It can be smart to be dumb

Transcription

1 It can be smart to be dumb Davd H. Wolpert, Mchael Harre 2 MS 269-, NASA Ames Research Center, Moffett Feld, CA, 94035, USA 2 The Centre for the Mnd and The School of Informaton Technologes, Sydney Unversty, Australa To whom correspondence should be addressed; E-mal: dhw@emal.arc.nasa.gov. An mportant problem n game theory s why bounded ratonalty occurs, e.g., why altrusm to non-kn occurs. Prevous explanatons nvolved computatonal lmtatons of the players, repeated play among the players etc. As an alternatve, we ntroduce a pre-play step n whch each player chooses a persona,.e., a fctcous utlty functon she commts to use n the game. By adoptng dfferent personas, player nduces dfferent moves by the other players n the game. Sometmes by adoptng a bounded ratonal persona, player nduces moves by the other players that ncrease s true utlty functon. In such cases, player s actng n a bounded ratonal way can be optmal for. Ths phenomenon can explan some expermental observatons concernng the prsoner s dlemma, the ultmatum game, and the traveler s dlemma. Summary: We show how bounded ratonal behavor n general, and altrusm n partcular, can be utlty-maxmzng n non-repeated, anonymous games.

2 One of the endurng problems of game theory ( 3) s explanng why human and anmal players so often exhbt bounded ratonalty, n that they seem not to adopt the strategy that optmzes ther objectve (4 7). As a partcular example, outsde of the bology-specfc phenomenon of kn selecton, the wde-spread phenomena of altrusm and cooperaton often appears to be bounded ratonal. Many models have been offered to explan bounded ratonalty n general and non-knselecton altrusm n partcular (,2,4,5,8,9,9 6). Some of these rely on ad hoc assumptons of bounded computatonal capabltes of the players. Other nvolve repeated nteractons among the players, where the players have some ablty to recognze ther opponents from one nteracton to the next, at least mplctly. The folk theorems of game theory can be vewed as the earlest examples of these repeated-nteracton explanatons. More recent examples are models that nvolve reputaton effects, punshment, loners, negotatng processes, etc. Here we present a smple and broadly-applcable framework that provdes an alternatve explanaton for bounded ratonalty. Ths framework s based on the observaton that socal organsms often have defnte personas that they adopt for ther nteractons wth one another. For example, someone mght act dumb n some socal stuatons but not others. More generally, we act lke a dfferent person when we nteract wth (.e., engage n a game wth) our boss, wth our spouse, wth a chld, etc. Often we even conscously decde how to act wth those people before the nteracton. Say that all players are free to choose such a persona before the start of play. Then for many games, adoptng a persona that s bounded ratonal (e.g., actng dumb) actually results n larger expected utlty when the game s played than does adoptng a fully ratonal persona. Ths s true even for anonymous, non-repeated games, and even when there s no explct pre-play sgnalng of personas among the players. In ths sense, bounded ratonalty can actually be 2

3 utlty-maxmzng. We show below how ths framework can explan a lot of behavoral game theory expermental data, focusng on the famous Traveler s Dlemma (TD) and Ultmatum Game (UG). We also explore the possble mplcatons of our framework for how to optmally desgn dstrbuted systems of adaptve agents. Next we show that our framework contans phase transtons, n whch nfntesmal changes n a player s choce of persona can completely change the equlbrum of the game. We end wth a dscusson of the possble relaton between our framework and phenomena lke socal ntellgence and culture gaps. Before presentng our framework, we frst revew game theory usng the example of the famous Prsoner s Dlemma (PD) (0, 7, 8). Say we have two players, Row and Col, each of whom can choose one of two moves (also know as pure strateges ). Wrte those sets of allowed moves as {Top, Down} for Row, and {Le f t, Rght} for Col. Both players have a utlty functon, whch maps any jont move by both players nto a real number. For example, n the PD the utlty functon pars (u Row, u Col ) for the four possble jont moves can be wrtten as the matrx [ (6, 0) (4, 4) (5, 5) (0, 6) ] () So for example, f Row plays T whle Col plays L, then Row s utlty functon equals 6 and Col s equals 0. To play a game each player {Row, Col} ndependently chooses a mxed strategy,.e., a probablty dstrbuton P (x ) over her par of allowed moves. So the expected utlty for player s E P (u ) = x,x P (x )P (x )u(x, x ), where P (x ) s the mxed strategy of s opponent. We say that a par of mxed strateges (P Row, P Col ) s a Nash Equlbrum (NE) of the game f for all players, E P (u ) cannot ncrease f P changes whle P stays the same. Intutvely, at a NE, nether player could beneft by changng her mxed strategy, n lght of her opponent s mxed 3

4 strategy. If ether player volates ths condton, she s sad to exhbt bounded ratonalty. As an example, n the PD, there s a unque NE, where Row plays T wth probablty.0 and Col plays R wth probablty.0. (Gven the mxed strategy of Row, Col s expected utlty would decrease f she played L wth non-zero probablty, and gven the mxed strategy of Col, Row s expected utlty would decrease f she played D wth non-zero probablty.) Note though that at the (non-ne) jont move (D, L), both players have hgher expected utlty than at the NE. So f they could only both be nduced to cooperate wth one another and choose that move and n dong so both be bounded ratonal both of the players would beneft. In lght of ths, often the move T by Row and the move R by Col are referred to as the Defect move, whle ther other moves are referred to as Cooperate. Now say that rather than beng ratonal n the PD, Col were perfectly rratonal. Ths means that she commts to choosng unformly randomly between the two columns, wth no evdent concern for the resultant value of her utlty functon, and n partcular wth no concern for what strategy Row adopts. Gven such rratonalty of Col, Row would have expected utlty of 5 for playng T, and of 2.5 for playng D. So f Row were ratonal, gven that Col s rratonal, Row would stll play T wth probablty.0. Gven that Col plays both columns wth equal probablty, ths n turn would mean that E(u C ) = 2. Snce f Col were ratonal her expected utlty would be 4, beng rratonal rather than ratonal would hurt her n the PD. Now however modfy the PD to have the followng utlty functons (u R, u C ): [ (0, 0) (6, ) (5, 5) (4, 6) ] (2) Agan the pure strategy (T, R) s the only NE. At that NE, E(u C ) =.0. Now though f Col were rratonal, Row would have expected utlty of 3 for playng T, and of 4.5 for playng D. So f Row were ratonal, gven that Col s rratonal, Row would play D wth probablty.0. Gven that Col plays both columns wth equal probablty, ths n turn would mean that E(u C ) =

5 So by beng rratonal rather than ratonal, Col has mproved her expected utlty from.0 to 5.5. Loosely speakng, such rratonalty by Col allows Row to play a move that Row otherwse wouldn t be able to play, and that ends up helpng Col. Ths s true even though Col would ncrease her expected utlty by actng ratonally rather than rratonally f Row s mxed strategy were fxed (at D). The mportant pont s that f Col were to act ratonally rather than rratonally whle Row s ratonalty were fxed (at full ratonalty), then Col would decrease her expected utlty. Ths phenomenon can be seen as a model of the common real-world scenaro n whch someone acts dumber than they are (by not beng fully ratonal), and benefts by dong so. Smlar phenomena can model other types of real-world bounded ratonalty. To llustrate ths, further modfy our game to have the followng utlty functons: [ (0, 0) (6, ) (5, 5) (0, 6) ] (3) As before, (T, R) s the sole Nash equlbrum of the game. Agan as before, assume that Row s perfectly ratonal. Then Col doesn t beneft from beng rratonal, snce dong that won t nduce Row to flp from T to D. But now say that Col were ant-ratonal, n that she always chooses the strategy that mnmzes E(u c ) (gven the strategy of Row), rather than maxmzes t. Snce R s domnant for Col (.e., gves hgher utlty regardless of Row s move), ths antratonalty means that Col always chooses move L. Ths n turn causes Row to flp from T to D, whch benefts Col (E(u Col ) goes from.0 to 5.0). Loosely speakng, n ths game, gven that Row s ratonal, Col would beneft from beng her own worst enemy,.e., from preferentally choosng whatever move s worst for her gven Row s move. It s mportant to realze that the potental beneft of beng ant-ratonal s not the same as the beneft that non-credble threats can provde n certan extensve form games (, 2). In essence, a player makng a non-credble threat says to her opponent, If you do α, I ll do somethng 5

6 that wll hurt me but wll also hurt you. So you must not do α, and I wll explot that. In contrast, a player wth β = says No matter what you do, I ll do somethng that wll hurt me, and n certan games, derves beneft from that. In fact, such a beneft can arse even f we restrct attenton to subgame perfect NE, whch preclude non-credble threats (see supplemental onlne materals). We can model all of these phenomena wth a sequence of two games. In the frst, ratonalty game, all players decde how ratonal to be. That jont ratonalty choce specfes the subsequent realzed game that the players then play wth one another. It s the NE of that subsequent realzed game that determnes the ultmate jont strategy P(x) of the players. Therefore t s the NE of the realzed game that ultmately provdes the utlty to the players of the ratonalty game. As an example, ndcate the choce of full ratonalty as β = and ant-ratonalty as β =. Let γ (X, X 2, u, u 2 ) be the orgnal, concrete game, where X s player s strategy space (e.g., X Row = {T, D}), and u s her utlty functon. The ratonalty choces made by the two players n the ratonalty game determne how to modfy the concrete game to construct the realzed game: the realzed game assocated wth a par of ratonalty choces {β Row, β Col } s γ (X, X 2, u, u 2 ) where for both players, u = u f β =, and u = u f β =. Any such realzed game γ has a NE jont strategy P γ (x). (Note that for any, n general a change to u changes P γ.) In turn, that strategy determnes E P γ (u ) for every player. Ths s the expected utlty of each player of the ratonalty game. So the goal of each player n the ratonalty game s to choose a ratonalty β such that, gven the ratonalty choce of her opponent, the NE of the assocated realzed game γ maxmzes s expected utlty. (See the supplemental onlne materals for a more detaled formal defnton of ratonalty games.) To llustrate ths, let the concrete γ be the game ndcated n Table 3. If β R = β C =, then γ = γ. So the NE of γ n ths case s (T, R), the NE of γ dscussed above. For ths jont 6

7 ratonalty, the utltes to the players s (6, ). We can do an analogous calculaton of utltes for the remanng three possble jont ratonaltes. For ths concrete game, dong ths generates the followng utlty bmatrx to the players of the ratonalty game for ther four possble jont ratonalty choces: Col ratonalty + Row ratonalty (0, 0) (0, 6) (4) + (5, 5) (6, ) (Note that ths s just a rearrangement of the entres n the concrete game γ.) There s a domnant NE of ths ratonalty game at the pure strategy (+, ). Accordngly, f two humans play the ratonalty game and resultant realzed game rather than the sngle concrete game, then Col wll elect to be ant-ratonal, and Row wll elect to be ratonal; nether would beneft from choosng a dfferent ratonalty, gven the choce of her opponent. Alternatvely, magne that behavor of two speces playng ths game wth each other has been determned va repeated plays of the game conducted under natural selecton pressures. If those pressures acted on genes whose alleles code for the concrete game strateges, then typcally the frequency of those alleles wll evolve to the NE of the concrete game (3,5). However f natural selecton nstead acts on genes whose alleles code for one of the two ratonaltes {, + }, then n general the frequency of those alleles wll evolve to the NE of the ratonalty game. Ths would manfest tself as an equlbrum strategy over the realzed game n whch members of the Col speces always choose L, and members of the Row speces always plays D; an outcome that dffers from the NE of the concrete game. It s mportant to note that when a player commts to some ratonalty, she fxes nether her strategy n the realzed game nor the strateges of her opponents n that game. These strateges are only determned by the realzed game players after all of the ratonalty game players have 7

8 made ther choces. Now modfy our two-player concrete game by allowng each player four possble moves rather than two, and have the utlty functons (u R, u C ) be: (0, 6) (4, 7) (, 5) (4, 4) (, 6) (5, 5) (2, 3) (7, 4) ( 2, ) (3, 2) (0, 0) (5, ) (, ) (6, 0) (, 2) (6, ) (5) A NE of ths game s the jont pure strategy where Row plays her bottom-most move, and Col plays her left-most move. An ant-ratonal equlbrum, where both players try to mnmze ther utlty functons, occurs f Row plays the top-most row and Col plays the rght-most column. The remanng two possble jont ratonaltes of the players correspond to the remanng two entres on the skew-dagonal of the matrx. Ths results n the followng ratonalty game: Col ratonalty + Row ratonalty (4, 4) (3, 2) (6) + (2, 3) (, ) The jont ratonalty (, ) of ths game s Pareto superor to (+, + ),.e., f both players play ant-ratonally rather than ratonally, then both players beneft. Moreover, (, ) s a (domnant) NE of the ratonalty game. At that jont ratonalty, nether player would beneft from changng to ratonal behavor, gven that her opponent were ant-ratonal. Note n partcular that (+, + ) s not a NE of the ratonalty game. So f the players are sophstcated enough to play the ratonalty game wth each other rather than the concrete game, they wll both act ant-ratonally, and wll thereby both beneft. There are also concrete games where the NE of the assocated ratonalty game s (, ) but ths s not optmal for ether player. An example s the followng concrete game: 8

9 (3, 3) (6, 2) (, 0) (4, ) (2, ) (5, 0) ( 2, 2) (3, 3) (, 8) (8, 7) (0, 5) (2, 6) (0, 3) (7, 4) ( 3, 2) (, ) (7) The four possble jont ratonaltes have equlbra lyng on the man dagonal of ths matrx, so the assocated ratonalty game s: Col ratonalty + Row ratonalty (, ) (5, 0) (8) + (0, 5) (3, 3) Ths par of utlty matrces s just the PD ntroduced above (up to rrelevant rescalngs, etc.) wth defect-defect dentfed as (, ), and cooperate-cooperate as (+, + ). So for that concrete game, the players would not beneft by beng sophstcated enough to play the ratonalty game. Rather they would be better off smply playng the (NE of the) concrete game. Many dstrbuted engneered systems can be vewed as games among the dstrbuted subsystems where, loosely speakng, the engneer has the ablty to set some aspects of the utlty functons of the players and/or of how ratonal the players are. Examples nvolvng purely artfcal players nclude dstrbuted adaptve control, dstrbuted renforcement learnng (e.g., such systems nvolvng multple autonomous adaptve rovers on Mars or multple adaptve telecommuncatons routers), and more generally mult-agent systems nvolvng adaptve agents (9 22). In other nstances of such engneered systems some of the players are human bengs. Examples here nclude ar-traffc management (23), mult-dscplnary optmzaton (24, 25), and n a certan sense, much of mechansm desgn, and n partcular desgn of auctons (, 2, 26). The mplcatons of the ratonalty games based on the concrete game n Table 2 suggests that the performance of some of these engneered systems could be mproved f the players 9

10 were mpeded from playng ratonally (e.g., by corruptng ther sensor nput). The ratonalty game assocated wth the concrete games n Table 3 and 5 suggests that some of the players mght even mprove ther performance f they were nduced to always act aganst ther own best nterests (e.g., by approprate transformaton of ther reward sgnals from ther envronment). The focus of the dscusson so far has been on pure strategy equlbra of ratonalty games. At such equlbra, though a player may not be playng ratonally n the realzed game, her ratonalty has a sngle, defnte value. In contrast, when the equlbrum of the ratonalty game s mxed, some players chooses how ratonal to be n the realzed game by randomly samplng ther mxed strateges. Intutvely, such players beneft by beng caprcous, or flghty, and keepng ther opponents on ther toes as to whether they wll be ratonal or not. Such a mxed strategy equlbrum of the ratonalty game arses when the concrete game s the famous Traveller s Dlemma (TD) (27 32). The TD models a stuaton where two travelers fly on the same arlne wth an dentcal antque n ther baggage, and the arlne accdentally destroys both antques. The arlne asks them separately how much the antque was worth, allowng them the answers {2, 3,..., 0}. To try to nduce honesty n ther clams, the arlne tells the travelers that t wll compensate both of them wth the lower of ther two clams, wth a bonus of R for the maker of the lower of the two clams, and a penalty of R for the maker of the hgher of the two clams. To formalze the TD, let Θ(z) be the Heavsde step functon, Θ(z) = {0, /2, } for z < 0, z = 0 and z > 0, respectvely. Then for both players, the utlty functon n the TD concrete game s u (x, x ) = (x + R)Θ(x x ) + (x R)Θ(x x ) where R = 2 s the reward/penalty (for makng a low/hgh clam), x s the monetary clam made by player, and x s the monetary clam made by the other player. The NE of ths game s (2, 2), snce whatever s opponent clams, t wll beneft to undercut that clam by. However n experments (not to menton common sense), ths NE never arses. 0

11 Even when game theoretcans play the TD wth one another for real stakes, they tend to make clams that are not much lower than 0, and almost never make clams of 2. For smplcty, we consder a ratonalty game based on the TD concrete game where both players can only choose ether to be ratonal or rratonal. When both players are fully ratonal, the expected utlty to both s 2,.e., E(u β =, β 2 = ) = 2 for both players. Now say that player s ratonal whle the other player s rratonal. Ths results n the expected utlty ([ E(u x, β = 0) = x 00 y=2 (y 2)] + x + [ 0 y=x +(x + 2) ]) for all of s possble concrete game moves x. The (nteger) maxmum of ths s at x {97, 98}. The assocated expected utlty s E(u β =, β = 0) 49.6 (see supplemental onlne materals). Contnung n ths way gves ratonalty game utlty functons wth the followng (rounded) values: Player 2 ratonalty 0 + Player ratonalty 0 (34.8, 34.8) (53.3, 49.6) (9) + (49.6, 53.3) (2, 2) Ths ratonalty game has two pure strategy NE, (β, β 2 ) = (0, ) and (β, β 2 ) = (, 0). The assocated dstrbuton P(x ) for the frst of these ratonalty NE s unform. The assocated P(x 2 ) nstead has half ts mass on x 2 = 97, and half on x 2 = 98. The two dstrbutons for the other pure strategy ratonalty NE are dentcal, just wth P(x ) and P(x 2 ) flpped. (As an asde, note that f one of the players s rratonal and the other ratonal, t s better to be the rratonal one of the two players rather than the ratonal one.) There s also a symmetrc mxed strategy NE of the ratonalty game, at whch both ratonalty players choose β = 0 wth probablty.78. The assocated margnal dstrbutons P(x ) are dentcal for both s: P(x = 2) 5.8%, P(x = 97) = P(x = 98) 9.5%, and P(x ) 0.8% for all other values of x. (Note that because P(β, β 2 ) s not a delta functon, P(x, x 2 ) P(x )P(x 2 ).) Unformly averagng over these three NE gves a P(x) that s hghly

12 based to large values of x, n agreement wth expermental data. We can do the same analyss for other values of R besdes 2. When R grows, the mxed strategy equlbrum of the ratonalty game places more weght on the persona. Ths makes P(x) become more weghted towards low values. In fact, when R gets larger than 38.2, the two pure strategy NE of the ratonalty game dsappear, and the mxed strategy NE reduces to the pure strategy where both players are fully ratonal. So for such values of R, the players are fully ratonal. These results agree wth expermental data (28) on what happens as R changes. In some real-world scenaros, the ratonalty that every player adopts s made known to all the other players before play of the realzed game. In other stuatons however, each player n the realzed game only knows her own ratonalty, together wth the fact that her opponents are humans. When the ratonalty game has a mxed strategy NE (as does the TD), ths means that no player of the realzed game explctly knows the ratonaltes of any player other than herself. The framework outlned above has to be extended to deal wth such scenaros. Formally, ths extenson s smlar to Bayesan games (, 2). The dea s that each player explots common knowledge (n the game theory sense, of knowng the possble moves of one another, etc.) to nfer the ratonalty mxed strateges of the other players. Ths extenson s presented n the supplemental onlne materals along wth a related one that addresses the possblty of multple ratonalty game NE, as occurs for examples n the TD concrete game. In the scenaros consdered so far the set of personas that a player can commt to before start of the realzed game s some subset of the three ratonaltes {, 0, + }. There s lttle reason to beleve that real human bengs are so severely lmted n the set of personas they can choose among. One slghtly rcher space of personas arses by extendng the concept of ratonalty to all real numbers: f player chooses the ratonalty β R, then she commts to play the mxed strategy P (x ) exp[β E P (u x )] x (0) 2

13 n the realzed game. As an example, the ratonalty β = 0 means that player s rratonal n the realzed game and the lmt β + means she s fully ratonal. (More precsely, one of the NE of the concrete game arses n the realzed game n the lmt of β + for all players (33).) Smlarly β corresponds to ant-ratonalty. The set of values β allowed to player can be any (potentally nfnte) subset of R, and such sets may vary among the players. Gven any vector β of such real-values for all of N players n a game, there s always at least one jont mxed strategy P(x) that satsfes Eq. 0 smultaneously for all players. Such a P s known as a (logt) Quantal Response Equlbrum (QRE) (32, 33, 35, 36). So each β R N specfes a QRE P(x), whch n turn specfes expected utltes for all of the players. We can wrte those expected utltes as E(u β), where ranges over the players. In ths way the QRE concept gves us N mappngs from R N nto R, f : β E(u β). Just as wth the ratonalty games dscussed above, we can vew these N functons f as the utltes for N ratonalty game players, each of whom sets a separate component of the vector β. The ratonalty games dscussed above are just the specal case of such a generalzed ratonalty game where each β s restrcted to some assocated subset of {, 0, }. As an example, consder agan the two-player game wth utlty functons gven n Table 2. If the set of allowed β s all of R for both players, one can show that there s a unque pure strategy equlbrum of the ratonalty game at the par β = (ln(5/2) ɛ, ) where ɛ s arbtrarly small (see supplemental onlne materals). At that β, nether player can mprove her assocated expected utlty by changng β. 2 See (34) for an nvestgaton of the relaton between Eq. 0 and a model-ndependent measure of the degree of ratonalty of a player who adopts a partcular mxed strategy when the other players adopt a specfed partcular (potentally non-equlbrum) mxed strategy. Also see those references for a dscusson of the relatonshp between Eq. 0 and the canoncal ensemble of statstcal physcs. Intutvely speakng, the parameter β n Eq. 0 can be vewed as the recprocal of a temperature. In ths, t s formally true that a ratonal person s cold ; they have low temperature. Smlarly an rratonal person s hot. In optmzaton usually such a temperature s vewed as determnng a player s exploraton / explotaton tradeoff as she searches for what strategy to adopt. (For example ths s the case n smulated annealng.) However games lke those n Table 3 and 5 ndcate temperature plays other roles n optmzaton as well. 2 As an asde, such ratonalty games can have phase transtons, even for concrete games havng two players 3

14 As another example of ths rcher set of personas, we now consder the famous Ultmatum Game (UG) (4, 6, 37 39). The UG s a two-player game where the frst player, the Buyer (B) selects some x [0, ] and offers t to the Seller (S ). S can ether accept or refuse B s offer. If she accepts, S gets a utlty of x, and B gets a utlty of x. If S refuses the offer, both players get a utlty of 0. The NE for ths game s for S to accept whatever offer B makes, and therefore for B to make the mnmal possble offer. In experments though, often S refuses an offer f t s too low, n essence sayng I d prefer nothng to such an nsultngly low offer. In keepng wth ths, typcally B makes an offer substantally hgher than the mnmum. For smplcty, we wll consder a ratonalty game based on the UG concrete game where B s always perfectly ratonal, but S can choose essentally any postve ratonalty value β. Intutvely, f S s suffcently rratonal (.e., β s low enough), then for her probablty of acceptng some partcular offer x to be sgnfcantly greater than /2, that offer must be substantally greater than 0. So f B knows that S has that suffcent rratonalty, B has to make an offer substantally greater than 0 to have a probablty sgnfcantly greater than /2 of gettng any non-zero utlty at all. Snce S prefers offers that are as large as possble, ths means that t s n S s nterests to be suffcently rratonal, and therefore for B to make an offer substantally greater than 0. Carryng through the calculaton more formally, we fnd that at the equlbrum of the ratonalty game S adopts the pure strategy β = Any value of β gves a unque assocated offer value that s optmal for B, whch we wrte as x(β); x(β ) = 0.2. The assocated probablty that S accepts the offer s Whle t s hard to drectly compare these results to expermental data (due to dosyncrases n the protocols n the experments n the lterature), they are broadly wth two moves each. More precsely, the log of the normalzaton constant for the QRE dstrbuton (.e., the log of the QRE analog of a partton functon ) can be dscontnuous as a functon of β (see supplemental onlne materals). Whle n condensed matter physcs phase transtons only occur for nfnte-partcle systems, n game theory they can arse even for only two players. 4

15 consstent wth that data. One explanaton that has been offered for expermental results concernng the UG s that B has a farness bas bult nto her utlty functon. Other explanatons nvolve ad hoc cogntve models that restrct the capabltes of the players. In contrast, the explanaton offered by ratonalty games does not post modfcatons to the utlty functons of the players, nor lmts ther cogntve capabltes. In addton, the focus shfts from B to S, sayng, n essence, that S would beneft by beng spteful and therefore wll be, whch fact B must account for. Note though that n the ratonalty game consdered above, all of S s allowed personas only concern her own expected utlty (va her ratonaltes). Introspecton suggests that n the real world, S sometmes adopts personas that also concern B s expected utlty. (Ths may be why n experments S sometmes rejects offers as too low, n conflct wth the predcton of the ratonalty game consdered above.) For example, a farness bas can be modeled wth such a persona. Ths rases the queston of whether there are games where a player can beneft by adoptng a persona whch nvolves the expected utltes of that player s opponents, just as there are games where she can beneft by adoptng a fnte ratonalty persona. Such personas nvolvng the expected utltes of one s opponents mght mplement many types of other-regardng preference, ncludng altrusm as well as farness bases. If a player benefts by adoptng a persona wth such an other-regardng preference n a partcular game, then that other-regardng preference s actually optmal for purely sel f -regardng reasons. To nvestgate ths, let {u j : j =,... N} be the utlty functons of the orgnal N-player concrete game. Then rather than have the possble personas of player be parameterzed by a set of ratonalty real numbers, {β }, have them be parameterzed by a set of dstrbutons {ρ } each of the form (ρ, ρ2,..., ρn ). By adoptng persona ρ, player commts to playng the realzed game wth a utlty functon j ρ j u j rather than u. So pure selfshness for player s 5

16 ρ j = δ(, j). Altrusm s a ρ j that places probablty mass on more than one j. (Farness s a slghtly more elaborate persona than these lnear combnatons of utltes, e.g., the commtment to play the realzed game wth a utlty functon [(N )u j u j ] 2.) To dstngush them from persona games n general, we refer to persona games where some personas nvolve the (concrete game) utltes of multple playes as other-regardng games. To llustrate ths, consder the two-player two-move concrete game wth the followng utlty functons: [ (2, 0) (, ) (3, 2) (0, 3) ] () There s one pure strategy NE of ths game, at (T, R). Say that both players n the assocated other-regardng game only have 2 possble pure strateges, ρ j δ, j and ρ j δ, j, whch we refer to as selfsh and santly, respectvely. So for example, f Row chooses selfsh whle Col chooses santly, then the equlbrum s (D, L). Ths gves both players a hgher utlty than f both were selfsh. Contnung ths way, we get the followng par of other-regardng player utlty functons: Col ρ E A Row ρ E (, ) (3, 2) (2) A (0, 3) (3, 2) The pure strategy NE of ths other-regardng game s (E, A),.e., the optmal persona for Row s to be selfsh, and for Col s to be santly. Note that both players beneft by havng Col be santly. So for example Row would be wllng to pay up to 2.0 to nduce Col to be santly. Perhaps more surprsngly, Col would be wllng to pay up to 2.0 to be allowed to completely gnore her own utlty functon, and work purely n Row s nterests. In the case of the PD concrete game, other-regardng personas can lead the players n the realzed game to cooperate. For example, say that each player can choose ether to be selfsh, or 6

17 to be chartable, meanng that ρ s unform, so that the player has equal concern for her own utlty and her opponent s utlty. Then for the PD concrete game n Table, the equlbrum of the assocated other-regardng game s for both players to be chartable. That n turn leads them to cooperate n the realzed game. They do ths for purely self-centered reasons, n a game they play only once. Such effects mght account for some of the expermental data showng a substantal probablty for real-world humans to cooperate (40). (A detaled exploraton of other-regardng games can be found n (4).) Dscusson. Persona games hghlght the dstncton between games aganst Nature and games wth other objectve-maxmzng players. In partcular, when the concrete game s a game aganst Nature, the ratonalty equlbrum s always β =, regardless of the detals of the concrete game. Ths s not the case when the concrete game nvolves other players. In addton to havng other players n the concrete game, another necessary condton for a player to adopt a persona other than perfect ratonalty s that she beleves that the other players are aware that she can do that. The smple computer objectve-maxmzng algorthms currently used n game theory experments do not have such awareness. Accordngly, f a human knows she s playng aganst such an algorthm, she should always play perfectly ratonally, n contrast to her behavor when playng aganst humans. Ths dstncton between behavor when playng computers and playng humans agrees wth much expermental data, e.g., concernng the UG game. Typcally that calculatng a persona equlbrum nvolves far more computatonal work than calculatng the equlbra of the assocated concrete game. (For every possble jont persona, one has to calculate the assocated realzed game equlbra, and only then can one calculate the persona game equlbra.) Hence, one would expect persona games only n members of a speces wth advanced cogntve capabltes, who have a lot of nteractons wth other anmals 7

18 that can also play persona games. A member of such a speces who plays persona games well has hgh socal ntellgence. For smlar reasons, one would expect the persona set of any anmal not to be too large. (A large set both ncreases the burden on the player wth that set, and wth the other players she plays aganst.) Computatonal consderatons mght also lead an anmal to use a smlar persona set across those games that she s lkely to encounter n her lfe. Phenomenologcally, such a persona s what s smlar to what s colloqually called a personalty. What happens f the players msconstrue the persona (or more generally persona sets) adopted by one another? Intutvely, one would expect that the players would feel frustrated when that happens, snce n the realzed game they each do what would be optmal f ther opponents were usng that msconstrued persona but they aren t. Ths can be vewed as a rough model of what s colloqually called a culture gap, or a generaton gap, as the case mght be. (See supplemental onlne materals for futher speculaton on anthropologcal and evolutonary mplcatons of persona games.) There are many open ssues, both expermental and formal, assocated wth the current persona game framework (see the supplementary onlne materal). Future extensons of the persona game framework wll nvolve ntegratng t wth prevous work on bounded ratonalty nvolvng computatonal restrctons, repeated games n whch there are reputaton effects, games on networks, punshment, etc. But even based only on the prelmnary results recounted above, t appears that persona games may provde the formalzaton, recently called for by Basu (27), of the dea of behavor generated by ratonally rejectng ratonal behavor... (whch s necessary) to solve the paradoxes that plague game theory. Supplemental Onlne materal 8

19 In these supplemental onlne materals we present formal defntons of persona games, and then make the calculatons mentoned n the text concernng the UG, TD, and phase transtons. As mght be expected gven the breadth of scenaros that persona games can model, ther formal defnton takes some care. We start by ntroducng our notaton. Notaton. Defne N {, 2,...}, fx a postve nteger N, and defne N as the ntegers {,..., N}. We wll occasonally use curly brackets to ndcate a set of ndexed elements where the ndex set s mplct, beng all N. For any set Z, Z ndcates the cardnalty of Z. We use P(.) to ndcate a probablty dstrbuton (or densty functon as the case mght be). An upper case argument of P(.) ndcates the entre dstrbuton (.e., the assocated random varable), and a lower case argument ndcates the dstrbuton evaluated at a partcular value. When defnng a functon the symbol ndcates that the defnton holds for all values of the lsted arguments. So for example, f (a, b) dc r(a)s(b, c) means that the defnton holds for all values of a and b. We wll use the ntegral symbol wth the measure mplct. So for example for fnte X, dx X mplctly uses a pont-mass measure and therefore means a sum. Smlarly, we wll be loose n dstngushng between probablty dstrbutons and probablty densty functons, usng the term probablty dstrbuton to mean both concepts, wth the context makng the precse meanng clear f only one of the concepts s meant. The unt smplex of possble dstrbutons over a space Z s wrtten Z. The nteror of a set A (wth mplct topology) s wrtten A 0, e.g., the nteror of Z s 0 Z. Gven two spaces A, B, we wrte A B to mean the unt smplex over the Cartesan product A B. Smlarly, A B ndcates the set of all functons from B nto A,.e, the set of all condtonal dstrbutons P(A B). Gven a set of N fnte spaces, {X }, we wrte X N X and for any x X, use x to ndcate the th component of x. We use a mnus sgn before a set of subscrpts of a vector 9

20 to ndcate all components of the vector other than the ndcated one(s). For example, we wrte X j N : j X j and use x to ndcate the ordered lst of all components of x except for x. We defne X as the set of dstrbutons n q X that are product dstrbutons,.e., that are of the form q(x) N q (x ). Smlarly, we defne X as the set of product dstrbutons n X. In the usual way, for any q X and N, we defne the dstrbuton q X as j N : j q j. Fnally, gven any fnte Z and p Z, we wrte the Shannon entropy of p as S (p) Z p(z)ln[p(z)]. Objectve games. Say we are gven a set of N (pure) strategy fnte spaces, {X }. Then we refer to any functon that maps X R as an objectve functon for X. As an example, for a fxed utlty functon u : X R, the expected value of u under q X, E q (u), s an objectve functon. Any par of a set of fnte strategy spaces and an assocated set of one objectve functon for each stategy space s called a (strategc form) objectve game. Often we leave the ndces on the elements of an objectve game mplct, for example referrng to (X, U) rather than ({X }, {U }). Objectve games provde a clean way to model scenaros n whch some of the players use a conventonal (expected utlty) best response, and the others nstead use a quantal response. Note n ths regard that one cannot model an rratonal player who always plays a unform mxed strategy as an expected utlty best-responder for some approprate utlty functon. In partcular, f a utlty functon s ndependent of x, then any dstrbuton over x can arse at equlbrum, dependng on the strateges of the other players; optmzng ths utlty functon does not always pck out the unform dstrbuton assocated wth rratonalty. On the other hand, a player wth a free utlty objectve (.e., a quantal response player) where β = 0 always plays a unform dstrbuton over x, exactly as desred. Best responses n objectve games, extensve form objectve games, Nash equlbra of ob- 20

21 jectve games, and (tremblng hand) perfect equlbra of objectve games are defned n the obvous way. We wrte the Nash equlbra of an objectve game (X, U) as E(X, U) {q X : N, q X, U(q, q ) U(q, q )}. (3) We wll sometmes be loose wth the termnology and refer to player as makng move q X, even though her pure strategy space s X, not X. As an example, let each objectve functon U be an expected utlty E q (u ). Then E({X }, {U }) s the set of Nash equlbra of the N-player noncooperatve game wth pure strategy spaces {X } and utlty functons {u }. We call such an objectve game a utlty game. As another example, assocate wth each player a non-zero real number β. Defne each player s objectve functon U as the free utlty, E q (u )+β S (q ). Then E(X, U) s the set of (logt) QRE s of the N-player game wth pure strategy spaces {X } and logt exponents {β } (20, 33, 42, 43). 3 As a fnal example, an N-player Bayesan objectve game s any trple (X, A, P A ) where {X } s a set of N fnte strategy spaces, each of the N separate A s a set of objectve functons wth doman X, and P A s a product dstrbuton over A, P A (a) N P A (a ). For smplcty, n ths paper we wll always take any A to be fnte. An equlbrum of a Bayesan objectve game s any set of condtonal dstrbutons {P (X A ) X A : N } where for all players and assocated objectves a such that P A (a ) 0, there s no alternatve condtonal dstrbuton P (X A ) for whch da P (a A )a [P (X a )P (X a )] > da P (a A )a [P (X a )P (X a )] 3 There are at least two ways to ncorporate refnements of Nash equlbra nto the defnton of objectve games. Frst, many refnements of objectve game equlbra can be expressed smply by modfyng the objectve functons. For example, f for every player, U (q) E q (u ), then we can mpose many refnements by modfyng U (q) to have some value less than mn x X [u (x)] f q volates the refnement condton. As an alternatve, one could defne an equlbrum concept as a correspondence takng any par (X, U) to a subset of X, and then have the specfcaton of the equlbrum concept be part of the specfcaton of an objectve game. E s one possble choce for such a concept, but by choosng other ones, we could have the objectve game equlbra correspond be any desred refnement of the Nash equlbra. 2 (4)

22 So long as all objectve functons a obey certan bengn propertes, Kakutan s fxed pont theorem can be appled to the assocated best-response correspondences n the usual way to establsh that there s always such an equlbrum. In partcular, f for all players each a A s the expected value of some utlty functon over X, then Eq. 4 reduces to the defnton of a conventonal Bayesan game. So for such sets {A }, there s always an equlbrum of the assocated Bayesan objectve game. Persona sample games. The foregong provdes the machnery for us to defne and analyze several knds of persona game. Here we present four such knds of game. (As usual, whch type of game to use to model a partcular experment depends on the precse detals of that experment.) Frst, defne a persona world as any trple ({X : N }, {U : N }, {A : N }) (5) where {X } s a set of N fnte strategy spaces, {U } s an assocated set of N objectve functons wth doman X, and each A s a set of objectve functons wth doman X. For smplcty, n ths paper we wll always take any A to be fnte. We refer to an a A as a persona of the th player. Note that any such persona s a mappng from χ nto R. 4 We refer to any N-tuple a = (a,..., a N ) A A... A N of every player s persona as a jont persona of the players. We wrte A to mean the members of A that are product dstrbutons,.e., that are of the form P A (a) = N P A (a ). We defne A smlarly. We also defne X A to mean the members of X A that are product dstrbutons,.e., that are of the form P(x a) = N P(x a ) for all a A, and make a smlar defnton for X A. 4 At the expense of more notaton, we could extend the domans of the objectve functons {U } to be X,A. Ths would allow us to model scenaros n whch a player of the persona game has a pror preferences over her possble personas. 22

23 Gven a persona world and a player N, an assocated summarzer w s a real-valued measure of s desre for the realzed game specfed by any possble jont persona a A. In ths paper we restrct attenton to summarzers w that can be expressed as w (a) w (Q(a)) where Q s a mappng from a nto the assocated set of equlbra of (X, a) for some pre-fxed equlbrum concept. An example of such a w s the unform average of the values of U evaluated at the NE of the realzed game (X, a). We refer to such a summarzer as the equlbrumaveragng summarzer. A varaton, motvated by the entropc pror and ts use n predctve game theory (42), s to weght the value of U at each NE q n ths average by e αs (q) /Z, where α s the entropc pror constant and Z s a normalzaton constant. More generally, w (a) could also reflect concerns lke the computatonal cost to of evaluatng those NE. As shorthand, we often abbrevate ({X : N }, {U : N }, {A : N }, {w : N }) as (X, U, A, w). From now on, we mplctly assume that for any persona world that we wll consder, (X, U, A), for every a A, E(X, a) s non-empty. (As an example, ths s true when ether every persona objectve functon a s ether an expected utlty or a free utlty.) Accordngly, gven a persona world (X, U, A) and summarzer w, every jont persona a specfes a real number for each player accordng to the rule W (a) = w [E(X, a)]. (6) As an example, say that each a A s the free utlty for some assocated parameter β a and some fxed utlty u. (So A s parameterzed by a set of non-zero real numbers {β a : a A }). Then the jont persona of the players can be vewed as an N-tuple of logt exponents, n the sense that the equlbrum for the realzed game assocated wth a jont persona a s a QRE wth logt exponents gven by β a (β a : N ). Say we also have each each U (q) be the expected value under q of a utlty functon u : X R. Also have w be the equlbrum-averagng summarzer. Then W (a) s the unform average of the expected utltes of player over the 23

24 QRE s of the game for the vector of logt exponents β a. The summary persona sample game for persona world (X, U, A) and summarzer w s the N-player (noncooperatve) utlty game where each player s space of pure strateges s A and her utlty functon s W. Snce we are assumng that for every a A the assocated strategc form game (X, a) has a NE, the summary persona sample game s a well-defned strategc form noncooperatve game, and therefore always has a NE. Such a NE s a product dstrbuton over jont personas, P A (a) N = PA (a ). We refer to (X, U) as a concrete game, and any (X, a) as a realzed game. Summary persona sample games based on any partcular refnement of the NE concept for the realzed game are defned n the obvous way. Note that f each w s an equlbrum-averagng summarzer, then at any equlbrum P A there are two averages defnng the assocated expected value of each W : The average over a accordng to P A (a), and then for each a, the average over all NE q(x) for that a. In addton, f U s an expectaton of a utlty functon u (x), then for each NE q there s yet another average, of the values of u (x) dstrbuted accordng to q(x). Typcally a summary sample equlbrum P A s nvarant under any affne transformaton of A for any N. To llustrate ths, wrte such a transform as a Ca + D a A, and consder agan the example above where each a A s a free utlty. Applyng our affne transform to such an A s equvalent to multplyng both u and [β a ] for each a A by C, and then addng D to u. Dong ths won t affect the value of e β a [E q (u x ) E q (u x ) for any x, x, q. Accordngly such a transform won t affect the equlbrum q(s) assocated wth any jont set of free utltes, a,.e., t won t affect E(X, a) for any jont persona a. Therefore t won t affect any functon W j, and so won t change the summary sample equlbrum P A. (The other types of persona equlbra dscussed below also typcally obey such an nvarance under an affne transform of any A.) Just as persona games are a meta verson of concrete games, so one could have a meta 24

25 verson of a persona game. Ths would mean no longer assumng that each player s perfectly ratonal n the persona game, but allowng other personas for the play of that game as well. In partcular, f the persona set of each player s the same sze as her move space X, then the persona game consttutes a dual verson of the concrete game. One can nvestgate how teratvely takng the dual of a dual of such a game moves one through the space of all games. Say we have an expermental scenaro n whch, ntutvely speakng, a player s able to adopt a persona that conssts of an objectve functon a combned wth the restrcton that can only choose a strategy from some set X X, no matter what personas are adopted by the other players. We can model such scenaros as summary persona sample games. To do ths, we redefne the value of a (q) for all q X such that q (x ) 0 for some x X. The only thng requred of ths redefnton s that t ensures that a (q) < a (q ) for every q X that obeys q (x ) = 0 x X. (Ths redefnton mplctly assumes that the orgnal functon a s bounded.) Such a redefnton means that persona a wll place zero probablty mass on x n the game (X, (a, a )), no matter what dstrbuton q the other personas a jontly choose. The most natural way to mplement a summary persona sample game for a persona world (X, U, A) and equlbrum-averagng summarzer w s wth a two-step process. Frst, the N players play the persona game, based on common knowledge n the usual way. Ths produces a Nash equlbrum,.e., a product dstrbuton over jont personas, P A (a). Next, that Nash equlbrum s sampled to produce a jont persona, a. At the end of ths frst step, each player adopts persona a. The resultant jont persona s then made known to all the players (hence the name sample persona game ), so that that jont persona becomes common knowledge. (See the secton below on communcaton of personas and natural selecton.) In the second step, n the usual way the N players play a realzed objectve game wth ther objectve functons set by a : (X, a ). That produces a set of possble equlbrum jont product dstrbutons E(X, a ). Fnally, each player receves payoff w (E(X, a ). Typcally, ths payoff 25

26 s actually an expectaton value, over all q E(X, a ), of the assocated value U (q). The goal of each player n such a persona game s to choose P A(a ) so that, gven the choces P A (a ) of the other players, the expected value of the payoff she receves at the end of the second step s as large as possble. Persona dstrbuton games. In the verson of the two-step process presented above, each player commts both to adopt her persona choce n the second step, and to communcate that persona choce to all the other players just before the second step. It s presumed that these two commtments are common knowledge. (Ths common knowledge s n addton to the usual common knowledge of the objectve functons and persona sets of all the players.) However when there s only one equlbrum P A and t s a pure strategy, these two commtments can be combned nto a sngle, weaker commtment. For such a stuaton, say that each player only commts to adopt that persona choce specfed n the persona sample equlbrum P A, whatever that persona s. As before, assume that all commtments are common knowledge. Also as before, assume that the objectve functons and persona sets of all the players are common knowledge. Ths means that every player j can calculate the equlbrum persona a of player. Accordngly, ths mnmal addtonal common knowledge beyond the objectve functons and persona sets that commts to adopt her equlbrum persona allows the other players to deduce what persona player wll adopt n the second step. Importantly, they get ths nformaton of what persona player wll adopt wthout any explct communcaton from player of what persona she adopts. The stuaton when the equlbrum P A s not a pure strategy s more subtle. In such stuatons, our addtonal common knowledge would only tell the other players what dstrbuton P A commts to samplng, not what actual sample of that dstrbuton s generated. Ths means that those other players would have to play a Bayesan best response to P A. We refer to such 26

27 a scenaro where the players all commt to play the Bayesan objectve game wth P A set to an equlbrum of the summary persona sample game as a summary persona dstrbuton game. We refer to the Bayesan objectve game arsng n a summary persona dstrbuton game as a realzed (Bayesan) game. Note that there s no role for a summarzer n the realzed Bayesan game that s played after the jont persona dstrbuton s set n a summary persona dstrbuton game. (The summarzer s only used to set persona dstrbutons, wth no role once those dstrbutons are set.) Also, note that f the equlbrum P A of a summary persona dstrbuton equlbrum s a pure strategy, then the realzed Bayesan game s just a conventonal objectve game, where players have fxed objectve functons. That means that the entre two-step process of the summary persona dstrbuton game s equvalent to the summary sample game. (So n partcular, all the examples of persona games n the text nvolvng pure strategy P A s are both summary persona sample and summary persona dstrbuton equlbra.) It s only when where there are mxed persona equlbra of a summary persona sample game that there may be a dfference n outcomes between that game and the assocated summary persona dstrbuton game. A summary persona dstrbuton game can be vewed as a model of many dfferent physcal scenaros. One s where all N players thnk they are playng a summary persona sample game, and determne ther jont dstrbuton P A accordngly. Each player then samples P A, to get an assocated persona a. However at the next, persona revelaton step, the players all balk, and refuse to communcate ther personas. (Or alternatvely, the expermenter prevents them from communcatng ther personas). Accordngly, all the players have to play the realzed Bayesan game gven by P A. As another example, say that before play each player communcates to all other players a sngle bt: whether she wll or wll not adopt the summary persona sample equlbrum P A. (Implctly, to not adopt that equlbrum P A means she nstead adopts the full ratonalty per- 27