Best response equivalence

Transcription

1 Gmes nd Economc Behvor ) Best response equvlence Stephen Morrs,, Tksh U b Deprtment of Economcs, Yle Unversty, USA b Fculty of Economcs, Yokohm Ntonl Unversty, Jpn Receved 2 August 2002 Avlble onlne 15 Aprl 2004 Abstrct Two gmes re best-response equvlent f they hve the sme best-response correspondence. We provde chrcterzton of when two gmes re best-response equvlent. The chrcterztons explot dul reltonshp between pyoff dfferences nd belefs. Some potentl gme rguments [Gmes Econ. Behv ) 124] rely only on the property tht potentl gmes re best-response equvlent to dentcl nterest gmes. Our results show tht lrge clss of gmes re best-response equvlent to dentcl nterest gmes, but re not potentl gmes. Thus we show how some exstng potentl gme rguments cn be extended Elsever Inc. All rghts reserved. JEL clssfcton: C72 Keywords: Best response equvlence; Dulty; Frks Lemm; Potentl gmes 1. Introducton We consder three progressvely stronger equvlence reltons on gmes nd chrcterze ech of them. Two gmes re best-response equvlent f they hve the sme best-response correspondence. Two gmes re better-response equvlent f, for every pr of strteges, they gree when one strtegy s better thn the other. * Correspondng uthor s ddress: Cowles Foundton, P.O. Box , New Hven, CT , USA. E-ml ddresses: stephen.morrs@yle.edu S. Morrs), ou@ynu.c.jp T. U) /$ see front mtter 2004 Elsever Inc. All rghts reserved. do: /j.geb

2 S. Morrs, T. U / Gmes nd Economc Behvor ) Two gmes re von Neumnn Morgenstern equvlent VNM-equvlent) f, for ech plyer, the pyoff functon n one gme s equl to constnt tmes the pyoff functon n the other gme, plus functon tht depends only on the opponents strteges. Two gmes re VNM-equvlent f nd only f, for ech plyer, there s constnt w > 0 such tht the rto of pyoff dfferences from swtchng between one strtegy to nother strtegy s lwys w. The constnt w s thus ndependent of the strteges beng compred. Two gmes re better-response equvlent f nd only f they hve the sme domnnce reltons nd, for ech plyer nd ech pr of strteges nd such tht nether strtegy strctly domntes the other, there exsts constnt w > 0 such tht the rto of pyoff dfferences from swtchng between nd s lwys w. In generl, ths s weker requrement thn VNM-equvlence. It s weker both becuse the proportonl pyoff dfferences property s no longer requred to hold between some strtegy prs, nd becuse the weght w s not necessrly ndependent of the strtegy pr. But f the gme does not hve domnted strteges, the weghts cn no longer depend on the strteges beng compred, nd better-response equvlence collpses to VNM-equvlence. Two gmes re best-response equvlent f nd only f, for ech plyer nd ech pr of strteges nd such tht both strteges re best response to some belef, there exsts constnt w > 0 such tht the rto of pyoff dfferences from swtchng between nd s lwys w. Even f gme hs no domnted strteges, ths s weker requrement thn VNM-equvlence. In gmes wth dmnshng mrgnl returns, best-response equvlence s lwys strctly weker requrement thn VNM-equvlence. Exmples re gven n the pper. The most extensve dscusson nd pplctons of these reltons hve come n the lterture on potentl gmes. Monderer nd Shpley 1996b) sd tht gme ws potentl gme f there exsts potentl functon, defned on the strtegy spce, wth the property tht the chnge n ny plyer s pyoff functon from swtchng between ny two of hs strteges holdng other plyers strteges fxed) ws equl to the chnge n the potentl functon. 1 A gme s weghted potentl gme, f the pyoff chnges re proportonl for ech plyer. Thus gme s weghted potentl gme f nd only f t s VNM-equvlent to gme wth dentcl pyoff functons. Whle some results usng potentl or weghted potentl gme rguments re usng the VNM-equvlence to dentcl nterest gmes, other rguments re just usng the better-response equvlence nd even only best-response equvlence mplctons of VNM-equvlence. 2 Any pper tht dels only wth equlbr s usng only best-response equvlence e.g., Neymn, 1997; U, 2001; Morrs nd U, 2002). Smlrly, fcttous ply only uses the best-response 1 See lso U 2000) for chrcterzton nd exmples of potentl gmes. 2 Arguments tht explot potentl rguments to prove the exstence of pure strtegy equlbrum e.g., Rosenthl, 1973) only use ordnl propertes of pyoffs. Monderer nd Shpley 1996b) ntroduced ordnl potentl gmes nd Voorneveld 2000) nd Dubey et l. 2002) showed how ordnl potentl gmes cn be wekened to only requre pure strtegy best-response equvlence.

3 262 S. Morrs, T. U / Gmes nd Economc Behvor ) propertes of the gme Monderer nd Shpley, 1996). 3 An pplcton usng only better-response equvlence but not the VNM-equvlence ppers n Morrs 1999). Some ppers studyng quntl responses or stochstc best responses n potentl gmes use the full power of VNM-equvlence e.g., Blume, 1993; Brock nd Durluf, 2001; Anderson et l., 2001; U, 2002). 4 The fct tht VNM-equvlence s the sme s better-response equvlence n the bsence of domnted strteges nd my be dfferent n the presence of domnted strteges hs been noted n number of contexts see Sel, 1992; Blume, 1993, p. 409; Monderer nd Shpley, 1996b, footnote 9; Mskn nd Trole, 2001, p. 209). However, our chrcterztons of better-response equvlence n the presence of domnted strteges nd of the sgnfcnt gp between better-response equvlence nd best-response equvlence fll gp n the lterture. 5 The pper s orgnzed s follows. In Secton 2, we descrbe our notons of equvlence nd gve n exmple llustrtng the dfferences. In Secton 3, we report our chrcterztons. In Secton 4, we restrct ttenton to clss of gmes where best-response equvlence s strctly weker requrement thn VNM-equvlence nd chrcterze tht clss. We lso dscuss n extenson to gmes wth nfnte strtegy spces nd ts pplcton. Secton 5 brefly dscusses better-response nd best-response equvlence n the mxed strtegy extenson of gme. 2. Equvlence propertes of gmes A gme conssts of fnte set of plyers N nd fnte strtegy set A for N, nd pyoff functon g : A R for N where A = N A. We wrte A = j A j nd = j ) j A. We smply denote gme by g = g ) N. Throughout the pper, we regrd g, ) : A R s vector n R A. We wrte g, ) g, ) f g, )>g, ) for ll A,ndg, ) g, ) f g, ) g, ) for ll A. For N, let A ) denote the set of ll probblty dstrbutons over A. We cll ech element of A ) plyer s belef. For X A,letΛ,X g ) A ) be set of plyer s belefs such tht plyer wth pyoff functon g nd belef λ Λ,X g ) wekly prefers to ny strtegy n X : Λ,X g ) { } )) = λ A ) λ ) g, ) g, 0forll X. When X s sngleton,.e., X ={ }, we wrte Λ, g ) nsted of Λ, { } g ). 3 Sel 1999) estblshes convergence of fcttous ply n clss of one-gnst-ll gmes. These gmes re best-response equvlent to dentcl nterest gmes, but not potentl gmes. 4 More precsely, they use the full power of VNM-equvlence such tht the constnt w s the sme for ll the plyers. 5 Mertens 1987) studed vrous notons of best-response equvlence, but wth hs more bstrct strtegy spces nd focus on dmssble best responses, there s lttle overlp wth the mterl n ths pper.

4 S. Morrs, T. U / Gmes nd Economc Behvor ) We re nterested n chrcterzng two equvlence reltons on gmes cptured by these sets of belefs by whch plyers prefer one prtculr strtegy. Defnton 1. Agmeg s better-response equvlent to g = g ) N f, for ech N, Λ, g ) = Λ, ) g for ll, A. Defnton 2. Agmeg s best-response equvlent to g = g ) N f, for ech N, Λ,A g ) = Λ,A g ) for ll A. If g s better-response equvlent to g,theng s best-response equvlent to g,snce Λ,A g ) = Λ, g ). A An esy suffcent condton for better-response equvlence s the followng. 6 Defnton 3. Agmeg s VNM-equvlent to g = g ) N f, for ech N, there exsts postve constnt w > 0 nd functon Q : A R such tht g, ) = w g, ) + Q ) for ll A. It s strghtforwrd to see tht f g s VNM-equvlent to g,then g, ) g, ) = w g, ) g, )) for ll, A. Conversely, f ths s true, then functon Q : A R such tht Q ) = g, ) w g, ) s well defned, nd thus g s VNM-equvlent to g. Thus, we hve the followng lemm. Lemm 1. A gme g s VNM-equvlent to g f nd only f, for ech N, there exsts w such tht g, ) g, ) = w g, ) g, )) 1) for ll, A. It s strghtforwrd to see tht VNM-equvlence s suffcent for better-response equvlence. In fct, 1) mples tht 6 Blume 1993) clled ths property strongly best-response equvlent.

5 264 S. Morrs, T. U / Gmes nd Economc Behvor ) )) λ ) g, ) g, = w λ ) g, ) g )), for ll λ A ) nd thus Λ, g ) = Λ, g ) for ll, A. Best-response, better-response, nd VNM-equvlence re equvlence reltons. Thus, they defne n equvlence clss of gmes. For exmple, weghted potentl gmes Monderer nd Shpley, 1996b) wth weghted potentl functon f : A R re regrded s VNM-equvlence clss of n dentcl nterest gme f = f ) N wth f = f for ll N. Ths s cler by Lemm 1 nd the followng orgnl defnton of weghted potentl gmes. Defnton 4. Agmeg = g ) N s weghted potentl gme f there exsts weghted potentl functon f : A R nd w > 0 for ech N such tht g, ) g, ) = w f, ) f, )) for ll, A.Ifw = 1forll N, g s clled potentl gme nd f s clled potentl functon. As the concept of VNM-equvlence leds us to the defnton of weghted potentl gmes, the concept of better-response equvlence nd tht of best-response equvlence led us to the followng defntons of new clsses of gmes. Defnton 5. Agmeg = g ) N s better-response potentl gme f t s betterresponse equvlent to n dentcl nterest gme f = f ) N wth f = f for ll N. A functon f s clled better-response potentl functon. Defnton 6. Agmeg = g ) N s best-response potentl gme f t s best-response equvlent to n dentcl nterest gme f = f ) N wth f = f for ll N. A functon f s clled best-response potentl functon. Voorneveld 2000) clled gme best-response potentl gme f ts best-response correspondence concdes wth tht of n dentcl nterest gme over the clss of belefs such tht λ ) = 0 or 1. Thus, best-response potentl gmes n ths pper form specl clss of those n Voorneveld 2000). Exstng potentl gme results tht rely only on better-response equvlence or bestresponse equvlence, such s those mentoned n the ntroducton, utomtclly hold for the lrger clss of better-response potentl gmes or tht of best-response potentl gmes. Thus, we re nterested n exctly when nd to wht extent better-response nd best-response equvlence re weker requrements thn VNM-equvlence. Notce tht best-response nd better-response equvlence re clerly weker requrements thn VNM-equvlence, becuse the ltter mposes too mny constrnts on pyoffs from domnted strteges. Moreover, best-response equvlence s sgnfcntly weker thn better-response equvlence, s shown by the followng exmple.

6 S. Morrs, T. U / Gmes nd Economc Behvor ) Consder two plyer, three strtegy, symmetrc pyoff gme gx, y) prmeterzed by x, y) R 2 ++, where ech plyer s pyoffs re gven by the followng pyoff mtrx where the plyer s own strteges re represented by rows nd hs opponent s strteges re represented by columns): x x 2x y y y. In the specl cse where x = y = 1, we hve gme g1, 1) wth the followng pyoff mtrx: If row plyer hs belef λ k) = π k for k {1, 2, 3}, he prefers strtegy 1 to strtegy 2 f nd only f π 1 π 2 + 2π 3 ; he prefers strtegy 1 to strtegy 3 f nd only f x + 2y)π 1 x y)π 2 + 2x + y)π 3 ; he prefers strtegy 3 to strtegy 2 f nd only f π 3 π 2 + 2π 1. Thus the regon of ndfference between strteges 1 nd 2, nd between strtegy 2 nd 3, does not depend on x nd y. Moreover, whenever strtegy 1 or 3) s preferred to strtegy 2, t s lso preferred to strtegy 3 or 1). Thus the best response regons for ths gme re s n Fg. 1, for ny x, y) R Thus gx, y) s best-response equvlent to g1, 1) for ny Fg. 1. The best response regons.

7 266 S. Morrs, T. U / Gmes nd Economc Behvor ) x, y) R On the other hnd, the regon of ndfference between strteges 1 nd 3 does depend on x nd y: n prtculr, gx, y) s better-response equvlent to g1, 1) f nd only f x = y. We wll dscuss ths exmple gn n Secton Results 3.1. Generc propertes of gmes We wll ppel to some generc propertes of gmes,.e., propertes tht wll hold for ll but Lebesgue mesure zero set of pyoffs s long s ech plyer hs t lest two ctons). G1. For ll N, fg, ) g, ),theng, ) g, ) for dstnct, A. G2. For ll N, vectors g, ) g, ) nd g, ) g, ) re lnerly ndependent for dstnct,, A. G3. For ll N, fλ,a g ) Λ,A g ),then Λ,A \ { } g ) \Λ, g ) for dstnct, A Better-response equvlence Strtegy strctly domntes n gme g we wrte g )fg, ) g, ), or, equvlently, Λ, g ) =. Strteges nd re better-response comprble we wrte g ) f nether g nor g. Proposton 1. If gmes g nd g stsfy generc property G1, then g s better-response equvlent to g f nd only f, for ech N, ) they hve the sme domnnce reltons g = g ) nd b) whenever s better-response comprble to g ), there exsts w, )>0 such tht g, ) g, ) = w, ) g, ) g, )). 2) Frks Lemm 7 plys centrl role n the proofs. Lemm 2 Frks Lemm). For vectors 0, 1,..., m R n, the followng two condtons re equvlent. If 1.y),..., m.y) 0 for y R n,then 0.y) 0. 8 There exsts x 1,...,x m 0 such tht x x m m = 0. 7 See textbook of convex nlyss such s the recent one by Hrrt-Urruty nd Lemréchl 2001), or the clssc one by Rockfellr 1970). 8 I.e., f n j=1 j y j 0 for ech = 1,...,m,then n j=1 0j y j 0.

8 S. Morrs, T. U / Gmes nd Economc Behvor ) Proof of Proposton 1. We frst show tht ) nd b) re suffcent for the better-response equvlence of g nd g.if g, then b) mples tht )) λ ) g, ) g, = w, ) g, ) g )), λ ) nd thus Λ, g ) = Λ, ) g. If g,then Λ, g ) = Λ, ) g = A ). If g,then Λ, g ) = Λ, ) g =. To prove necessty, suppose tht g s better-response equvlent to g.snce Λ, g ) = Λ, ) g, we hve g ) Λ, g = Λ, g ) = g nd thus ) holds. To prove b), suppose tht g. We know tht g.letλ A ) be such tht )) λ ) g, ) g, 0. Snce λ Λ, g ) = Λ, g ), λ ) g, ) g )), 0. Ths mples tht f y ) A R A s such tht )) g, ) g, 0, then y y 0 forll A, y g, ) g )), 0. By Frks Lemm, there exst x 0ndz 0for A such tht x g, ) g, )) z δ ) = g, ) g, ))

9 268 S. Morrs, T. U / Gmes nd Economc Behvor ) where δ : A R s such tht δ ) = 1f = nd δ ) = 0otherwse. Thus, x g, ) g, )) g, ) g, ). If x = 0, then g, ) g, ) 0. However, ths s mpossble snce g mples tht does not strctly domnte n g nd G1 requres tht f does not strctly domnte, then t s not the cse tht g, ) g, ) 0. Thus, x > 0. Symmetrclly, we hve x g, ) g, ) ) g, ) g, ) where x > 0. Thus, ) x x g, ) g, )) 0. If x x > 0, then g, ) g, ) 0, nd f x g, ) 0, whch we lredy noted re mpossble. Thus, x x g, ) g, )) = g, ) g, ). x < 0, then g, ) = x, whch mples tht Ths proves b). If g hs no domnted strtegy, then 2) s true for every, A.Ifw, ) s the sme for every, A, then better-response equvlence mples VNM-equvlence. However, Proposton 1 does not sy nythng bout whether w, ) does depend upon, A. Thus, we re nterested n when better-response equvlence mples VNMequvlence. The followng proposton provdes suffcent condton for the equvlence of better-response equvlence nd VNM-equvlence. Proposton 2. Suppose tht gmes g nd g stsfy generc propertes G1 nd G2, nd tht, for ech N nd for ny, A, there exsts sequence { k}m k=1 such tht 1 =, m =, k g k+1 for k = 1,...,m 1, nd k g k+2 for k = 1,...,m 2. Then g s better-response equvlent to g f nd only f g s VNM-equvlent to g. Note tht the bove condton concernng g s trvlly stsfed f no strtegy s domnted,.e., g s the complete relton. So, the proposton mmedtely hs the followng corollry. Corollry 3. If g nd g stsfy generc propertes G1 nd G2 nd hve no strctly domnted strteges, then g s better-response equvlent to g f nd only f g s VNMequvlent to g.

10 S. Morrs, T. U / Gmes nd Economc Behvor ) It should be emphszed tht the suffcent condton of Proposton 2 s sometmes stsfed even when there re strctly domnted strteges n the gme. For exmple, consder the followng two plyer gme, where only the row plyer s pyoffs re shown: Consder strteges of the row plyer. We hve 1 g 2, 2 g 3, 3 g 4, 1 g 3, 2 g 4 s n Fg. 2, stsfyng the condton of Proposton 2, whle strtegy 1 strctly domntes strtegy 4. To prove the proposton, we use the followng lemm. Lemm 3. Suppose tht g nd g stsfy generc property G2. For some A A,fthere exsts w, )>0 such tht g, ) g, ) = w, ) g, ) g, )) for ll, A,thenw, ) s the sme for ll, A. Proof. Wthout loss of generlty, ssume tht A 3. For dstnct,b,c A,there exst w,b ), w b,c ), w,c )>0 such tht g, ) g b, ) = w,b ) g, ) g b, ) ), g b, ) g c, ) = w b,c ) g b, ) g c, ) ), g, ) g c, ) = w,c ) g, ) g c, ) ). We frst show tht w,b ) = w b,c ) = w,c ).Snce w,b ) g, ) g b, ) ) + w b,c ) g b, ) g c, ) ) = g, ) g b, ) + g b, ) g c, ) = g, ) g c, ) = w,c ) g, ) g c, ) ) = w,c ) g, ) g b, ) ) + w,c ) g b, ) g c, ) ), Fg. 2. The grph of g.

11 270 S. Morrs, T. U / Gmes nd Economc Behvor ) we hve w,b ) w,c ) ) g, ) g b, ) ) + w b,c ) w,c ) ) g b, ) g c, ) ) = 0. By G2, g, ) g b, ) nd g b, ) g c, ) re lnerly ndependent nd thus t must be true tht w,b ) = w b,c ) = w,c ). Smlrly, for dstnct b,c,d A, w b,c ) = w c,d ) = w b,d ). Therefore, w,b ) = w c,d ) for ny,b,c,d A, whch completes the proof. We now report the proof of Proposton 2. Proof of Proposton 2. We show tht f g s better-response equvlent to g then g s VNM-equvlent to g. By G1 nd Proposton 1, f g,thereexstw, )>0 such tht g, ) g, ) = w, ) g, ) g, )). If A =2, ths completes the proof by Lemm 1. Suppose tht A 3. For, A, let { k}m k=1 be sequence such tht 1 =, m =, k g k+1 for k = 1,...,m 1, nd k g k+2 for k = 1,...,m 2. There exst x k,y k > 0 such tht g k, ) g k+1, ) = x k g k, ) g k+1, )), g k+1, ) g k+2, ) = x k+1 g k+1, ) g k+2, )), g k, ) g k+2, ) = y k g k, ) g k+2, )). By Lemm 3, x k = x k+1 = y k for ll k m 2. By lettng x k = w, ),wehve m 1 g, ) g, ) = g k, ) g k+1, )) k=1 m 1 = k=1 x k g k, ) g k+1, )) = w, ) g, ) g, )). To summrze, for ll, A, there exsts w, )>0 stsfyng the bove equton. By Lemm 3, w, ) s the sme for ll, A. By Lemm 1, g s VNM-equvlent to g, whch completes the proof Best-response equvlence Strteges nd re best-response comprble we wrte g ) f both strteges re best responses t some belef,.e., Λ,A g ) Λ,A g ). Note tht g f nd only f Λ,A g ).

12 S. Morrs, T. U / Gmes nd Economc Behvor ) Proposton 4. If gmes g nd g stsfy generc property G3, then g s best-response equvlent to g f nd only f, for ech N, ) they hve the sme best-response comprblty relton g = g ) nd b) whenever s best-response comprble to g ), there exsts w, )>0 such tht g, ) g, ) = w, ) g, ) g, )). Proof. We frst show tht ) nd b) re suffcent for the best-response equvlence of g nd g.ifλ,a g ) =,thenλ,a g ) = Λ,A g ) = becuse Λ,A g ) = mples tht g s not true nd thus ) mples tht g s not true. If Λ,A g ),then{ g }, nd we must hve Λ,A g ) = Λ, g ) = Λ, g ). 3) A { g } Clerly, 3) s true when { g }=A. To see tht 3) s true when { g } A, suppose otherwse. Then, Λ, g ) Λ, g ), A { g } nd thus there exsts / { g } such tht Λ, g ) Λ, g ). A A \ { } However, ths mples tht g, whch s contrdcton. Thus, 3) must be true. If g, then b) mples tht )) λ ) g, ) g, = w, ) g, ) g )),, nd thus λ ) Λ, g ) = Λ, g ). 4) Therefore, by ), 3), nd 4), we hve Λ,A g ) = Λ,A g ). Ths completes the proof of suffcency. To prove necessty, suppose tht g s best-response equvlent to g.snce Λ,A g ) = Λ,A g ), we hve ) Λ,A g ) Λ,A g = Λ,A g ) Λ,A g ) nd thus g = g.thsproves).

13 272 S. Morrs, T. U / Gmes nd Economc Behvor ) If g, then there exsts λ A ) such tht λ ) g, ) g, )) 0 forll A, ) λ ) g, g, )) 0 forll A \{ }. Snce λ Λ,A g ) = Λ,A g ), λ ) g, ) g )), 0. The bove mples tht, f y ) A R A s such tht )) g, ) g, 0, then y y y g, ) g, )) 0 forll A \{, }, ) g, g, )) 0 forll A \{, }, y 0 forll A, y g, ) g )), 0. By Frks Lemm, there exst x 0, γ : A R,ndδ : A R such tht, )) where wth u x g, ) g, )) γ ) δ ) = g, ) g γ ) =,,v 0nd δ ) = z δ ) wth z 0. Thus, x u g, ) g, )) +, g, ) g, )) + γ ) g, ) g, ). We show x > 0. Suppose tht x λ Λ,A \ { } g ) \Λ, g ), v g, ) g, )) = 0,.e., γ ) g, ) g, ).Let

14 S. Morrs, T. U / Gmes nd Economc Behvor ) whch exsts by g nd G3. Snce λ Λ,A \{ } g ) Λ,A \{ } g ), λ )γ ) = u λ ) g, ) g, )), + v λ ) ) g, g, )) 0., Snce λ Λ,A g ) = Λ,A g ) nd λ / Λ,A g ) = Λ,A g ), λ ) g, ) g )), < 0. Ths s contrdcton. Thus, we must hve x > 0. We hve x g, ) g, )) + γ ) g, ) g nd symmetrclly, ) x g, ) g, ) ) + γ ) g, ) g, ) where x,x > 0. Addng both, x x ) g, ) g, )) + γ We show x x = 0. Suppose tht x x > 0. Let ) ) λ Λ,A \{ } g \Λ, g Λ,A \ { } ) ) g Λ,A \{ } g. ) + γ ) 0. 5) Then, the expectton of the left-hnd sde of 5) s postve becuse ) x x )) λ ) g, ) g, > 0 nd ) λ ) γ ) + γ ) = ) u + v, λ ) g, ) g, )) + ) v + u ) λ ) g, g, )) 0.,

15 274 S. Morrs, T. U / Gmes nd Economc Behvor ) Ths s contrdcton. Symmetrclly, f x x < 0, then we hve the symmetrc contrdcton. Thus, x x = 0, nd 5) s reduced to γ ) + γ ) 0. 6) We show γ ) = γ ) = 0. Suppose tht ether γ ) 0orγ ) 0strue.Let λ,λ A ) be such tht ) ) λ Λ,A \{ } g \Λ, g Λ,A \ { } ) ) g Λ,A \{ } g, λ Λ,A \ { } ) g \Λ, g ) Λ,A \ { } ) ) g Λ,A \{ } g. Consder λ + λ )/2 A ). Then, the expectton of the left-hnd sde of 6) s postve becuse λ ) + λ 2 =, + u,, + v u, v ) γ ) + v ) ) + γ ) λ ) + λ 2 ) + u + v ) λ ) + λ 2 ) λ ) 2 + u ) λ ) Ths s contrdcton. Thus, γ ) = γ ) = 0. Summrzng the bove, we hve x g, ) g, )) = g, ) g where x > 0. Ths proves b). 2 ) g, ) g, )) ) g, g, )) g, ) g, )), ) g, ) g, )) > 0. The followng proposton nd corollry follow by exctly the sme rguments n Proposton 2 nd Corollry 3 n the prevous subsecton for better-response equvlence. Proposton 5. Suppose tht gmes g nd g stsfy generc propertes G2 nd G3, nd tht, for ech N nd for ny, A, there exsts sequence { k}m k=1 such tht

16 S. Morrs, T. U / Gmes nd Economc Behvor ) =, m =, k g k+1 for k = 1,...,m 1, k g k+2 for k = 1,...,m 2.Then g s best-response equvlent to g f nd only f g s VNM-equvlent to g. Corollry 6. If g nd g stsfy generc propertes G2 nd G3 nd g s the complete relton, then g s best-response equvlent to g f nd only f g s VNM-equvlent to g. 4. Gmes wth own-strtegy unmodlty Best-response equvlence relton s n equvlence relton. It wll be useful f, s closed form, we cn descrbe the best-response equvlence clss of gme n whch best-response equvlence s strctly weker requrement thn VNM-equvlence. Let A be lnerly ordered such tht A ={1,...,K } wth K 3. For q : A R nd w : A \{K } R ++,letq,w ) g : A R be such tht q,w ) g 1, ) = q ), 1 q,w ) g, ) = q ) + w k) g k + 1, ) g k, ) ) for 2. k=1 Let D g ) be clss of pyoff functons of plyer obtned by ths trnsformton: D g ) = { g : A R g = q },w ) g,q : A R, w : A \{K } R ++. It s strghtforwrd to see tht g D g ) f nd only f there exsts w : A \{K } R ++ such tht g + 1, ) g, ) = w ) g + 1, ) g, ) ) 7) for ll A \{K }. Note tht g D g ), g D g ) mples g D g ),ndg D g ) wth g D g ) mples g D g ). Thus, D g ) defnes n equvlence clss of pyoff functons of plyer. We wrte Dg) = { g = g ) N g D g ) for ll N }. For exmple, consder prmetrzed clss of gmes {gx, y)} x,y) R 2 dscussed n ++ Secton 2. We hve tht {gx, y)} x,y) R 2 Dg1, 1)). To see ths, we wrte gx, y) = ++ g x,y)) {1,2}. Then, for ny x, y) R 2 ++ nd j, g 1, j x,y) = q j ), g 2, j x,y) = q j ) + x g 2, j 1, 1) g 1, j 1, 1) ), g 3, j x,y) = q j ) + x g 2, j 1, 1) g 1, j 1, 1) ) + y g 3, j 1, 1) g 2, j 1, 1) ) where q : {1, 2, 3} R s such tht q 1) = x, q 2) = x, ndq 3) = 2x. Remember tht, for ny x, y) R 2 ++, gx, y) s best-response equvlent to g1, 1). It s esy to see tht every gme n Dg1, 1)) s VNM-equvlent to gx, y) for some x, y) R Thus, every gme n Dg1, 1)) s best-response equvlent to g1, 1).

17 276 S. Morrs, T. U / Gmes nd Economc Behvor ) Ths observton leds us to the queston when every gme n Dg) s best-response equvlentto g. We provde necessry nd suffcent condton for t. We sy tht g s own-strtegy unmodl f, for ll λ A ), there exsts k A such tht, λ ) g, ) g 1, ) ) 0 f k nd k > 1, λ ) g, ) g + 1, ) ) 8) 0 f k nd k <K. Note tht f g s own-strtegy unmodl, then 8) s true f nd only f λ Λ k,a g ). Clerly, by 7), g s own-strtegy unmodl f nd only f g D g ) s own-strtegy unmodl. We sy tht g s own-strtegy concve f g, ) : A R s concve,.e., g + 1, ) g, ) s decresng n for ll A. Lemm 4. Suppose tht g + 1, ) g, ) for ll A \{K } nd A, nd tht there s no wekly domnted strtegy. Then, g s own-strtegy unmodl f nd only f there exsts g D g ) such tht g s own-strtegy concve. Proof. Suppose tht g D g ) s own-strtegy concve. Then, g + 1, ) g, ) s decresng n for ll A. Thus, λ ) g + 1, ) g, )) s lso decresng n for ll λ A ). Ths mmedtely mples tht g D g ) s own-strtegy unmodl. Snce λ ) g + 1, ) g, ) ) = 1 w ) λ ) g + 1, ) g, ) ), g s lso own-strtegy unmodl. Suppose tht g s own-strtegy unmodl. We prove the exstence of n own-strtegy concve pyoff functon g = q,w ) g by constructon. Lter, we wll show tht there exsts C k > 0 such tht g k + 1, ) g k, ) C k g k + 2, ) g k + 1, ) ). 9) For C k stsfyng 9), we let w : A R ++ be such tht w 1) = 1ndw ) = 1 k=1 C k for 2, nd q : A R be such tht q ) = 0forll A.Snce g + 1, ) g, ) = w ) g + 1, ) g, ), we hve 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, ) ).

18 S. Morrs, T. U / Gmes nd Economc Behvor ) By ths nd 9), we hve g k + 1, ) g k, ) g k + 2, ) g k + 1, ), whch mples tht g s own-strtegy concve. We prove the exstence of C k stsfyng 9) by Frks Lemm. Before dong t, we must frst observe tht f λ ) g k + 1, ) g k, ) ) = 0 10) then λ ) g k + 2, ) g k + 1, ) ) 0. To see ths, suppose otherwse. Then, there exsts λ A ) stsfyng both 10) nd λ ) g k + 2, ) g k + 1, ) ) > 0. Snce g k + 1, ) g k, ) 0forll A, 10) mples tht there exst, A such tht 0 <λ )<1 wth g k + 1, ) g k, )>0nd 0 <λ )<1 wth g k + 1, ) g k, )<0. Let ε>0 be suffcently smll. More precsely, let ε>0 be such tht { } ) ) ε<mn λ, 1 λ, A λ )g k + 2, ) g k + 1, )). 2 mx A g k + 2, ) g k + 1, ) Let λ A ) be such tht λ ) ε f = λ, ) = λ ) + ε f =, λ ) otherwse. Then, we hve λ ) g k + 1, ) g k, ) ) = λ ) g k + 1, ) g k, ) ) + ε ) )) ) )) g k + 1, g k, ε g k + 1, g k, = ε ) ) )) g k + 1, g k, )) ε g k + 1, g k, < 0, λ ) g k + 2, ) g k + 1, ) ) = λ ) g k + 2, ) g k + 1, ) ) + ε ) )) ) )) g k + 2, g k + 1, ε g k + 2, g k + 1,

19 278 S. Morrs, T. U / Gmes nd Economc Behvor ) λ ) g k + 2, ) g k + 1, ) ) 2ε mx g k + 2, ) g k + 1, ) > 0, whch contrdcts to the ssumpton tht g s own-strtegy unmodl. Now, we know tht, f g s own-strtegy unmodl nd stsfes the ssumptons, then t must be true tht f λ ) g k + 1, ) g k, ) ) 0, then λ ) g k + 2, ) g k + 1, ) ) 0. Ths mples tht f y ) A R A s such tht g k + 1, ) g k, ) ) 0, then y y 0 forll A, y g k + 2, ) g k + 1, ) ) 0. By Frks Lemm, there exst x k 0ndz 0for A such tht x k g k + 1, ) g k, ) ) z δ ) = g k + 2, ) g k + 1, ). 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 wekly domnted strtegy. Thus, x k > 0. By lettng C k = 1/x k, 11) mples 9). Consder gn {gx, y)} x,y) R 2 ++ Dg1, 1)). In generl, g x,y) s not lwys own-strtegy concve. However, g 1, 1) s own-strtegy concve. Thus, Lemm 4 sys tht g x,y) s own-strtegy unmodl. We clm tht, generclly, Dg) s best-response equvlence clss f nd only f g s own-strtegy unmodl for ll N. Proposton 7. Suppose tht g hs no domnted strtegy. Every gme n Dg) s bestresponse equvlent to g f nd only f g s own-strtegy unmodl for ll N. Ifg s own-strtegy unmodl for ll N nd g stsfes generc property G3, then every gme best-response equvlent to g nd stsfyng G3 s n Dg).

20 S. Morrs, T. U / Gmes nd Economc Behvor ) Proof. Suppose tht g s own-strtegy unmodl for ll N. We show tht f g Dg) then g s best-response equvlent to g.letλ Λ,A g ). Then, 8) mples tht λ ) g, ) g 1, ) ) 0 f nd > 1, λ ) g, ) g + 1, ) ) 0 f nd <K 12). By 7), ths s true f nd only f λ ) g, ) g 1, ) ) 0 f nd > 1, λ ) g, ) g + 1, ) ) 0 f nd <K. Thus, λ Λ,A g ).Conversely,letλ Λ,A g ).Snceg s own-strtegy unmodl, we hve 13), whch s true f nd only f 12) s true. Thus, λ Λ,A g ). Therefore, Λ,A g ) = Λ,A g ) nd thus g s best-response equvlent to g. Conversely, suppose tht every gme n Dg) s best-response equvlent to g.weshow tht g s own-strtegy unmodl for ll N. Seekng contrdcton, suppose otherwse. Then, there exst, ã A nd λ Λ,A g ) such tht ether of the followng s true: < ã nd λ ) g ã, ) g ã 1, ) ) > 0, 14) > ã nd λ ) g ã, ) g ã + 1, ) ) > 0. 15) When 14) s true, let g = q,w ) g D g ) be such tht q ) = 0nd { w ) = L f =ã 1, 1 otherwse. Then, we hve λ ) g ã, ) g, ) ) = λ ) g ã, ) g ã 1, ) ) + λ ) g ã 1, ) g, ) ) = L λ ) g ã, ) g ã 1, ) ) + λ ) g ã 1, ) g, ) ). 13)

21 280 S. Morrs, T. U / Gmes nd Economc Behvor ) By choosng very lrge L>0, we hve λ ) g ã, ) g, ) ) > 0 nd thus Λ,A g ) Λ,A g ). When 15) s true, we lso hve Λ,A g ) Λ,A g ) by the smlr rgument. Ths mples tht some gme n Dg) s not best-response equvlent to g, whch completes the proof of the frst hlf of the proposton. We prove the lst hlf of the proposton. Suppose tht g s own-strtegy unmodl for ll N nd tht g stsfes generc property G3. Let g be best-response equvlent to g nd stsfy G3. We show g Dg). We frst observe tht g +1forll A \{K }. To see ths, let λ k Λ k, A g ) for k A, whch exsts snce g hs no domnted strtegy. Note tht f λ = λ k or λ = λ k+1 then λ )g k, ) λ )g, ) for ll k, 16) λ )g k + 1, ) λ )g, ) for ll k + 1. Let t [0, 1] nd λ k,t = tλ k + 1 t)λk+1 A ) be such tht )g k, ) = )g k + 1, ). 17) λ k,t λ k,t Then, 16) mples tht )g k, ) λ k,t λ k,t λ k,t )g k + 1, ) )g, ) for ll k, λ k,t )g, ) for ll k + 1. By 17), we hve λ k,t Λ k, A g ) Λ k + 1,A g ). Thsmplestht g + 1 for ll A \{K }. Snce g nd g stsfy G3 nd re best-response equvlent, we cn use Proposton 4, whch sys tht there exsts w : A \{K } R ++ such tht g + 1, ) g, ) = w ) g + 1, ) g, ) ). Ths mples tht g D g ) nd thus g Dg). A weker, but smlr clm s true for gmes such tht strtegy sets re ntervls of rel numbers nd pyoff functons re dfferentble, whch hs couple of pplctons. In the remnder of ths secton, we dscuss ths ssue. Abusng notton, we gve defnton of best-response equvlence for clss of gmes wth contnuum of ctons. Let A be closed ntervl of R for ll N. Assume tht g : A R s bounded nd contnuously dfferentble. Let A ) be the set of ll probblty mesures over A nd Λ,X g ) be such tht

22 S. Morrs, T. U / Gmes nd Economc Behvor ) Λ,X g ) { = λ A ) g, ) g, )) dλ ) 0forll X }. A The defnton of best-response equvlence s the sme s tht for fnte gmes: we sy tht g s best-response equvlent to g f, for ech N, Λ,A g ) = Λ,A g ) for ll A. We sy tht g s own-strtegy unmodl f, for ny λ A ), there exsts x such tht g, ) dλ ) 0 f x nd x > mn A, A 18) g, ) dλ ) 0 f x nd x < mx A. A Note tht f g s own-strtegy unmodl, then 18) s true f nd only f λ Λ x,a g ). Snce g, ) g, ) dλ ) = dλ ), A A g s own-strtegy unmodl f g s own-strtegy concve,.e., g, )/ s decresng n for ll A. For mesurble functons q : A R nd w : A R ++,letq,w ) g : A R be such tht, for A nd A, Let q,w ) g, ) = q ) + x w x) g x, ) x D g ) = { g : A R g = q },w ) g,q : A R, w : A R ++, Dg) = { g = g ) N g D g ) }. Proposton 8. Suppose tht g s own-strtegy unmodl for ll N. Then, every gme n Dg) s best-response equvlent to g. Proof. Let g Dg). Snceg s own-strtegy unmodl, for ll λ A ), there exsts A such tht g, ) dλ ) 0 f nd > mn A, A 19) g, ) dλ ) 0 f nd < mx A. A dx.

23 282 S. Morrs, T. U / Gmes nd Economc Behvor ) Snce g, ) = w ) g, ), 19) s true f nd only f g, ) dλ ) 0 f nd > mn A, A g, ) dλ ) 0 f nd < mx A. A Thus, g s lso own-strtegy unmodl. Snce 19) s true f nd only f λ Λ,A g ) nd 20) s true f nd only f λ Λ,A g ),wemusthveλ,a g ) = Λ,A g ), whch completes the proof. Ths proposton hs useful pplcton concernng the unqueness of correlted equlbr. Neymn 1997) showed tht f g hs contnuously dfferentble nd strctly concve potentl functon, 9 then the potentl mxmzer s the unque correlted equlbrum of g. The set of correlted equlbr s the sme for two gmes f the two gmes re best-response equvlent. Thus, we clm the followng. Corollry 9. Suppose tht g hs contnuously dfferentble nd strctly concve potentl functon f. Then, the potentl mxmzer s the unque correlted equlbrum of every gme n Dg). Note tht gme n Dg) s not necessrly potentl gme nd pyoff functons re not necessrly concve. 20) 5. Mxed extensons of equvlence We hve focused on plyers preferences over pure strteges, gven nondegenerte conjectures bout ther opponents behvor. But we could sk the sme queston n the mxed strtegy extenson of the orgnl gme; equvlently, we could look t plyers preferences over mxed strteges. 10 The nturl queston s whether or not our dscusson so fr must be modfed by the mxed extenson of equvlence. For N,let A ) denote the set of ll mxed strteges of plyer. Abusng notton, we wrte g p, ) = A p )g, ) for p A ). By the mxed extenson of Λ, we cn nturlly defne Λ p,x g ) for p A ) nd X A ): Λ p,x g ) 9 The defnton of potentl functons of ths clss of gmes s the sme s those of fnte gmes. 10 The ssocte edtor suggested the observtons n ths secton.

24 { = λ A ) S. Morrs, T. U / Gmes nd Economc Behvor ) } )) λ ) g p, ) g p, 0forllp X. We use the sme rule Λ p,p g ) = Λ p, {p } g ) s before. We consder the followng equvlence reltons of gmes. Defnton 7. Agmeg s mxed better-response equvlent to g f, for ech N, Λ p,p g ) = Λ p,p ) g for ll p,p A ). Defnton 8. Agmeg s mxed best-response equvlent to g f, for ech N, ) Λ p, A ) g = Λ p, A ) g ) for ll p A ). Note tht VNM-equvlence s suffcent for both mxed better-response equvlence nd mxed best-response equvlence. Note lso tht mxed better-response equvlence s suffcent for better-response equvlence, nd tht mxed best-response equvlence s suffcent for best-response equvlence. It s esy to see tht mxed best-response equvlence s not only suffcent but lso necessry for best-response equvlence. Lemm 5. A gme g s mxed best-response equvlent to g f nd only f g s best-response equvlent to g. Proof. Note tht λ Λ p, A ) g ) ) p rg mx λ )g p, p A ) ) p )>0 mples rg mx λ )g, A p )>0 mples λ Λ,A g ). Thus, f Λ,A g ) = Λ,A g ) for ll A,thenΛ p, A ) g ) = Λ p, A ) g ) for ll p A ). Ths completes the proof. Ths lemm mples tht the chrcterzton of mxed best-response equvlence s reduced to tht of best-response equvlence. On the other hnd, mxed better-response equvlence s strctly stronger requrement thn better-response equvlence. Consder two plyer, three strtegy, symmetrc pyoff gmes g nd g, where ech plyer s pyoffs re gven by the followng pyoff mtrces

25 284 S. Morrs, T. U / Gmes nd Economc Behvor ) where the plyer s own strteges re represented by rows nd hs opponent s strteges re represented by columns): g g x 2x 2x y y y We ssume tht x,y > 0ndx/2 <y<2x. Then, 1 g 2, 2 g 3, 3 g 1, nd g = g. We lso hve xg 1, ) g 2, )) = g 1, ) g 2, ) nd yg 2, ) g 3, )) = g 2, ) g 3, ). Thus, by Proposton 1, g s better-response equvlent to g. However, we cn show tht g s mxed better-response equvlent to g only f x = y. To see ths, suppose tht row plyer beleves tht the column plyer never chooses 1: row plyer hs belef λ wth λ 1) = 0. Consder row plyer s mxed strtegy p such tht p 1) = p nd p 2) = 1 p. Ing, he prefers strtegy p to strtegy 3 f nd only f 2p 1,.e., p 1/2. In g, he prefers strtegy p to strtegy 3 f nd only f 2xp y,.e., p y/2x. In order for g to be mxed better-response equvlent to g,tmustbetruetht 1/2 = y/2x,.e., x = y. In ths cse, g s VNM-equvlent to g ndthusmxedbetterresponse equvlent to g. In the bove exmple, the relton g genertes connected grph snce 1 g 2 nd 2 g 3. Thus, under the connectedness of g, better-response equvlence does not necessrly mply VNM-equvlence, but mxed better-response equvlence my mply VNM-equvlence. The nturl queston s whether ths s true. Remember tht Proposton 2 provdes condton to ensure the equvlence of better-response equvlence nd VNM-equvlence. The condton ncludes the connectedness of g.but the connectedness s not suffcent s demonstrted by the bove exmple. In contrst, the followng proposton sserts tht the connectedness of g ensures the equvlence of mxed better-response equvlence nd VNM-equvlence. Proposton 10. Suppose tht gmes g nd g stsfy generc propertes G1 nd G2, nd tht, for ech N, g genertes connected grph on A.Theng s mxed betterresponse equvlent to g f nd only f g s VNM-equvlent to g. To prove the proposton, we use the followng lemm. Lemm 6. Suppose tht g nd g stsfy generc propertes G1 nd G2, nd tht g s mxed better-response equvlent to g. For dstnct,b,c A,f g b nd b g c,then there exsts w > 0 such tht g, ) g b, ) = w g, ) g b, ) ), g b, ) g c, ) = w g b, ) g c, ) ). Proof. By G1 nd Proposton 1, there exst w,b ), w b,c )>0 such tht

26 S. Morrs, T. U / Gmes nd Economc Behvor ) g, ) g b, ) = w,b ) g, ) g b, ) ), g b, ) g c, ) = w b,c ) g b, ) g c, ) ). We show tht w,b ) = w b,c ). Ether g c, g c,orc g s true. If g c, there exsts w,c )>0such tht g, ) g c, ) = w,c ) g, ) g c, ) ) by Proposton 1. Thus, by Lemm 3, w,b ) = w b,c ) = w,c ). Suppose tht g c. Note tht g = g by Proposton 1. Let λ A ) be such tht λ ) g b, ) g c, ) ) < 0, 21) whch exsts snce b g c nd G1. The relton g c mples tht λ ) g, ) g c, ) ) > 0. 22) By the weghted verge of 21) nd 22), we cn choose p A ) such tht p ) = p, p b ) = 1 p, nd λ ) g p, ) g c, ) ) = 0. 23) Mxed better-response equvlence mples tht λ ) g p, ) g c, ) ) = 0. 24) Now clculte g p, ) g c, ) = pg, ) + 1 p)g b, ) g c, ) = p g, ) g b, ) ) + g b, ) g c, ) = w,b )p g, ) g b, ) ) + w b,c ) g b, ) g c, ) ) = w,b ) pg, ) + 1 p)g b, ) g c, ) ) + w b,c ) w,b ) ) g b, ) g c, ) ) = w,b ) g p, ) g c, ) ) + w b,c ) w,b ) ) g b, ) g c, ) ). By the expecttons wth respect to λ for both sdes of the equton, nd by 23) nd 24), we hve w b,c ) w,b ) ) λ ) g b, ) g c, ) ) = 0.

27 286 S. Morrs, T. U / Gmes nd Economc Behvor ) By 21), we must hve w,b ) = w b,c ). Smlrly, f c g,wemusthve w,b ) = w b,c ). Ths completes the proof. We now report the proof of Proposton 10. Proof of Proposton 10. We show tht f g s mxed better-response equvlent to g then g s VNM-equvlent to g. By G1 nd Proposton 1, f g, there exsts w, )>0 such tht g, ) g, ) = w, ) g, ) g, )). If A =2, ths completes the proof by Lemm 1. Suppose tht A 3. For, A, let { k}m k=1 be sequence such tht 1 =, m =, k g k+1 for k = 1,...,m 1, whch exsts by the connectedness of g. There exsts x k > 0 such tht g k, ) g k+1, ) = x k g k, ) g k+1, )). By Lemm 6, x k = x k+1 for ll k m 1. By lettng x k = w, ),wehve m 1 g, ) g, ) = g k, ) g k+1, )) k=1 m 1 = k=1 x k g k, ) g k+1, )) = w, ) g, ) g, )). To summrze, for ll, A, there exsts w, )>0 stsfyng the bove equton. By Lemm 3, w, ) s the sme for ll, A. By Lemm 1, g s VNM-equvlent to g, whch completes the proof. Acknowledgments We re very grteful for vluble nput from Lrry Blume, George Mlth nd Phlp Reny. U cknowledges fnncl support by the Mnstry of Educton, Scence, Sports nd Culture, Grnt-n-Ad for Scentfc Reserch. References Anderson, S.P., Jcob, K., Holt, C.A., Mnmum-effort coordnton gmes: stochstc potentl nd logt equlbrum. Gmes Econ. Behv. 34, Blume, L., The sttstcl mechncs of strtegc ntercton. Gmes Econ. Behv. 5, Brock, W., Durluf, S., Dscrete choce wth socl nterctons. Rev. Econ. Stud. 68, Dubey, P., Hmnko, O., Zpechelnyuk, A., Strtegc substtutes nd potentl gmes. Mmeo. SUNY t Stony Brook. Hrrt-Urruty, J.-B., Lemréchl, C., Fundmentls of Convex Anlyss. Sprnger-Verlg, New York. Mskn, E., Trole, J., Mrkov perfect equlbrum. J. Econ. Theory 100,

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