents. On the other hand, n General Equlbrum, ths concept s drectly related to ndvdual constraned optmzatons by agents of an economy. In ths framework, the payoff functons depend only on the behavour of the owner of ths payoff. However, agents are not completely free n solvng those ndvdual problems. The market constrants create an ndrect dependency among the actons of agents and ther opponents utltes. In the lterature, there are a number of studes such as Shafer and Sonnenschen (1975) that establsh close relatonshps among dfferent equlbrum concepts of Game Theory and Economcs. The objectve of ths paper s to extend some recent results n related areas, whch deal wth the exstence of equlbra. 1

11 Frst result that s obtaned s nspred by the well-known paper of Dasgupta and Maskn (1986). We relax some of the contnuty assumptons and get an exstence result for ε-nash equlbrum as Dasgupta and Maskn (1986) guaranteed the exstence of Nash equlbrum wth some stronger condtons. Another result that s presented n ths study s about the nfntely repeated games. In the lterature such as Abreu and Rubnsten (1988), Abreu (1988) and Abreu et al (1994), the basc form of preference relaton that captures the combned effect of payoffs obtaned per perod durng the nfnte tme horzon s dscounted crteron whch dscounts the future payoffs by a real number δ (0,1) that s constant over tme. However, t seems unrealstc to assume the constancy of the dscount rate over tme. It may be changed n a certan range by external effects. Therefore, n a more realstc framework, assumng the varablty of δ, some Folk Theorems are restated. Our fnal result concerns wth a slghtly dverse subject, the General Equlbrum. In a recent paper, Danlov et al. (2001) provded suffcent condtons of the exstence of equlbra for producton economes wth ndvsble goods. They assumed that cost functons depend only on the level of output that would be produced. However, accordng to General Equlbrum approach, costs also depend on prce levels of outputs. As prces ncrease, cost of producton tends to ncrease for a fxed level of output and vce versa. The exstence theorem s restated wth more plausble cost functons dependng on output and prce levels. 2

12 Ths paper s organzed as follows. In Chapter 2, the strategc games are ntroduced wth some examples and two fundamental exstence theorems are formulated as Theorem 1 and Theorem 2. Also, an nterestng unqueness result due to Rosen (1965) s stated wthout ts proof. At the end of ths chapter, an orgnal result of exstence of ε-nash equlbrum s establshed as Proposton 3. In Chapter 3, the mportant theorems of Dasgupta and Maskn (1986) that guarantee the exstence of Nash equlbrum n dscontnuous games are stated and another orgnal and more advanced exstence result of ε-nash equlbrum s establshed as Theorem 6. In Chapter 4, we revew the extensve games wth perfect nformaton to prepare ourselves to the concept of nfntely repeated games. Such games are ntroduced n the followng Chapter n whch the Folk Theorems are formulated and they are restated as Proposton 6 and Proposton 7 wth the tme-varyng dscounted crteron that s mentoned n the precedng paragraphs. Fnally, n Chapter 6, the model of producton economes wth ndvsble goods s presented and the exstence theorem s stated as Theorem 9 wth more realstc producton functons, namely, the ones dependng on the prce and the level of output. Concludng remarks are made n Chapter 7. 3

13 CHAPTER II: NASH EQUILIBRIUM IN STRATEGIC GAMES 2.1 STRATEGIC GAMES A strategc game conssts of a fnte number of players (N), the strategy sets for each player (S, N) and the payoff functons defned on strategy profles (u, N). The payoff functons gve the von Neumann-Morgenstern utlty u (s) for each s = ( s1, s2,..., s N ) of strateges to each player. A strategy profle s also denoted as ( s, s ) where s = ( s1, s2,..., s 1, s + 1,..., s ), and S - s the N set of all s -. Here, abusng notaton, the set of players {1, 2,..., N } s denoted by N. If the strategy set of each player s fnte, the strategc game s sad to be fnte. Such games can be shown usng matrces. In the Fgure 1, one of the most famous games, the prsoner s dlemma, s shown n a 2 2 matrx. The rows are the strateges of player 1 whle the columns are of player 2, and C, D are the labels of strateges of each player such that C and D mean confess and don t confess, respectvely. The story behnd ths game s that two people are arrested because of a crme, and they are put nto seperate cells so that they have no chance to communcate. They are seperately told that they wll be sentenced to three years f they both confess whereas one year f they both do not confess. If only one of them confesses, as a reward he wll be freed and the other wll be sentenced to four years. The mnus sgns of the numbers n Fgure 1 emphasze that the best 4

14 C D C -3,-3 0,-4 D -4, 0-1,-1 Fgure 1 job they could do s to be free, otherwse the utlty n a prson wll always be negatve. Formally, ths game can be formulated as follows: 1 2 { } N = 2, S = S = C, D u ( C, C) = u ( C, C) = u ( C, D) = 0, u ( C, D) = u ( D, C) = 4, u ( D, C) = u ( D, D) = u ( D, D) = The crucal pont of the story above s that they cannot communcate, whch s the base of strategc games. Namely, a player s not nformed about what the others play, what she knows s just the possble strateges that opponents can choose (the strategy sets). Ths s summarzed accurately n Fudenberg as follows: It s helpful to thnk of players strateges as correspondng to varous buttons on a computer keyboard. The players are thought of as beng n separate rooms, and beng asked to choose a button wthout communcaton wth each other. In a strategc game, the am of a player s to maxmze her payoff. Certanly, she wll try to do ths by choosng her strategy, however, solely her 5

15 b 1 b 2 b 3 b 4 a 1 3, 7 1, 6 7, 5 6, 3 a 2 0, 1 4, 4 5, 1 7, 2 a 3 7, 0 1, 5 0, 6 1, 2 a 4 3, 1 0, 0 2, 1 8, 0 Fgure 2 strategy does not suffce. In Fgure 1, let player 1 play C n two dfferent games. If her opponent responses by playng C n the frst and D n the second game, the payoffs of player 1 wll be 3 and 4 respectvely though her own strategy does not change. Therefore, the strategy s chosen by each player consderng the opponents strateges assumng that all the players are ratonal(wllng to maxmze ther payoffs). Whch of these strateges are ratonal to be played? Frst of all, a strategy wll not be chosen by any of the players f t s not the best response aganst a strategy of the opponent. Such a bad strategy s worthless deservng to be elmnated. For nstance, consder the game shown n Fgure 2. There are two players wth the strategy sets S = { a, a, a, a } and S { b, b, b, b } =. As seen, b 4 never gves the best payoff for player 2 aganst any of the strateges of player 1 so that t s not ratonal to play b 4 n ths sense. The second stage derves from the fact that all the players know the rratonalty of playng b 4. Thus, player 1 never plays a 4 whch s only a best response to the bad strategy b 4 of player 2. On the other hand, player 1 plays a 1 f she thnks that her opponent wll play b 3, player 2 plays b 3 f she thnks that player 1 wll play a 3, player 1 plays a 3 f she thnks 6

16 player 2 wll play b 1, player 2 plays b 1 f she thnks that player 1 wll play a 1, and so on. In fact, the strateges n ths chan does not lead to a steady state because both players are never content n ths chan. Here, the steady state wll occur when the thoughts of the players agree wth each other whch leads to Nash Equlbrum. Before ntroducng ths new concept, let us defne best response formally. Defnton 1 In a strategc game [,( ),( )] player to hs opponents strateges for all s S. N S u, strategy s f u ( s, s ) u ( s, s ) Then, best response correspondence for player s defned as { } BR ( s ) = s S : u ( s, s ) u ( s, s ), s S. s s a best response for Example 1 Consder the game depcted n Fgure 2. Some best response correspondences are BR1( b1) = a3, BR1( b2) a2 =, BR ( a ) { b, b } = NASH EQUILIBRIUM A steady state s better understood wth the formal defnton of best response correspondence. For nstance, for the game of Fgure 2, a 2 BR 1 ( b 2 ) and b2 BR2 a2 ( ). Thus, both players are content f the chosen strateges are a 2 7

17 and b 2 for player 1 and player 2, respectvely. Namely, f a pror compromse were allowed, the agreement would be on playng a 2 and b 2 (Though no pror nformaton about the opponents actons s stpulated, t does not volate ths condton to assume nformal agreements before the start of the game). Defnton 2 A strategy profle ( s1, s2,..., s ) s a Nash equlbrum of the game [,( ),( )] N S u f s s a best response to the other players strateges Equvalently, ( s1, s2,..., s ) s a Nash equlbrum f for every N, for all s S. N N s for all. u ( s, s ) u ( s, s ) (1) An mportant pont about ths soluton concept s that devaton s dscouraged n a Nash equlbrum. Namely, f any player herself ntends to devate from her strategy, then (1) stpulates that she loses. Ths pont s clarfed n the followng examples of Nash equlbra n varous games. Example 2 (The Prsoner s Dlemma) In the game of Fgure 1, (C,C) s the unque Nash equlbrum. Each of the players gets 4 by devatng from C whle gettng 1 at the Nash equlbrum. The other profles do not match ths nondevaton rule. For nstance, examne the profle (D,C). If player 1 devates from playng D, she gets 3 nstead of 4, so devaton s proftable. Ths shows that (D,C) s not a Nash Equlbrum. 8

18 X Y X 10, 10 0, 0 Y 0, 0 1, 1 Fgure 3 Example 3 The strategy profle (, ) n Fgure 2. a b s a Nash equlbrum of the game shown 2 2 Example 4 In the game depcted n Fgure 3, (X,X) and (Y,Y) are both Nash equlbra even though (X,X) seems more proftable than (Y,Y) for both players. In these examples, the strateges of Nash equlbra are all determnstc. Such strateges are called pure strateges. However, an alternatve noton of equlbrum s possble. Players may choose ther strateges, for nstance, by lotteres so that the strateges are not certan, what s known may be just the probablty dstrbuton over the strategy set. Such strateges allow us to defne mxed extensons of strategc games. 9

19 2.3 MIXED NASH EQUILIBRIUM Let [,( ),( )] N S u be a strategc game. Then the set of probablty dstrbutons 1 over S s denoted by (S ), and any element of ths set s referred to as a mxed strategy of player. Defnton 3 The mxed extenson of the strategc game [,( ),( )] [ N, ( S ),( u )] N S u s the game n whch (S ) s the set of all probablty dstrbutons for player over S, and u s defned over ( S ) as [ ] u ( σ, σ,..., σ ) = σ ( s ) σ ( s )... σ ( s ) u ( s, s,..., s ) (2) 1 2 N N N 1 2 N s S where σ s the probablty dstrbuton of player, and σ ( s ) s the contrbuton of the pure strategy s to σ. Namely, σ ( s) = 1 for every player N. s S strateges. The equlbrum concept may be generalzed to nclude the mxed 1 Such probablty dstrbuton s formed by player by randomsng over her own pure strateges, and ths s ndependent of the other players such randomsatons f any. 10

20 H T H 1,-1-1, 1 T -1, 1 1,-1 Fgure 4 Defnton 4 A mxed strategy profle σ s a Nash equlbrum f for all players, for every σ ( S ). u ( σ, σ ) u ( σ, σ ) Example 5 A smple example s the Matchng Pennes game shown n Fgure 4. H and T are head and tal, respectvely. It s easy to see that ths game has no pure strategy equlbrum. However, f both players randomze on ther own 1 = =, 2 strateges wth equal probabltes, namely; σ ( H) σ ( T) for all { 1,2} such mxed strateges consttute a mxed strategy Nash equlbrum. In fact, t s the unque Nash equlbrum. As n the prevous example, when any player chooses the mxed strategy 1 σ( H) = σ( T) = of Nash equlbrum, the opponent becomes ndfferent 2 between playng H or T, the pure strateges. In fact, t s a general property of mxed strategy Nash equlbrum as stated n Proposton 1. Proposton 1 Let [ N,( S),( u )] be a fnte strategc game and S + S be the set 11

21 of strateges that contrbute postvely to the mxed strategy of the profle ( ) σ = σ1, σ2..., σ N for all N. Then σ s a mxed strategy Nash equlbrum f and only f for every player, s s a best response toσ for all s S +. Proof Let σ ( σ σ σ ) = 1, 2..., N be a mxed strategy Nash equlbrum. Suppose that there exsts N, and s S + such that s s not a best response to σ, that s, there exsts s S such that u( s, σ ) > u( s, σ ). Then player can strctly ncrease her payoff by playng s nstead of s by (2). Conversely, suppose that for every player, s s a best response to σ for all s S +, and σ s not a mxed strategy Nash equlbrum. Then there exsts N and σ ( S ) such that u ( σ, σ ) u ( σ, σ ) (3) By (3), there must be pure strateges s S + and s S + that contrbute postvely to σ and contradcton. σ respectvely such that s gves a hgher payoff than s whch s a Example 4 (contnued) The game shown n Fgure 3 has two pure strategy Nash equlbra as t s ponted out n Example 4. Let ( σ1, σ 2) be a mxed strategy Nash equlbrum, by Proposton 1, 10. σ ( X) = 1. σ ( Y) = 1 σ ( X) (4)

22 1 10 Then σ 1( X ) =, smlarly σ 1( Y ) = Here, (4) follows from the fact that both X and Y are best responses of player 2 to σ 1 f σ 2 s not a degenerate mxed Nash equlbrum. 2 As a result, choosng X wth probabltes 1 11 for both players s a Nash equlbrum. 2.4 EXISTENCE OF A NASH EQUILIBRIUM In the prevous sectons all the games n the examples had at least one Nash equlbrum. Indeed, t s not a chance. The common property of these examples, the fnteness of the games, leads to such a concluson as stated n the followng theorem. Theorem 1 (Nash, 1950) Every fnte strategc game has a mxed strategy Nash equlbrum. 3 What can be sad f the strategy sets are not fnte? In fact, fnteness s a strong assumpton. In most applcatons of Nash Theory to economc theory and other areas the strategy sets are not fnte, there may be a contnuum of strateges. For nstance, n determnaton of prce of a good by an agent, the set of strateges, 2 At least two of the pure strateges contrbute postvely to ths mxed strategy. 3 For the well known proofs of Theorem 1 and Theorem 2 that use akutan s Theorem, the readers are referred to Sundaram (1996 ) and Mouln (1986). 13

23 the nterval of possbly chosen prce, s not fnte. The followng theorem gves a result for exstence of Nash equlbrum for pure strateges where the strategy sets are not fnte. But before the theorem, let us make a defnton. Defnton 5 Let f : D R where D s a convex subset of n R. Let U f ( a ) denote the upper-contour set of f at a R such that U f ( a) { x D f( x) a} =. Then the functon f s sad to be quas-concave on D f U ( a ) s a convex set for each a. f Theorem 2 (Debreu, 1952; Glcksberg, 1952; Fan, 1952) Suppose that for all N, the strategy set S s nonempty, convex and compact, the utlty functon u s contnuous over S = S1 S2... SN and quas-concave wth respect to s. Then the game [ N,( S ),( u )] has a pure strategy Nash equlbrum. Theorem 1 s a specal case of ths theorem. The set of mxed strateges over a fnte set S s a compact, convex set, and the utlty functons whch are lnear polynomals are trvally contnuous over S and quas-concave wth respect to σ. The followng are corollares that are drectly derved from Theorem 2. The frst one s the applcaton for symmetrc games. The latter one s the exstence of mxed strategy Nash equlbrum for the same assumptons except the convexty of S s and the quas-concavty of u s. 14

24 Corollary 1 Let [ N,( S),( u )] be a symmetrc game ( x 1 =... = xn and u () s = u ( s ) f s s deduced from s by exchangng s and s j ) wth convex, j compact strategy sets, and a utlty functon u for each whch s contnuous over S and quas-concave wth respect to ts varable s. Then ths game has a symmetrc Nash equlbrum s, that s, s 1 = s 2 =... = sn. Corollary 2 (Glcksberg, 1952) Let [ N,( S ),( u )] be a strategc game n whch strategy sets (S ) are nonempty, compact, and utlty functons (u ) are contnuous over S for all N. Then there exsts a mxed strategy Nash equlbrum. These theorems do not say anythng about the unqueness of Nash equlbra. One of the most mportant results, whch holds under really strong assumptons, s the followng. Theorem 3 (Rosen, 1965) Let X [ a, b] = be a compact real nterval for all and X = X1... XN. Let u be a C 2 functon defned on X satsfyng 2 u ( ) 0 2 x < x for all x X. Denote by the n n matrx wth (,j) entry 2 ( u)/( x xj). If + t s negatve defnte for all x X, then the game [ N,( X ),( u )] has a unque Nash equlbrum. 15

25 2.5 ε-nash EQUILIBRIUM In a Nash equlbrum, each player ensures that she maxmzes her utlty assumng that the opponents play ther Nash equlbrum strateges. She does not beneft from devatng from a Nash equlbrum because of ths maxmzaton. However, n some crcumstances, players do not want to leave a steady state, though t s not a Nash equlbrum, for a small amount of gan. Such an approach leads to ε-nash equlbrum. Defnton 6 In a strategc game [ N,( S ),( u )], a strategy profle σ s an ε-nash equlbrum where ε > 0 f for every player N, for all σ ( S ). u ( σ, σ ) u ( σ, σ ) ε (5) In such a strategy profle, no player can gan more than ε by devatng from the profle. It s a drect consequence that any Nash equlbrum of a strategc game satsfes ths condton. Proposton 2 In a strategc game, a Nash equlbrum s always an ε-nash equlbrum for all ε 0. 16

26 Proof Let σ ( σ σ σ ) every σ ( S ), = 1, 2..., n be a Nash equlbrum. Then for every player and u ( σ, σ ) u ( σ, σ ) It follows that (5) s satsfed for any ε>0 whch completes the proof. Let us defne upper sem-contnuty and lower sem-contnuty that we wll use frequently n ths study. Defnton 7 Let Θ and S be subsets of R m and R n, respectvely. A correspondence Φ: Θ PS ( ) s called upper sem-contnuous at a pont θ Θ f for all open sets V such that Φ( θ ) V, there exsts an open set U contanng θ such that θ U Θ mples Φ( θ ) V. Φ s called upper sem-contnuous on Θ f Φ s upper sem-contnuous at each θ Θ. In partcular, a functon n f : D R R s sad to be upper sem-contnuous at x D f for all sequences xk x, lmsup f( x ) f( x). k k Defnton 8 A correspondence Φ: Θ PS ( ) s called lower sem-contnuous at a pont θ Θ f for all open sets V such that Φ( θ ) V, there exsts an open set U contanng θ such that θ U Θ mples Φ( θ ) V. Φ s called lower sem-contnuous on Θ f Φ s lower sem-contnuous at each θ Θ. In n partcular, a functon f : D R R s sad to be lower sem-contnuous at 17

27 x D f for all sequences x k x, lmnf f( x ) f( x),.e., f f s upper k k sem-contnuous at x. The exstence theorems are all applcable to ε-nash equlbrum. Moreover, some assumptons made for Nash equlbrum can be weakened whle applyng to ε-nash equlbrum. Proposton 3 Let [ N,( S ),( u )] be a strategc game n whch the strategy sets S are nonempty, convex and compact, and let the utlty functons u be contnuous n s, upper sem-contnuous n s and quas-concave wth respect to s for every. Then there exsts an ε-nash equlbrum. Lemma 1 4 Let ϕ : X 2 Y where X and Y are convex subsets of Eucldean spaces. If ϕ s nonempty, convex valued and lower sem contnuous, then there exsts a contnuous functon f : X Y such that f( x) ϕ( x) for all x X. Lemma 2 5 (Brouwer s Fxed Pont Theorem) Let X n R be compact and convex, and f : X X a contnuous functon. Then f has a fxed pont, that s, there exsts x X such that f( x) = x. 4 Partal case of Theorem 3.1 of E. Mchael (1956, p.368). 5 For the proof the readers are referred to Smart (1974). 18

28 Remark 1 Proposton 3 and Lemma 1 together mply a verson of akutan s Fxed Pont Theorem (stated as Lemma 3 n Chapter III) n whch lower semcontnuty replaces upper sem-contnuty. Proof (of Proposton 3) Let us defne ε R ( s ) = s S u( s, s ) > sup u( s, s ) ε s S Moreover, defne R ( s) = R ( s ) R ( s )... R ( s ). ε ε ε ε N N R ( s ) s nonempty by defnton of supremum and convex by quas-concavty of ε ε u wth respect to s for any s S. Then the correspondence R : S 2 S s ε nonempty-, convex-valued. Let s O R ( s ), that s, u ( s, s ) > sup u ( s, s ) ε and s O. s S Now, we wll show that ( s ) = sup u ( s, s ) s upper sem-contnuous. Let ϕ s S s k s and let k s be such that k k k 1 u( s, s ) > ϕ( s ). (6) k Then wthout loss of generalty, s k s snce S s compact. Then by (6) and usng the upper sem-contnuty of u, lmsup ϕ ( s k ) lmsup u ( s k, s k ) u ( s, s ) ϕ ( s ) k and so ϕ s upper sem-contnuous. k 19

29 Denote a = u ( s, s ) [ ϕ ( s ) ε] > 0. Snce ϕ s upper sem-contnuous, there exsts a neghbourhood of s, say V 1, such that a ϕ( s ) < ϕ( s ) + for s V1. (7) 3 On the other hand, snce u s contnuous n s - there exsts a neghbourhood of say V 2, such that s, a u( s, s ) > u( s, s ) for s V2. (8) 3 Now, by (7) and (8) we have a a a a u( s, s ) > u( s, s ) > ϕ( s ) ε + a > ϕ( s ) + a ε ϕ( s ) ε ε Then s R ( s ) for s V = V 1 V. So, ε ( ) 2 s O R s for every s V. Hence, R ε s lower sem-contnuous. Therefore, we have that R ε s a lower sem-contnuous correspondence. Now, we can apply Lemma 1 to ths correspondence, namely, there exsts a contnuous functon θ : S S such that θ( s) R ε ( s) for all s S. Then, by Lemma 2, there exsts a fxed pont of the functon θ whch s a fxed pont of the correspondence R ε, and ths pont s a Nash equlbrum of the game. Another theorem for exstence of ε-nash equlbrum wth weaker condtons s stated n Chapter V. 20

30 CHAPTER III: EXISTENCE OF EQUILIBRIUM IN DISCONTINUOUS GAMES Whle consderng the exstence of Nash equlbrum n strategc games, contnuty of payoff functons over payoff profles s generally assumed. Nevertheless ths s a strong assumpton. In many crcumstances, games have dscontnuous payoff functons. In a famous paper of Dasgupta and Maskn(1986a) two exstence theorems for dscontnuous games are presented. Frst, they provde condtons that are weaker than contnuty and allow the use of akutan s Theorem to guarantee the exstence of a pure-strategy Nash equlbrum. Second, they provde condtons for the exstence of a mxed strategy Nash equlbrum n games wthout quas-concave payoff functons. These two theorems wll be stated n Theorem 4 and Theorem 5 wthout ther proofs. Theorem 4 6 Let S be a nonempty, convex and compact subset of a fnte dmensonal Eucldean space for all. Let u be quas-concave n s, upper sem- contnuous n s, and have a contnuous maxmum, that s, u ( s ) = max u ( s, s ) s S s lower sem-contnuous n s -. Then there exsts a pure strategy Nash equlbrum. 21

31 Theorem 5 7 Let S be a closed nterval of R. Let S ** () denote the set of s such that u s dscontnuous at s and { } S ( s ) = s S ( s, s ) S ( ) ** ** Let, for any two players and j, D() be a postve nteger and for each nteger d d wth 1 d D( ), let there exst a fnte number of functons f : S S, that are one-to-one and contnuous such that for each j j d { j j } ** * S S = s S j d d D s = f s () (), 1 () s.t. ( ) Suppose that u s contnuous except on a subset S ** () of S * (), u ( s) s upper sem-contnuous, and u ( s ) s bounded. Suppose also that u s weakly lower sem- contnuous n s, that s, for all s S there exsts λ [0,1] such that for all s S ** ( s ), N λlmnf u ( s, s ) + (1 λ)lmnf u ( s, s ) u ( s, s ) s s s s Then the game has a mxed strategy Nash equlbrum. The concepts quas upper sem-contnuty and ε-lower sem-contnuty wll be orgnally defned to be used n the man theorem of ths chapter. 6,7 For the proofs of Theorem 4 and Theorem 5, the readers are referred to Dasgupta and Maskn(1986). 22

32 Fgure 5 Defnton 9 A functon f : X R s sad to be quas upper sem-contnuous f x X, k x x, f( x) lmnf f( x ). k k Clearly upper sem-contnuty s a stronger condton than quas upper sem-contnuty. Example 6 Let f : R R be a functon defned as f( x ) = x+ 1 f x< 0 1 f x = f x > 0. As t can be seen from the Fgure 5, maxmum s not attaned by the functon f whereas any upper sem-contnuous functon always attans ts maxmum. 23

33 Therefore, f s not upper sem-contnuous. It s easy to check that f s quas upper sem-contnuous. Defnton 10 A functon f : k lmnf f( x ) > f( x) ε for all x A and k x x A R s sad to be ε-lower sem-contnuous f k x A where k x x. In the followng theorem, the exstence of ε-nash equlbrum s consdered nstead whch leads to some weaker assumptons compared to Theorem 4, that s, for nstance, maxmum ε-lower sem-contnuty s assumed nstead of maxmum contnuty, and quas upper sem-contnuty n s wth upper sem-contnuty n s s assumed nstead of upper sem-contnuty n s. But before statng the theorem, let us recall the akutan s Fxed Pont Theorem. Lemma 3 (akutan s Fxed Pont Theorem) Suppose that N A R s nonempty, compact and convex. Let f : A A be an upper sem-contnuous correspondence and f( x) A be nonempty and convex for every x A. Then f (.) has a fxed pont; that s, there s an x A such that x f( x). Theorem 6 8 Let n the game [N, (S ),(u )], S be nonempty, convex, compact subsets n Eucldean spaces, u be quas-concave and upper sem-contnuous n s for all, and quas upper sem-contnuous n s. Moreover, assume that for all, 8 Ths theorem s orgnally stated and proved by Farhad Hussenov. 24

34 the maxmum functon u * ( s ) = max u ( s, s ) s ε-lower sem-contnuous. Then s S for every ε > ε there exsts an ε -Nash equlbrum. ε ε Proof Fx ε > ε and put η =. Consder η-best response correspondence 2 * { η} η R ( s ) = s S u ( s, s ) u ( s ). Snce u s quas-concave and upper sem-contnuous n s, R η s a nonempty-, compact-, convex-valued correspondence. Denote R η the closure of R η, that s, η η ( ) Gr ( R ) Gr R =. Frst we wll show that η ε R ( s ) R ( s ), s S (9) k Let s Gr( R η ) and s k ε s. Then we must show that s R ( s ), N. We have u s u s η for all k. Wth quas upper sem-contnuty of u and k * k ( ) ( ) ε-lower sem-contnuty of the maxmum functon, ths mples u s u s u s u s ε. k * k * ( ) lmnf ( ) lmnf ( ) η ( ) k k ε η Therefore s R ( s ) and (9) s proved. Now, for all s S, R ( s ) s a ε closed set n the compact convex set R ( s ). Hence, ( ) η ε co R ( s ) R ( s ), N For the correspondence cor η = cor N η, all assumptons of akutan s Fxed Pont Theorem stated as Lemma 3 are satsfed. By ths theorem, there exsts s S such that 25

35 η s ( cor )( s ) η ε Therefore, s cor ( s ) for all N. Ths together wth (9) gves s R ( s ) for all N, so s s an ε -Nash equlbrum. The followng consequence of Theorem 6 s obvous. Corollary 3 Let n Theorem 6, the maxmum functon u * (.) be a lower semcontnuous for all. Then for an arbtrary ε > 0, there exsts an ε-nash equlbrum. 26

36 CHAPTER IV: EQUILIBRIUM IN EXTENSIVE GAMES WITH PERFECT INFORMATION In strategc games, a one shot game s played, the strateges are chosen smultaneously for once, and the game s fnshed. However, n most of the game theory related stuatons there s tme dmenson. Players may act several tmes observng partally or completely opponents past actons. Ths s a dynamc stuaton as opposed to the statc stuaton n strategc games. Such knd of games are sad to be extensve games. 4.1 EXTENSIVE GAMES An extensve game s a detaled descrpton of the sequental structure of the decson problems encountered by the players n a strategc stuaton (Osborne and Rubnsten, 1994). The extensve form descrbes the order n whch players move and what each player knows about the opponents moves when makng each of her decsons. Ths knowledge may be partal or complete as mentoned n the ntroducng paragraph. If every player knows the prevous moves completely, these games are called extensve games wth perfect nformaton. If some of the players do not know some nformaton about the actons of the other players taken prevously or a player forgets the prevous moves of herself or she s uncertan 27

37 Fgure 6 whether another player has acted, such knd of games are called extensve games wth mperfect nformaton. Extensve games are llustrated usng game trees n whch nodes and the lne segments represent players and ther actons, respectvely. In the followng examples the concept game tree s llustrated. Example 7 In the extensve game wth perfect nformaton depcted n Fgure 6(a), the players move sequentally rather than smultaneously. Player 1 has two choces of move: or L. Learnng her actual move, the player 2 has three choces of move: m, n and p f player 1 plays, and two choces of move: r and s f player 1 plays L. For ths reason, player 2 has two decson nodes. Moreover, f player 1 and player 2 decde to play, for nstance, and n sequentally, ther payoffs wll be 1 and 4, respectvely. 28

38 Example 8 The game depcted n Fgure 6(b) s a smple example to extensve games wth mperfect nformaton. Imperfecton comes from the nformaton set shown by the dots. The meanng of ths nformaton set s that when t s player 1 s second turn to move, she does not know whch of these two nodes she s at snce she could not observe the prevous acton of player 2. In ths game, the player 1 has to decde twce. In fact, each sngle node may be nterpreted as an nformaton set ncludng one node. Hence, every player acts at each nformaton set belongng to herself. Ths s why player 1 acts twce nstead of three tmes. In the extensve games wth mperfect nformaton, generally Nash equlbrum (and subgame perfect Nash equlbrum that wll be defned n Secton 4.3) s not suffcently powerful so that some other soluton concepts such as sequental equlbrum, perfect Bayesan equlbrum that are all specal cases of Nash equlbrum are defned and mostly used. Therefore, we wll not study further the extensve games wth mperfect nformaton. 4.2 EXTENSIVE GAMES WITH PERFECT INFORMATION We start ths secton wth formal defnton of extensve games wth perfect nformaton. Defnton 11 An extensve game wth perfect nformaton conssts of the followng components: 29

39 . A set N (the set of players). A set H of sequences (fnte or nfnte) that satsfes the followng propertes: H (the empty sequence belongs to H) If ( a ) 1,..., ( a ) 1,..., k k M H (where may be nfnte) and M <, then = k k H. = If ( a ) = 1,2,... s an nfnte sequence and ( a ) = 1,..., H for every postve k k nteger L, then ( a ) = 1,2,... H. k k k k L (Each member of H s called hstory, and a hstory s composed of actons taken by the players. A hstory ( a ) 1,..., H s called termnal hstory f t = k k M s an nfnte sequence or f ( a ) = 1,..., + 1 denoted by T. ). A functon P: H / T of N. ) k k M H, the set of termnal nodes are N (t assgns to each nontermnal hstory an element v. The payoff functons U : T R for each N. The set of all possble actons of a player after a hstory h s denoted as ( ) { (, ) } Ah = a ha H. Example 7 (contnued) Let us ndcate the components of the game mentoned at Fgure 6(a). There are two players, N = { 1, 2}. The set of hstores s {,, L, m, n, p, Lr, Ls} H = where, for nstance, p s a sequence of actons 30

40 and p. Besdes, T { m, n, p, Lr, Ls} defned as P( ) P( ) P( L) =. The functon P: H / T N s = 1, = = 2. Fnally, the functons U : T R for { 1, 2} are the payoff functons gvng the outputs (2,3), (1,4), For example, ( ) ( ) U n = 1, U n = 4. Hence, ths game s seen formally to be an extensve 1 2 game wth perfect nformaton. Example 9 Chess s one of the most famous extensve games wth perfect nformaton. There are two players W (whte) and B (black). However, t s very complex and almost mpossble to ndcate completely the components of ths game. For nstance, ( E4, E5, F3) H. E4 and F3 are the actons of player W, and E5 s the acton of player B. The functon P s that f the last acton of a hstory belongs to player W and the game contnues, then P assgns ths hstory to the player B, and vce versa. The last component, the payoff functons U B may be the followng: U W = 1 f W wns the game 1 f a draw s occured 2 0 f B wns the game. U W and U B = 1 f B wns the game 1 f a draw s occured 2 0 f W wns the game. 31

41 Fgure 7 At playng chess, can t be a strategy to start a game wth E4? It can be at most a part of a strategy though t s an acton of player W. Indeed, strategy s somethng dfferent from acton. Roughly speakng, t s an overall plan of a game whereas an acton s an nstant plan. Defnton 12 Let H { h H P( h) } = = and A = U A( h) be the set of all h H actons for player. A pure strategy for player s a map s : H A wth s( h) A( h) for all h H. We denote the set of all pure strateges of player as S, and S = S1... SN s the set of all strategy profles. Example 10 In the game depcted n Fgure 7, the pure strateges of player 1 are { acc acd adc add bcc bcd bdc bdd} S1 =,,,,,,,. There are three nodes belongng to 32

42 player 1, and she has an acton at each of these nodes. acd means that a s the acton decded at the frst node, c and d are actons decded at the second and the thrd nodes of player 1. On the other hand, player 2 has only two nodes at her 2 =,,,. Moreover, f s 1 own, and S { xx xy yx yy} = acd and s 2 = xy, then the output s the one specfed by the path startng from the frst node sgned wth arrows n Fgure 7. Hence, u1( s1, s 2) = 4 and u2( s1, s 2) = 2. Two results can be obtaned from the last example. Frst, the number of the pure strateges for player s #( Ah ( )) whch s easy to derve h H arthmetcally. Second and more mportant one s that, a strategy often specfes actons for a player at her nodes that may not be reached due to these actons or durng the actual play of ths game. For nstance, n Example 10, adc S1, however playng a n the frst node, player 1 wll never reach the thrd node of herself at the rght n Fgure 7 though she has to specfy an acton (the acton c due to the strategy adc) for ths node. As n the strategc games, mxed strategy of player s defned as a probablty dstrbuton over the set of pure strateges of player. Ths concept s valuable mostly for extensve games wth mperfect nformaton so that we wll not go further n detal n ths drecton. 33

43 4.3 SUBGAME PERFECT NASH EQUILIBRIUM We start ths secton defnng Nash equlbrum for extensve games wth perfect nformaton. Defnton 13 Let [N,H,P,(U )] be an extensve game wth perfect nformaton. A strategy profle s = ( s1, s2,..., s n ) s a Nash equlbrum for ths game f for every N, U ( s, s ) U ( s, s ) for all strateges s of player. In general, t s not easy to fnd the set of Nash equlbra from the extensve form of a game. For nstance, the game mentoned n Example 10 probably has several Nash equlbra, however from the fgure t s hard to understand whch of the strategy profles are Nash equlbra. Now we wll state the strategc form of an extensve game wth perfect nformaton. Defnton 14 Let [N,H,P,(U )] be an extensve game wth perfect nformaton. The strategc form of ths game s the strategc game [N,(S ),(u )] n whch S s the strategy set of player n the game [N,H,P,(U )] for each and u ( s) = U ( s) for every player and s S1... SN = S. The games descrbed n Defnton 14 are not equvalent ndeed. The order of the actons for the frst game dsappears as expressng t n strategc form 34

44 Xx xy yx Yy acc 4,2 4,2 0,3 0,3 acd 4,2 4,2 0,3 0,3 adc 3,4 3,4 0,3 0,3 add 3,4 3,4 0,3 0,3 bcc 1,3 2,1 1,3 2,1 bcd 0,0 2,1 0,0 2,1 bdc 1,3 2,1 1,3 2,1 bdd 0,0 2,1 0,0 2,1 Fgure 8 because the strategc game s a one-shot game due to ts characterzaton. But the mportant common property of these two games s that the sets of Nash equlbra of two games mentoned n Defnton 14 concde. The followng example llustrates ths common property. Example 10 (contnued) In Fgure 8 the strategc form of the extensve game depcted n Fgure 7 s shown. Usng Fgure 8 t s easy to see that the set of Nash equlbra s E {( bcc, yx),( bcd, yy),( bdc, yx),( bdd, yy) } =. Pck (bdd,yy). Accordng to Fgure 7, the decsons of player 1 do not seem to be plausble at the nodes followed by the hstores (a,x) and (b,x). It s more plausble to choose c whch ncreases the payoff at each node. In fact, (bcd,yy) and (bdc,yx) also have such mplausbltes. 35

45 A new soluton concept, the subgame perfect Nash equlbrum, elmnates these undesrable Nash equlbra n the prevous example. But before ths concept let us see what a subgame s. Defnton 15 A subgame of an extensve game wth perfect nformaton [N,H,P,(U )] s a subset of the game wth the followng propertes: (a) Ths subset begns wth a non-termnal node x, (b) It contans the nodes that are successors (both mmedate and later) of the node x, and contans no other node. Then the game tself s also a subgame. The subgames excludng the game tself are called proper subgames. The subgames ntatng from the nodes whose successors are only the termnal nodes are sad to be fnal subgames. Example 10 (contnued) The game shown n Fgure 7 has four proper subgames two of whch start from the nodes of player 2, and the other two start from the second and the thrd nodes of player 1. The latter two subgames are the fnal subgames of the game. Wth the game tself, ths game has fve subgames. As t s seen n the prevous example, each non-termnal node of an extensve form game wth perfect nformaton ntates a dfferent subgame. If we consder a subgame n solaton, t s a game tself wth the payoffs of the orgnal game. Therefore, the dea of Nash equlbrum can be appled to 36

46 the new game. We say that a strategy profle of an extensve game wth perfect nformaton nduces a Nash equlbrum n a subgame f the restrcton of each player s strategy nto ths subgame consttutes a Nash equlbrum when ths game s consdered n solaton. Defnton 16 A strategy profle of an extensve game wth perfect nformaton s called subgame perfect Nash equlbrum f t nduces a Nash equlbrum n every subgame. Clearly, a subgame perfect Nash equlbrum nduces a Nash equlbrum n tself so that every subgame perfect Nash equlbrum s a Nash equlbrum whereas the converse s not true, n general. Example 10 (contnued) As determned before, the set of Nash equlbra for the =. For game depcted n Fgure 7 s E {( bcc, yx),( bcd, yy),( bdc, yx),( bdd, yy) } nstance, playng c at the fnal subgame at the left s the unque Nash equlbrum f ths subgame s consdered n solaton. However ( bdd, yy) E, and the strategy bdd of the player 1 does not nduce Nash equlbrum n ths fnal subgame. Hence, ( bdd, yy ) s not subgame perfect Nash equlbrum whereas t s Nash equlbrum. To determne the set of subgame perfect Nash equlbra n a fnte extensve game wth perfect nformaton there s a useful procedure called backward nducton. Frst, the optmal actons at the fnal decson nodes(those 37

47 for whch the only successor nodes are termnal nodes) are determned. Then, gven that these are the actons taken at the fnal decson nodes, we can proceed to the next-to-last decson nodes and determne the optmal actons to be taken there by players that antcpate correctly the actons that wll follow at the fnal decson nodes, and so on backward through the game tree. By ths procedure, the followng result s easly derved. Proposton 4 9 Every fnte extensve game wth perfect nformaton has a subgame perfect Nash equlbrum. Moreover, f no player has the same payoffs at any two termnal nodes, then there s a unque subgame perfect Nash equlbrum. Example 10 (contnued) By Proposton 4, the game depcted n Fgure 7 has unque subgame perfect Nash equlbrum. The optmal actons are shown by the arrows n Fgure 7. Therefore, the subgame perfect Nash equlbrum s ( bcc, yx ). 9 For the proof the readers are referred to MasColell, Whnston and Green(1995). 38

48 CHAPTER V: EQUILIBRIUM IN REPEATED GAMES In many crcumstances such as economcs, poltcs, socology etc, a game s played between the agents, governments or people not once but many tmes. For nstance, at each stage, frms adjust the prces of goods they sell regardng the demand of consumers or the profts they plan to get so that a knd of game s repeated sometmes wth mprovements n each stage. Such games are called mult-stage game. The players know all prevous actons, but they do not know the actons whch are dsplayed by the opponents at the current stage, namely at each stage a smultaneous move game s played. Ths setup makes possble for the players to condton ther actons on the prevous actons of the opponents. A specal case of mult-stage games s repeated games n whch the smultaneous game s played n every stage. The game that s repeated s called stage game. The mportant feature about the repeated games s that past actons do not affect the set of possble actons or payoff functons at current stage. In repeated games some nterestng equlbra may be observed whch do not arse when the stage game s played once. These equlbra are caused by punshments that wll be mentoned n detal n ths chapter. Due to the length of horzon of a game, repeated games are classfed as fnte and nfnte repeated games. As we wll see, the behavour of players n two classes of games s sgnfcantly dfferent. The dfference s summarzed n 39

49 Fudenberg and Trole (1991) essentally as follows: The nfnte-horzon case s a better descrpton of stuatons where the players always thnk the game extends one more perod wth hgh probablty; the fnte-horzon model descrbes a stuaton where the termnal date s well-known and commonly foreseen. 5.1 INFINITELY REPEATED GAMES Before defnng an nfntely repeated game formally, t s necessary to ndcate that throughout the games, the acton set of each player s compact and the payoff functon of each player s contnuous. It s obvous that a repeated game s an extensve game. Though t does not have perfect nformaton we model t as f t s an extensve game wth perfect nformaton. 10 Defnton 17 Let Γ= [N,(A ),(u )] be a strategc game and let A= A1... AN. An nfntely repeated game of Γ s an extensve game wth perfect nformaton [N,H,P,(U )] n whch =. H { } t U A where s the ntal hstory, and t = 1 t A s the set of the a sequences ( ) = 1 r t r of acton profles n Γ wth length t.. Ph ( ) = Nfor each non-termnal hstory h H. 10 Why t does not have perfect nformaton s perceved by the dfference of Defnton 11 from Defnton

50 . U s the payoff functon defned on the set A (the set of termnal hstores) of nfnte sequences ( a ) = 1 r r of acton profles n Γ. In ths chapter we wll use the term strategy for the pure strateges, that s, for smplcty the strategy set wll be composed of the pure strateges only. Recall that a vector n w R s a payoff profle of Γ f there s a strategy profle a A of Γ such that w = u ( a) for every N. A vector w s called a feasble payoff profle f t s a convex combnaton of the payoff profles n A, that s, there exsts { 1,..., } a a A such that for each, w = k β u ( a ) k= 1 k where βk = 1 and β k s a nonnegatve ratonal number for every k= 1 { 1,..., } k. We choose k ndependent of. β s ratonal for smplcty. 11 Notce that ( ) β s a a A There are several alternatve specfcatons of payoff functons for the nfntely repeated games. We wll focus mostly on the case n whch the players dscount future utltes wth the dscount factor δ < 1. In ths specfcaton, the payoff functon of player s t 1 t δ δ w (10) t= 1 (1 ) 11 In general, these coeffcents are assumed to be real numbers. 41

51 where w t s the payoff profle of the stage game n perod t. The term (1-δ) n (10) s to normalze the summaton so that the gan of a player recevng 1 per perod s one. Then (1 δ) t 1 t δ w s a payoff profle of the δ-dscounted nfntely t= 1 N repeated game. We assume that A s compact for each, ths s why the value n (10) s always fnte. Under dscounted crteron, the value of a gven gan dmnshes over tme. Any player s expected payoff from perod t on, whch s the payoff of the proper subgame that begns at perod t, can be computed. We call ths the contnuaton payoff and formulate t as follows: where w τ s defned as n (10). (1 δ) δ τ = t τ t w τ Another specfcaton of payoff functon s sad to be tme-average crteron n whch the perods are treated equally so that the dscount factor s thought as f δ=1. The payoff functon of the player n ths crteron s T 1 t lm nf w (11) T T t= 1 We take lmt nfmum of the summaton n (11) snce some nfnte sequences have no well defned average values. 42

52 C D C 0, 0 2,-1 D -1, 2 1, 1 Fgure 9 In the tme-average nfntely repeated games the players are unconcerned not only about the tmng of payoffs but also about the payoffs for fnte number of perods. For nstance, the sequences (0,0,1,1,1,0, ) and (0,0,0, ) are equally preferred havng the same average zero. Another specfcaton, overtakng crteron, has the advantage of treatng all the perods equally and consderng the changes mportantly n fnte number of perods. However, ths specfcaton cannot be specfed by a payoff functon. In ths crteron, the sequence ( w t ) s preferred to ( y t ) f and only f T t= 1 t t ( w y) lm nf > 0. T Example 11 The game shown n Fgure 9 has unque Nash equlbrum whch s (C,C). Ths strategy profle gves zero to each player. Despte ths, both players are better off when they play D. In the repeated verson of ths game, playng (D,D) may be an equlbrum f the players beleve that a devaton wll termnate playng (D,D) resultng n a long-term loss for them that outweghs ther shortterm gans. 43

53 player s Suppose that both players play D at each perod. The payoff of each (1 δ) δ t= 1 t 1 t whch s equal to 1, where w = 1 for all t and {1, 2}. Suppose that the output s (C,D) at the odd perods and s (D,C) at the even perods. Then the payoff of player 1 s 1 1 (1 ) t t δ δ.2 + (1 δ) δ.( 1) t odd t even whch s equal to 2 δ. On the other hand, the payoff of player 2 s 1+ δ 1 1 (1 ) t t δ δ.( 1) + (1 δ) δ.2 t odd t even whch s equal to 2 δ δ What can be sad about the Nash equlbra of nfntely repeated games? The followng result shows that f the stage game has some Nash equlbra, the nfntely repeated game has trval subgame perfect Nash equlbra. Proposton 5 12 Let E S be the set of Nash equlbrum strategy profles of the stage game [N,(A ),(u )], then any strategy profle s: H \ T E s a subgame perfect Nash equlbrum of the nfntely repeated game. 44