What to Maximize if You Must


 Laureen Grace Bailey
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1 What to Maxmze f You Must Avad Hefetz Chrs Shannon Yoss Spegel Ths verson: July 2004 Abstract The assumpton that decson makers choose actons to maxmze ther preferences s a central tenet n economcs. Ths assumpton s often justfed ether formally or nformally by appealng to evolutonary arguments. In contrast, we show that n almost every game and for almost every famly of dstortons of a player s actual payoffs, some degree of ths dstorton s benefcal to the player because of the resultng effect on opponents play. Consequently, such dstortons wll not be drven out by any evolutonary process nvolvng payoffmonotonc selecton dynamcs, n whch agents wth hgher actual payoffs prolferate at the expense of less successful agents. In partcular, under any such selecton dynamcs, the populaton wll not converge to payoffmaxmzng behavor. We also show that payoffmaxmzng behavor need not preval even when preferences are mperfectly observed. We are grateful for valuable comments from Joerg Oechssler, Bob Anderson, Bll Zame, Edde Dekel, Youngse Km, Menachem Yaar, three referees and the assocate edtor, and partcpants of the 11th European Workshop n General Equlbrum Theory. The Economcs and Management Department, The Open Unversty of Israel, correspondng author: please drect correspondence to Department of Economcs, Unversty of Calforna, Berkeley, The Faculty of Management, Tel Avv Unversty, 1
2 1 Introducton The assumpton that decson makers choose actons to maxmze ther preferences s a central tenet n economcs. Ths assumpton s often justfed ether formally or nformally by appealng to evolutonary arguments. For example, n ther classc work, Alchan (1950) and Fredman (1953) argue that proft maxmzaton s a reasonable assumpton for characterzng outcomes n compettve markets because only frms behavng n a manner consstent wth proft maxmzaton wll survve n the long run. Under ths argument, frmsfalngtoactsoastomaxmzeprofts wll be drven out of the market by more proftable rvals, even f none of these frms delberately maxmzes profts or s even aware of ts cost or revenue functons. Smlar arguments that consumers behave as f maxmzng preferences due to myrad market forces that explot nonoptmal behavor are pervasve. More recently, Sandron (2000) gves such a justfcaton for ratonal expectatons equlbra, showng that a market populated by agents who ntally dffer n the accuracy of ther predctons wll nonetheless converge to a compettve ratonal expectatons equlbrum as those agents who make naccurate predctons are drven out of the market by those who are more accurate. In contrast, ths paper shows that n almost every strategc nteracton, payoff maxmzaton cannot be justfed by appealng to evolutonary arguments. Specfcally, we show that n almost every game and for almost every famly of dstortons of a player s actual payoffs, some degree of ths dstorton s benefcal to the player because of the resultng effect on opponents play. Consequently, we show that such dstortons wll not be drven out by any evolutonary process nvolvng payoffmonotonc selecton dynamcs, n whch agents wth hgher actual payoffs prolferate at the expense of less successful agents. In partcular, under any such selecton dynamcs, the populaton wll not converge to payoff maxmzng behavor. The dea that n strategc stuatons players may gan an advantage from havng an objectve functon dfferent from actual payoff maxmzaton dates back at least to Schellng (1960), and hs dscusson of the commtment value of decson rules. Related deas run through work rangng from Stackelberg s (1934) classc work on tmng n olgopoly to the theores of reputaton n Kreps and Wlson (1982), and Mlgrom and Roberts (1982). For smlar reasons, Frank (1987, 1988) argues that emotons may be a benefcal commtment devce. Recently, a large and growng lterature has emerged that formalzes some of these deas by explctly studyng the evoluton of preferences. Ths work shows that n strategc nteractons, a wde array of dstortons of actual payoffs, representng features such as altrusm, spte, overconfdence, farness, and recprocty, that bas ndvduals objectves away from actual payoff maxmzaton, may be evolutonarly stable. 1 1 For a bref overvew of ths lterature, see Samuelson (2001). Examples nclude Güth and Yaar (1992), Huck and Oechssler (1999), Fershtman and Wess (1997, 1998), Fershtman and Hefetz (2002), Rotemberg (1994), Bester and Güth (1998), Possajennkov (2000), Bolle (2000), Bergman and Bergman (2000), Koçkesen, Ok, and Seth (2000a, 2000b), Guttman (2000), Seth and Somanathan (2001), Kyle and Wang (1997), Benos (1998), Hefetz and Segev (2003), and Hefetz, Segev and Talley (2004). 2
3 Unlke most standard evolutonary game theory, n whch ndvduals are essentally treated as machnes programmed to play a specfc acton, the work on the evoluton of preferences treats ndvduals as decson makers who choose actons to maxmze ther preferences, and then studes how the dstrbuton of these preferences evolves over tme. Preferences that are dstortons of true payoffs or dspostons drve a wedge between an ndvdual s objectves and actual payoffs. Dspostons may nonetheless be evolutonarly stable because the resultng bas n a player s objectves may nduce favorable behavor n rvals that may more than compensate for the loss stemmng from departures from actual payoff maxmzaton. Thus the lterature on the evoluton of preferences llustrates the pont that n a varety of strategc nteractons, ndvduals who fal to maxmze ther true payoffs due to the bas created by varous dspostons may actually end up wth hgher payoffs than ndvduals who are unbased. Such benefcal dspostons would then not be weeded out by any selecton dynamcs n whch more successful behavor prolferates at the expense of less successful behavor, where success s measured n terms of actual payoffs. Much of the work on the evoluton of preferences, however, focuses on specfc knds of dspostons, such as altrusm or recprocty, and addresses these questons usng specfc functonal forms for both the ndvduals payoffs and dspostons. Such results then provde condtons on the parameters of the partcular model at hand that guarantee that some nonzero degree of ths dsposton wll survve evolutonary pressures. Our results generalze ths work n an mportant way by solatng the general prncple drvng these results and by showng that the evolutonary emergence of dspostons s n fact generc. Our genercty results are farly ntutve. Havng a dsposton affects a player s payoff n two ways: drectly, through the player s own actons, and ndrectly, by nfluencng other players actons. A crucal observaton s that a some small nonzero degree of dsposton leads to a slght devaton from payoffoptmzng behavor, and therefore has only a neglgble drect effect on the player s payoff. The crux of our argument s that for generc combnatons of games and dspostons, the ndrect effect on the player s payoff resultng from such a small degree of the dsposton s not neglgble. Interestngly, ths result also mples that, genercally, players can gan strategc advantage over opponents by hrng delegates whose preferences dffer from thers to play the game on ther behalf. Ths mples n turn that earler results obtaned n the strategc delegaton lterature n the context of specfc models (e.g. Green 1992; Fershtman and Judd, 1987; Fershtman, Judd and Kala 1991; Katz1991) are n fact generc. Central to our results are approprate parameterzatons of games and dspostons. Snce our analyss s based on frstorder condtons, we restrct attenton to purestrategy equlbra n games wth contnuous acton sets. Because we are nterested n the evolutonary vablty of payoff maxmzaton rather than the emergence of one partcular type of bas, such as altrusm or overconfdence, we consder a dsposton to encompass a famly of bases ndexed by a degree that can be postve, negatve, or zero. In ths 3
4 framework a zero degree means that the player s unbased and nterested n maxmzng hs actual payoff. The nterpretaton of a postve or negatve degree wll typcally depend on the specfcaton of the gven famly of dspostons; for example, the dsposton mght reflect otherregardng preferences, wth a postve degree correspondng to altrusm and a negatve degree correspondng to spte. For a generc set of payoff functons and dspostons, however, some nonzero degree of the dsposton has a postve ndrect effect. Ths guarantees that such dspostons wll not be elmnated from the populaton under any payoff monotonc selecton dynamcs. We frst prove ths result for a class of fntedmensonal manfolds of payoff and dsposton functons, and then generalze t to the nfntedmensonal famles of all payoff and dsposton functons. Our man results are derved under the assumpton that players preferences are perfectly observable. We then show that dspostons may reman evolutonarly vable even when the players preferences are only mperfectly observed. Here the natural soluton concept gven the mperfect observablty of preferences s Bayesan equlbrum. Ths hghlghts a techncal obstacle surroundng results about the evolutonary vablty of dspostons. Unlke Nash equlbra wth perfect observablty, Bayesan equlbra are typcally not locally unque (see, e.g., Lennger, Lnhart, and Radner, 1989). In such cases an equlbrum selecton s not welldefned even locally, and dfferent selectons from the equlbrum correspondence may result n contradctory conclusons regardng the effects of dspostons. Whle ths precludes a general analyss of mperfect observablty, n the context of an example wth a unque Bayesan equlbrum we show that the populaton does not converge to payoffmaxmzng behavor even f preferences are observed wth nose. The paper proceeds as follows. Secton 2 contans the development of our framework and our man results, showng genercally that dspostons do not become asymptotcally extnct under payoffmonotonc selecton dynamcs. We prove ths result both n the case where the payoff and dsposton functons vary over a partcular class of fntedmensonal sets, as well as for the case where they vary over the nfntedmensonal set of all payoff and dsposton functons. In Secton 3 we relax the assumpton that types are perfectly observed and assume nstead that they are observed wth nose. We show, by means of a specfc example, that our man results carry over to ths settng. All proofs are collected n the Appendx. 2 The genercty of dspostons 2.1 Payoffs and dspostons Two players, and j, engage n strategc nteracton. The strategy spaces of the two players, X and X j,areopensubsetsofr M and R N, respectvely, where, wthout loss 4
5 of generalty, M N. 2 Typcal strateges are denoted x =(x 1,...,x M) and x j = (x j 1,...,x j N ). The payoffs of the two players are gven by the C3 functons Π, Π j : X X j R. In what follows we denote the partal dervatves of Π by µ Π Π D Π =,..., Π and Π x 1 x j D j Π = M 2 Π x 1 xj 1 2 Π x M xj 1 2 Π x 1 xj N... 2 Π x M xj N. The partal dervatves of Π j and of other functons are denoted smlarly. In the course of ther strategc nteracton, the players perceve ther payoffs tobe where U (x,x j,τ) Π (x,x j )+B (x,x j,τ), (2.1) U j (x,x j,θ) Π j (x,x j )+B j (x,x j,θ), B : X X j E R B j : X X j E j R are the dspostons of players and j, and τ and θ are the players (onedmensonal) types,whcharedrawnfromdomanse,e j R each contanng a neghborhood of 0. The ntroducton of dspostons then drves a wedge between the objectves of the players, whch are to maxmze ther perceved payoffs U and U j, and ther eventual realzed payoffs Π and Π j.weassumethatb and B j are C 3. Moreover, as a normalzaton we assume that when τ or θ s zero, the players perceved payoffs concde wth ther actual payoffs: B (x,x j, 0) B j (x,x j, 0) 0. (2.2) That s, a type 0 player has no dsposton and smply chooses actons to maxmze hs actual payoff. 3 Our framework captures a wde range of stuatons. For nstance, the players mght be altrustc or spteful, and thus care not only about ther own payoffs but also about ther rval s payoffs. To model ths dea we can, as n Bester and Güth (1998) and Possajennkov (2000), wrte the players dspostons as B (x,x j,τ)=τπ j (x,x j ) and 2 The restrcton to two players s just for notatonal convenence; all of our results carry over drectly for games wth an arbtrary number of players. For games wth more players and more general strategy sets, see Remarks 2 and 3 below. 3 Notce that ths formulaton n terms of an addtve dsposton term s equvalent to specfyng nstead that a player has preferences gven by a utlty functon U (x,x j,τ) such that U (x,x j, 0) Π (x,x j ). To see ths, gven such a utlty functon smply set B (x,x j,τ) U (x,x j,τ) Π (x,x j ). 5
6 B j (x,x j,θ)=θπ (x,x j ).Whenτ and θ are postve, the players are altrustc as they attach postve weghts to ther rval s payoff, whlewhenτ and θ are negatve the players are spteful. Another example of ths framework s concern about socal status. Here suppose that M = N = 1 (the strateges of the two players are onedmensonal) and let Π and Π j represent the monetary payoffs of the two players. Then, as n Fershtman and Wess (1998), we can wrte the dspostons as B (x,x j,τ)=τσ(x x e ) and B j (x,x j,θ)= θσ(x j x e ), where σ s ether a postve or a negatve parameter and x e s the average acton n the populaton. Here the revealed preferences of the players are to maxmze the sum of ther monetary payoffs and ther socal status, where the latter s lnked to the gap between the players own actons and the average acton n the populaton. The players types, τ and θ, represent the weghts that the players attach to socal status. 2.2 The evoluton of dspostons Let Γ =(X,X j, Π, Π j,b,b j ) denote the game n whch players and j choose actons from X and X j, respectvely, to maxmze ther perceved payoffs, U (,τ) and U j (,θ), and obtan true payoffs Π and Π j. If for (τ,θ) thegamehasapurestrategynash equlbrum, let (y (τ,θ),y j (τ,θ)) denote such an equlbrum. 4 We assume for ths dscusson that the selecton (y (τ,θ),y j (τ,θ)) from the Nash equlbrum correspondence s contnuously dfferentable at (τ,θ)=(0, 0). 5 The true payoffs ofplayers and j n ths Nash equlbrum are f (τ,θ) Π y (τ,θ),y j (τ,θ) and f j (τ,θ) Π j y (τ,θ),y j (τ,θ). (2.3) Snce we cast our analyss n an evolutonary settng, these equlbrum payoffs, f and f j, wll represent ftness. Ths formulaton leads drectly to a natural selecton process among dfferent types n the populaton. To assess the evolutonary vablty of varous dspostons, we begn by askng whch dspostons are benefcal to a player. Snce we are nterested n characterzng whether havng no dsposton (.e., maxmzng true payoffs) can survve evolutonary pressures, we ntroduce the followng noton: Defnton 1 (Unlaterally benefcal dspostons) The dsposton B (B j )ssadtobe unlaterally benefcal for player (player j) nthegameγ f there exsts τ 6= 0(θ 6= 0) such that f (τ,0) >f (0, 0) (f j (0,θ) >f j (0, 0)). 4 Snce the strategy spaces X and X j are open, the equlbrum s nteror. For a dscusson of the ssues of exstence and nterorty of pure strategy equlbra, see Remarks 1 and 3. 5 We show n the Appendx that such a selecton s feasble for generc games. 6
7 It s mportant to note that ths defnton says that a dsposton s unlaterally benefcal for player f, gven that player j has no dsposton (.e., θ =0), there exsts some nonzero type of player whose ftness s hgher than the ftness of type 0. In partcular, the defnton does not requre ths property to hold for all types of player : a unlaterally benefcal dsposton mght be benefcal for some types of player but harmful for others. 6 To study how dspostons evolve, suppose that there are two large populatons of ndvduals, one for each player, and wth a contnuum of ndvduals of each type. At each pont t 0 n tme, ths par of populatons s characterzed by the par of dstrbutons (T t, Θ t ) (E ) (E j ) of (τ,θ), where (E ) and (E j ) denote the set of Borel probablty dstrbutons over E and E j. WeassumethatT 0 has full support over E and Θ 0 has full support over E j. At each nstance n tme, an ndvdual n one populaton s randomly matched wth an ndvdual of the other populaton to play the game Γ. The average ftness levels of the ndvduals of types τ and θ at tme t are gven by Z Z f (τ,θ)dθ t and f j (τ,θ)dt t. (2.4) We assume that the selecton dynamcs are monotoncally ncreasng n average ftness. That s, we assume that the dstrbutons of types evolve as follows: d dt T t(a )= R g (τ,θ A t )dt t, d Θ dt t(a j )= R g j (T A j t,θ)dθ t, A R Borel measurable, A j R Borel measurable, (2.5) where g and g j are contnuous growthrate functons that satsfy Z Z g (τ,θ t ) > g ( τ,θ t ) f (τ,θ)dθ t > f ( τ,θ)dθ t, (2.6) Z Z g j (T t,θ) > g j (T t, θ) f j (τ,θ)dt t > f j (τ, θ)dt t. To ensure that T t and Θ t reman probablty measures for each t, we also assume that g and g j satsfy Z Z g (τ,θ t )dt t =0, and g j (T t,θ)dθ t =0 for each t. (2.7) Equatons (2.5)(2.7) reflect the dea that the proporton of more successful types n the populaton ncreases from one nstance or perod to another at the expense of less successful types. Ths may be due to the fact that more successful ndvduals have more 6 Consder for nstance the altrusm/spte example mentoned above. Suppose that f τ (0, 0) 6= 0. Then f a small degree of altrusm (τ >0) sbenefcal, a small degree of spte (τ <0) would be harmful and vce versa. 7
8 descendants, who then nhert ther parents preferences ether genetcally or by educaton. An alternatve explanaton s that the decson rules of more successful ndvduals are mtated more often. The same mathematcal formulaton s also compatble wth the assumpton that successful types translate nto stronger nfluence rather than numercal prolferaton. Under ths nterpretaton, not all ndvduals are matched to play n each nstance of tme, and more successful ndvduals take part n a larger share of the economc nteractons, and so are matched to play wth a hgher probablty. To guarantee that the system of dfferental equatons (2.5) has a welldefned soluton, we requre some addtonal regularty condtons on the selecton dynamcs as follows. Defnton 2 (Regular dynamcs) Payoffmonotonc selecton dynamcs are called regular f g and g j canbeextendedtothedomanr Y, where Y s the set of sgned Borel measures wth varatonal norm smaller than 2, and on ths extended doman, g and g j are unformly bounded and unformly Lpschtz contnuous. That s, sup g (τ,θ t ) < M, sup g (τ,θ t ) g (τ, e Θ t ) <K Θt e Θ t, Θt, Θ f t Y, τ E τ R sup gj (T t,θ) < M j, sup g j (T t,θ) g j ( e T t,θ) <K j Tt e T t, Tt, T e t Y, θ E j θ R for some constants M,M j,k,k j > 0, where kµk = sup RR hdµ s the varatonal h 1 norm of the sgned measure µ. Oechssler and Redel (2001, Lemma 3) show that regularty of the dynamcs guarantees that the map (T t, Θ t ) 7 R R g (τ,θ t )dt t, g j (T t,θ)dθ t s bounded and Lpschtz contnuous n the varatonal norm, whch mples that for any ntal dstrbutons (T 0, Θ 0 ), the dfferental equaton (2.5) has a unque soluton. 7 To characterze the asymptotc propertes of the dstrbutons (T t, Θ t ) we wll use the followng noton. Defnton 3 (Asymptotc extncton) The dspostons (B,B j ) become asymptotcally extnct n the game Γ f (T t, Θ t ) converges weakly to a unt mass at (τ,θ)=(0, 0) as t. 7 In addton, the boundedness of g and g j guarantees that any set havng postve probablty under the ntal dstrbutons T 0 or Θ 0, wll have postve probablty under T t or Θ t for all t. In partcular, snceweassumedthatt 0 and Θ 0 have full support on the domans E and E j,sodot t and Θ t for all t. 8
9 Theorems 1 and 2 below show that genercally dspostons do not become asymptotcally extnct under any regular payoffmonotonc selecton dynamcs. Theorem 1 apples to fntedmensonal manfolds of payoff and dsposton functons. Here we allow payoff and dsposton functons to vary over an arbtrary fntedmensonal manfold provded t contans a suffcently rch class of functons. We use these fntedmensonal results to show n Theorem 2 that the same result holds when varyng over the entre nfntedmensonal famles of all thrce contnuously dfferentable payoff and dsposton functons. 2.3 Fntedmensonal manfolds Let G denote the space of all pars of C 3 payoff functons (Π, Π j ),andlet B denote the space of all pars of C 3 dsposton functons (B,B j ). We endow G and B wth the Whtney C 3 topology, and G B wth the natural product topology. 8 In what follows, we wll often make use of a partcular class of payoff functons correspondng to games n whch each pure strategy equlbrum s locally unque. We wll slghtly abuse termnology by referrng to a par of payoff functons (Π, Π j ) as a game (the strategy spaces X,X j reman fxed throughout). Defnton 4 (Regular games) A game s called regular f at each of ts Nash equlbra (y,y j ),the(m + N) (M + N) matrx µ Π (y,y j ) Π j(y,y j ) Π j j(y,y j ) Π j jj(y,y j ) has full rank. We start by consderng a fntedmensonal, boundaryless submanfold G of G that s rch enough to allow us to perturb each payoff functon n each of the drectons x m,x j n and x mx j m ndependently and obtan a new par of payoff functons n G. To formalze ths dea, let p = p 1,p 2,p 3 = p 1 1 M,...,p1, p 2 1,...,pN 2, p 3 1,...,pM 3 R M+N+M, q = q 1,q 2,q 3 = q1,...,qm 1 1, q 2 1,...,qN 2, q 3 1,...,qM 3 R M+N+M. 8 Roughly, the Whtney C k topology s the topology n whch two C k functons are close f ther values, and the values of all of ther dervatves of orders up to and ncludng k, are unformly close. For a formal descrpton and dscusson, see e.g. Golubtsky and Gullemn (1973). Ths s the approprate topology for our problem because t guarantees that all of the maps we work wth, such as the frst order condtons for Nash equlbra, are contnuous as we vary the payoff and dsposton functons. 9
10 Gven a par of payoff functons (Π, Π j ),defne Π (x,x j,p) Π x,x j + Π j (x,x j,q) Π j x,x j + MX p 1 mx m + m=1 NX p 2 nx j n + n=1 MX p 3 mx mx j m, (2.8) m=1 MX N qm 1 x m + X M qn 2 xj n + X qm 3 x m xj m. m=1 Usng ths notaton, we assume that the manfold G s such that for every par of payoff functons (Π, Π j ) G there exst open neghborhoods P, Q R M+N+M of zero such that ( Π (,,p), Π j (,,q)) G for every (p, q) P Q. Smlarly, let v =(v 1,...,v M ) R M and w =(w 1,...,w N ) R N. Gven a par of dspostons (B,B j ),defne n=1 B (x,x j,τ,v) B x,x j,τ + τ B j (x,x j,θ,w) B j x,x j,θ + θ m=1 MX v m x m, (2.9) m=1 NX w n x j n. We consder a fntedmensonal submanfold B of B such that for every (B,B j ) B, there exst neghborhoods V R M,W R N of zero such that for every (v, w) V W, ( B (,,,v), B j (,,,w)) B. Whle ths framework and the resultng theorem allow for general combnatons of sets of payoff functons G and sets of dspostons B, notcethatwecouldrestrctattenton to manfolds G and B such that for each (Π, Π j ) G and for each (B,B j ) B, the resultng game Γ has pure strategy Nash equlbra for all type profles (τ,θ) n some neghborhood of (0, 0) (see also Remark 1 below). 9 In ths fntedmensonal settng, the natural noton of genercty s as follows. Defnton 5 (Genercty)Apropertyssadtoholdforgenerccombnatonsofpars of payoff functons n G and dspostons n B f there s an open, fullmeasure subset A of the product manfold G Bsuch that the property obtans for all (Π, Π j,b,b j ) A. We can now state the frst verson of our man result. n=1 Theorem 1 For generc combnatons of pars of payoff functons (Π, Π j ) G and dspostons (B,B j ) B: 9 Because the set of regular games havng pure strategy equlbra s open, such combnatons of sets of payoff functons and sets of dspostons exst. 10
11 () The dsposton B s unlaterally benefcal for player and the dsposton B j s unlaterally benefcal for player j. () The dspostons (B,B j ) do not asymptotcally become extnct under any regular payoffmonotonc selecton dynamcs. The basc dea behnd ths result can be summarzed as follows. Suppose that both players do not have dspostons, so that τ = θ =0. The resultng Nash equlbrum of the game Γ s therefore (y (0, 0),y j (0, 0)). Introducng a slght dsposton for player would then change the player s ftness at the rate fτ (0, 0) = Π y (0, 0),y j (0, 0) yτ (0, 0) + Π j y (0, 0),y j (0, 0) yτ j (0, 0). (2.10) The frst term s the drect effect on s equlbrum payoff due to the change n s own behavor. The second term s the ndrect effect caused by the change n j s equlbrum behavor. For generc pars of payoffs and dspostons, yτ(0, 0) and yτ(0, j 0) are welldefned. As (y (0, 0),y j (0, 0)) s an nteror Nash equlbrum of Γ, t follows that Π (y (0, 0),y j (0, 0)) = 0. (2.11) Therefore the frst, drect effect vanshes. The essence of the proof s then to show that for generc combnatons of payoff and dsposton functons, a perturbaton n s dsposton ensures that the second, ndrect effect does not vansh. That s, fτ (0, 0) = Π j y (0, 0),y j (0, 0) yτ j (0, 0) 6= 0. (2.12) Ths mples n turn that payoffmonotonc selecton dynamcs cannot converge to a unt mass at (τ,θ)=(0, 0). If nstead the dstrbuton of player j s type were to become concentrated around θ =0, the fact that f τ(0, 0) 6= 0means that some small nonzero value of τ (postve or negatve, dependng on the sgn of f τ(0, 0)) ncreasestheftness of player. Ths n turn mples that a nonzero type of player would fare better than atypezeroplayer, and would therefore ncrease n number at the expense of the type zero player. Thus n the lmt the dspostons wll not become extnct. 10 Several remarks about Theorem 1 are now n order. Remark 1: Theorem 1 s stated for general fntedmensonal manfolds of games and dspostons, whch may nclude games that do not have pure strategy equlbra. Notce that n ths case propertes () and () hold vacuously. As we dscussed above, the theorem nstead could be stated for collectons of games and dspostons for whch selectons of 10 For symmetrc games, Güth and Peleg (2001) dentfedtheanalogueof(2.12) as a necessary condton for evolutonary stablty (n contrast wth the fully dynamc analyss of the current paper). However, Güth and Peleg dd not nvestgate the genercty of ths condton. 11
12 pure strategy equlbra exst n a neghborhood of (0,0). We state the result as above for ease of use n extendng the result to the general class of games, where the ssues nvolved n restrctng attenton to games wth pure strategy equlbra are slghtly more complcated. We dscuss ths n more detal below. Remark 2: Theorem 1 can be easly generalzed to games wth fntely many players. In that case, the proof of the theorem apples verbatm wth the ndex j beng nterpreted as the vector of all players but, and wth N beng the dmenson of the product of the strategy spaces of all players but. Remark 3: The proof of Theorem 1 reles on the frstorder necessary condtons that obtan at nteror Nash equlbra of Γ. If we allow the strategy spaces of the players, X and X j, to be closed subsets of R M and R N, then some Nash equlbra may be on the boundary. In such a case, the analyss carres over when restrctng attenton to the set of drectons for whch the frstorder condtons do hold at equlbrum. 11 No frstorder condtons need to hold at Nash equlbrum strateges that are extreme ponts n the strategy sets X and X j, however. Ths wll be the case for nstance for pure strategy Nash equlbra when X and X j are smplces of mxed strateges. Such extreme equlbra are not perturbed when the game s perturbed wth a slght dsposton, so the margnal analyss n the proof does not apply n ths case. In such games, types wth small dspostons may have the same ftness as zero types wth no dsposton. Our genercty analyss s also napproprate for pure strategy Nash equlbra n games wth fntely many pure strateges. For such games a global analyss rather than a margnal one s approprate for characterzng equlbra. Nonetheless, smlar results may hold n some such games. For example, n symmetrc games wth fntely many pure strateges, Dekel et al. (1998) show that for any symmetrc Nash equlbrum dfferent from the payoffmaxmzng symmetrc outcome (as, for example, n the prsoners dlemma), the lack of dspostons s not evolutonarly vable. Remark 4: A smlar result holds when the strategy spaces X and X j are nfntedmensonal. Unfortunately, n the most obvous examples of such games, such as nfntely repeated games or games wth ncomplete nformaton, Nash equlbra are typcally not locally unque. For nfntely repeated games ths follows from the Folk Theorem, whle ncomplete nformaton games typcally have a contnuum of BayesanNash equlbra (see e.g., Lennger, Lnhart, and Radner, 1989). In such cases, an equlbrum selecton s not welldefned even locally, so when small dspostons are ntroduced t s unclear whch equlbrum to consder. Dfferent selectons from the equlbrum correspondence may result n contradctory conclusons regardng the effects of the dspos 11 Dubey (1986) and Anderson and Zame (2001) employ a smlar approach to demonstrate the generc Paretoneffcency of nonvertex Nash equlbra. 12
13 tons. 12 We wsh to emphasze however that ths problem arses not from any nherent lmtaton of the argument tself; rather, the evolutonary analyss ceases to be predctve because the equlbrum s not locally unque. Remark 5: Theorem 1 has an nterestng mplcaton for the strategc delegaton lterature. Ths lterature has demonstrated that players can gan strategc advantage over rvals by hrng a delegate whose preferences dffer from thers to play the game on ther behalf (e.g. Green 1992; Fershtman and Judd, 1987; Fershtman, Judd and Kala 1991; Katz 1991). Vewng the perceved payoff functon of player as representng the preferences of a delegate hred by player to play the game on player s behalf, part () of Theorem 1 mples that earler results obtaned n the strategc delegaton lterature are n fact generc. That s, n almost every strategc nteracton hrng a delegate whose preferences dffer from the player s own preferences s benefcal to the player because of ts resultng effect on opponents play. 2.4 All games and dspostons The genercty result establshed n the prevous subsecton mght appear to be somewhat lmted n scope because of ts restrcton to certan fntedmensonal submanfolds G and B. Next we show that an analogous result holds when we vary over the nfntedmensonal sets of all possble pars of payoff functons and dspostons. To extend our genercty results to the space of all payoff and dstrbuton functons, we wll need a noton of genercty that s sutable n an nfntedmensonal settng. Unfortunately, there s no natural analogue of Lebesgue measure n an nfntedmensonal space, and standard topologcal notons of almost all such as open and dense or resdual are not entrely satsfactory, partcularly n problems lke ours n whch almost all s loosely nterpreted n a probablstc sense as a statement about the lkelhood of partcular events. For example, open and dense sets n R n can have arbtrarly small measure, and resdual sets can have measure 0. Nevertheless, Chrstensen (1974) and Hunt, Sauer, and Yorke (1992) have developed measuretheoretc analogues of Lebesgue measure 0 and full Lebesgue measure for nfntedmensonal spaces, called shyness and prevalence. Defnton 6 (Shyness and prevalence) Let Y be a topologcal vector space. A unversally measurable subset E Y s shy f there s a regular Borel probablty measure µ on Y wth compact support such that µ(e +y) =0for every y Y. 13 A (not necessarly unversally measurable) subset F Y s shy f t s contaned n a shy unversally measurable set. AsubsetE Y s prevalent f ts complement Y \ E s shy. 12 In specfc cases, however, there may be more natural canddates for such selectons; see for example the analyss n Secton 3 below and n Hefetz and Segev (2003). 13 AsetE Y s unversally measurable f for every Borel measure η on Y, E belongs to the completon wth respect to η of the sgma algebra of Borel sets. 13
14 If A Y s open, then a set E A s relatvely shy n A f E sshy,andaset F A s relatvely prevalent n A f A \ Y s relatvely shy n A. Chrstensen (1974) and Hunt, Sauer and Yorke (1992) show that shyness and prevalence have the propertes we ought to requre of measuretheoretc notons of smallness and largeness. In partcular, the countable unon of shy sets s shy, no relatvely open subset s shy, prevalent sets are dense, and a subset of R n s shy n R n fandonlyfthas Lebesgue measure 0. It s straghtforward to show that the correspondng propertes hold for relatvely shy and relatvely prevalent subsets of an open set as well. Hunt, Sauer, and Yorke (1992) also provde smple suffcent condtons for ther notons of shyness and prevalence (here we adopt the somewhat more descrptve termnology from Anderson and Zame 2001). 14 Defnton 7 (Fnte shyness and fnte prevalence) Let Y be a topologcal vector space. A unversally measurable set E Y s fntely shy f there s a fnte dmensonal subspace V Y such that (E y) V has Lebesgue measure 0 n V for every y Y.Aunversally measurable set E Y s fntely prevalent f ts complement Y \ E s fntely shy. Sets that are fntely shy are shy, hence sets that are fntely prevalent are prevalent. Usng ths fact together wth the results we establshed for fntedmensonal submanfolds wll yeld a general verson of our results when payoffs and dspostons vary over theentrenfntedmensonal spaces G and B. We can now state a second verson of our man result. Theorem 2 There exsts an open, prevalent subset P of G B such that for each (Π, Π j,b,b j ) P, () The dsposton B s unlaterally benefcal for player and the dsposton B j s unlaterally benefcal for player j. () The dspostons (B,B j ) do not asymptotcally become extnct under any regular payoffmonotonc dynamcs. In partcular, let Rp G be the set of regular games wth pure strategy equlbra. Then R p B contans an open, relatvely prevalent subset satsfyng () and (). 14 Anderson and Zame (2001) have extended the work of Hunt, Sauer and Yorke (1992) and Chrstensen (1974) by defnng prevalence and shyness relatve to a convex subset that may be a shy subset of the ambent space. Ther extenson s useful n many applcatons, partcularly n economcs, n whch the relevant parameters are drawn not from the whole space but from some subset, such as a convex cone or an order nterval, that may tself be a shy subset of the ambent space. Here we use the orgnal noton as formulated n Hunt, Sauer and Yorke (1992). 14
15 As wth Theorem 1, here too we could gve other versons of ths result restrcted to games wth pure strategy Nash equlbra. Ths becomes somewhat more delcate, however, due to the fact that the subset of G B for whch each game has pure strategy equlbra s not necessarly open, nor necessarly convex. The dffculty les n extendng the noton of prevalence to a relatve one. Anderson and Zame (2001) provde one such extenson, but, crucally, they requre the doman to be convex. To restrct to games wth pure strategy equlbra, we have taken the smplest approach by consderng the subset R p G of regular games wth pure strategy equlbra, whch s open. Then t follows mmedately that R p B contans an open, relatvely prevalent subset satsfyng () and () above. Alternatvely, gven any convex subset C G p, one can show that there exsts a relatvely prevalent subset of C B satsfyng () and (). Justfyng a restrcton to a convex set of games wth pure strategy equlbra seems dffcult, however. 3 Nosy observablty of dspostons Thus far, we have assumed that players and j play a Nash equlbrum gven ther perceved payoff functons. One justfcaton for ths assumpton s that players perceved payoffs are perfectly observed. Of course, by standard arguments, Nash equlbrum play does not necessarly requre observablty of payoffs. If the nteracton lasts several rounds, n mportant classes of games play can converge to a Nash equlbrum even f players have very lmted knowledge or adapt ther behavor myopcally, for nstance by followng some verson of fcttous play (see e.g. Fudenberg and Levne, 1998). In ths secton, we pursue further the possblty that preferences may not be perfectly observed. Specfcally, we assume that players observe each other s preferences wth some randomly dstrbuted nose. The natural soluton concept for ths settng s Bayesan equlbrum. Unfortunately, as we dscussed n the ntroducton, Bayesan equlbra are typcally not locally unque; consequently, t s mpossble to generalze Theorems 1 and 2 to ths settng. Nonetheless, usng a specfc example that gves rse to a unque Bayesan equlbrum for any gven dstrbuton (T, Θ) of types, we show that n the absence of ths techncal obstacle, the evolutonary vablty of dspostons s mantaned. Qualtatvely smlar results would obtan for any other example that admts a unque Bayesan equlbrum at least n some weak neghborhood of the unt mass at (τ,θ)= (0, 0). Suppose that the strategy spaces of the players are X = X j = R, andtheactual payoff functons are Π (x,x j )=(α bx j x )x, Π j (x,x j )=(α bx x j )x j, (3.1) where α>0, andb ( 1, 1). Moreover, suppose that the dspostons of the players are gven by: B (x,x j,τ)=τx, B j (x,x j,θ)=θx j, τ,θ R. (3.2) 15
16 Usng these payoff and dsposton functons, the perceved payoff functons are gven by U (x,x j,τ)=π (x,x j )+B (x,x j,τ)=(α + τ bx j x )x, U j (x,x j,θ)=π j (x,x j )+B (x,x j,θ)=(α + θ bx x j )x j (3.3). From (3.3) t s clear that the dspostons can be nterpreted as selfesteem bases reflectng over and underconfdence. Here the players ether overestmate the return to ther own actons, f τ and θ are postve, or underestmate these returns, f τ and θ are negatve. Ths example can be used to llustrate our more general results. Here, f perceved payoff functons are completely observable, then any regular payoff monotonc dynamcs results n a dstrbuton of types that converges to a unt mass at a type that s postve as long as b 6= 0, that s, as long as the game s one wth nontrval strategc nteracton. We prove ths, along wth some more general results, n Hefetz, Shannon and Spegel (2004). To extend these results to a settng wth partal observablty, we assume that the observaton of opponents perceved payoffs s subject to some randomly dstrbuted nose. Specfcally, we assume that before choosng actons players and j receve the followng sgnals about each other s types: s = τ +ν, s j = θ + ν, (3.4) where ν s a random varable dstrbuted on the support [ r, r] accordngtoacumulatve dstrbuton functon N wth a postve densty. The assumpton that the support of ν s symmetrc around 0 s not essental; however, the assumpton that the support s bounded s mportant as t makes t possble for players to dstngush between zero and nonzero types. 15 Gven the sgnals, s and s j, the players update ther belefs about each other s preferences, and then play a Bayesan equlbrum gven these updated belefs. In ths settng we now prove the followng result: Proposton 1. Suppose that the players have the perceved payoff functons specfed n (3.3) and they receve the sgnals s and s j specfed n (3.4), and moreover, the ntal dstrbutons of both τ and θ have full support. Then the dspostons do not asymptotcally become extnct under any regular payoffmonotonc selecton dynamcs. In the workng paper verson (Hefetz, Shannon and Spegel, 2003) we also establsh some postve convergence results for ths game wth nosy observablty. When the supports of the ntal bas dstrbutons T 0, Θ 0 are confned to some large enough compact 15 In dfferent but related models, Dekel et al. (1998), Ely and Ylankaya (2001), Ok and Vega Redondo (2001) and Güth and Peleg (2001) show that payoffmaxmzaton s evolutonarly stable f preferences are completely unobservable. In our settng, ths would correspond to the lmt case n whch the nose s dstrbuted wth an mproper unform pror on the entre real lne. 16
17 nterval, then under any regular payoffmonotonc dynamcs the dstrbutons T t, Θ t converge weakly to a unt mass at τ = θ = b2 α. In partcular, as n the case of full 4+2b b 2 observablty, ths value s nonzero as long as b 6= 0, thus as long as there s nontrval strategc nteracton. Smlar results hold f the preferences U,U j are unobserved n some fracton ρ of the nteractons (n whch case the correspondng Bayesan equlbrum s played). Fnally, smlar results obtan n a verson of ths model ncorporatng costly sgnalng of types. Here player j observes a sgnal m of player s type τ, where player ncurs ftness cost c (m τ) 2 whchsconvexnthedstancebetweenthesgnalm and thetruetypeτ; and analogously for player j. Now the dstrbutons of typesgnal pars (τ,m ) and (θ, m j ) evolve accordng to some regular payoffmonotonc dynamcs. Then these dstrbutons converge to a unt mass at values that are nonzero as long as b 6= 0. For detals and more dscusson of all of these results, see Hefetz, Shannon and Spegel (2004). 4 Concluson The lterature on the evoluton of preferences, whle successful n provdng foundatons for varous types of dspostons and bases, s often crtczed on two mportant grounds (see e.g., Samuelson, 2001). Frst, specfc results typcally consder preferences and dspostons that are carefully talored to the partcular game of nterest, whch rases the queston of how robust such specfc examples are and whether they extend to more general types of preferences and dspostons. Second, most of the exstng work modelng the evoluton of preferences assumes that preferences are perfectly observed, whle t s unclear whether ths assumpton s reasonable or whether the results obtaned stll hold f ths assumpton s relaxed. Our work addresses both of these questons. Under the assumpton that preferences are observable, we show that n almost every game and for almost every type of dstorton of a player s actual payoffs, some postve or negatve extent of ths dstorton s benefcal to the player because of the resultng effect on opponents play. Hence, any standard evolutonary process n whch selecton dynamcs are monotone n payoffs wll not elmnate such dstortons; n partcular, under any such selecton dynamcs, the populaton wll not converge to payoff maxmzng behavor. Ths mples n turn that the evolutonary vablty of dspostons s generc, and ndependent of the partcular parametrc models employed n most of the lterature. We also show that the vablty of dspostons may be robust to nosy observablty of preferences. Although the lack of local unqueness of Bayesan equlbra n models n whch preferences are observed wth nose precludes a general extenson of our results, when the Bayesan equlbrum s unque, dspostons reman evolutonarly vable n such settngs n the sense that the populaton stll does not converge to payoff maxmzng behavor. 17
18 5 Appendx In order to prove Theorems 1 and2weproceedwthasequenceoflemmata. Wemake repeated use of the followng standard defnton and theorem, whch we nclude here for completeness. 16 Defnton 8 (regular value) Let X and S be boundaryless, C r manfolds, and G : X S R K be a C r functon, where r 1. An element y R K s a regular value of G f for all (x, s) such that G(x, s) =y, the dervatve D x,s G(x, s) has rank K. In partcular, notce that f there are no ponts (x, s) such that G(x, s) =y, theny s trvally a regular value of G. Remark 6: In the arguments below we wll frequently need to show that zero s a regular value of varous maps. To ths end we wll rely on two useful observatons. Frst, we wll repeatedly use the assumpton that these manfolds contan an open set around each pont consstng of a partcular type of perturbaton. More precsely, fx (Π, Π j ) G and recall that we assume that there exst open neghborhoods P, Q R M+N+M of zero such that ( Π (,,p), Π j (,,q)) G for each (p, q) P Q, where Π and Π j are gven n (2.8). Now let h : X X j G R K be an arbtrary C 1 functon. Then zero s aregularvalueofh provded Dh(x,x j, Π, Π j ) has rank K (.e., s surjectve) for each (x,x j, Π, Π j ) h 1 (0). Gven our assumptons about G, to show that Dh(x,x j, Π, Π j ) has rank K t then suffces to show that has rank K. Second, f the dervatve D p,q h(x,x j, Π (x,x j, 0), Π j (x,x j, 0)) D,j h(x,x j, Π, Π j ) does not have rank K for any (x,x j ) X X j, then zero can be a regular value of h(,, Π, Π j ) only f h(x,x j, Π, Π j ) 6= 0for all (x,x j ) X X j. Theorem 3 (The transversalty theorem). Let X and S be fntedmensonal, boundaryless, C r manfolds and G : X S R K be a C r functon, where r>max {0, dm X K}. For each s S let G(,s) be the restrcton of G to X {s}. If y R K s a regular value of G, then for almost every s S, y s a regular value of G(,s). In addton, f s 7 G(,s) s contnuous n the Whtney C r topology, then {s S : s s a regular value of G(,s)} s open. 16 For example, see Hrsch (1976). 18
19 The frst step n our argument s to show that equlbra are locally unque n almost all games. Ths follows from the genercty of regular games, establshed n Lemma 1, and the local unqueness of equlbra n regular games, establshed n Lemma 2. Lemma 1 The set of regular games R s an open, fullmeasure subset of G. Proof. Fx a game (Π, Π j ) G. Snce the strategy spaces X,X j are open, Nash equlbra of the game are nteror. Thus, at each Nash equlbrum (y,y j ) of the game, the followng system of M + N frst order condtons holds: µ Π (y,y j ) Π j j(y,y j =0. ) Defne the map φ : X X j G R M+N by µ Π φ(,, Π, Π j )= (, ) Π j j (, ) Consder the dervatve D p 1,q 2φ(y,y j, Π (,, 0), Π j (,, 0)) =. µ IM 0, 0 I N where I M and I N are the M M and N N dentty matrces. Snce the matrx has rank M + N for each (y,y j ), t follows from Remark 6 that zero s a regular value of φ. Therefore, the transversalty theorem mples that there s a set of full measure R G such that zero s a regular value of φ(,, Π, Π j ) for each game (Π, Π j ) R. For each (Π, Π j ) R, the defnton of regular value and the fact that zero s a regular value of φ(,, Π, Π j ) mples that the dervatve µ Π D,j φ(y,y j, Π, Π j )= (y,y j ) Π j(y,y j ) Π j j (y,y j ) Π j jj (y,y j ) has full rank M + N at each Nash equlbrum (y,y j ) of (Π, Π j ). Thus, usng the defnton of a regular game, a game (Π, Π j ) G s regular f and only f 0 s a regular value of φ(,, Π, Π j ),thats,r = R. ThusR has full measure. Fnally, snce the map (Π, Π j ) 7 φ(,, Π, Π j ) s contnuous n the Whtney C 1 topology, R s open by the transversalty theorem. The next lemma shows that n a regular game, the Nash equlbrum correspondence s locally snglevalued n a neghborhood of zero. Ths feature allows us to study the effects of small dspostons on the true equlbrum payoffs nawelldefned manner. 19
20 Lemma 2 Consder a regular game (Π, Π j ) and let (y,y j ) be a Nash equlbrum of the game. For any par of dspostons (B,B j ) B, there s a neghborhood V 0 of τ =0 and a unque C 1 functon Z( ) (y (, 0),y j (, 0)) : V 0 X X j, such that (y (0, 0),y j (0, 0)) = (y,y j ) and (y (τ,0),y j (τ,0)) sanashequlbrumofthe game (Π + B, Π j ) when τ V o. Moreover, µ µ µ Π (y,y j ) Π j(y,y j ) y τ (0, 0) B Π j j(y,y j ) Π j jj(y,y j ) yτ(0, j = τ (y,y j, 0). (A.1) 0) 0 Proof. Suppose that θ =0(player j has no dsposton), so that B j (,, 0) 0. Then a Nash equlbrum (y (τ,0),y j (τ,0)) of the game (Π + B, Π j ) satsfes the followng system of M + N frst order condtons µ Π (y,y j )+B(y,y j,τ) Π j j (y,y j =0. (A.2) ) Snce B (,, 0) 0, B(y,y j, 0) 0, henceatτ =0ths system becomes µ Π (y,y j ) Π j j (y,y j =0. ) Snce the game (Π, Π j ) s regular, zero s a regular value of the map µ Π (, ) Π j : R M+N R M+N. j(, ) The mplct functon theorem then mples that the Nash equlbrum map Z( ) (y (, 0),y j (, 0)) s locally defned and C 1 n a neghborhood V 0 of τ =0. Fnally, snce B (,, 0) 0, B (y,y j, 0) = B j(y,y j, 0) 0. Then(A.1) follows by dfferentatng (A.2) wth respect to τ and evaluatng at τ =0. Now let U = G Bbethemanfoldofpercevedpayoff functons, so U = U,U j =(Π + B, Π j + B j ):X X j R R 2 (Π, Π j ) G, (B,B j ) B ª. (A.3) Snce, B (x,x j, 0) B j (x,x j, 0) 0, the projecton Pr G : U G maps (U,U j ) to the correspondng game Pr G (U,U j ) U (,, 0),U j (,, 0), whle the projecton Pr B : U B maps (U,U j ) to the correspondng dspostons Pr B (U,U j ) U U (,, 0),U j U j (,, 0). By Lemma 1, thesetu R R B s an open, fullmeasure subset of U. 20
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