Selfconfirming Equilibrium and Model Uncertainty

Transcription

1 Selfconfrmng Equlbrum and Model Uncertanty P. Battgall S. Cerrea-Voglo F. Maccheron M. Marnacc IGIER and Department of Decson Scences Unverstà Boccon June 20, 2014 Part of ths research was done whle the frst author was vstng the Stern School of Busness of New York Unversty, whch he thanks for ts hosptalty. We thank Larry Samuelson (the coedtor) and two anonymous referees for suggestons that lead to a sgnfcant mprovement of the paper. We also thank Veronca Cappell, Ncodemo De Vto, Ignaco Esponda, Eduardo Fengold, Alejandro Francetch, Faruk Gul, Johannes Hörner, Yuchro Kamada, Margaret Meyer, Sujoy Mukerj, Wolfgang Pesendorfer, and Bruno Strulovc for useful dscussons, as well as semnar audences at CISEI Capr, D-TEA 2012, Games 2012, RUD 2012, as well as Chcago, Duke, Georgetown, Gothemburg, Lugano, Mlano-Bcocca, MIT, Napol, NYU, Oxford, Penn, Prnceton, UC Davs, UCLA, UCSD, and Yale. The authors gratefully acknowledge the fnancal support of the European Research Councl (BRSCDP - TEA - GA , STRATEMOTIONS - GA ) and of the AXA Research Fund. 1

2 We analyze a noton of selfconfrmng equlbrum wth non-neutral ambguty atttudes that generalzes the tradtonal concept. We show that the set of equlbra expands as ambguty averson ncreases. The ntuton s qute smple: By playng the same strategy n a statonary envronment, an agent learns the mpled dstrbuton of payoffs, but alternatve strateges yeld payoffs wth unknown dstrbutons; ncreased averson to ambguty makes such strateges less appealng. In sum, a knd of status quo bas emerges: In the long run, the uncertanty related to tested strateges dsappears, but the uncertanty mpled by the untested ones does not. Keywords: Selfconfrmng equlbrum, conjectural equlbrum, model uncertanty, smooth ambguty. JEL classfcaton: C72, D81. 2

3 Ch lasca la va veccha per la va nuova, sa quel che perde ma non sa quel che trova 1 1 Introducton In a stuaton of model uncertanty, or ambguty, the decson maker does not know the probablstc model for the varables affectng the consequences of choces. Such uncertanty s nherent n stuatons of strategc nteracton. Ths s qute obvous when such stuatons have been faced only a few tmes. In ths paper, we argue that uncertanty s pervasve also n games played recurrently where agents have had the opportunty to collect a large set of observatons and the system has settled nto a steady state. Such a stuaton s captured by the selfconfrmng equlbrum concept (also called conjectural equlbrum). In a selfconfrmng equlbrum (henceforth, SCE), agents best respond to confrmed probablstc belefs, where confrmed means that ther belefs are consstent wth the evdence they can collect, gven the strateges they adopt. Of course, ths evdence depends on how everybody else plays. We analyze SCE and model uncertanty jontly and show that they are conceptually complementary: The SCE condtons endogenously determne the extent of uncertanty, and uncertanty averson nduces a knd of status quo bas that expands the set of selfconfrmng patterns of behavor. The SCE concept can be framed wthn dfferent scenaros. A benchmark scenaro s just a repeated game wth a fxed set of players n whch there are no ntertemporal strategc lnks between the plays. That s, the ndvduals who play the game many tmes are concerned only wth ther nstantaneous payoff, and gnore the effects of ther current actons on the other players future behavor; they smply best respond to ther updated belefs about the current perod strateges of the opponents. Although all our results apply to ths stuaton, our presentaton s framed nto the so called large populatons (or Nash s mass acton) scenaro: There s a large socety of ndvduals 1 Italan proverb Those who leave the old road for a new one, know what they leave but do not know what they wll fnd. 3

4 who play recurrently a gven game G, possbly a sequental game wth chance moves: for each player/role n G (male or female, buyer or seller, etc.), there s a large populaton of agents who play n role. Agents are drawn at random and matched to play G. Then, they are separated and re-matched to play G wth (almost certanly) dfferent opponents, and so on. After each play of a game n whch he was nvolved, an agent obtans some evdence on how the game was played. The accumulated evdence s the data set used by the agent to evaluate the outcome dstrbuton assocated wth each choce. Note, there s an ntrnsc lmtaton to the evdence that an agent can obtan: If the game has sequental moves, he can observe at most the termnal node reached, but often he wll observe even less, e.g., only hs monetary payoffs (and not those of hs opponents). Each agent s nterested n the dstrbuton of strategy profles adopted by the opponents wth whom he s matched, because t determnes the expected payoffs of hs alternatve strateges. Typcally, ths dstrbuton s not unquely dentfed by the long-run frequences of the agent s observatons. Ths defnes the fundamental nference problem he faces, and explans why model uncertanty s pervasve also n steady states. The key dfference between SCE and Nash equlbrum s that, n a SCE, agents may have ncorrect belefs because many possble underlyng dstrbutons are consstent wth the emprcal frequences they observe (see Battgall and Guatol 1988, Fudenberg and Levne 1993a, Fudenberg and Kreps 1995). Partal dentfcaton of the true dstrbuton and awareness of the possble ncorrectness of belefs form the natural doman for ambguty averson. Yet, accordng to the tradtonal SCE concept, agents are Bayesan subjectve expected utlty maxmzers and hence ambguty neutral. Here we modfy the noton of SCE to allow for non-neutral atttudes toward model uncertanty (see Glboa and Marnacc, 2013, for a recent revew on the topc). The decson theoretc work whch s more germane to our approach dstngushes between objectve and subjectve uncertanty. Gven a set S of states, there s a set Σ (S) of possble probablstc models that the agent posts. 2 Each model 2 In ths context, we call objectve probabltes the possble probablty models (dstrbutons) over a state space S. These are not to be confused wth the objectve probabltes 4

5 σ Σ specfes the objectve probabltes of states and, for each acton a of the decson maker, t determnes a von Neumann-Morgenstern expected utlty evaluaton U(a, σ); the decson maker s uncertan about the true model σ (see Cerrea-Voglo et al., 2013a,b). In our framework, a s the acton, or strategy, of an agent playng n role, σ s a dstrbuton of strateges n the populaton of opponents (or a profle of such dstrbutons n n-person games), and Σ s the set of dstrbutons consstent wth the database of the agent. Roughly, an agent s uncertanty averse f he dslkes the uncertanty about U(a, σ) mpled by the uncertanty about the true probablty model σ Σ. We nterchangeably refer to such feature of preferences wth the expresson averson to model uncertanty or the shorter ambguty averson. A now classcal descrpton of ambguty averson s the maxmn crteron of Glboa and Schmedler (1989), where actons are chosen to solve the problem max a mn σ U(a, σ). In ths paper, we span a large set of ambguty atttudes usng the smooth ambguty model of Klbanoff, Marnacc and Mukerj (2005, henceforth KMM). Ths latter crteron admts the maxmn crteron as a lmt case, and the Bayesan subjectve expected utlty crteron as a specal case. In a Smooth SCE, agents n each role best respond to belefs consstent wth ther database, choosng actons wth the hghest smooth-ambguty value, and ther database s the one that obtans under the true data generatng process correspondng to the actual strategy dstrbuton. The followng example shows how our noton of SCE dffers from the tradtonal, or Bayesan, SCE. stemmng from an Anscombe and Aumann settng. For a dscusson, see Cerrea-Voglo et al. (2013b). 5

6 1 (0 < ε < 0.5) O 1 + ε MP 1 h 1 t 1 H T MP 2 h 2 t 2 H T Fgure 1: Matchng Pennes wth ncreasng stakes In the zero-sum game 3 of Fgure 1, the frst player chooses between an outsde opton O and two Matchng-Pennes subgames, say MP 1 and MP 2. Subgame MP 2 has hgher stakes than MP 1 : It has a hgher (mxed) maxmn value (2 > 1.5), but a lower mnmum payoff (0 < 1). In ths game, there s only one Bayesan SCE outcome, 4 whch must be the unque Nash outcome: MP 2 s reached wth probablty 1 and half of the agents n each populaton play Head. But we argue nformally that moderate averson to uncertanty makes the lowstakes subgame MP 1 reachable, and hgh averson to uncertanty makes the outsde opton O also possble. 5 Specfcally, let µ k denote the subjectve probablty assgned by an ambguty neutral agent n role 1 to h k, wth k = 1, 2. Gong to the low-stakes subgame MP 1 has subjectve value max{ µ 1 + 1, 2 µ 1 )} 1.5 and gong to the hgh-stakes subgame MP 2 has subjectve value max{4 µ 2, 4(1 µ 2 )} 2. Thus, O s never an ambguty-neutral best reply and cannot be played by a postve fracton of agents n a Bayesan SCE. Furthermore, also the low-stakes subgame MP 1 cannot be played n a Bayesan SCE. For suppose by way of contradcton that a postve fracton of agents n populaton 1 played MP 1. In the long run, each one of these agents, 3 The zero-sum feature smplfes the example, but t s nessental. 4 We call outcome a dstrbuton on termnal nodes. 5 See Secton 4 for a rgorous analyss. 6

7 and all agents n populaton 2, would learn the relatve frequences of Head and Tal. Snce n a SCE agents best respond to confrmed belefs, the relatve frequences of Head and Tal should be the same n equlbrum,.e., the agents n populaton 1 playng MP 1 would learn that ts objectve expected utlty s 1.5 < 2 and would devate to MP 2 to maxmze ther SEU. On the other hand, for agents who are (at least) moderately averse to model uncertanty and keep playng MP 1, havng learned the rsks nvolved wth the low-stakes subgame confers to reduced-form 6 strateges H 1 and T 1 a knd of status quo advantage : The objectve expected utlty of the untred strateges H 2 and T 2 s unknown, and therefore they are penalzed. Thus, the low-stakes subgame MP 1 can be played by a postve fracton of agents f they are suffcently averse to model uncertanty. Fnally, also the outsde opton O can be played by a postve fracton of agents n a SCE f they are extremely averse to model uncertanty, as represented by the maxmn crteron. 7 If an agent keeps playng O, he cannot learn anythng about the opponents strategy dstrbuton, hence he deems possble every dstrbuton, or model, σ 2. Therefore, the mnmum expected utlty of H 1 (resp. T 1 ) s 1 and the mnmum expected utlty of H 2 (resp. T 2 ) s zero, justfyng O as a maxmn best reply. 8 The example shows that, by combnng the SCE and ambguty averson deas, a knd of status quo bas emerges: In the long run, uncertanty about the expected utlty of tested strateges dsappears, but uncertanty about the expected utlty of the untested ones does not. Therefore, ambguty averse agents have weaker ncentves to devate than ambguty neutral agents. More generally, hgher ambguty averson mples a weaker ncentve to devate from an equlbrum strategy. Ths explans the man result of the paper: The set of SCE s expands as ambguty averson ncreases. We make ths precse by adoptng the smooth ambguty model of KMM, whch convenently sepa- 6 H k (resp. T k ) corresponds to the class of realzaton-equvalent strateges that choose subgame MP k and then select H k (resp. T k ). 7 As antcpated, the maxmn crteron s a lmt case of the smooth one, therefore the same result holds for very hgh degrees of ambguty averson. 8 Note that we are excludng the possblty of mxng through randomzaton, an ssue addressed n Secton 5. 7

8 rates the endogenous subjectve belefs about the true strategy dstrbuton from the exogenous ambguty atttudes, so that the latter can be partally ordered by an ntutve more ambguty averse than relaton. Wth ths, we provde a defnton of Smooth SCE whereby agents smooth best respond to belefs about strategy dstrbutons consstent wth ther long-run frequences of observatons. The tradtonal SCE concept s obtaned when agents are ambguty neutral, whle a Maxmn SCE concept obtans as a lmt case when agents are nfntely ambguty averse. By our comparatve statcs result, these equlbrum concepts are ntutvely nested from fner to coarser: Each Bayesan SCE s also a Smooth SCE, whch n turn s also Maxmn SCE. Fnally, we show how our results for Smooth SCE extend to other robust decson crtera. The rest of the paper s structured as follows. Secton 2 gves the setup and our defnton of SCE. In Secton 3, the core of the paper, we present a comparatve statcs result and analyze the relatonshps between equlbrum concepts. Secton 4 llustrates our concepts and results wth a detaled analyss of a generalzed verson of the game of Fgure 1. Secton 5 concludes the paper wth a dscusson of some mportant theoretcal ssues and of the related lterature. In the man text we provde nformal ntutons for our results. All proofs are collected n the Appendx. 2 Recurrent games and selfconfrmng equlbrum 2.1 Games wth feedback We consder a fnte game played recurrently between agents drawn at random from large populatons, one populaton for each player role. The game may be dynamc, but n ths case we assume that the agents play ts strategc form; that s, they smultaneously and rreversbly choose a pure strategy, whch s then mechancally mplemented by some devce. 8

9 The rules of the game determne a game form wth feedback (I, (S, M, F ) I ), where: I = {1,...n} s the set of player roles, and we call player the agent who n a gven nstance of the game plays n role I; S s the fnte set of strateges of I; wth ths, we let S = Π I S and S = Π j S j denote the set of all strategy profles and of s opponents strategy profles, respectvely; M s a set of messages that player may receve ex post (at the end of the game); F : S M s a feedback functon. For each player role I, there s a correspondng populaton of agents. Agents playng n dfferent roles are drawn at random, hence ndependently, from the correspondng populatons, whch do not overlap. Once the game s played by the agents matched at random, the resultng strategy profle s generates a message m = F (s) for each player I. Ths message encodes all the nformaton about play that player receves ex post. Ths nformaton typcally ncludes, but needs not be lmted to, the materal consequences of nteracton observed by, such as hs consumpton. If the game s dynamc, a player s feedback s a functon of the termnal node ζ(s) Z reached under strategy profle s. In ths case, F (s) = f (ζ(s)) where f : Z M s the extensve-form feedback functon for player. Example 1 Three natural specal cases are: For every I and s S, F (s) = ζ(s), each player observes the termnal node (reached under the realzed strategy profle), that s, f s the dentty on Z; F (s) = g (ζ(s)), each player observes everybody s materal consequences at the termnal node, that s, f s the consequence functon g; 9 9 The consequence functon g : Z Π I C assocates profles of consequences wth termnal nodes, where C denotes the set of all materal consequences that player may face at the end of the game. 9

10 F (s) = g (ζ(s)), each player observes hs own materal consequences at the termnal node, that s, f s the -th projecton of g. Note that, whle n the frst two cases all agents obtan the same feedback, n the thrd one feedback s personal. We mplctly assume that player knows the feedback functon F and remembers the strategy s he just played. Hence, upon playng s and recevng message m, he nfers that the strategy profle played by hs opponents must belong to the set {s S : F (s, s ) = m } = F 1,s (m ), where F,s : S M denotes the secton at s of the feedback functon F. 10 To streamlne notaton, and nspred by the mportant specal cases n whch F = F does not depend on, we wrte F s nstead of F,s. Wth ths, every strategy s gves rse to an ex post nformaton partton of the set of opponents strategy profles: F s = {F 1 s (m ) : m M }. Example 2 In the game of Fgure 1, assumng that player 1 observes only hs monetary payoff, the ex post nformaton partton depends on s 1 as follows: 11 F O = {S 2 }, F H 1 = F T 1 = {{h 1.h 2, h 1.t 2 }, {t 1.h 2, t 1.t 2 }}, F H 2 = F T 2 = {{h 1.h 2, t 1.h 2 }, {h 1.t 2, t 1.t 2 }}, where a 1.a 2 denotes the strategy of player 2 that chooses acton a 1 {h 1, t 1 } (resp. a 2 {h 2, t 2 }) n subgame MP 1 (resp. MP 2 ). Summng up, F s1 depends on s 1 and t never fully reveals the strategy played by the opponent. A game form wth feedback (I, (S, M, F ) I ) satsfes own-strategy ndependence of feedback f the ex post nformaton partton F s 10 That s, F,s (s ) = F (s, s ) for every s S. 11 We are coalescng realzaton-equvalent strateges of player 1. s ndependent 10

11 of s for every I. 12 Ths property s very strong and s volated n many nterestng cases. For example, the property fals whenever the strategc game form s derved from a non trval extensve game form where agents nfer ex post the termnal node reached, such as the game dscussed above. 2.2 Players preferences Next we descrbe agents personal features. We assume for notatonal smplcty that all agents n any gven populaton have the same atttudes toward rsk and the same atttudes toward uncertanty (or ambguty). The former are represented, as usual, by a von Neumann-Morgenstern payoff functon U : S R. We say that game G has observable payoffs whenever the payoff of every player only depends on hs ex post nformaton about play. Our man results rely on ths mantaned assumpton, whch can be formalzed as follows: For each I, each s S, and every s, s S such that F s ( s ) = Fs ( s ), we have U (s, s ) = U (s, s ). Contrapostvely, ths means that, upon playng a fxed strategy and obtanng dfferent utltes, the agent would detect a dfference n hs opponents counter strateges (receve a dfferent feedback). 13 We call game wth feedback the tuple G = (I, (S, M, F, U ) I ) where agents payoffs are specfed. For each I, the atttudes toward uncertanty, or ambguty atttudes, of agents n populaton are represented by a strctly ncreasng and contnuous functon φ : U R, where U = [mn s S U (s), max s S U (s)]. Suppose that player s uncertan about the true dstrbuton σ (S ) of strateges n 12 Ths property s called non manpulablty of nformaton by Battgall et al. (1992) and Azrel (2009), and own-strategy ndependence by Fudenberg and Kamada (2011). 13 Mathematcally, ths amounts to F s -measurablty of each secton U s = U,s of U. 11

12 the populaton of potental opponents, 14 and that hs uncertanty s expressed by some pror belef µ wth support on a posted subset Σ of (S ). 15 Then, the value to player of playng strategy s S s gven by the KMM smooth ambguty crteron: where V φ ( ) (s, µ ) = φ 1 φ (U (s, σ ))µ (dσ ), (1) suppµ U (s, σ ) = s S U (s, s )σ (s ) s the von Neumann-Morgenstern expected utlty of s under σ, so that (1) s a certanty equvalent expressed n utls. The standard Bayesan SEU crteron V d (s, µ ) = U (s, σ )µ (dσ ), suppµ (2) corresponds to an affne φ ; 16 whle a robust crteron à la Glboa and Schmedler V ω (s, µ ) = mn U (s, σ ), (3) σ suppµ can be obtaned as a lmt of (1) when the measure of ambguty averson φ /φ converges pontwse to nfnty (see KMM for detals). robust preferences are dscussed n Secton 5. Alternatve We call game wth feedback and ambguty atttudes a par (G, φ), where G s a game wth feedback and φ = (φ ) I s a profle of ambguty atttudes. We adopt the conventonal equalty φ = ω for some or all n order to encompass preferences represented as n (3). Note that the latter preferences are fully characterzed by U and the set suppµ of opponents strategy dstrbutons 14 In games wth three or more players, s facng a profle of strategy dstrbutons (σ j ) j Π j (S j ). The random matchng structure mples that the objectve probablty of strategy profle s s σ (s ) = j σ j(s j ). Thus, σ (S ) s actually a product dstrbuton. 15 The smplex (S ) n R S s endowed wth the Borel sgma-algebra. 16 Snce ths s by far the most well known functonal form, the superscrpt d wll sometmes be omtted. 12

13 that agent deems plausble. 2.3 Partal dentfcaton Next we descrbe how an agent who keeps playng a fxed strategy n a statonary envronment can partally dentfy the opponents strategy dstrbutons, and, f payoffs are observable, he can learn n the long run the expected payoff of the fxed strategy tself. The probablty of observng a gven message m for a player that chooses s and faces populatons of opponents descrbed by σ s σ ({s S : F (s, s ) = m }) = σ (F 1 s (m )). The correspondng dstrbuton of messages σ Fs 1 (M ) s denoted ˆF s (σ ). Therefore, f plays the pure strategy s and observes the long-run frequency dstrbuton of messages ν (M ), then he can nfer that the set of (product) strategy dstrbutons of the opponents that may have generated ν s 17 {σ Π j (S j ) : ˆF s (σ ) = ν }. If σ = Π j σ j s the true strategy dstrbuton of hs opponents, the long-run frequency dstrbuton of messages observed by when playng s s (almost certanly) the one nduced by the objectve dstrbuton σ, that s, ν = ˆF s ( σ ). 18 The set of possble dstrbutons from s (long-run emprcst) perspectve s thus ˆΣ (s, σ ) = = {σ Π j (S j ) : ˆF s (σ ) = ˆF s ( σ ) } {σ Π j (S j ) : σ Fs = σ Fs }. Ths s, the set of all product probablty measures on S that concde wth 17 Wth a slght abuse of notaton we are dentfyng the product set Π j (S j ) wth the correspondng set of product dstrbutons on S. 18 As common n steady state analyss, we are heurstcally relyng on a law-of-largenumbers argument. 13

14 σ on the nformaton partton F s : Although σ remans unknown, ts restrcton to F s s learned n the lmt. The dentfcaton correspondence ˆΣ (s, ) s nonempty (snce σ ˆΣ (s, σ )) and compact valued; t s also convex-valued n two-person games. Our defnton of ˆΣ (s, σ ) reflects the nformal assumpton that each agent n populaton knows he s matched at random wth agents from other populatons. Hence, he knows that condtonal on the true profle of strategy dstrbutons the strategy played by the agent drawn from populaton j s ndependent of the strategy played by the agent drawn from populaton k. Therefore, ˆΣ (s, σ ) need not be convex n games wth three or more players. 19 If payoffs are observable, then can learn ther tme average and, n the long run, ther expectaton. 20 Lemma 3 If payoffs are observable n the game wth feedback G, then, for every, s, and σ, U (s, σ ) = U (s, σ ) σ ˆΣ (s, σ ). In contrast, f a dfferent strategy s s s consdered, the value of U (s, σ ) as σ ranges n ˆΣ (s, σ ) remans uncertan: The set {U (s, σ ) : σ ˆΣ (s, σ )} s not, n general, a sngleton. 21 Ths s the feature that, under ambguty averson, wll generate a knd of status-quo bas n favor of the strategy s that has been played for a long tme. As a matter of nterpretaton, we assume that each agent n populaton knows I, S, M, F, U, and φ, but he may not know F, U, and φ. In Secton 5 we comment extensvely on the lmtatons and possble extensons of our framework. 19 If we assumed total gnorance about the matchng process, then the partally dentfed set would be convex, as n the two person case: ˆΣ (s, σ ) = { σ (S ) : ˆF s (σ ) = ˆF s ( σ ) }. 20 Agan, by a law-of-large-numbers heurstc. 21 Because U s : S R s F s -measurable and not, n general, F s -measurable. 14

15 2.4 Selfconfrmng equlbrum Next we gve our defnton of selfconfrmng equlbrum wth non-neutral atttudes toward uncertanty. Recall that we restrct agents to choose pure strateges, so that mxed strateges arse only as dstrbutons of pure strateges wthn populatons of agents. Defnton 4 A profle of strategy dstrbutons σ = (σ ) I s a smooth selfconfrmng equlbrum (SSCE) of a game wth feedback and ambguty atttudes (G, φ) f, for each I and each s suppσ, there s a pror µ s wth support contaned n ˆΣ (s, σ ) such that V φ ( ) s, µ s V φ ( ) s, µ s s S. (4) The confrmed ratonalty condton (4) requres that every pure strategy s that a postve fracton σ (s ) of agents keep playng must be a best response wthn S to the evdence, that s, the statstcal dstrbuton of messages ˆF s (σ ) (M ) generated by playng s aganst the strategy dstrbuton σ. If all φ s are affne, we obtan a defnton of Bayesan selfconfrmng equlbrum (BSCE) that subsumes the earler defntons of conjectural and selfconfrmng equlbrum. Fnally, we also consder the correspondng classcal (as opposed to Bayesan) case of maxmn selfconfrmng equlbrum. Defnton 5 A profle of strategy dstrbutons σ = (σ ) I s a maxmn selfconfrmng equlbrum (MSCE) of a game wth feedback G f, for each I and each s suppσ, mn U (s σ ˆΣ (s,σ ), σ ) mn U (s, σ ) s S. (5) σ ˆΣ (s,σ ) Formally, ths defnton s a specal case of the prevous one. In fact, an MSCE s a SSCE of a game (G, φ) wth φ ω under the addtonal assumpton that, for each s played by a postve fracton of agents, the justfyng 15

16 pror µ s has full support on ˆΣ (s, σ ). However, we state t separately snce ths maxmn noton also admts a conceptually dfferent, classcal, statstcal nterpretaton n whch prors are absent and so agents are emprcal frequentsts. In Secton 4, we llustrate these defntons wth a detaled analyss of a generalzed verson of the game of Fgure 1. Here we consder a more symmetrc example. 1\2 O 2 H 2 T 2 O 1 1, 1 1, 2 1, 2 H 1 2, 1 4, 0 0, 4 T 1 2, 1 0, 4 4, 0 Fgure 2 Example 6 Fgure 2 gves the reduced strategc form of a sequental game where players unlaterally and smultaneously decde ether to stop and get out (O ) or contnue. If they both stop, they get 1 utl each; f only one of them does, the player who stops gets 1 utl, the other player gets 2 utls; f they both contnue, next they play a Matchng Pennes subgame. Suppose that each only observes hs own payoff, that s, F ( ) = U ( ). Then, an agent who stops cannot observe anythng, whle an agent who plays Head or Tals dentfes the strategy dstrbuton of the populaton of opponents: ˆΣ (O, σ ) = (S ) and ˆΣ (H, σ ) = ˆΣ (T, σ ) = {σ } for every {1, 2} and σ (S ). A necessary condton for σ to be a SCE s σ (O ) < 1 = σ (H ) = σ (T ), {1, 2}, because agents who do not stop dentfy the opponents dstrbuton and have to be ndfferent between Head and Tal. Next note that stoppng s never a best response for an ambguty neutral agent. Wth ths, t s easy to check that BSCE and NE concde: Nobody stops and the 16

17 two populatons splt evenly between Heads and Tals. But the set of SSCE s s much larger f agents are suffcently ambguty averse. Specfcally, t can be shown that the belef that mnmzes the ncentve for an ambguty averse agent to devate from O s µ = 1 2 δ H δ T. That s, agents wth such belef thnk that ether all agents n populaton play Head, or all of them play Tal, and that these two extreme dstrbutons are equally lkely. Let φ (U) = U 1/α wth α > 0 for each. Then, ( V φ (H, µ ) = V φ 1 (T, µ ) = 2 41/α + 1 ) α 2 01/α 1 α 2. Therefore, f α < 2, then O cannot be a best reply to any pror, and so SSCE = BSCE = NE; f α 2, then O s a best reply to µ, whch s trvally confrmed, and the necessary condton for a SCE s also suffcent: SSCE = {σ : {1, 2}, σ (O ) < 1 = σ (H ) = σ (T )} = {σ : {1, 2}, σ (H )(1 σ (O )) = σ (T )(1 σ (O ))}. We conclude that f agents are suffcently ambguty averse,.e. α 2, then they may stop n a SSCE. As antcpated above and dscussed n Secton 5, our defnton of Bayesan SCE subsumes earler defntons of conjectural and selfconfrmng equlbrum as specal cases. Lke these earler notons of SCE, our more general noton s motvated by a partal dentfcaton problem: The mappng from strategy dstrbutons to the dstrbutons of observatons avalable to an agent s not one to one. In fact, f for each agent dentfcaton s full that s, ˆΣ (s, σ ) = {σ } for all s and all σ condton (4) s easly seen to ( ( reduce to the standard Nash equlbrum condton U s, σ ) U s, σ ). In other words, f none of the agents features a partal dentfcaton problem, we are back to the Nash equlbrum noton (n ts mass acton nterpretaton). 17

18 3 Comparatve statcs and relatonshps In ths secton, we compare the equlbra of games wth dfferent ambguty atttudes. Ths allows us to nest the dfferent notons of SCE defned above. We also dentfy a specal case where they all collapse to Nash equlbrum. 3.1 Man result Ambguty atttudes are characterzed by the weghtng functons profle φ = (φ ) I. We say that φ s more ambguty averse than ψ f there s a concave and strctly ncreasng functon ϕ : ψ (U ) R such that φ = ϕ ψ (see KMM). 22 Game (G, φ) s more ambguty averse than (G, ψ) f, for each, φ s more ambguty averse than ψ. Game (G, φ) s ambguty averse f t s more ambguty averse than (G, d U1,..., d Un ), that s, f each functon φ s concave. Observe that we do not assume that the φ s are concave. Therefore, our comparson of ambguty atttudes does not hnge on ths assumpton. In other words, for the relaton of beng more ambguty averse, t only matters that profle φ be comparatvely more ambguty averse than profle ψ, somethng that can happen even f both are ambguty lovng. Buldng on the non ambguty of the expected payoff of the long-run strategy, establshed n Lemma 3, we can now turn to the man result of ths paper: The set of equlbra expands as ambguty averson ncreases. Theorem 7 If (G, φ) s more ambguty averse than (G, ψ), then the SSCE s of (G, ψ) are also SSCE s of (G, φ). Smlarly, the SSCE s of any game wth feedback and ambguty atttudes (G, φ) are also MSCE s of G. We provde ntuton for ths result n the Introducton. Now we can be more precse: Let σ be an SSCE of (G, ψ), the less ambguty averse game, and pck any strategy played by a postve fracton of agents, s suppσ ; then, there s a justfyng confrmed belef µ s such that s s a best reply to 22 Wth the conventon that φ = ω s more ambguty averse than any ψ, and that f φ s more ambguty averse than ω then φ = ω. 18

19 µ s gven ψ, that s, V ψ (s, µ s ) V ψ (s, µ s ) for all s. We nterpret µ s as the belef held n the long-run by an agent who keeps playng the longrun strategy s n the statonary envronment determned by σ. Such agent eventually learns the long-run frequences of the (observable) payoffs of s ; therefore, the value of s for ths agent converges to ts objectve expected utlty U(s, σ ), ndependently of hs ambguty atttudes (cf. Lemma 3). But the value of an untested strategy s s typcally depends on ambguty atttudes and, keepng belefs fxed, t s hgher when ambguty averson s lower, that s, V ψ V φ (s, µ s ) V φ (s, µ s ). Therefore (s, µ s ) = U(s, σ ) = V ψ (s, µ s ) V ψ (s, µ s ) V φ (s, µ s ) for all s. Ths means that t s possble to justfy σ as an SSCE of the more ambguty averse game (G, φ) usng the same profle of belefs justfyng σ as an SSCE of (G, ψ). 3.2 Relatonshps Theorem 7 mples, that under observable payoffs, () the set of BSCE s of G s contaned n the set of SSCE s of every (G, φ) wth ambguty averse players; () the set of SSCE s of every (G, φ) s contaned n the set of MSCE s of G. that In other words, under observable payoffs and ambguty averson, t holds BSCE SSCE MSCE. (6) The degree of ambguty averson determnes the sze of the set of selfconfrmng equlbra, wth the sets of Bayesan and Maxmn selfconfrmng equlbra beng, respectvely, the smallest and largest one But note that the nclusons BSCE MSCE and SSCE MSCE do not requre ambguty averson. Furthermore, one can show that, n two-person games, BSCE SSCE ndependently of the ambguty atttudes φ, due to the convex-valuedness of ˆΣ (s, ) n ths case (see Battgall et al., 2011). 19

20 It s well known that every Nash equlbrum σ s also a Bayesan SCE. The same relatonshp holds more generally for Nash and smooth selfconfrmng equlbra (also when agents are ambguty lovng). Intutvely, a Nash equlbrum s an SSCE wth correct (hence confrmed) belefs about strategy dstrbutons; snce correct belefs cannot exhbt any model uncertanty, they satsfy the equlbrum condtons ndependently of ambguty atttudes. Lemma 8 If a profle of dstrbutons σ s a Nash equlbrum of a game wth feedback G, then t s a SSCE of any game wth feedback and ambguty atttudes (G, φ). Snce the set NE of Nash equlbra s nonempty, we automatcally obtan exstence of SSCE for any φ. 24 In partcular, we can enrch the chan of nclusons n (6) as follows: NE BSCE SSCE MSCE under observable payoffs and ambguty averson. The next smple, but nstructve result establshes a partal converse. Recall that G has own-strategy ndependent feedback f what each player can nfer ex post about the strateges of other players s ndependent of hs own choce. The followng proposton llustrates the strength of ths assumpton. Proposton 9 In every game wth observable payoffs and own-strategy ndependent feedback, every type of SCE s equvalent to Nash equlbrum: NE = BSCE = SSCE = MSCE. The ntuton for ths result s qute smple: The strategc-form payoff functon U (s, ) : S R s constant on each cell Fs 1 (m ) of the partton F s = {Fs 1 (m )} m M (observablty of payoffs), but ths partton s ndependent of s (own-strategy ndependence of feedback). Ths means that, n the long run, an agent does not only learn the objectve probabltes of the payoffs 24 Hence, we also obtan exstence of MSCE, by Theorem 7. 20

21 assocated wth hs status quo strategy, but also the objectve probabltes of the payoffs assocated wth every other strategy. Hence, model uncertanty s rrelevant and he learns to play the best response to the true strategy dstrbutons of the other players/roles even f he does not exactly learn these dstrbutons. 25 Further results about the relatonshp between equlbrum concepts can be obtaned when G s derved from a game n extensve form under specfc assumptons about the nformaton structure (see Battgall et al., 2011). We conclude by emphaszng the key role played by payoff observablty n establshng the nclusons n (6). The followng example shows that, ndeed, these nclusons need not hold when payoffs are not observable. Example 10 Consder the zero-sum game of Fgure 1 of the Introducton, but now suppose that player 1 cannot observe hs payoff ex post (he only remembers hs actons). For example, the utlty values n Fgure 1 could be a negatve affne transformaton of the consumpton of player 2, reflectng a psychologcal preference of player 1 for decreasng the consumpton of player 2 (not observed by 1) even f the consumpton of 1 s ndependent of the actons taken n ths game. Then, even f 1 plays one of the Matchng Pennes subgames for a long tme, he gets no feedback: Under ths volaton of the observable payoff assumpton, ˆΣ 2 (s 1, σ 2 ) = (S 2 ) for all (s 1, σ 2 ). Snce u 1 (O) = 1 + ε s larger than the mnmum payoff of each subgame, the outsde opton O s the only MSCE choce of player 1 at the root. If φ 1 s suffcently concave, O s also an SSCE choce (justfed by a sutable pror). But, as already explaned, O cannot be an ambguty-neutral best reply. Furthermore, t can be verfed that every strategy s 1 s an SSCE strategy. Therefore, BSCE MSCE = and SSCE MSCE and so the nclusons of (6) here do not hold. 25 Related results are part of the folklore on SCE. See, for example, Battgall (1999) and Fudenberg and Kamada (2011). 21

22 4 A parametrzed example In ths secton, we analyze the SCE s of a zero-sum example parametrzed by the number of strateges. The zero-sum assumpton s nessental, but t smplfes the structure of the equlbrum set. The game s related to the Matchng Pennes example of the Introducton. We show how the SSCE set gradually expands from the BSCE set to the MSCE set as the degree of ambguty averson ncreases. To help ntuton, we frst consder a generalzaton of the game of Fgure 1: Player 1 chooses between an outsde opton O that yelds n 1+ε utls (0 < ε < 1/2) and n 2 Matchng-Pennes subgames aganst player 2. Subgames wth a hgher ndex k have hgher stakes, that s, a hgher mxed maxmn value, but a lower mnmum payoff (see Fgure 3). The game of Fgure 1 obtans for n = 2. 1 O n 1 + ε MP 1 MP k h k t k H k n + 2(k 1) n k T k n k n + 2(k 1) MP n Fgure 3: Fragment of zero-sum game In ths game, player 1 has (n+1) 2 n strateges and player 2 has 2 n strateges. To smplfy the notaton, we nstead analyze an equvalent extensve-form game Γ n obtaned by two transformatons. Frst, player 2 s replaced by a team of opponents 2.1,..., 2.n, one for each (zero-sum) subgame k. Second, the sequence of moves (k, H k ) of player 1 (go to subgame k then choose Head) 22

23 whch s common to 2 n 1 realzaton-equvalent strateges s coalesced nto the sngle strategy H k. Smlarly, (k, T k ) becomes T k. The new strategy set of player 1 has 2n+1 strateges: S 1 = {O, H 1, T 1,..., H n, T n }. If player 1 chooses H k or T k, player 2.k moves at nformaton set {H k, T k } (.e., wthout knowng whch of the two actons was chosen by player 1) and chooses between h k and t k ; hence S 2.k = {h k, t k }. See Fgure 4. 1 O H 1 T 1 H 2 T ε h 1 t 1 h 1 t 1 h 2 t 2 h 2 t Fgure 4: The case n = 2 We assume that players observe the termnal node, or equvalently that the game has observable payoffs (cf. Example 2). Although there are no proper subgames n Γ n, we slghtly abuse language and nformally refer to subgame k when player 1 chooses H k or T k, gvng the move to opponent 2.k. The game Γ n and the prevously descrbed game have somorphc sets of termnal nodes (wth cardnalty 4n + 1) and the same reduced normal form (once players 2.1,..., 2.n of the second game are coalesced nto a unque player 2). By standard arguments, these two games have equvalent sets of Nash equlbra, equvalent BSCE and MSCE sets, and equvalent SSCE sets for every φ Each profle σ = (σ 1, (σ 2.k ) n k=1 ) of the new n-person game can be mapped to an equv- 23

24 That sad, consder the game wth feedback G n derved from extensve-form game Γ n under the assumpton that the termnal node reached s observed ex post (or that payoffs are observable). It s easly seen that, for every profle of strategy dstrbutons σ2 = (σ2.k )n k=1, t holds that27 ˆΣ 2 (O, σ 2) = Π n k=1 (S 2.k ), (7) and ˆΣ 2 (H k, σ 2) = ˆΣ 2 (T k, σ 2) = {σ 2 : σ 2.k = σ 2.k}. (8) As a result, next we provde necessary SCE condtons that partally characterze the equlbrum strategy dstrbuton for player/role 1 and fully characterze the equlbrum strategy dstrbutons for the opponents. Lemma 11 For every (Bayesan, Smooth, Maxmn) SCE σ and every k = 1,..., n, σ 1(H k ) + σ 1(T k ) > 0 σ 1(H k ) σ 1(H k ) + σ 1(T k ) = 1 2 = σ 2.k(h k ). (9) Furthermore, for every (σ 1, σ 2) and σ 1, f (σ 1, σ 2) s a (Bayesan, Smooth, Maxmn) SCE, and suppσ1 Smooth, Maxmn) SCE. = supp σ 1, then ( σ 1, σ 2) s also a (Bayesan, Note that these necessary condtons do not restrct at all the set of strateges that can be played n equlbrum: For every s 1 {O, H 1, T 1,..., H n, T n } there s some dstrbuton profle σ such that σ 1(s 1 ) > 0 and (9) holds. The alent profle ( σ 1, σ 2 ) of the old two-person game and vceversa whle preservng the equlbrum propertes. Specfcally, (σ 2.k ) n k=1 s also a behavoral strategy of player 2 n the two-person game, whch corresponds to a realzaton-equvalent strategy dstrbuton σ 2 for player 2. Smlarly, any such dstrbuton σ 2 can be mapped to a realzaton-equvalent profle (σ 2.k ) n k=1. As for σ 1, for each s 1 n the new game, the probablty mass σ 1 (s 1 ) can be dstrbuted arbtrarly among the pure strateges of the old two-person game that select the correspondng sequence of moves (that s, ether (O), or (k, H k ) or (k, T k )), thus obtanng a realzaton-equvalent dstrbuton σ 1. In the opposte drecton, every σ 1 of the old game yelds a unque realzaton-equvalent σ 1 n the new game, where σ 1 (s 1 ) s the σ 1 -probablty of the set of (realzaton-equvalent) strateges that select the same sequence of moves as s For ease of notaton, n ths secton we denote ˆΣ 1 by ˆΣ 2. 24

25 formal proof of the lemma s straghtforward and left to the reader. Intutvely, f subgame k s played wth postve probablty, then each agent playng ths subgame learns the relatve frequences of Head and Tal n the opponent s populaton, and the best response condtons mply that a SCE reachng subgame k wth postve probablty must nduce a Nash equlbrum n ths Matchng- Pennes subgame. Thus, the σ 2-value to an agent n populaton 1 of playng the status quo strategy H k or T k (wth σ 1(H k ) + σ 1(T k ) > 0) s the mxed maxmn value of subgame k, n 1 + k/2. Wth ths, the value of devatng to another untested strategy depends on the exogenous atttudes toward model uncertanty, and on the subjectve belef µ 1 (ˆΣ 2 (H k, σ 2)), whch s only restrcted by σ2.k (eqs. (7) and (8)). As for the agents n roles 2.1,..., 2.n, ther atttudes toward uncertanty are rrelevant, because, f they play at all, they learn all that matters to them, that s, the relatve frequences of H k and T k. Suppose that a postve fracton of agents n populaton 1 play H k or T k, wth k < n. By Lemma 11, n a SCE, the value that they assgn to ther strategy s ts von Neumann-Morgenstern expected utlty gven that opponent 2.k mxes ffty-ffty, that s, n 1 + k/2. But, f they are ambguty neutral, the subjectve value of devatng to subgame n s at least the mxed maxmn value n 1 + n/2 > n 1 + k/2. Furthermore, the outsde opton O s never an ambguty-neutral best reply. 28 Ths explans the followng: Proposton 12 The BSCE set of G n concdes wth the set of Nash equlbra. Specfcally, BSCE = NE = { σ Σ : σ1(h n ) = σ1(t n ) = σ2.n(h n ) = 1 }. 2 Next we analyze the SSCE s assumng that agents are ambguty averse n the KMM sense. The followng prelmnary result, whch has some ndependent nterest, specfes the belefs about opponents strategy dstrbutons that mnmze the subjectve value of devatng from a gven strategy s 1 to any subgame j. 28 Indeed, O s strctly domnated by every mxed strategy 1 2 Hk T k. 25

26 Lemma 13 Let φ 1 be concave. For all j = 1,..., n, µ 1, ν 1 (Π n k=1 (S 2.k)), f mrg (S2.j )ν 1 = 1 2 δ h j δ t j, then max{v φ 1 1 (H j, µ 1 ), V φ 1 1 (T j, µ 1 )} V φ 1 1 (H j, ν 1 ) = V φ 1 1 (T j, ν 1 ). Intutvely, an ambguty averse agent dslkes devatng to subgame j the most when hs subjectve pror assgns postve weght only to the hghest and lowest among the possble objectve expected utlty values,.e., when ts margnal on (S 2.j ) has the form xδ h j + (1 x)δ t j. By symmetry of the 2 2 payoff matrx of subgame k, he would pck, wthn {H k, T k }, the strategy correspondng to the hghest subjectve weght (H k f x > 1/2). Hence, the subjectve value of devatng to subgame j s mnmzed when the two Drac measures δ h j and δ t j have the same weght x = 1/2. To analyze how the SSCE set changes wth the degree of ambguty averson of player 1, we consder the one-parameter famly of negatve exponental weghtng functons φ α 1 (U) = e αu, where α > 0 s the coeffcent of ambguty averson (see KMM p. 1865). Let SSCE(α) denote the set of SSCE s of (G n, φ α 1, φ 2,...φ n ). To characterze the equlbrum correspondence α SSCE(α), we use the followng transformaton of φ α 1 (U): ( 1 M(α, x, y) = (φ α 1 ) 1 2 φα 1 (x) + 1 ) 2 φα 1 (y). By Lemma 13, ths s the mnmum value of devatng to a subgame characterzed by payoffs x and y. The followng known result states that ths value s decreasng n the coeffcent of ambguty averson α, t converges to the mxed maxmn value as α 0 (approxmatng the ambguty neutral case), and t converges to the mnmum payoff as α +. 26

27 Lemma 14 For all x y, M(, x, y) s strctly decreasng, contnuous, and satsfes lm M (α, x, y) = 1 α 0 2 x + 1 y and lm M (α, x, y) = mn {x, y}. (10) 2 α + By Lemma 11, to analyze the SSCE(α) correspondence, we only have to determne the strateges s 1 that can be played by a postve fracton of agents n equlbrum, or conversely the strateges s 1 that must have measure zero. Let us start from very small values of α,.e., approxmately ambguty neutral agents. By Lemmas 13 and 14, the subjectve value of devatng to the hgheststakes subgame n s approxmately bounded below by n 1 + n/2 > u 1 (O). Therefore, the outsde opton O cannot be a best reply. Furthermore, suppose by way of contradcton that H k or T k (k < n) are played by a postve fracton of agents. By Lemma 11, the value of playng subgame k s the vnm expected utlty n 1 + k/2 < n 1 + n/2. Hence all agents playng ths game would devate to the hghest-stakes subgame n. Thus, for α small, SSCE(α) = BSCE. By Lemma 14, as α ncreases, the mnmum value of devatng to subgame n decreases, convergng to zero for α +. More generally, the mnmum value M(α, n j, n+2(j 1)) of devatng to subgame j converges to n j for α +. Snce n j < u 1 (O) < n 1+k/2, ths means that, as α ncreases, t becomes easer to support an arbtrary strategy s 1 as an SSCE strategy. Therefore, there must be thresholds 0 < α 1 <... < α n such that only the hgher-stakes subgames k + 1,...n can be played by a postve fracton of agents n equlbrum f α < α n k, and every strategy (ncludng the outsde opton O) can be played by a postve fracton of agents for some α α n k. In partcular, for α suffcently large, SSCE(α) concdes wth the set of Maxmn SCE s, whch s just the set Σ = {σ Σ : eq. (9) holds} of dstrbuton profles satsfyng the necessary condtons of Lemma To 29 Ths characterzaton holds for every parametrzed famly of dstrbutons that satsfes, at every expected utlty value Ū, propertes analogous to those of Lemma 14, wth α 27

28 summarze, by the propertes of the functon M(α, x, y) stated n Lemma 14, we can defne strctly postve thresholds α 1 < α 2 <... < α n so that the followng ndfference condtons hold max M(α n k, n j, n + 2(j 1)) = n 1 + k, k = 1,..., n 1, (11) j {k+1,...,n} 2 max M(α n, n j, n + 2(j 1)) = n 1 + ε, (12) j {1,...,n} and SSCE(α) expands as α ncreases, makng subgame k playable n equlbrum as soon as α reaches α n k, expandng to MSCE and makng the outsde opton O playable as soon as α reaches α n. Formally: Proposton 15 Let α 1 <... < α n be the strctly postve thresholds defned by (11) and (12). For every α and k = 1,...n 1, α < α n k = SSCE(α) = { σ Σ : σ 1({O, L 1, T 1,..., H k, T k }) = 0 } and α < α n = SSCE(α) = {σ Σ : σ 1(O) = 0}. Furthermore SSCE(α) = Σ = MSCE, α α n k and SSCE(α) = BSCE = NE f α < α 1, whle SSCE(α) = MSCE f α α n. 5 Concludng remarks and related lterature The SCE concept characterzes stable patterns of behavor n games played recurrently. We analyze a noton of SCE wth agents who have non-neutral atttudes toward uncertanty about the true steady-state data generatng process. We showed that ths uncertanty comes from a partal dentfcaton replaced by the coeffcent of ambguty averson φ 1(Ū)/φ (Ū). 28

29 problem: The mappng from strategy dstrbutons to the dstrbutons of observatons avalable to an agent s not one to one. We use as our workhorse the KMM smooth-ambguty model, whch separates endogenous belefs from exogenous ambguty atttudes. Ths makes our setup partcularly well suted to connect wth the prevous lterature on SCE and to analyze how the set of equlbra changes wth the degree of ambguty averson. Assumng observablty of payoffs, we show that the set of smooth SCE s expands when agents become more ambguty averse. The reason s that agents learn the expected utlty values of the strateges played n equlbrum, but not those of the strateges they can devate to, whch are thus penalzed by hgher ambguty averson. Ths allows us to derve ntutve relatonshps between dfferent notons of SCE. Nash equlbrum s a refnement of all of them, whch guarantees exstence. All notons of SCE collapse to Nash equlbrum under the addtonal assumpton of own-strategy ndependence of feedback. We develop our theoretcal nsghts n the framework of populaton games played recurrently, but smlar ntutons apply to dfferent strategc contexts, such as repeated games, or dynamc games wth a statonary Markov structure. Our nsghts are lkely to have consequences for more appled work. For example, the SCE and ambguty averson deas have been appled n macroeconomcs to analyze, respectvely, learnng n polcy makng (see Sargent, 1999, and the references n Cho and Sargent, 2008) and robust control (Hansen and Sargent, 2008). Our analyss suggests that these two approaches can be frutfully merged. Fershtman and Pakes (2012) put forward a concept of experence based equlbrum akn to SCE to provde a framework for the theoretcal and emprcal analyss of dynamc olgopoles. They argue that equlbrum condtons are, n prncple, testable when agents belefs are determned (f only partally) by emprcal frequences, as n ther equlbrum concept and n SCE. Ther model features observable payoffs because frms observe profts; therefore a verson of our man result apples: Ambguty averson expands the set of equlbra. In the remander of ths secton we consder some lmtatons and possble extensons of our analyss, and we brefly dscuss the related lterature. We 29

30 refer the reader to the workng paper verson (Battgall et al. 2011) and to Battgall et al. (2014) for a more detaled dscusson. More on robust preferences crteron correspondng to φ (t) = e t α ambguty averson coeffcent α > 0, can be wrtten as V φ It s well known that the smooth ambguty for all t R, wth constant absolute ( ) (s, µ ) = nf U (s, σ ) ν (dσ ) + αh (ν µ ). ν µ suppν Here H s the Kullback-Lebler dvergence; thus the correspondng smooth crteron s akn to the multpler crteron of Hansen and Sargent (2001). Ths suggests consderng robust preferences of the form V Φ ( ) (s, µ ) = nf U (s, σ ) ν (dσ ) + Φ (ν µ ), (13) ν µ suppν where Φ s a generc dvergence between prors, that s, a functon Φ : ( (S )) ( (S )) [0, ] such that Φ ( µ ) s convex and Φ (µ µ ) = 0 for every µ. Maccheron, Marnacc and Rustchn (2006) and Cerrea-Voglo, Maccheron, Marnacc and Montruccho (2013a) show how Φ ( ) captures ambguty atttudes n a smple way: Φ s more ambguty averse than Ψ f Φ ( µ ) Ψ ( µ ) for every µ ( (S )). It can be shown that all results n Secton 3 hold when the smooth crteron (1) s replaced wth the robust crteron (13). Dynamc consstency and condtonal belefs To avod dynamc consstency ssues, we assume that agents play the strategc form of the recurrent game,.e., an essentally smultaneous stage game. But when agents really play a game wth sequental moves, not ts strategc form, they cannot commt to any contngent plan. A strategy for an agent s just a plan that allows hm to evaluate the lkely consequences of takng actons at any nformaton set. The plan s credble and can be mplemented only f t prescrbes, at each possble 30

31 nformaton set, an acton that has the hghest value, gven the agent s condtonal belefs and planned contnuaton. The plans wth ths unmprovablty property are obtaned by means of a foldng back procedure on the subjectve decson tree mpled by the agent s belefs. We sketch how we can make ths precse n the context of the smooth-ambguty model, and thus provde a noton of dynamcally consstent SSCE. Next we dscuss the propertes of ths concept. We assume that agents feedback functons satsfy ex post perfect recall, that s, after playng the game agents remember the nformaton sets they crossed and the actons chosen at such nformaton sets. For an n-depth analyss wth proofs of clams, see Battgall et al. (2014). Each agent n role has a system of belefs µ ( ) about dstrbutons σ gven by a pror µ ( (S )) and a posteror µ ( h ) at each nformaton set h of. The predctve probablty of reachng nformaton set h gven that the agent chooses the actons leadng to h s P µ (h ) = (S ) σ (S (h )) µ (dσ ), where S (h ) denotes the set of strategy profles s consstent wth h. If P µ (h ) > 0, the posteror belef µ ( h ) s derved from the pror usng Bayes rule; 30 otherwse, µ ( h ) s derved from µ ( h ), where h s the nformaton set closest to the root such that P µ (h h ) > 0 (note, t may be h = h ). Such system of belefs yelds a condtonal probablty system on S (S ) gven the collecton of condtonng cylndrcal events S (h) (S ) (cf. Battgall and Snscalch, 1999). A plan s s a sequental best reply to µ ( ) f, at each nformaton set h of, t selects an acton maxmzng the KMM value V φ, gven µ ( h ) and the s - contnuaton after h. A profle of dstrbutons σ s a dynamcally consstent SSCE, for brevty SSCE DC, f each s wth σ (s ) > 0 s a sequental best reply to some µ s ( ) such that the pror µ s suppµ s ˆΣ (s, σ ). satsfes the confrmaton condton By the dynamc consstency of SEU maxmzaton, SSCE DC s realzaton- 30 That s, µ (E h ) = for every Borel set E (S ). 1 σ (S (h )) µ (dσ ) P µ (h ) E 31