COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY

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1 Best Response Equvalence by Stephen Morrs and Takash U July 2002 COWLES FOUNDATION DISCUSSION PAPER NO COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box New Haven, Connectcut

2 Best Response Equvalence Stephen Morrs Department of Economcs Yale Unversty Takash U Faculty of Economcs Yokohama Natonal Unversty ou@ynu.ac.jp July 2002 Abstract Two games are best-response equvalent f they have the same best-response correspondence. We provde a characterzaton of when two games are best-response equvalent. The characterzatons explot a dual relatonshp between payoff dfferences and belefs. Some potental game arguments cf. Monderer and Shapley, 1996, Games Econ. Behav. 14, ) rely only on the property that potental games are best-response equvalent to dentcal nterest games. Our results show that a large class of games are best-response equvalent to dentcal nterest games, but are not potental games. Thus we show how some exstng potental game arguments can be extended. Keywords: best response equvalence; dualty; Farkas Lemma; potental games. Suggested Runnng Ttle: Best Response Equvalence. We are very grateful for valuable nput from Larry Blume, George Malath and Phlp Reny. Correspondng author s address: Stephen Morrs, Cowles Foundaton, P.O.Box , New Haven, CT , U.S.A. Phone: Fax:

3 1 Introducton We consder three progressvely stronger equvalence relatons on games and characterze each of them. Two games are best-response equvalent f they have the same best-response correspondence. Two games are better-response equvalent f, for every par of strateges, they agree when one strategy s better than the other. Two games are von Neumann-Morgenstern equvalent VNM-equvalent) f, for each player, the payoff functon n one game s equal to a constant tmes the payoff functon n the other game, plus a functon that depends only on the opponents strateges. Two games are VNM-equvalent f and only f, for each player, there s a constant w > 0 such that the rato of payoff dfferences from swtchng between one strategy to another strategy s always w. The constant w s thus ndependent of the strateges beng compared. Two games are better-response equvalent f and only f they have the same domnance relatons and, for each player and each par of strateges a and a such that nether strategy strctly domnates the other, there exsts a constant w > 0 such that the rato of payoff dfferences from swtchng between a and a s always w. In general, ths s a weaker requrement than VNM-equvalence. It s weaker both because the proportonal payoff dfferences property s no longer requred to hold between some strategy pars, and because the weght w s not necessarly ndependent of the strategy par. But f the game does not have domnated strateges, the weghts can no longer depend on the strateges beng compared, and better-response equvalence collapses to VNM-equvalence. Two games are best-response equvalent f and only f, for each player and each par of strateges a and a such that both strateges are a best response to some belef, there exsts a constant w > 0 such that the rato of payoff dfferences from swtchng between 2

4 a and a s always w. Even f a game has no domnated strateges, ths s a weaker requrement than VNM-equvalence. In games wth dmnshng margnal returns, bestresponse equvalence s always a strctly weaker requrement than VNM-equvalence. Examples are gven n the paper. The most extensve dscusson and applcatons of these relatons has come n the lterature on potental games. Monderer and Shapley [10] sad that a game was a potental game f there exsts a potental functon, defned on the strategy space, wth the property that the change n any player s payoff functon from swtchng between any two of hs strateges holdng other players strateges fxed) was equal to the change n the potental functon. 1 A game s weghted potental game, f the payoff changes are proportonal for each player. Thus a game s a weghted potental game f and only f t s VNM-equvalent to a game wth dentcal payoff functons. Whle some results usng potental or weghted potental game arguments are usng the VNM-equvalence to dentcal nterest games, other arguments are just usng the better-response equvalence and even only best-response equvalence mplcatons of VNM-equvalence. 2 Any paper that deals only wth equlbrum s usng only best-response equvalence e.g., Neyman [13], U [19], Morrs and U [12]). Smlarly, fcttous play only uses the best-response propertes of the game Monderer and Shapley [9]). 3 An applcaton usng only betterresponse equvalence but not the VNM-equvalence appears n Morrs [11]. Some papers studyng quantal responses or stochastc best responses n potental games use the full power of VNM-equvalence e.g., Blume [2], Brock and Durlauf [3], Anderson et al. [1], U [20]). 4 1 See also U [18] for a characterzaton and examples of potental games. 2 Arguments that explot potental arguments to prove the exstence of a pure strategy equlbrum e.g., Rosenthal [15]) only use ordnal propertes of payoffs. Monderer and Shapley [10] ntroduced ordnal potental games and Voorneveld [21] and Dubey et al. [4] showed how ordnal potental games can be weakened to only requre pure strategy best-response equvalence. 3 Sela [17] establshes convergence of fcttous play n a class of One-Aganst-All games. These are games best-response equvalent to dentcal nterest games, but not potental games. 4 More precsely, they use the full power of VNM-equvalence such that the constant w s the same for all the players. 3

5 The fact that VNM-equvalence s the same as better-response equvalence n the absence of domnated strateges and may be dfferent n the presence of domnated strateges has been noted n a number of contexts see Sela [16], Blume [2] p409, Monderer and Shapley [10] footnote 9, and Maskn and Trole [6] p209). However, our characterzatons of better-response equvalence n the presence of domnated strateges and of the sgnfcant gap between better-response equvalence and best-response equvalence fll a gap n the lterature. 5 The paper s organzed as follows. In secton 2, we descrbe our notons of equvalence and gve an example llustratng the dfferences. In secton 3, we report our characterzatons. In secton 4, we restrct attenton to a class of games where best-response equvalence s a strctly weaker requrement than VNM-equvalence and characterze the class of games. We also dscuss an extenson to games wth nfnte strategy spaces and ts applcaton. 2 Equvalence Propertes of Games A game conssts of a fnte set of players N and a fnte strategy set A for N, and a payoff functon g : A R for N where A = N A. We wrte A = j A j and a =a j ) j A. We smply denote a game by g =g ) N. Throughout the paper, we regard g a, ) :A R as a vector n R A. We wrte g a, ) g a, ) f g a,a ) >g a,a ) for all a A, and g a, ) g a, ) fg a,a ) g a,a ) for all a A. For N, let A ) denote the set of all probablty dstrbutons over A. We call each element of A ) player s belef. For X A, let Λ a,x g ) A ) be a set of player s belefs such that player wth a payoff functon g and a belef 5 Mertens [8] studed varous notons of best-response equvalence, but wth hs more abstract strategy spaces and focus on admssble best responses, there s lttle overlap wth the materal n ths paper. 4

6 λ Λ a,x g ) weakly prefers a to any strategy n X : Λ a,x g ) = {λ A ) λ a ) g a,a ) g a,a ) ) 0 for all a X }. a A When X s a sngleton,.e., X = {a }, we wrte Λ a,a g ) nstead of Λ a, {a } g ). We are nterested n characterzng two equvalence relatons on games captured by these sets of belefs by whch players prefer one partcular strategy. Defnton 1 A game g s better-response equvalent to g =g ) N f, for each N, Λ a,a g )=Λ a,a g ) for all a,a A. Defnton 2 A game g s best-response equvalent to g =g ) N f, for each N, for all a A. Λ a,a g )=Λ a,a g ) If g s better-response equvalent to g, then g s best-response equvalent to g, snce Λ a,a g )= Λ a,a g ). a A An easy suffcent condton for better-response equvalence s the followng. 6 Defnton 3 A game g s VNM-equvalent to g =g ) N f, for each N, there exsts a postve constant w > 0 and a functon Q : A R such that g a, ) =w g a, )+Q ). 6 Blume [2] called ths property strongly best-response equvalent. 5

7 It s straghtforward to see that f g s VNM-equvalent to g, then g a, ) g a, ) =w g a, ) g a, ) ) for all a,a A. Conversely, f ths s true, then a functon Q : A R such that Q ) =g a, ) w g a, ) s well defned, and thus g s VNM-equvalent to g. Thus, we have the followng lemma. Lemma 1 A game g s VNM-equvalent to g f and only f, for each N, there exsts w such that g a, ) g a, ) =w g a, ) g a, ) ) 1) for all a,a A. It s straghtforward to see that VNM-equvalence s suffcent for better-response equvalence. In fact, 1) mples that λ a ) g a,a ) g a,a ) ) a A = w λ a ) g a,a ) g a,a ) ) a A for all λ A ) and thus Λ a,a g )=Λ a,a g ) for all a,a A. Best-response, better-response, and VNM-equvalence are equvalence relatons. Thus, they defne an equvalence class of games. For example, weghted potental games Monderer and Shapley [9]) wth a weghted potental functon f : A R are regarded as a VNM-equvalence class of an dentcal nterest game f = f ) N wth f = f for all N. Ths s clear by Lemma 1 and the followng orgnal defnton of weghted potental games. Defnton 4 A game g =g ) N s a weghted potental game f there exsts a weghted potental functon f : A R and w > 0 for each N such that g a, ) g a, ) =w fa, ) fa, ) ) 6

8 for all a,a A.Ifw = 1 for all N, g s called a potental game and f s called a potental functon. As the concept of VNM-equvalence leads us to the defnton of weghted potental games, the concept of better-response equvalence and that of best-response equvalence lead us to the defnton of the followng new classes of games. Defnton 5 A game g =g ) N s a better-response potental game f t s betterresponse equvalent to an dentcal nterest game f =f ) N wth f = f for all N. A functon f s called a better-response potental functon. Defnton 6 A game g =g ) N s a best-response potental game f t s best-response equvalent to an dentcal nterest game f =f ) N wth f = f for all N. A functon f s called a best-response potental functon. Voorneveld [21] called a game a best-response potental game f ts best-response correspondence concdes wth that of an dentcal nterest game over the class of belefs such that λ a ) = 0 or 1. Thus, best-response potental potental games n ths paper form a specal class of those n Voorneveld [21]. Exstng potental game results that rely only on better-response equvalence or bestresponse equvalence, such as those mentoned n the ntroducton, automatcally hold for the larger class of better-response potental games or that of best-response potental games. Thus, we are nterested n exactly when and to what extent better-response and best-response equvalence are weaker requrements than VNM-equvalence. Notce that best-response and better-response equvalence are clearly weaker requrements than VNM-equvalence, because the latter mposes too many constrants on payoffs from domnated strategy. Moreover, best-response equvalence s sgnfcantly weaker than better-response equvalence, as shown by the followng example. Consder a two player, three strategy, symmetrc payoff game g x, y) parameterzed by x, y) R 2 ++, where each player s payoffs are gven by the followng payoff matrx where the player s own strateges are represented by rows and hs opponent s strateges 7

9 are represented by columns) x x 2x y y y In the specal case where x = y = 1, we have game g 1, 1) wth the followng payoff matrx If a row player has a belef λ k) =π k for k {1, 2, 3}, he prefers strategy 1 to strategy 2 f and only f π 1 π 2 +2π 3 ; he prefers strategy 1 to strategy 3 f and only f x +2y) π 1 x y) π 2 +2x + y) π 3 ; he prefers strategy 3 to strategy 2 f and only f π 3 π 2 +2π 1. Thus the regon of ndfference between strateges 1 and 2, and between strategy 2 and 3, does not depend on x and y. Moreover, whenever strategy 1 or 3) s preferred to strategy 2, t s also preferred to strategy 3 or 1). Thus the best response regons for ths game are as n fgure 1, for any x, y) R Thusg x, y) s best-response equvalent to g 1, 1) for any x, y) R On the other hand, the regon of ndfference between strateges 1 and 3 does depend on x and y: n partcular, g x, y) s better-response equvalent to g 1, 1) f and only f x = y. We wll dscuss ths example agan n secton 4. 8

10 Strategy 1 π 1 =1 π 2 =1 Strategy 2 s best response Strategy 3 π3 =1 Fgure 1: The best response regons 3 Results 3.1 Generc Propertes of Games We wll appeal to some generc propertes of games,.e., propertes that wll hold for all but a Lebesgue measure zero set of payoffs. G1: For all N, fg a, ) g a, ), then g a, ) g a, ) for dstnct a,a A. G2: For all N, vectors g a, ) g a, ) and g a, ) g, ) are lnearly ndependent for dstnct a,a,a A. G3: For all N, fλ a,a g ) Λ a,a g ), then Λ a,a \{a } g )\Λ a,a g ) for dstnct a,a A. 3.2 Better-Response Equvalence Strategy a strctly domnates a n game g we wrte a g a )fg a, ) g a, ), or, equvalently, Λ a,a g )=. Strateges a and a are better-response comparable we wrte a g a ) f nether a g a nor a g a. Proposton 1 If games g and g satsfy generc property G1, then g s better-response equvalent to g f and only f, for each N, a) they have the same domnance relatons g = g ) and b) whenever a s better-response comparable to a a g a ), there exsts w a,a ) > 0 such that g a, ) g a, ) =w a,a ) g a, ) g a, ) ). 2) 9

11 Farkas Lemma 7 plays a central role n the proofs. Lemma 2 Farkas Lemma) For vectors a 0, a 1,...,a m R n, the followng two condtons are equvalent. If a 1, y),...,a m, y) 0 for y R n, then a 0, y) 0. There exsts x 1,...,x m 0 such that x 1 a x m a m = a 0. Proof of Proposton 1. We frst show that a) and b) are suffcent for the betterresponse equvalence of g and g.ifa g a, then b) mples that λ a ) g a,a ) g a,a ) ) a A = w a,a ) λ a ) g a,a ) g a,a ) ) a A and thus If a g a, then If a g a, then Λ a,a g )=Λ a,a g ). Λ a,a g )=Λ a,a g )= A ). Λ a,a g )=Λ a,a g )=. To prove necessty, suppose that g s better-response equvalent to g. Snce we have Λ a,a g )=Λ a,a g ), a g a Λ a,a g )=Λ a,a g )= a g a 7 See a textbook of convex analyss such as recent one by Hrart-Urruty and Lamaréchal [5], or classc one by Rockafellar [14]. 10

12 and thus a) holds. To prove b), suppose that a g a. We know that a g a. Let λ A )be such that λ a ) g a,a ) g a,a ) ) 0. a A Snce λ Λ a,a g )=Λ a,a g ), λ a ) g a,a ) g a,a ) ) 0. a A Ths mples that f y a ) a A R A a A y a s such that g a,a ) g a,a ) ) 0, y a 0 for all a A, then g a,a ) g a,a ) ) 0. a A y a By Farkas Lemma, there exst x a a 0 and z a 0 for a A such that x a a g a, ) g a, ) ) z a δ a ) = g a, ) g a, ) ) a A where δ a : A R s such that δ a a ) = 1 f a = a and δ a a ) = 0 otherwse. Thus, x a a g a, ) g a, )) g a, ) g a, ). If x a = 0, then g a a, ) g a, ) 0. However, ths s mpossble snce a g a mples that a does not strctly domnate a n g and G1 requres that f a does not strctly domnate a, then t s not the case that g a, ) g a, ) 0. Thus, xa > 0. a Symmetrcally, we have x a a g a, ) g a, ) ) g a, ) g a, ) 11

13 where x a a > 0. Thus, ) x a g x a a a a, ) g a, )) 0. If x a x a a a > 0, then g a, ) g a, ) 0, and f xa x a a a < 0, then g a, ) g a, ) 0, whch we already noted are mpossble. Thus, x a = x a a a, whch mples that x a a g a, ) g a, )) = g a, ) g a, ). Ths proves b). If g has no domnated strategy, then 2) s true for every a,a A.Ifw a,a ) s the same for every a,a A, then better-response equvalence mples VNM-equvalence. However, Proposton 1 does not say anythng about whether w a,a ) does depend upon a,a A. Thus, we are nterested n when better-response equvalence mples VNMequvalence. The followng proposton provdes a suffcent condton for the equvalence of better-response equvalence and VNM-equvalence. Proposton 2 Suppose that games g and g satsfy generc propertes G1 and G2, and that, for each N, a) they have the same domnance relatons g = g ), b) g generates a connected graph on A, and c) for any a,a,a,a A such that a g a and a g wth a, there exsts a sequence {ak }m k=1 such that a1 = a,a 2 = a,am 1 =,am =, ak g a k+1 for k =1,...,m 1, a k g a k+2 for k =1,...,m 2. Then g s better-response equvalent to g f and only f g s VNMequvalent to g. Note that c) s trvally satsfed f no strategy s domnated,.e., g s the complete relaton. So, the proposton mmedately has the followng corollary. Corollary 3 If g and g satsfy generc propertes G1 and G2 and have no strctly domnated strateges, then g s better-response equvalent to g f and only f g s VNMequvalent to g. 12

14 Strategy 4 Strategy 3 Strategy 2 Strategy 1 Fgure 2: The graph of g It should be emphaszed that the suffcent condton of Proposton 2 s sometmes satsfed even when there are strctly domnated strateges n the game. For example, consder the followng two player game, where only the row player s payoffs are shown Consder strateges of the row player. We have 1 g 2, 2 g 3, 3 g 4, 1 g 3, 2 g 4as n fgure 2, satsfyng the condton of Proposton 2, whle strategy 1 strctly domnates strategy 4. Proof of Proposton 2. We show that f g s better-response equvalent to g then g s VNM-equvalent to g. Note that, by Proposton 1, f a g wth a, there exst xa a = x a a x a a g a, ) g a, )) = g a, ) g a, ). If A = 2, ths completes the proof by Lemma 1. Suppose that A 3 and let a g a and a g wth a. Then there exsts a sequence {ak }m k=1 satsfyng 13

15 the condtons n c). Thus, ) x ak+2 g a k+1 a k+2, ) g a k+1, ) + x ak+1 a k ) = g ak+2, ) g ak+1, ) + ) g a k+1, ) g a k, ) ) g ak+1, ) g ak, ) = g ak+2, ) g ak, ) ) = x ak+2 g a k a k+2, ) g a k, ) ) ) = x ak+2 g a k a k+2, ) g a k+1, ) + x ak+2 g a k a k+1, ) g a k, ) and x ak+2 a k+1 ) x ak+2 g a k a k+2, ) g a k+1, ) + x ak+1 a k x ak+2 a k ) ) ) g a k+1, ) g a k, ) =0. By G2, g a k+2, ) g a k+1 t must be true that x ak+2 a k+1 that x a a = x a a,a A wth a g a., ) and g a k+1 = x ak+1 a k = x ak+2 a k, ) g a k, ) are lnearly ndependent and thus for k =1,...,m 2. Thus, t must be true. In other words, there exsts a constant c>0 such that x a a = c for any In addton, snce g generates a connected graph on A, for any a, A wth a, there exsts a,a and {a k }m k=1 satsfyng the condtons n c). Thus, c g, ) g a, ) ) m 1 ) = c g a k+1, ) g a k, ) To summarze, for any a,a A, k=1 m 1 = k=1 ) g ak+1, ) g ak, ) = g, ) g a, ). c g a, ) g a, )) = g a, ) g a, ). Ths mples that g s VNM-equvalent to g by Lemma 1. 14

16 3.3 Best-Response Equvalence Strateges a and a are best-response comparable we wrte a g a ) f both strateges are best responses at some belef,.e., Λ a,a g ) Λ a,a g ). Note that a g a f and only f Λ a,a g ). Proposton 4 If games g and g satsfy generc property G3, then g s best-response equvalent to g f and only f, for each N, a) they have the same best-response comparablty relaton g = g ) and b) whenever a s best-response comparable to a a g a ), there exsts w a,a ) > 0 such that g a, ) g a, ) =w a,a ) g a, ) g a, )). Proof. We frst show that a) and b) are suffcent for the best-response equvalence of g and g.ifλ a,a g )=, then Λ a,a g )=Λ a,a g )= because Λ a,a g )= mples that a g a s not true and thus a) mples that a g a s not true. If Λ a,a g ), then { a a g } a, and we must have Λ a,a g )= a A Λ a,a g )= {a a g a } Λ a,a g ). 3) Clearly, 3) s true when { a a g } a = A. To see that 3) s true when { a a g } a A, suppose otherwse. Then, Λ a,a g ) Λ a,a g ), a A {a a g a } and thus there exsts { a a g } a such that a A Λ a,a g ) a A \{ } Λ a,a g ). 15

17 However, ths mples that a g, whch s a contradcton. Thus, 3) must be true. If a g a, then b) mples that λ a ) g a,a ) g a,a ) ) a A = w a,a ) λ a ) g a,a ) g a,a ) ), a A and thus Λ a,a g )=Λ a,a g ). 4) Therefore, by a), 3), and 4), we have Λ a,a g )=Λ a,a g ). Ths completes the proof of suffcency. To prove necessty, suppose that g s best-response equvalent to g. Snce we have Λ a,a g )=Λ a,a g ), Λ a,a g ) Λ a,a g )=Λ a,a g ) Λ a,a g ) and thus g = g. Ths proves a). If a g a, then there exsts λ A ) such that λ a ) a A λ a ) a A g a,a ) g,a ) ) 0 for all A, g a,a ) g,a ) ) 0 for all A \{a }. Snce λ Λ a,a g )=Λ a,a g ), λ a ) g a,a ) g a,a ) ) 0. a A 16

18 The above mples that, f y a ) a A R A s such that a A y a a A y a a A y a g a,a ) g a,a ) ) 0, g a,a ) g,a ) ) 0 for all A \ { a,a }, g a,a ) g,a ) ) 0 for all A \ { a,a }, y a 0 for all a A, then g a,a ) g a,a ) ) 0. a A y a By Farkas Lemma, there exst x a 0, γ a a : A a R, and δ a : A a R such that x a a g a, ) g a, )) γ a a ) δ a ) = g a a, ) g a, )) where γ a ) = a a,a u a g a, ) g, ) ) + a,a v a g a, ) g, ) ) wth u a,v a 0 and δ a ) = z a a δ a ) a A wth z a 0. Thus, x a a g a, ) g a, )) + γ a ) g a a, ) g a, ). We show x a > 0. Suppose that x a a = 0,.e., γ a a ) g a a, ) g a, ). Let λ Λ a,a \{a } g )\Λ a,a g ), 17

19 whch exsts by a g a and G3. Snce λ Λ a,a \{a } g ) Λ a,a \{a } g ), λ a )γ a a a )= u a λ a ) g a,a ) g,a ) ) a A a,a a A + v a λ a ) g a,a ) g,a ) ) 0. a A a,a Snce λ Λ a,a g )=Λ a,a g ) and λ Λ a,a g )=Λ a,a g ), λ a ) g a,a ) g a,a ) ) < 0. a A Ths s a contradcton. Thus, we must have x a > 0. a We have x a a g a, ) g a, )) + γ a ) g a a, ) g a, ) and symmetrcally where x a,x a a a > 0. Addng both, x a a x a x a a g a, ) g a, ) ) + γ a a ) g a, ) g a, ) a ) g a, ) g a, ) ) + γ a a )+γ a a ) 0. 5) We show x a x a a a = 0. Suppose that x a x a a a > 0. Let λ Λ a,a \{a } g )\Λ a,a g ) Λ a,a \{a } g ) Λ a,a \{a } g ). Then, the expectaton of the left-hand sde of 5) s postve because ) x a x a a a λ a ) g a,a ) g a,a ) ) > 0 a A and ) λ a ) γ a a a )+γ a a a ) a A = u a + v a ) λ a,a a ) a A + v a + u a ) λ a ) a A a,a g a,a ) g,a ) ) g a,a ) g,a ) ) 0. 18

20 Ths s a contradcton. Symmetrcally, f x a x a a a < 0, then we have the symmetrc contradcton. Thus, x a a x a a = 0, and 5) s reduced to We show γ a ) =γ a a λ,λ A ) be such that γ a )+γ a a a ) 0. 6) a ) = 0. Suppose that ether γ a a ) 0orγ a a ) 0 s true. Let λ Λ a,a \{a } g )\Λ a,a g ) Λ a,a \{a } g ) Λ a,a \{a } g ), λ Λ a,a \{a } g )\Λ a,a g ) Λ a,a \{a } g ) Λ a,a \{a } g ). Consder λ + λ )/2 A ). Then, the expectaton of the left-hand sde of 6) s postve because λ a )+λ 2 a A = a,a + a,a a,a + a,a a ) u a v a u a v a γ a a ) a )+γ a a a ) + v a λ a ) )+λ 2 a A λ + u a a a ) )+λ 2 a A + v a ) + u a ) λ a ) 2 a A λ a ) 2 a A a ) a ) g a,a ) g,a ) ) g a,a ) g,a ) ) g a,a ) g,a ) ) g a,a ) g,a ) ) > 0. Ths s a contradcton. Thus, γ a a ) =γ a a ) =0. Summarzng the above, we have x a a g a, ) g a, )) = g a, ) g a, ) where x a a > 0. Ths proves b). The followng proposton and corollary follow by exactly the same arguments n Proposton 2 and Corollary 3 n the prevous subsecton for better-response equvalence. 19

21 Proposton 5 Suppose that games g and g satsfy generc propertes G2 and G3, and that, for each N, a) they have the same best-response comparablty relaton g = g ), b) g such that a g a generates a connected graph on A, and c) for any a,a,a,a A and a g a wth a, there exsts a sequence {ak }m k=1 such that a 1 = a,a 2 = a,am 1 =,am =, ak g a k+1 for k =1,...,m 1, a k g a k+2 for k =1,...,m 2. Then g s best-response equvalent to g f and only f g s VNMequvalent to g. Corollary 6 If g and g satsfy generc propertes G2 and G3 and g s the complete relaton, then g s best-response equvalent to g f and only f g s VNM-equvalent to g. 4 Games wth Own-strategy Unmodalty Best-response equvalence relaton s an equvalence relaton. It wll be useful f, as a closed form, we can descrbe the best-response equvalence class of a game n whch best-response equvalence s a strctly weaker requrement than VNM-equvalence. Let A be lnearly ordered such that A = {1,..., K } wth K 3. For q : A R and w : A \{K } R ++, let q,w ) g : A R be such that q,w ) g 1, ) =q ), a 1 q,w ) g a, ) =q )+ w k)g k +1, ) g k, )) for a 2. k=1 Let D g ) be a class of payoff functons of player obtaned by ths transformaton: D g )={g : A R g =q,w ) g,q : A R, w : A R ++ }. It s straghtforward to see that g D g ) f and only f there exsts w : A \{K } R ++ such that g a +1, ) g a, ) =w a )g a +1, ) g a, )) 7) for all a A \{K }. Note that g D g ), g D g ) mples g D g ), and g D g ) wth g D g ) mples g D g ). Thus, D g ) defnes an equvalence 20

22 class of payoff functons of player. We wrte Dg) ={g =g ) N g D g ) for all N}. For example, consder a parametrzed class of games {gx, y)} x,y) R 2 ++ dscussed n secton 2. We have {gx, y)} x,y) R 2 ++ Dg1, 1)). To see ths, we wrte gx, y) = g x, y)) {1,2}. Then, for any x, y) R 2 ++ and j, g 1,a j x, y) =q a j ), g 2,a j x, y) =q a j )+x g 2,a j 1, 1) g 1,a j 1, 1)), g 3,a j x, y) =q a j )+x g 2,a j 1, 1) g 1,a j 1, 1)) + y g 3,a j 1, 1) g 2,a j 1, 1)) where q : {1, 2, 3} R s such that q 1) = x, q 2) = x, and q 3) = 2x. Remember that, for any x, y) R 2 ++, gx, y) s best-response equvalent to g1, 1). It s easy to see that every game n Dg1, 1)) s VNM-equvalent to gx, y) for some x, y) R Thus, every game n Dg1, 1)) s best-response equvalent to g1, 1). Ths observaton leads us to the queston when every game n Dg) s best-response equvalent to g. We provde a necessary and suffcent condton for t. We say that g s own-strategy unmodal f, for all λ A ), there exsts k A such that, a A λ a )g a,a ) g a 1,a )) 0fa k, a A λ a )g a,a ) g a +1,a )) 0fa k. Note that f g s own-strategy unmodal, then 8) s true f and only f λ Λ k,a g ). Clearly, by 7), g s own-strategy unmodal f and only f g D g ) s own-strategy unmodal. We say that g s own-strategy concave f g,a ):A R s concave,.e., g a + 1,a ) g a,a ) s decreasng n a for all a A. Lemma 3 Suppose that g a +1,a ) g a,a ) for all a A \{K } and a A, and that there s no weakly domnated strategy. Then, g s own-strategy unmodal f and only f there exsts g D g ) such that g s own-strategy concave. 21 8)

23 Proof. Suppose that g D g ) s own-strategy concave. Then, g a +1,a ) g a,a ) s decreasng n a for all a A. Thus, a A λ a ) g a +1,a ) g a,a )) s also decreasng n a for all λ A ). Ths mmedately mples that g D g )s own-strategy unmodal. Snce λ a )g a +1,a ) g a,a )) a A = 1 w a ) a A λ a ) g a +1,a ) g a,a )), g s also own-strategy unmodal. Suppose that g s own-strategy unmodal. We prove the exstence of an own-strategy concave payoff functon g =q,w ) g by constructon. Later, we wll show that there exsts C k > 0 such that g k +1, ) g k, ) C k g k +2, ) g k +1, )). 9) For C k satsfyng 9), we let w : A R ++ be such that w 1) = 1 and w a )= a 1 k=1 C k for a 2, and q : A R be such that q a ) = 0 for all a A. Snce we have By ths and 9), we have g a +1, ) g a, ) =w a )g a +1, ) g a, )), g k +1, ) g k, ) =w k)g k +1, ) g k, )), g k +2, ) g k +1, ) =C k w k)g k +2, ) g k +1, )). g k +1, ) g k, ) g k +2, ) g k +1, ), whch mples that g s own-strategy concave. We prove the exstence of C k satsfyng 9) by Farkas Lemma. Before dong t, we must frst observe that f λ a )g k +1,a ) g k,a )) = 0 10) a A 22

24 then a A λ a )g k +2,a ) g k +1,a )) 0. To see ths, suppose otherwse. Then, there exsts λ A ) satsfyng both 10) and λ a )g k +2,a ) g k +1,a )) > 0. a A Snce g k +1,a ) g k,a ) 0 for all a A, 10) mples that there exst a,a A such that 0 < λ a ) < 1 wth g k +1,a ) g k,a ) > 0 and 0 <λ ) < 1 wth g k +1, ) g k, ) < 0. Let ε>0 be suffcently small. More precsely, let ε>0 be such that } ε<mn {λ a ), 1 λ ), a A λ a )g k +2,a ) g k +1,a )). 2 max a A g k +2,a ) g k +1,a ) Let λ A ) be such that Then, we have λ a ) ε f a = a, λ a ) = λ a )+ε f a = a, λ a ) otherwse. λ a )g k +1,a ) g k,a )) a A = λ a )g k +1,a ) g k,a )) a A + ε g k +1, ) g k, )) ε g k +1,a ) g k,a )) = ε g k +1, ) g k, )) ε g k +1,a ) g k,a )) < 0, 23

25 λ a )g k +2,a ) g k +1,a )) a A = λ a )g k +2,a ) g k +1,a )) a A + ε g k +2, ) g k +1, )) ε g k +2,a ) g k +1,a )) λ a )g k +2,a ) g k +1,a )) a A 2ε max g k +2,a ) g k +1,a ) > 0, a A whch contradcts to the assumpton that g s own-strategy unmodal. Now, we know that, f g s own-strategy unmodal and satsfes the assumptons, then t must be true that f λ a )g k +1,a ) g k,a )) 0, a A then a A λ a )g k +2,a ) g k +1,a )) 0. Ths mples that f y a ) a A R A s such that y a g k +1,a ) g k,a )) 0, a A y a 0 for all a A, then a A y a g k +2,a ) g k +1,a )) 0. By Farkas Lemma, there exst x k 0 and z a 0 for a A such that x k g k +1, ) g k, )) z a δ a ) =g k +2, ) g k +1, ). a A 24

26 Thus, x k g k +1, ) g k, )) g k +2, ) g k +1, ). 11) If x k = 0, then g k +2, ) g k +1, ) 0. However, ths s mpossble snce there s no weakly domnated strategy. Thus, x k > 0. By lettng C k =1/x k, 11) mples 9). Consder agan {gx, y)} x,y) R 2 ++ Dg1, 1)). In general, g x, y) s not always own-strategy concave. However, g 1, 1) s own-strategy concave. Thus, Lemma 3 says that g x, y) s own-strategy unmodal. We clam that, genercally, Dg) s a best-response equvalence class f and only f g s own-strategy unmodal for all N. Proposton 7 Suppose that g has no domnated strategy. Every game n Dg) s bestresponse equvalent to g f and only f g s own-strategy unmodal for all N. If g s own-strategy unmodal for all N and g satsfes generc property G3, then every game best-response equvalent to g and satsfyng G3 s n Dg). Proof. Suppose that g s own-strategy unmodal for all N. We show that f g Dg) then g s best-response equvalent to g. Let λ Λ a,a g ). Then, 8) mples that λ a )g a,a ) g a 1,a )) 0fa a, a A 12) λ a )g a,a ) g a +1,a )) 0fa a. a A By 7), ths s true f and only f λ a ) g a,a ) g a 1,a ) ) 0fa a, a A λ a ) g a,a ) g a +1,a ) ) 0fa a. a A 13) Thus, λ Λ a,a g ). Conversely, let λ Λ a,a g ). Snce g s own-strategy unmodal, we have 13), whch s true f and only f 12) s true. Thus, λ Λ a,a g ). Therefore, Λ a,a g )=Λ a,a g ) and thus g s best-response equvalent to g. 25

27 Conversely, suppose that every game n Dg) s best-response equvalent to g. We show that g s own-strategy unmodal for all N. Seekng a contradcton, suppose otherwse. Then, there exst a, ã A and λ Λ a,a g ) such that ether of the followng s true: a < ã and λ a )g ã,a ) g ã 1,a )) > 0, 14) a A a > ã and λ a )g ã,a ) g ã +1,a )) > 0. 15) a A When 14) s true, let g =q,w ) g D g ) be such that q ) = 0 and { L f a =ã 1, w a )= 1 otherwse. Then, we have λ a ) g ã,a ) g a,a ) ) a A = λ a ) g ã,a ) g ã 1,a ) ) a A + λ a ) g ã 1,a ) g a,a ) ) a A = L λ a )g ã,a ) g ã 1,a )) a A + λ a )g ã 1,a ) g a,a )). a A By choosng very large L>0, we have λ a ) g ã,a ) g a,a ) ) > 0 a A and thus Λ a,a g ) Λ a,a g ). When 15) s true, we also have Λ a,a g ) Λ a,a g ) by the smlar argument. Ths mples that some game n Dg) s not bestresponse equvalent to g, whch completes the proof of the frst half of the proposton. 26

28 We prove the last half of the proposton. Suppose that g s own-strategy unmodal for all N and that g satsfes generc property G3. Let g be best-response equvalent to g and satsfy G3. We show g Dg). We frst observe that a g a + 1 for all a A \{K }. To see ths, let λ k Λ k,a g ) for k A, whch exsts snce g has no domnated strategy. Note that f λ = λ k or λ = λ k+1 then λ a )g k,a ) λ a )g a,a ) for all a k, a A a A 16) λ a )g k +1,a ) λ a )g a,a ) for all a k +1. a A a A Let t [0, 1] and λ k,t = tλ k a A λ k,t +1 t)λk+1 a )g k,a )= Then, 16) mples that a )g k,a ) a A λ k,t a A λ k,t a )g k +1,a ) A ) be such that a A λ k,t a A λ k,t a A λ k,t a )g k +1,a ). 17) a )g a,a ) for all a k, a )g a,a ) for all a k +1. By 17), we have λ k,t Λ k,a g ) Λ k +1,A g ). Ths mples that a g a + 1 for all a A \{K }. Snce g and g satsfy G3 and are best-response equvalent, we can use Proposton 4, whch says that there exsts w : A \{K } R ++ such that g a +1, ) g a, ) =w a )g a +1, ) g a, )). Ths mples that g D g ) and thus g Dg). A weaker, but smlar clam s true for games such that strategy sets are ntervals of real numbers and payoff functons are dfferentable, whch has a couple of applcatons. In the remander of ths secton, we dscuss ths ssue. 27

29 Abusng notatons, we gve a defnton of best-response equvalence of the class of games. Let A be a closed nterval of R for all N. Assume that g : A R s bounded and contnuously dfferentable. Let A ) be the set of all probablty measures over A and Λ a,x g ) be such that Λ a,x g ) = {λ A ) g a,a ) g a,a ) ) dλ a ) 0 for all a X }. A The defnton of best-response equvalence s the same as that for fnte games: we say that g s best-response equvalent to g f, for each N, Λ a,a g )=Λ a,a g ) for all a A. We say that g s own-strategy unmodal f, for any λ A ), there exsts x such that g a,a )dλ a ) 0fa x, a A 18) g a,a )dλ a ) 0fa x. a A Note that f g s own-strategy unmodal, then 18) s true f and only f λ Λ x,a g ). Snce g a,a ) g a,a )dλ a )= dλ a ), a A A a g s own-strategy unmodal f g s own-strategy concave,.e., g a,a )/ a s decreasng n a for all a A. For measurable functons q : A R and w : A R ++, let q,w ) g : A R be such that, for a A and a A, q,w ) g a,a )=q a )+ w x) g x, a ) dx. x a x Let D g )={g : A R g =q,w ) g, q : A R, w : A R ++ }, Dg) ={g =g ) N g D g )}. 28

30 Proposton 8 Suppose that g s own-strategy unmodal for all N. Then, every game n Dg) s best-response equvalent to g. Proof. Let g Dg). Snce g s own-strategy unmodal, for all λ A ), there exsts a A such that g a,a )dλ a ) 0fa a, a A 19) g a,a )dλ a ) 0fa a. a A Snce g a,a ) = w a ) g a,a ), a a 19) s true f and only f g a a,a )dλ a ) 0fa a, A 20) g a a,a )dλ a ) 0fa a. A Thus, g s also own-strategy unmodal. Snce 19) s true f and only f λ Λ a,a g ) and 20) s true f and only f λ Λ a,a g ), we must have Λ a,a g )=Λ a,a g ), whch completes the proof. Ths proposton has a useful applcaton concernng the unqueness of correlated equlbra. Neyman [13] showed that f g has a contnuously dfferentable and strctly concave potental functon, 8 then the potental maxmzer s the unque correlated equlbrum of g. The set of correlated equlbra s the same for two games f the two games are best-response equvalent. Thus, we clam the followng. Corollary 9 Suppose that g has a contnuously dfferentable and strctly concave potental functon f. Then, the potental maxmzer s the unque correlated equlbrum of every game n Dg). Note that a game n Dg) s not necessarly a potental game and payoff functons are not necessarly concave. 8 The defnton of potental functons of ths class of games s the same as those of fnte games. 29

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