Download - A COGNITIVE HIERARCHY MODEL OF GAMES* Most theories of

A COGNITIVE HIERARCHY MODEL OF GAMES

COLIN F CAMERER

TECK-HUA HO

JUIN-KUAN CHONG

Players in a game are ldquoin equilibriumrdquo if they are rational and accuratelypredict other playersrsquo strategies In many experiments however players are notin equilibrium An alternative is ldquocognitive hierarchyrdquo (CH) theory where eachplayer assumes that his strategy is the most sophisticated The CH model hasinductively defined strategic categories step 0 players randomize and step kthinkers best-respond assuming that other players are distributed over step 0through step k 1 This model fits empirical data and explains why equilibriumtheory predicts behavior well in some games and poorly in others An average of15 steps fits data from many games

I INTRODUCTION

Most theories of behavior in games assume that playersthink strategically meaning that they form beliefs by analyzingwhat others might do and choose rational responses given theirbeliefs But these assumptions by themselves cannot preciselypredict outcomes This is because players can act rationally giventheir beliefs but have mistaken beliefs about what others will doThus game theorists add the assumption that players are mutu-ally consistent that is each playerrsquos belief is consistent with whatthe other players actually do Taken together mutual rationalityand mutual consistency define equilibrium

In many real-world games however such as the stock mar-ket some players believe incorrectly and overconfidently thatthe other participants are not doing as much thinking as theythemselves are In these situations the players are not in equi-librium because some playersrsquo beliefs are mistaken

In his book General Theory of Employment Interest andMoney Keynes [1936] likens the stock market to a newspaper

This research was supported by NSF grants SES-0078911 andSES-0078853 Thanks to C Monica Capra Haitao Cui Paul Glimcher and RogerMyerson Sara Robinson was a helpful wordsmith Former math nerd MatthewRabin directed our attention to the golden ratio Meghana Bhatt Ming Hsu Ye LiBrian Rogers and Lisa Wang provided excellent research assistance Usefulcomments were received from seminars at University of California at BerkeleyCalifornia Institute of Technology University of Chicago Carnegie Mellon Uni-versity Columbia University Harvard University New York University Univer-sity of Pittsburgh Stanford University the Nobel Symposium in Sweden (Decem-ber 2001) and from four referees and an unusually helpful editor (EdwardGlaeser)

copy 2004 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics August 2004

861

contest in which people guess which faces others will judge to bethe most beautiful ldquoIt is not the case of choosing those which tothe best of onersquos judgment are really the prettiest nor even thosewhich average opinion genuinely thinks the prettiest We havereached the third degree where we devote our intelligences toanticipating what average opinion expects the average opinion tobe And there are some I believe who practice the fourth fifthand higher degreesrdquo [p 156]

The essence of Keynesrsquos observation is captured in theldquobeauty contestrdquo game in which players are asked to pick num-bers from 0 to 100 and the player whose number is closest to 2frasl3of the average wins a prize Equilibrium theory predicts eachcontestant will reason as follows ldquoEven if all the other playersguess 100 I should guess no more than 2frasl3 times 100 or 67Assuming that the other contestants reason similarly however Ishould guess no more than 45 rdquo and so on finally concludingthat the only rational and consistent choice for all the players iszero

When the beauty contest game is played in experimentalsettings the group average is typically between 20 and 35 Ap-parently some players are not able to reason their way to theequilibrium value or they assume that others are unlikely to doso If the game is played multiple times with the same group theaverage moves close to 0

There are other games where players are surprisingly closeto equilibrium however Consider a stylized business entry gamein which n players decide simultaneously whether to enter amarket with known demand d where d n The payoffs aresuch that players prefer to enter if a total of d or fewer playersenter and they prefer to stay out otherwise Equilibrium theorypredicts the total amount of entry will be very close to the demandd and this is exactly what happens even in one-shot experimentswhere the usual forces that might lead to equilibration (such aslearning) have not yet had a chance to operate

In this paper we propose an alternative to equilibrium the-orymdasha cognitive hierarchy modelmdashthat explains empirical behav-ior in both of these games In cognitive hierarchy theories eachplayer believes he understands the game better than the otherplayers Specifically CH models posit decision rules that reflectan iterated process of strategic thinking (see Binmore [1988])The iteration formalizes Seltenrsquos [1998 p 421] intuition that ldquothenatural way of looking at game situations is not based on

862 QUARTERLY JOURNAL OF ECONOMICS

circular concepts but rather on a step-by-step reasoningprocedurerdquo

The CH model consists of iterative decision rules for playersdoing k steps of thinking and the frequency distribution f(k)(assumed to be Poisson) of step k players The iterative processbegins with ldquostep 0rdquo types who do not assume anything abouttheir opponents and merely choose according to some probabilitydistribution (for simplicity we assume uniform) ldquoStep krdquo think-ers assume that their opponents are distributed according to anormalized Poisson distribution from step 0 to step k 1 thatis they accurately predict the relative frequencies of playersdoing fewer steps of thinking but ignore the possibility that someplayers may be doing as much or more Step 2 players of thebeauty contest game for example assume the other players are acombination of step 0 players (whose average guess is 50) andstep 1 players (who guess 2frasl3 times 50)

A Poisson distribution is described by a single parameter which is the mean and variance1 In 24 beauty contest data setsthe median estimate is 161 This value explains why theconvergence process stops at an average around 30 in the beautycontest game rather than converging to the equilibrium of zeroOur model with a similar value also offers an explanation ofthe ldquoinstant equilibrationrdquo that occurs in business entry gamesIndeed values of between 1 and 2 explain empirical results fornearly 100 games suggesting that assuming a value of 15 couldgive reliable predictions for many other games as well

The paper is organized as follows The next section describesthe model Section III collects some theoretical results Section IVreports estimation of the parameter from six classes of gamesSection V explores the ldquoeconomic valuerdquo of CH and other theoriesSection VI notes how the Poisson-CH model can help account fortwo patterns of broad economic interest-speculation and moneyillusion Section VII concludes and sketches future researchMore details are in our longer paper [Camerer Ho and Chong2002]

1 Future work could endogenize the distribution f(k) based on trade-offsbetween benefits and costs of thinking harder

863A COGNITIVE HIERARCHY MODEL

II THE POISSON COGNITIVE HIERARCHY (CH) MODEL

IIA Decision Rules

Denote player irsquos jth strategy by s ij and assume that i has

finitely many (mi) strategies The iterative rules for our modelstart with 0-step players who choose according to a probabilitydistribution that is not derived from strategic thinking A conve-nient special case used in this paper is uniform randomization (aplaceholder assumption which could easily be relaxed in laterresearch2) Assuming uniform randomization the 0-step think-ersrsquo choice probabilities are P0(s i

j ) 1mi jDenote a k-step playerrsquos belief about the proportion of h-step

players by gk(h) We assume that players doing k 1 steps donot realize that others are using more than k steps of thinking(that is gk(h) 0 h k 1) This is plausible because thebrain has limits (such as working memory in reasoning throughcomplex games) and also does not always understand its ownlimits We also assume that people are overconfident and do notrealize there are others using exactly as many thinking steps asthey are (ie gk(k) 0) This is consistent with psychologicalevidence of persistent overconfidence about relative skill in manydomains (eg Camerer and Lovallo [1999]) Both assumptionsimply for example that a 1-step player optimizes against per-ceived random response3

We assume that k-step players have an accurate guess aboutthe relative proportions of players who are doing less thinkingthan they are They normalize these actual frequencies to formtheir beliefs about the competition so that gk(h) f(h)yenl0

k1 f(l )

2 In maximum-likelihood estimation algorithms find a value of the modelfree parameter (in this case ) which implies predicted probabilities that maxi-mize the product of the predicted likelihoods of all the strategies that are actuallychosen by the subjects If there is any chosen strategy that is predicted to havezero probability then the product of all the likelihoods is zero This is a problembecause one parameter value might yield much more accurate predictions thananother parameter value but if the more accurate model includes a single zeroprobability the product is zero Assuming that 0-step thinkers randomize acrossall possible strategies means that the predicted probability of each strategy is atleast f(0)mi (if there are mi strategies) Therefore no chosen strategies have zeropredicted probability and the zero-likelihood problem does not arise Readersmore familiar with game theory will appreciate that having all strategies chosenwith positive probability also solves two familiar theoretical problemsmdasheliminat-ing noncredible threats (since all threats are ldquotestedrdquo by randomizing 0-stepthinkers) as subgame perfection does and eliminating ad hoc rules for Bayesianupdating after zero probability events (since all events have probability greaterthan zero)

3 This is the familiar ldquoprinciple of insufficient reasonrdquo of Laplace (see BanksCamerer and Porter [1994] and Haruvy and Stahl [1998])


h k This specification exhibits ldquoincreasingly rational expec-tationsrdquo As k increases the total absolute deviation between theactual frequencies f(h) and the beliefs gk(h) shrinks This alge-braic property implies that as k grows large players doing k andk 1 steps of thinking will in the limit have the same beliefsmake the same choices and have the same expected payoffsThus for a large k there is no marginal benefit for a k-step playerto think harder This could prove useful for establishing limitedthinking through some kind of cost-benefit analysis (see for ex-ample Gabaix and Laibson [2000] Gabaix et al [2003] andChen Iyer and Pazgal [2003])

Denote another player irsquos strategy by s ij and player irsquos

payoffs from choosing s ij when the other player chooses s i

j byi(s i

j s ij ) Given the k-step thinkerrsquos beliefs the expected payoff

to a k-step thinker from choosing strategy s ij is Ek(i(s i

j)) yenj1mi

i(s ij s i

j )yenh0k1 gk(h) Ph(s i

j ) For simplicity we assume thatplayers best-respond (Pk(si) 1 iff si argmax s i

j Ek(i(s ij)))

and randomize equally if two or more strategies have identicalexpected payoffs4 Given a specific distribution f(k) the modelcan be solved recursively starting with 0-step player behaviorand iterating to compute P1(s i

j) P2(s ij) (In practice we

truncate the recursion at a k large enough that the remainingfrequencies f(k) for k k are tiny)

Our Poisson-CH model has some distinct advantages overalternative CH models such as making gk(k 1) 1 that isk-step players think all others do only k 1 steps of thinking (seeNagel [1995] Stahl and Wilson [1995] Ho Camerer and Weigelt[1998] Costa-Gomes Crawford and Broseta [2001a] and Costa-Gomes and Crawford [2004]) This alternative model fits dataabout as well as our specification but exhibits increasingly irra-tional expectationsmdashie gk(h) gets farther from the true f(h) ask grows rather than closermdashand makes implausible predictions

4 Allowing stochastic response gives rise to other models (and could easily beincorporated into CH) In quantal response equilibrium [Rosenthal 1989McKelvey and Palfrey 1995 1998] playersrsquo beliefs and choices of other players areconsistent are stochastic The Poisson CH model retains best-response (except for0-step thinkers) and weakens equilibrium (ie belief-choice consistency) QREretains equilibrium but weakens best-response Weiszacker [2003] allowsthe degree of stochastic response i and beliefs about stochastic response of othersj to be different (i and j 0 is 1-step thinking) Caprarsquos ldquothinkingtreerdquo [1999] uses one parameter that simultaneously weakens best-response andequilibrium


in some games5 Assuming that players respond stochasticallyinstead of best-responding is obviously a plausible alternativetoo but requires an extra parameter and generally improves fitonly a little in most games we have studied

Another possibility is to assume that k-step players realizethere are other k-step thinkers ( gk(k) 0) Self-awareness ofthis sort is a step away from the goal of creating a precisedisequilibrium theory because it imposes consistency of beliefsand choices within each group of same-step thinkers Such amodel is also more difficult to solve since finding a solutionrequires finding a fixed point at each step of thinking Moreoverwhen evaluated relative to the gk(k) 0 specification for fivedata sets (in our working paper) gk(k) 0 achieves a worse fit

IIB The Distribution f(k)

One way of getting a rough idea of a natural distribution ofthinking steps f(k) is to let f(0) f(1) f(k) be free parametersup to some reasonable k then use data to estimate each f(k)separately using maximum likelihood (cf Stahl and Wilson[1995] Ho Camerer and Weigelt [1998a] and Bosch-Domenechet al [2002]) Estimation of this sort reveals substantial frequen-cies of levels 0ndash2 (see our working paper) and medians of 1ndash26

To determine a precise parametric distribution f(k) we firstoutline a list of properties such a distribution should have sincethe thinking steps are integers a discrete distribution of f(k) isnatural (see also Stahl [1998]) The decision rules described abovealso require more and more steps of computation as k risesbecause a k-step thinker does all the computations the lower-stepthinkers do and then combines the results to calculate her ownexpected payoffs If this process is sharply constrained by work-ing memory it is plausible that as k rises fewer and fewer

5 In the entry game described in more detail in subsection IIIC below thegk(k 1) 1 specification leads to cycles in which e(id) entry functions predictentry for d 5 and staying out for d 5 for i even and the opposite pattern fori odd Averaging across these entry functions gives a step function E(kd) with asingle step at d 5 This is too simple because entry frequencies are smoothlymonotonic across d rather than jumping at d 5

6 This fact is consistent with the last part of Keynesrsquos passage on newspaperbeauty contests and the stock market ldquothere are some I believe who practise thefourth fifth and higher degrees [of reasoning about reasoning]rdquo Keynesrsquos word-ing suggests he believed that few investors do more than three steps an intuitioncorroborated by these experimental estimates 50 years later


players do the next step of thinking beyond k7 A reduced-formway to express this constraint is that the relative proportionf(k)f(k 1) declines with k If this decline is captured byassuming that f(k)f(k 1) is proportional to 1k then f(k) is thePoisson distribution f(k) ekk which is characterized byone parameter (both its mean and variance) Values of can befurther pinned down by restrictions on particular f(k) values(eg if k 1 is the mode then (12))8

We focus on the one-parameter Poisson distribution for f(k)because the simpler one-parameter Poisson form fits almost aswell as a seven-parameter model (with frequencies f(k) up to k 7)mdashallowing each f(k) to be independent results in less than a 1percent decrease in log likelihoodmdashin four of the five data sets weexamined The Poisson model is also easier to compute and esti-mate and easier to work with theoretically (see Section III)

III SOME THEORETICAL PROPERTIES OF THE POISSON-CH MODEL

The combination of optimizing decision rules and the one-parameter Poisson structure makes the Poisson-CH model rela-tively easy to work with theoretically This section illustratessome of its properties

IIIA Dominance-Solvable Games

When f(k) is Poisson-distributed the relative proportions oftypes one step below and two steps below a k-step thinker f(k 1)f(k 2) (k 1) puts overwhelming weight on the k 1types if is very large (ie k ) In that case a k-step thinkeracts as if almost all others are using k 1 steps This property ofthe Poisson distribution provides a simple way to link thinking

7 Devetag and Warglien [2003] report evidence that working memory playsa role in strategic thinking They measure the amount of working memory usinga classic ldquodigit spanrdquo task (ie how many digits a person can remember from along string) Working memory is modestly correlated with the tendency toeliminate iteratedly dominated strategies Other evidence of limited thinking isreported by Hedden and Zhang [2002]

8 If f(1) is maximized compared with the neighboring frequencies f(0) andf(2) or if 0- and 2-step thinking are equally common then 2 If f(0) f(1) 2f(2) then (5 1)2 1618 a remarkable constant known as theldquogolden ratiordquo (see Livio [2002]) The golden ratio is the limit of the ratios ofadjacent numbers in the Fibbonaci sequence It is often used in architecturebecause rectangles with golden ratio proportions are aesthetically pleasing It alsooccurs in spiral patterns of seashells and hawks circling their prey None of thesedifferent assumptions (and resulting ) are more compelling than the others Theyjust show how an exact value can be derived by adding a simple restriction to aone-parameter distribution


steps to iterated deletion of dominated strategies9 First note that1-step thinkers will never choose weakly dominated strategiesbecause those strategies are never best responses to the randomstrategies of 0-step types10 Now assume that is very large Then2-step thinkers act as if they are playing a mixture of (almost) all1-step thinkers who have deleted weakly dominated strategiesand a small percentage of 0-step thinkers who are random These2-step thinkers will not play strategies which are strictly domi-nated or strategies which are weakly dominated after deletingweakly dominated strategy play by others This logic can beextended to iteratively eliminate as many dominated strategiesas one likes because k-step thinkers will act as if almost all otherthinkers are one step below them (ie gk(k 1) 1 13) whenk is much smaller than

Another important property of our CH model is if a k-stepthinker plays a (pure) equilibrium strategy then all higher-stepthinkers will play that strategy too11 This means that once a typek reaches a pure equilibrium strategy all higher types will play ittoo

Thus as 3 the prediction of the Poisson-CH model willconverge to any Nash equilibrium which is reached by finitelymany iterated deletions of weakly dominated strategies It is notgenerally true however that CH converges to Nash in all gamesas 3

The Poisson-CH model makes an interesting predictionabout the beauty contest games that Nash equilibrium does notIn beauty contest games with two or more players the game isdominance-solvable and has a unique Nash equilibrium How-ever the two-person beauty contest game is special because it canbe solved by one step of weak dominance In the two-person game

9 This means first eliminating dominated strategies then eliminating anyremaining strategies which are dominated (assuming the strategies eliminated inthe first round will not be played) and so on Note that different outcomes mayresult depending on whether the eliminated strategies are strictly dominated (iealways yields a lower payoff than another strategy) or weakly dominated (ienever yields a better payoff than another strategy and sometimes yields a lowerpayoff)

10 This property holds even if 0-step thinkers do not randomize uniformlyas long as all their strategy choices have strictly positive probability

11 Simple proof the k-step thinker plays the equilibrium strategy call it seagainst a perceived mixture of types 0 to k 1 The k 1-step thinker faces aperceived mixture of types 0 to k 1 (with relative weight yenh0

k1 f(h)yenh0k f(h)

and type k (with relative weight f(k)yenh0k f(h)) But by definition se is a best-

response to the mixture of types 0 to k 1 and a best response to krsquos play of se(since it is a pure equilibrium strategy) By linearity of the expected payoffs se istherefore a best-response to the mixture of types from 0 to k 1 and type k


one player will always be high and one low and 2frasl3 times theaverage will be closer to the lower playerrsquos number Thereforerational players want to choose the lowest number possiblemdashzero In the Poisson-CH model (with any distribution f(k)) ap-plied to the two-person game all players using one or morethinking steps will choose 0 (ie the data should consist of ap-proximately 1 f(0) players choosing exactly 0) This is not truein the three-player game a smart player wants to choose anumber between the other two numbers if they are sufficiently farapart In experiments by Grosskopf and Nagel [2001] and newresults we report below there are more choices of 0 in two-playergames than in three-player games although not nearly as manyas the Poisson-CH model predicts

IIIB Coordination Games

Many interesting games and models of the macroeconomyhave multiple equilibria (eg Cooper [1999]) This raises animportant question of how players or an entire economy cancoordinate on an equilibrium and which types of equilibria aremost likely to arise A large game theory literature on ldquorefine-mentsrdquo has struggled with the problem of how to add furthermathematical restrictions in order to refine or limit the number ofplausible equilibria The holy grail being sought in this scientificprocess is a definition that would guarantee existence of a uniquerefined type of equilibrium No such definition has been discov-ered Our CH model goes in an opposite direction Since the CHmodel allows players to have incorrect beliefs about each other itcan be seen as a behavioral refinement that makes a preciseprediction about what will happen in coordination games12

Multiple equilibria typically arise because of the mutual con-sistency assumption For instance there are many economic sit-uations where there is one equilibrium that is Pareto- or payoff-dominant (ie better for everybody) but seems intuitively riskierthan another Pareto-inferior equilibrium (eg Cooper [1999] andCamerer [2003 chapter 7]) Players will enter a Pareto-inferiorequilibrium if each correctly believes that the others will play theless risky strategy even though it is not optimal

12 In an extensive-form game each equilibrium may require a different levelof cognitive effort from the players Ho and Weigelt [1996] show that players usesimplicity as the refinement criterion and choose the equilibrium that needs theleast cognitive effort The multiple equilibria in this paper require the same levelof cognitive effort so the simplicity criterion does not help


A model of this type of situation is the ldquostag huntrdquo (or ldquoas-surancerdquo) games Table I shows a stag hunt game in which eachof n players simultaneously choose either L or H The groupchoice is H if everyone chooses H and L otherwise (ie if at leastone person picks L) The row player earns 1 if everyone chooses Hearns 0 if she chooses H and the group outcome is L and earns x(0 x 1) if she chooses L (and the group choice is therefore L)Everyone choosing H is a Pareto-dominant equilibrium butreaching it depends on everyone thinking everyone else is likelyto choose H Choosing L is also an equilibrium but pays less thanif players could somehow coordinate on everyone choosing H

Game theorists have developed concepts to refine intuitionsabout when the (LL) or (HH) equilibria are likely to arise OurCH model can replicate some of these intuitions and predicts animportant effect of group size which has been observed in experi-mentsmdashnamely that as the group size n increases the group ismore likely to get drawn into the inefficient (LL) equilibrium[Van Huyck Battalio and Beil 1990 Camerer 2003 chapter 7]

In the two-player stag hunt game 1-step thinkers choose H ifx 1frasl2 and choose L if x 1frasl2 because they optimize against0-step thinkers who randomize (If x 1frasl2 then all playersrandomize equally) Higher-step thinkers do exactly what the1-step thinkers do In the three-player game however a 1-stepplayer thinks she is facing two 0-step players who randomizeindependently so the chance of at least one L is 75 As a resultthe 1-step player (and higher-level players) choose H iff x 25Thus for values 25 x 5 the Poisson-CH model predictsmostly H play in 2-player games and mostly L-play in 3-playergames (the frequencies of H and L play respectively are 1 ( f(0)) 2 or 89 percent for 15) This is a simple way ofexpressing the idea that there is more strategic uncertainty in

TABLE IROW PLAYERrsquoS PAYOFFS IN AN n-PERSON STAG HUNT GAME

Row choice

Group outcome

L H

L x mdashH 0 1


games with more players and fits the experimental fact that theinefficient (LL) outcome occurs more often in larger groups

There is an interesting connection between the ldquoselectionprinciplerdquo of risk-dominance in 2 2 coordination games [Har-sanyi and Selten 1988] and our CH model In 2-player symmetriccoordination games CH predicts that all players (except half the0-step thinkers) will choose the risk-dominant equilibrium (seeour working paper for details) which matches a wide range ofexperimental data showing that players tend to choose risk-dominant strategies rather than payoff-dominant ones in thesegames [Camerer 2003 chapter 7]

IIIC Market Entry Games

In Section IV below we report experimental results from asimple business entry game In this game N entrants simulta-neously decide whether to enter a market or stay out (denoted 1and 0 respectively) Denote the market demand by d N (ex-pressed as a fraction of the number of potential entrants N so0 d 1) If d or fewer players enter (ie supply is equal to orless than demand) the entrants all earn a payoff of 1 If morethan d enter (ie supply is greater than demand) the entrantsearn 0 while staying out yields a certain payoff of 05 Fortheoretical simplicity assume that there are infinitely many at-omistic entrants (In our empirical estimation we do not makethis assumption) If entrants are atomistic and risk-neutral theyonly care about whether the fraction of others entering is above dor not if the fraction of others entering is below d they shouldenter if it is above they stay out Denote the entry function ofstep k players for demand d by e(kd) d 3 [0 1] this functionmaps the demand into a decision to enter (1) or stay out (0)Denote the interim total entry function for all steps up to andincluding k by E(kd) d 3 [0 1] The function E(kd) adds upthe entry functions of the types up to and including k andnormalizes (by dividing by yenh0

k f(h)) The prediction of the modelabout how many players will enter for each value of d is thelimiting case E(d) (which will depend on )

Appendix 1 shows that in the Poisson-CH model a particularthinking-step type k has a series of cutpoints which prescribevalues of d at which the k-step thinker will enter or not (whichnaturally depend on ) For example a 1-step thinker will stayout for d 5 and enter for d 5 (and is indifferent when d 5) the entry-function therefore has one cutpoint at 5 The Ap-


pendix proves that the cumulative entry function E(d) will beweakly monotonic in dmdashmarkets with bigger demand attractmore entrantsmdashiff 1 2 e or 12513

Figure I shows the predicted entry functions for CH playersusing 0 1 and 2 levels of reasoning (e(0d) e(1d) e(2d)) andthe interim cumulative entry function (E(2d)) for 15 Notehow the entry function e(2d) of the 2-step type ldquosmoothesrdquo thecumulative entry function E(2d) The 2-step thinkers only enterwhen they think they can exploit the fact that too few lower-0-and 1-step thinkers entered (for 5(1 ) d 5 and (5 )(1 ) d 1) and stay out when they think too many 0-and 1-step types entered (for 0 d 5(1 ) and 5 d (5 )(1 ))

Note that if the entry game were actually played sequen-

13 When is a little greater than 125 as it seems to be empirically in manyexperimental data sets there is a small nonmonotonicity in which there is moreentry at values just below d 5 than the rate of entry at higher demands justabove d 5 This is because 1-step thinkers never enter when d 5 and alwaysenter when d 5 Their anticipated entry function e(1d) invites all higher-stepthinkers to enter for values just below 5 and to stay out for values just above 5The cumulative effect of the higher-step thinkersrsquo entry is never overturned as kgrows large for low The downward-blip in overall entry just below and aboved 5 is probably a small effect empirically but also illustrates a sharp coun-terintuitive prediction that could be tested in experiments with a large entry poolN 100 and values of d bracketed closely around 5 like d 49 and d 51

FIGURE IBehaviors of Level 0 1 and 2 Players ( 15)


tially it is easy to see how perfect equilibration could occurexactly d of N players would enter because all the later entrantswould know how many earlier players had entered (For examplein a subgame perfect equilibrium the first d entrants would enterand all the subsequent entrants would stay out) Because thePoisson-CH model is recursive the game becomes effectivelyldquopseudo-sequentialrdquo higher-level players act as if they are mov-ing ldquoafterrdquo they have observed what other players do even thoughthey are actually playing simultaneously The pseudo-sequen-tiality created by the recursive structure of the CH model approxi-mates the equilibration that would occur if the game were actu-ally played sequentially

A wide variety of experimental data show that in entrygames like these the entry rate is usually remarkably monotonicin demand d even though players do not communicate and haveno way to organize their choices to enter with the correct fre-quency (eg Rapoport and Seale [in press] and Camerer [2003chapter 7]) Remarking on the surprising similarity between pre-dicted entry rates across values of d and actual entry in pilotexperiments he conducted Kahneman [1988] wrote that ldquoto apsychologist it looks like magicrdquo The Appendix proof shows howthe Poisson-CH model can produce entry that is monotonic in dand approximates equilibriummdashthe ldquomagicrdquo which surprisedKahneman However players also collectively overenter at lowvalues of d and underenter at high values of d so their behavioris not entirely in equilibrium The Poisson-CH model also ac-counts for overentry at low d and underentry at high d due to thelingering effect of 0-step thinkers who enter half the time regard-less of d The Poisson-CH can therefore explain the magic ofapproximate equilibrationmdashmonotonicity of entry with the de-mand dmdashas well as systematic departures from equilibrium ob-served in the data

IV ESTIMATION AND MODEL COMPARISON

This section estimates values of in the Poisson-CH modeland compares its fit to Nash equilibrium Exploring a wide rangeof games and models is useful in the early stage of a researchprogram Models that sound appealing (perhaps because they areconventional) may fit surprisingly badly thus redirecting atten-tion to novel ideas Fitting a wide range of games turns up cluesabout where models fail and how to improve them


Since our model is designed to be general it is particularlyimportant to check its robustness across different types of gamesand see how regular the best-fitting values of are Once isspecified the modelrsquos predictions about the distribution of choicescan be easily derived by iterating steps of thinking (and boundingthe procedure at a high value of k)

IVA The Beauty Contest Games

An empirical warm-up example is the beauty contest gamedescribed above in which the player whose number (from 0 to100) is closest to 2frasl3 times the average wins a fixed prize Table IIshows estimates of in 24 p-beauty contest games ( p is themultiplier which so far has been 2frasl3) which were chosen tominimize the (absolute) difference between the predicted andactual mean of chosen numbers (see our working paper) Thetable is ordered from top to bottom by the mean number chosenThe first seven lines show games in which the equilibrium is notzero in all the others the equilibrium is zero

The first four columns describe the game or subject pool thesource group size and total sample size The fifth and sixthcolumns show the Nash equilibrium and the difference betweenthe equilibrium and the average choice The middle three col-umns show the mean standard deviation and mode in the dataThe mean choices are generally far off from the equilibrium theychoose numbers that are too low when the equilibrium is high(first six rows) and numbers that are too high when the equilib-rium is low (lower rows) The rightmost six columns show theestimate of from the Poisson-CH model and the mean predic-tion error standard deviation and mode predicted by the best-fitting estimate of and the 90 percent confidence interval for estimated from a randomized resampling (bootstrap) procedure

There are several interesting patterns in Table II The pre-diction errors of the mean (column 13 ldquoerrorrdquo) are extremelysmall less than 6 in all but two cases This is no surprise since is estimated (separately in each row) to minimize this predictionerror The pleasant surprise is that the predicted standard devia-tions and modes which result from the error-minimizing estimateof are also fairly close (across rows the correlation of thepredicted and actual standard deviation is 72) even though rsquoswere not chosen to match these moments

The values of have a median and mean across rows of 130and 161 close to the golden ratio (1618 ) and 2 ( 141)


TA

BL

EII

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DC

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etal

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ess]


values derived from simple axioms mentioned above The confi-dence intervals have a range of about one in samples of reason-able size (50 subjects or more)

Note that nothing in the Poisson-CH model per se requires to be fixed across games or subject pools or across details of howgames are presented or choices are elicited14 Outlying low andhigh values of are instructive about how widely might varyand why Estimates of are quite low (0ndash1) for the p-beautycontest game when p 1 and consequently the equilibrium is atthe upper end of the range of possible choices (rows 1ndash2) In thesegames subjects seem to have trouble realizing that they shouldchoose very large numbers when p 1 (though they equilibraterapidly by learning see Ho Camerer and Weigelt [1998]) Lowrsquos are also estimated among the PCC (Pasadena City College)subjects playing two- and three-player games (rows 8 and 10)High values of ( 3ndash5) appear in games where the equilibriumis in the interior 72 (rows 7ndash10)mdashsmall incremental steps towardthe equilibrium in these games produce high values of High values are also estimated in games with an equilibrium of zerowhen subjects are professional stock market portfolio managers(row 19) Caltech students (row 20) game theorists (row 24) andsubjects self-selecting to enter newspaper contests (row 25) Thelatter subject pools show that in highly analytical and educatedsubject pools (especially with self-selection) can be much higherthan in other subject pools

A sensible intuition is that when stakes are higher subjectswill use more steps of reasoning (and may think others will thinkharder too) Rows 3 and 6 compare low stakes ($1 per person perperiod) and high stakes ($4) in games with an interior equilib-rium of 72 When stakes are higher is estimated to be twice aslarge (501 versus 251) which is a clue that some sort of cost-benefit analysis may underlie steps of reasoning

Notwithstanding these interesting outliers there is also sub-stantial regularity across very diverse subject pools and payofflevels About half the samples have confidence intervals thatinclude 15 Subsamples of corporate CEOs (row 13) high-functioning 80-year old spouses of memory-impaired patients

14 Our working paper notes that in games where beliefs about the choices ofothers are elicited (along with onersquos own choices) the number of thinking stepssometimes shifts upward Forcing somebody to articulate what they think otherswill do apparently gets them to think harder


[Kovalchik et al forthcoming row 15] and high school students(row 16) all have values from 11ndash17

Since CH-type models are ideally suited to capture limitedequilibration in dominance-solvable games like the beauty con-test game it is important to see how well the same model and values fit games with different structures So we fit five otherdata sets using maximum likelihood estimation (MLE) procedure(see Appendix 2 for a list) three sets of matrix games with 2ndash4strategies (33 games in total) the binary entry game describedabove with 12 players and demands d 246810 and 22games with mixed equilibria

The estimation aims to answer two questions is the esti-mated value of reasonably regular across games with verydifferent structures And how accurate is the Poisson-CH speci-fication compared with Nash equilibrium

IVB How Regular Is

Table III shows game-by-game MLE estimates of in thePoisson CH model and estimates when is constrained to becommon across games within each data set The interquartilerange across the 60 estimates is (98221) and the median is 155Five of 60 game-specific estimates are high (four or more) anda few are zero

Appendix 3 shows bootstrapped 95 percent confidence inter-vals for the estimates Most of the intervals have a range ofabout one The common estimates are roughly 1ndash2 a of around15 is enclosed in the 90 percent interval in three data sets and seems to be about one in the Cooper-Van Huyck and entry dataThese reasonably regular rsquos suggest that the Poisson-CH modelwith 15 can be used to reliably predict behaviors in newgames

In future work variation in estimates of could be useful insharpening a theory of how steps of thinking are chosen endog-enously While endogenizing thinking steps or is beyond thescope of this paper it is likely that some kind of model comparingperceived benefits of thinking further with thinking costs (con-strained by working memory and permitting individual differ-ences) will do better Three pieces of evidence point to the promiseof a cost-benefit endogenization (1) is estimated to be quitelarge in p-beauty contest games in subject pools with unusual


analytical skill (eg Caltech undergraduates) or special training(game theorists and computer scientists who study multiagentmachine learning) which is a clue that lowering thinking costsdue to skill or training leads to higher (2) Unpublished data wehave collected also show larger estimates of (about 5 stepsmore) in Caltech undergraduates than in comparable studentsfrom a nearby community college (more evidence of skill or lowercognitive cost as an important variable) (3) Unpublished datashow that in incomplete-information signaling games is esti-mated to be lower (less than 1) Bayesian updating on whatanother playerrsquos signal choice reveals about her likely thinking-step type presumably consumes more working memory than sim-ply computing expected payoffs (raising thinking costs) so lowerrsquos in these games are also consistent with cost-benefit calculus

TABLE IIIPARAMETER ESTIMATE FOR COGNITIVE HIERARCHY MODELS

Data setStahl and

WilsonCooper andVan Huyck

Costa-Gomeset al Mixed Entry

Game-specific Game 1 293 1590 228 098 070Game 2 000 107 227 171 085Game 3 140 018 229 086 mdashGame 4 234 128 126 385 073Game 5 201 052 180 108 070Game 6 000 082 167 113Game 7 537 096 088 329Game 8 000 154 218 184Game 9 135 189 106Game 10 1133 226 226Game 11 648 123 087Game 12 171 103 206Game 13 228 188Game 14 907Game 15 349Game 16 207Game 17 114Game 18 114Game 19 155Game 20 195Game 21 168Game 22 306

Median 186 101 189 177 071Common 154 082 173 148 073


IVC Which Models Fit Best

Table IV shows log likelihoods (LL) and mean-squared devia-tions for several models estimated game-by-game or with com-mon parameters across games in a data set15 This table answersseveral questions Focusing first on the Poisson-CH model game-specific estimates of fit almost as well as common within-columnestimates in most data sets (except for the Stahl-Wilson data)The Poisson-CH model also fits substantially better than Nash inevery case This shows that relaxing mutual consistency can be afruitful approach to building a descriptive theory of disequilib-rium behavior in games

A graphical comparison of how much the theoriesrsquo predictionsdeviate from the data gives a quick snapshot of how accurate theyare Each point in Figures IIndashIII represents a distinct strategy in

15 When the Stahl-Wilson games 2 6 8 are included the common is 0because these games swamp the other ten so we excluded those games in doingthe common- estimation Poisson-CH fits badly in those games because the Nashstrategy is not reached by any number of thinking steps but is frequently chosenThe best the model can do is to pick 0 so that 1frasl3 of the players are predictedto pick it (since there are three strategies) These games show boundary condi-tions under which the model fails badly Modifying the model so that a fraction of the 0-step players are actually choosing Nash (which will then lead 1-step typesto choose Nash if is large enough) would patch this problem Including someself-awareness would also explain these anomalies

TABLE IVMODEL FIT (LOG-LIKELIHOOD LL AND MEAN SQUARED DEVIATION MSD)

Data setStahl and



Log-likelihoodCognitive hierarchy

(Game-specific ) 360 838 264 824 150Cognitive hierarchy

(Common ) 458 868 274 872 150Nash equilibriuma 1823 5422 1819 1270 154

Mean squareddeviation

Cognitive hierarchy(Game-specific ) 00074 00090 00035 00097 00004

Cognitive hierarchy(Common ) 00327 00145 00097 00179 00005

Nash equilibrium 00882 02038 01367 00387 00049

a The Nash Equilibrium result is derived by allowing a nonzero mass of 00001 on nonequilibriumstrategies


FIGURE IIMean Absolute Deviation for Matrix Games Nash Versus Cognitive Hierarchy

(Common )

FIGURE IIIMean Absolute Deviation for Mixed Games Nash Versus Cognitive Hierarchy

(Common )


each of the 33 matrix games (Figure II) and 22 mixed games(Figure III) Each point represents the absolute deviation be-tween the Nash prediction and the data (on the x-axis) and thePoisson-CH prediction (using a common within each data set)and the data (on the y-axis) Points in the lower right of the graphrepresent strategies in which CH is more accurate than Nashpoints in the upper left represent strategies in which Nash ismore accurate than CH

The graphs enable us to answer an important question visu-ally when the Nash predictions are good approximations is Pois-son-CH almost as accurate The answer appears to be yes be-cause there are few points with low Nash deviations and highPoisson-CH deviations (ie few points in the upper left of thegraphs) And when the Nash predictions are poor approxima-tions are CH predictions usually more accurate The answer isalso Yes Figure II shows that particularly in the matrix games(where Nash often makes 0ndash1 pure strategy predictions) thereare many strategies in which the Nash prediction is off by morethan 50 For these strategies the Poisson-CH prediction is usu-ally off by less than 20 So Poisson-CH is able to correct thelargest mistakes made by the equilibrium prediction Figure IIIshows that both models are generally more accurate in mixedgames than in the Figure II matrix games and that Poisson-CHimproves only a little on equilibrium

IVD Predicting Across Games

Good theories should predict behavior accurately in new sit-uations A simple way to see how well Poisson-CH and equilib-rium models can do this is to estimate the value of on n 1 datasets and forecast behavior in each holdout data set separately

TABLE VCROSS-GAME FIT (LOG-LIKELIHOOD LL AND MEAN SQUARED DEVIATION MSD)

Data setStahl and




(Common ) 469 956 293 884 154Mean squared deviationCognitive hierarchy

(Common ) 00416 00335 00237 00215 00046


The result of this kind of cross-game estimation is reported inTable V Across games the Poisson-CH model fits only a little lessaccurately than when estimates are common within games Thissuggests that the Poisson-CH model has some promise for pre-dicting behavior in one set of games based on observations fromother games

IVE Two Examples

Two specific games help illustrate concretely how the Pois-son-CH model can explain where Nash predictions succeed andfail These games were chosen because they have the medianlikelihood ratio of Poisson-CH relative to Nash within their re-spective data sets so they are statistically representative of theoverall result and are not biased either for or against the CH andNash models

Table VI shows game 8 from Costa-Gomes Crawford andBroseta [2001] (numbered 9a in their paper) The Nash predictionis pure play of (TL) Most row players do choose T but a third ofthe column players choose R instead of the equilibrium responseL The Poisson-CH model (using the common- within the Costa-Gomes Crawford and Broseta data set) predicts 82 percent playof T because it is a dominant strategy and so all players usingmore than 0 steps are predicted to choose it It also predicts 45percent of players will choose R because half the 0-step playersand all the 1-step players choose it (though players using two ormore steps pick L) So CH is able to approximate the accurateNash prediction about dominant strategy play of T but correctsNash theoryrsquos mistaken prediction of how often R is played Thekey point is that 1-step thinkers play R Only column players

TABLE VIGAME 8 FROM COSTA-GOMES ET AL [2001]

L R Data Nash CH

T 45 66 82 31 92 1 82TM 22 14 57 55 0 0 06BM 30 42 28 37 0 0 06B 15 60 61 88 08 0 06Data 64 36Nash 1 0CH 55 45


doing two or more steps of thinking figure out that row playerswill pick T (and respond optimally with L) so the relative infre-quency of high-step thinkers explain why R is chosen so often

Table VII shows game 4 from Binmore Swierzbinski andProulx [2001] This game has a mixed equilibrium in which rowplayers are predicted to choose B and column players are pre-dicted to choose R both at 67 percent of the time

The column player prediction is plausible because R pays either1 or 0 and players actually chose it 83 percent of the time ThePoisson-CH model reproduces this finding very closely because allplayers doing one or more steps of thinking are predicted to chooseR an aggregate frequency of 84 percent The Nash prediction thatrow players choose B most often is less plausible because B has nopositive payoffs and in fact it is the strategy chosen least often Inequilibrium of course players are predicted to choose B becausethey guess correctly that column players often choose R In thePoisson-CH model however 1-step thinkers do not anticipate theplay of R and mix between T and M So the Poisson-CH modelpredicts 25 percent choice of each T and M which is closer to whatactually happens than the Nash prediction

These examples illustrate how the Poisson-CH model canmix limited thinking with strategic thinking (through the behav-ior of players doing two or more steps of thinking) and as a resultgenerally fit data from one-shot games better than equilibriummodels do16 Remember that these are statistically typical exam-ples they were not chosen to highlight where Poisson-CH doesparticularly well or poorly

16 A web-based calculator is available to provide Poisson-CH prediction and estimate to fit data from one-shot games The calculator is located at httpgroupshaasberkeleyedusimulationsCH

TABLE VIIGAME 4 FROM BINMORE ET AL [2001]

L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84


V ECONOMIC VALUE OF THEORIES

In 1960 Schelling [p 98] wrote ldquoA normative theory mustproduce strategies that are at least as good as what people can dowithout themrdquo Schellingrsquos definition suggests a simple measureof the value of an economic theory when applied to a particulargame how much greater a payoff do players earn when theybest-respond to a theoryrsquos forecast rather than respondingnaively

A reasonable way to measure the ldquoeconomic valuerdquo of a the-ory applied to a particular game is to take a set of experimentaldata and compute the difference between the expected payofffrom using the best response given by the theory and the averagepayoff subjects actually earned (see Camerer and Ho [2001])

If a playerrsquos beliefs and the choices of others players aremutually consistent then equilibrium theory predicts the gameexactly Thus for a game where the players are in equilibriumthe economic value of equilibrium theory will be 0 On the otherhand if players are in equilibrium then models that assume theyare not in equilibrium (such as the Poisson-CH model) will havenegative economic value Thus the economic value of equilibriumfor a game is a way of measuring the degree of equilibration

Furthermore if the Poisson-CH model is correct for a givengame then the best response the theory dictates corresponds towhat the highest-step thinkers do Thus the economic value ofPoisson-CH can be interpreted as the marginal payoff to usingmany steps of thinking compared with average steps If theeconomic value is lowmdashie the marginal payoff to thinking veryhard is lowmdashthis fact could be used as a justification for anevolutionary or ldquocognitive economicsrdquo explanation of why moreplayers do not think harder which could potentially endogenizethe limits of thinking

Table VIII reports the economic value of the Poisson-CH andNash models across several data sets The economic value ofPoisson-CH is derived using parameters estimated on n 1 datasets to forecast the remaining data set17 The payoffs from pre-

17 Economic value is only slightly higher when parameters are estimatedwithin each data set (see our working paper) One could argue that economic valuemeasured using within-game is upward biased because the model effectivelyhas access to data about the particular game which the typical subject does notThe cross-game estimation below does not have this upward bias It is true thatthe cross-game Poisson-CH forecast uses other data the subject did not seemdashnamely the behavior in the other four data setsmdashbut this is typically the case in


dicting ldquoclairvoyantlyrdquo (ie using the actual distribution of strat-egies chosen by all other subjects) are also reported becausethese represent an upper bound on economic value

The Poisson-CH approach adds value in all data sets from 20to 70 percent of the maximum possible economic value Nashequilibrium typically adds economic value although only abouthalf as much as Poisson-CH and subtracts value in one data setRecall that if players were in equilibrium the Nash predictionswould have zero economic value and disequilibrium models likeCH would have negative economic value The fact that this pat-tern is not observed is another way of saying players are not inequilibrium and economic value measures the ldquodegreerdquo ofdisequilibrium

VI ECONOMIC IMPLICATIONS OF LIMITED STRATEGIC THINKING

Models of iterated thinking can be applied to several inter-esting problems in economics including asset pricing specula-

forecasting (after all distillation of data is part of what people pay for when theybuy forecasts) Furthermore the subjects have ldquodatardquo (or insight) which the modeldoes not havemdashnamely how people like themselves and their fellow subjectsmight react to a particular game and how they may have behaved in dozens ofother experiments they participated in

TABLE VIIIECONOMIC VALUE OF VARIOUS THEORIES

Data setStahl and



Observed payoff 195 586 264 328 118Clairvoyance payoff 243 664 306 708 176Economic valueClairvoyance 48 78 42 380 58Cognitive hierarchy

(Common ) 13 55 22 132 10Nash equilibrium 5 30 15 17 2 Maximum economic

value achievedCognitive hierarchy

(Common ) 26 71 52 35 17Nash equilibrium 10 39 35 4 3

The economic value is the total value (in experimental payoffs) of all rounds that a ldquohypotheticalrdquo subjectwill earn using the respective model to predict otherrsquos behavior and best responds with the strategy thatyields the highest expected payoff in each round


tion competition neglect in business entry incentive contractsand macroeconomics (see our longer paper for some ideas alongthese lines) An example is Crawfordrsquos [2003] model of optimallying He shows that if ldquosome of the people can be fooled some ofthe timerdquo the presence of such foolable nonstrategic types influ-ences rational players to misrepresent their intentions (much asthe presence of 0-step players influence the behavior of higher-step thinkers) Similarly Cai and Wang [2003] explain the ex-perimental tendency for players to overcommunicate private in-formation with a model of limited thinking

We illustrate further with two economic applications thathave been studied experimentally speculation and money illu-sion The idea is to see whether the Poisson-CH model can help usunderstand something fundamental about economics

VIA Speculation

In 1982 Milgrom and Stokey proved a remarkable ldquoGrouchoMarx theoremrdquo If rationality is common knowledge risk-averseplayers should not make speculative bets with one another (un-less they have hedging motives) Of course speculation goes onconstantly in the form of sports betting and a large fraction oftrading in financial markets and other forums It is difficult toknow from field data which assumption of the Groucho MarxTheorem is violated is widespread speculation due to hedging(undoubtedly an important part of the operation of foreign ex-change and futures markets) Or to the extra fun from watchinga sports event after betting on it Or is speculation due to limitson knowledge of rationality Since the Poisson-CH model does notimpose common knowledge of rationality it contradicts the Grou-cho Marx theorem and predicts that speculation will occur evenwhen hedging and spectator fun do not matter In CH somedegree of betting comes immediately from the fact that 0-step and1-step players are not thinking strategically about how the bet-ting propensities of others depends on what those other playersknow (cf Eyster and Rabin [2000])

Table IX illustrates a betting game originally studied experi-mentally by Sonsino Erev and Gilat [2002] and replicated bySovik [2000] There are four equally likely states ABCD Afterthe state is determined players are privately informed about aset of possible states including the true one Player I learns thestate is either A or B (denoted (AB)) or C or D (ie(CD)) Player II learns the state with certainty if it is A or D or


learns (BC) Players then choose whether to bet with payoffsgiven in Table IX If both players bet they win or lose theamounts in the table column corresponding to the true state

In equilibrium there should be no mutual betting if rational-ity is common knowledge and players think strategically Theproof begins with the fact that rational player IIs will never betwhen they know the state is A and will always bet when theyknow the state is D If player I is rational and believes player IIsare too she will figure out that she can never win by betting whenher information is (A B) Iterating further player II should neverbet in (B C) which leads player I to not bet in information set (CD) Therefore players should never mutually bet

This counterintuitive no-betting result is also the predictionof the Poisson-CH model with 3 With the typical empiricalvalue of 15 however a different picture emerges One-stepplayer Is will bet when they know (AB) because they think theyare equally likely to win 32 and lose 28 (they have not figured outthat they will never win 32 because rational player Irsquos never betin state A) However two-step thinkers know that one-step playerIIs will not bet in A so they will not bet in (AB) The eventualresult of this iterated limited rationality across all informationstates is high betting rates in the information states (AB) and(CD) for player I and in (BC) for player II

Table IX shows predicted betting rates (for 15) and Sovik

TABLE IXPOISSON CH PREDICTION AND EMPIRICAL FREQUENCY

FOR BETTING GAME [SOVIK 2000]

State of the world A B C D

Player IPayoff for betting 32 28 20 16Information set (A or B) (C or D)Betting rateData 77 53Poisson CH 46 89

Player IIPayoff for betting 32 28 20 16Information set A (B or C) DBetting rateData 0 83 100Poisson CH 12 72 89


[2000]rsquos first-round data18 The model does not track differencesin betting rates across the three ambiguous information statesparticularly well but it is an obvious improvement on the Nashprediction of no mutual betting in any states Furthermore themodel makes testable comparative static predictions for exam-ple if the payoff in the C state is changed from 20 to 32 bettingrates in (BC) and (CD) should fall dramatically (from 72 percentand 89 percent to 12 percent and 46 percent) This example showshow a small change in parameters can turn the predicted CHresult from a gross violation of the Groucho Marx Theorem to areasonable approximation of it More generally the exampleshows how CH captures partial awareness of adverse selectionwhich may be useful in understanding consumer product mar-kets the winnerrsquos curse in common-value auctions and so forth

VIB Money Illusion

A long-running debate in macroeconomics concerns the ex-tent of ldquomoney illusionrdquo the failure to adjust incomes and pricesfor inflation Fehr and Tyran [2002] investigate money illusion intwo parallel pricing games In their games groups of four playerschoose integer prices from 1 to 30 In one game prices are stra-tegic substitutesmdashplayers earn more by pricing high when othersprice low and vice versa In the other game prices are strategiccomplementsmdashplayers earn more by matching prices of otherplayers Each playerrsquos nominal payoffs depend on his or her ownprice and on the (rounded) average price of the other three play-ers in his or her group These nominal payoffs are displayed in a30 30 table with the playerrsquos own prices in 30 rows and theaverage price of others in the 30 columns To compute their realpayoffs players had to divide the nominal numbers in the tablethey see in front of them by the average price of the other playersin the group (which is clearly shown at the top of each column)

The research question is whether players use the nominalpayoff or act as if they calculated real payoffs If players use realpayoffs and are in equilibrium they will either choose prices of 11and 14 (depending on which of two cost structures denoted x or ythey have) in both the substitutes and complements conditions(This is also the prediction of CH with infinite )

18 The results of Sonsino Erev and Gilat [2002] are even more consistentwith CH except that they report substantial betting and nonbetting rates byplayer IIs in states A and D (about 20 percent) which suggests that 1


Fehr and Tyran [2002] found a striking regularity in thesubstitutes condition players choose prices very close to the Nashpredictions of 11 and 14 But in the complements condition priceswere far from equilibrium a median of 22ndash23 This pattern leavesunresolved whether players have money illusion or not It ap-pears that their money illusion depends on the strategic structureof the game which is an unsatisfying conclusion

The Poisson-CH model with the typical value 15 canaccount for this pattern (with one important modification19) rea-sonably well The key is that in the complements case 1-stepthinkers have best responses which are prices in the 20s wellabove the Nash equilibria and 2- and higher-step thinkers alsochoose prices which are too high

Table X shows summary statistics of predicted and actualprices and the Poisson-CH predictions for 15 (which is not thebest-fitting value20) Median experimental prices are predictedexactly in three of four samples Repeating the main empirical

19 The Poisson-CH model fits best if we assume that the 1-step thinkersbelieve the 0rsquos choices are perfectly correlated ie the distribution of the averageprice is uniform A similar improvement in fit from assuming that lower leveltypesrsquo choices are correlated occurs in p-beauty contests (see Ho Camerer andWeigelt [1998]) and weak link games [Camerer 2003 chapter 7] The tendency forplayers to think that a single 0-step playerrsquos price distribution is representative oran exemplar of the average price is an example of the ldquorepresentativenessrdquoheuristic which is well documented in research on the psychology of judgment(eg Kahneman [2003])

20 The best-fitting values are different in the two conditions 6 in comple-ments and 26 in substitutes Fehr and Tyran [2002] note that this difference isconsistent with the fact that in the substitutes case misestimating what others dois a much more costly error than in the complements case This is another clue forhow the number of thinking steps may respond endogenously to incentives

TABLE XPOISSON CH AND NASH PREDICTION FOR PRICING GAME ON MONEY ILLUSION

[FEHR AND TYRAN 2002]

Nashprediction

Median MeanStandarddeviation

Actual 15 Actual 15 Actual 15

x-comps 21 21 1765 1975 423 473x plays 11 y-comps 225 26 2055 2395 635 618

x-subs 11 11 1017 912 293 730y plays 14 y-subs 14 14 125 1016 410 779


theme of this paper the model can explain when Nash equilib-rium is reached surprisingly quickly (in the substitutes treat-ment) and can also explain when behavior is far from equilib-rium (in the complements treatment) More importantly themodel provides a clear answer to the central research question ofwhether money illusion occurs The answer is that players doappear to have money illusion in both treatments but observeddifferences in equilibration in both cases may result from a com-mon process of limited strategic thinking21

VII CONCLUSION

This paper introduced a simple cognitive hierarchy (CH)model of games The model is designed to be as general andprecise as Nash equilibrium In fact it predicts players are un-likely to play Nash strategies that are refined away by subgameor trembling-hand perfection and always selects one statisticaldistribution when there are multiple Nash equilibria so it is evenmore precise than simple Nash equilibrium

This paper uses both axioms and estimation to restrict thefrequencies of players who stop thinking at various levels Mostplayers do some strategic thinking but the amount of strategicthinking is sharply constrained by working memory This is con-sistent with a Poisson distribution of thinking steps that can becharacterized by one parameter (the mean number of thinkingsteps and the variance) Plausible restrictions and estimatesfrom many experimental data sets suggest that the mean amountof thinking is between one and two The value 15 is a goodomnibus guess which makes the Poisson-CH theory parameter-free and is very likely to predict as accurately as Nash equilib-rium or more accurately in one-shot games

The main contribution is showing that the same model canexplain limited equilibration in dominance-solvable games (likep-beauty contests) and the surprising accuracy of Nash equilib-rium in some one-shot games such as simultaneous binary entrygames in which players choose whether to enter a market with afixed demand In one-shot games with no communication the rate

21 When the Poisson-CH model is applied to players with no money illusionit fits the data poorly (since it predicts Nash-like play in the complements case)It also fits poorly if 2- and higher-step players do not have money illusion butlower-step players do


of entry in these entry games is ldquomagicallyrdquo monotonic in thedemand d but there is reliable overentry at low values of d andunderentry at high values of d The Poisson-CH approach pre-dicts monotonicity (it is guaranteed when 125) and alsoexplains over- and underentry Furthermore the Poisson-CHapproach creates a kind of endogenous purification that explainhow a population mixture of players who use pure strategies (andperhaps regard mixing as nonsensical) can approximate a mixedequilibrium

Because players do not appear to be mutually consistent inone-shot games where there is no opportunity to learn a theory ofhow others are likely to play could have economic valuemdashieplayers could earn more if they used the model to recommendchoices compared with how much they actually earn In facteconomic value is always positive for the Poisson-CH modelwhether is estimated within a data set or across data setsEconomic value is 10ndash50 percent of the maximum possible eco-nomic value that could be achieved by knowing in advance thesample frequencies of how others actually play The Nash ap-proach adds less economic value and sometimes subtracts eco-nomic value (eg in p-beauty contests with p 1 players arebetter choosing on their own than picking the Nash recommen-dation of 0) Economic value provides a precise measure of Schell-ingrsquos [1960] definition of how ldquonormativerdquo a theory is and alsomeasures the degree of disequilibrium in economic terms

There are many challenges in future research An obviousone is to endogenize the mean number of thinking steps pre-sumably from some kind of cost-benefit analysis in which playersweigh the marginal benefits of thinking further against cognitiveconstraint In a cost-benefit approach the fact that beliefs (andhence choices) converge as the number of steps rises createsdiminishing marginal benefits which leads to a natural trunca-tion that limits the amount of thinking

Since the Poisson-CH model makes a prediction about thekinds of algorithms that players use in thinking about gamescognitive data other than choicesmdashlike prompting players to statebeliefs (which might shift 0-step thinkers to one or more steps)information lookups or brain imaging (eg Camerer Prelec andLoewenstein [forthcoming])ndashcan be used to test the model Forexample Rubinstein [2003] reports response times in large web-based experiments that are consistent with slower responses by


players who use more thinking steps though Chong Camererand Ho [in press] find no such relation Costas-Gomes and Craw-ford [2004] have a more detailed analysis using lookups

The model is easily adapted to incomplete information gamesbecause the 0-step players make choices that reach every infor-mation set This eliminates the need to impose delicate refine-ments to make predictions Explaining behavior in signalinggames and other extensive-form games with incomplete informa-tion is therefore workable and a high priority Extending themodel to extensive-form games is easy by assuming that 0-stepthinkers randomize independently at each information set andhigher-level types choose best responses at information sets usingbackward induction Other models that link limited thinkingabout other players to limited look-ahead in extensive-formgames could prove more interesting

Finally the ultimate goal of the laboratory honing of simplemodels is to explain behavior in the economy Models of iteratedthinking could prove useful in thinking about asset marketsspeculation and betting contract structure and other phenom-ena (cf Eyster and Rabin [2000])

APPENDIX 1 ENTRY GAME ANALYSIS FOR POISSON-CH

We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1

ekd 0 if Ek 1d d1 if Ek 1d d


In general E(kd) is a step function with the following cutpointvalues (at which steps begin or end) with increasing d for d 1frasl2

1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk

1 2 f0 f2 middot middot middot fk

Fk

The cutpoint values for d 1frasl2 are

1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk

1 2 f0 f1 f2 middot middot middot fk

Fk

(For d 1frasl2 atomistic entrants are all indifferent and randomizeso E(k5) 5 k)

These cutpoints imply two properties the cutpoints are al-ways (weakly) monotonically increasing in d for the d 1frasl2segment as long as f(k 1) f(k) k 2 For a Poisson f(k)this is equivalent to 2 Furthermore the last cutpoint for thed 1frasl2 segment is smaller than the first cutpoint of the d 1frasl2segment iff 1frasl2f(0) f(2) f(3) f(k 1) f(k) 1frasl2f(0) f(1) This is equivalent to f(1) f(2) f(3) f(k) which implies that f(1) 1 f(0) f(1) For Poisson thisimplies that (1 2) e or 125 Thus 125 impliesweak monotonicity throughout both the left (d 1frasl2) and right(d 1frasl2) segments of the entry function E(kd) (since 125satisfies the 2 condition and ensures monotonicity across thecrossover from the left to right halves of E(kd))

APPENDIX 2 DETAILS OF GAMES AND EXPERIMENTAL METHODS

The matrix games are twelve games from Stahl and Wilson[1995] eight games from Cooper and Van Huyck [2003] (used tocompare normal- and extensive-form play) and thirteen games


from Costa-Gomes Crawford and Broseta [2001] All thesegames were played only once without feedback with sample sizeslarge enough to permit reliable estimation

In our experiments subjects played five entry games with nofeedback If the number of entrants was above (less than or equalto) d the entrants earned 0 ($1) nonentrants earned $50 Sub-jects also played 22 matrix games with mixed equilibria The 22games with mixed-equilibria are taken from those reviewed byCamerer [2003 chapter 3] with payoffs rescaled so subjects winor lose about $1 in each game The 22 mixed games are (in orderof presentation to the subjects) Ochs [1995] (matching penniesplus games 1ndash3) Bloomfield [1994] Binmore Swierzbinski andProulx [2001] game 4 Rapoport and Amaldoss [2000] BinmoreSwierzbinski and Proulx [2001] games 1ndash3 Tang [2001] games1ndash3 Goeree Holt and Palfrey [2000] games 2ndash3 Mookerjee andSopher [1997] games 1ndash2 Rapoport and Boebel [1992] Messick[1967] Lieberman [1962] OrsquoNeill [1987] Goeree Holt and Pal-frey [2000] game 1 Four games were perturbed from the originalpayoffs the row upper left payoff in Ochsrsquos original game 1 waschanged to 2 the Rapoport and Amaldoss [2000] game was com-puted for r 15 the middle row payoff in Binmore Swierzbin-ski and Proulx [2001] game 2 was 30 rather than 30 and thelower left row payoff in Goeree Holt and Palfreyrsquos [2000] game 3was 16 rather than 37 Original payoffs in games were multipliedby the following conversion factors 10 10 10 10 05 10 5 1010 10 1 1 1 025 01 30 30 30 5 3 10 025 Currency unitswere then equal to $10

The entry and mixed-equilibrium games were run in fourexperimental sessions of twelve subjects each Each game wasplayed (with no feedback) against a random opponent in the samesession and earnings accumulated Two sessions used undergradu-ates from Caltech and two used undergraduates from PasadenaCity College (PCC) which is near Caltech (Individual-level esti-mation in progress suggests that the PCC subjects do about 5steps of thinking fewer than Caltech students men do about 2steps more thinking than women and the average number ofthinking steps drifts up by about 7 and the first and secondhalves of 22 games) Mixed equilibrium games were run on theldquoplaying in the darkrdquo software developed by McKelvey and Pal-frey The entry games and some beauty contest games were runon software Taizan Chan wrote which is available from us


CALIFORNIA INSTITUTE OF TECHNOLOGY

HAAS SCHOOL OF BUSINESS UNIVERSITY OF CALIFORNIA BERKELEY

SUNGKYUNKWAN UNIVERSITY AND NATIONAL UNIVERSITY OF SINGAPORE

REFERENCES

Banks Jeffrey Colin F Camerer and David Porter ldquoExperimental Tests of NashRefinements in Signaling Gamesrdquo Games and Economic Behavior VI (1994)1ndash31

Binmore Kenneth ldquoModeling Rational Players Part IIrdquo Economics and Philos-ophy IV (1988) 9ndash55

Binmore Kenneth Joe Swierzbinski and Chris Proulx ldquoDoes Maximin Work AnExperimental Studyrdquo Economic Journal CXI (2001) 445ndash464

Bloomfield Robert ldquoLearning a Mixed Strategy Equilibrium in the LaboratoryrdquoJournal of Economic Behavior and Organization XXV (1994) 411ndash436

Bosch-Domenech Antoni Jose G Montalvo Rosemarie Nagel and Albert Sa-torra ldquoOne Two (Three) Infinity Newspaper and Lab Beauty-ContestExperimentsrdquo American Economic Review XCII (2002) 1687ndash1701

Cai Hongbin and Joseph Wang ldquoOver-communication and Bounded Rationality

APPENDIX 3 95 Percent Confidence Interval for the Parameter Estimate of Cognitive Hierarchy Models

Data set

Stahl andWilson

Cooper andVan Huyck


Lower Upper Lower Upper Lower Upper Lower Upper Lower Upper

Game-specific Game 1 240 365 1540 1671 158 304 067 122 021 143Game 2 000 000 083 127 144 280 098 237 073 088Game 3 075 173 011 030 166 318 057 137 mdash mdashGame 4 234 245 101 148 091 184 265 426 056 109Game 5 161 245 036 067 122 230 070 162 026 158Game 6 000 000 064 094 089 226 087 177Game 7 520 562 075 123 040 141 245 385Game 8 000 000 116 172 148 267 121 209Game 9 106 169 128 268 062 164Game 10 1129 1137 167 306 134 358Game 11 581 756 075 185 064 123Game 12 149 202 055 146 140 235Game 13 175 316 164 215Game 14 661 1084Game 15 246 525Game 16 145 264Game 17 082 152Game 18 078 160Game 19 100 215Game 20 128 259Game 21 095 221Game 22 170 363

Common 139 167 074 087 153 213 130 178 042 107


in Strategic Information Transmission Games An Experimental Investiga-tionrdquo University of California Los Angeles Department of Economics 2003httpeconwebsscnetuclaedu

Camerer Colin F Behavioral Game Theory Experiments on Strategic Interaction(Princeton NJ Princeton University Press 2003)

Camerer Colin F and Teck-Hua Ho ldquoStrategic Learning and Teachingrdquo in S JHoch and H C Kunreuther eds Wharton on Making Decisions (New YorkJohn Wiley and Sons 2001) pp 159ndash176

Camerer Colin F Teck-Hua Ho and Juin-Kuan Chong ldquoA Cognitive HierarchyTheory of One-Shot Games Some Preliminary Resultsrdquo California Instituteof Technology Working Paper 2002 httpwwwhsscaltecheducamererthinking2002pdf

Camerer Colin F and Daniel Lovallo ldquoOverconfidence and Excess Entry AnExperimental Approachrdquo American Economic Review LXXXIX (1999)306ndash318

Camerer Colin F Drazen Prelec and George Loewenstein ldquoNeuroeconomicsrdquoScandinavian Journal of Economics httpwwwhsscaltecheducamererwebmaterialscanjecon61pdf forthcoming

Capra C Monica ldquoNoisy Expectation Formation in One-shot Gamesrdquo PhDthesis 1999

Chen Yuxin Ganesh Iyer and Amit Pazgal ldquoLimited Memory and MarketCompetitionrdquo Working Paper University of California Berkeley 2003

Chong Kuan Colin F Camerer and Teck Ho ldquoA Cognitive Hierarchy Theory ofOne-Shot Gamesrdquo In R Zwick Advances in Experimental Business ResearchVolume II (Dordrecht Kluwer Academic Publishers in press)

Cooper David and John Van Huyck ldquoEvidence on the Equivalence of the Stra-tegic and Extensive Form Representation of Gamesrdquo Journal of EconomicTheory CX (2003) 290ndash308

Cooper Russell W Coordination Games Complementarities and Macroeconomics(Cambridge UK Cambridge University Press 1999)

Costa-Gomes Miguel and Vincent Crawford ldquoCognition and Behavior in Two-Person Guessing Games An Experimental Studyrdquo University of CaliforniaSan Diego Working Paper 2004 httpweberucsdeduvcrawforGuess

Costa-Gomes Miguel Vincent Crawford and Bruno Broseta ldquoCognition andBehavior in Normal-Form Games An Experimental Studyrdquo EconometricaLXIX (2001) 1193ndash1235

Crawford Vincent ldquoLying for Strategic Advantage Rational and Boundedly Ra-tional Misrepresentation of Intentionsrdquo American Economic Review XCIII(2003) 133ndash149

Devetag Giovanna and Massimo Warglien ldquoGames and Phone Numbers DoShort-Term Memory Bounds Affect Strategic Behaviorrdquo Journal of EconomicPsychology XXIV (2003) 189

Eyster Erik and Matthew Rabin ldquoCursed Equilibriumrdquo University of CaliforniaBerkeley Working Paper 2000 httpwwwnuffoxacukUsersEysterpaperseurocursepdf

Fehr Ernst and Jean-Robert Tyran ldquoLimited Rationality and Strategic Interac-tion The Impact of the Strategic Environment on Nominal Inertiardquo Institutefor Empirical Research in Economics No 130 University of Zurich 2002httpwwwiewunizhchwpiewwp130pdf

Gabaix Xavier and David Laibson ldquoA Boundedly Rational Decision AlgorithmrdquoAmerican Economic Review XC (May 2000) 433ndash438

Gabaix Xavier David Laibson Guillermo Moloche and Stephen Weinberg ldquoTheAllocation of Attention Theory and Evidencerdquo Massachusetts Institute ofTechnology Working Paper August 2003

Goeree Jacob Charles Holt and Thomas Palfrey ldquoRisk Aversion in Games withMixed Strategiesrdquo University of Virginia Department of Economics May2000

Grosskopf Brit and Rosemarie Nagel ldquoRational Reasoning or Adaptive Behav-ior Evidence from Two-Person Beauty Contest Gamesrdquo Harvard BusinessSchool NOM Research Paper No 01-09 2001

Harsanyi John C and Reinhard Selten A General Theory of Equilibrium Selec-tion in Games (Cambridge MA MIT Press 1988)


Haruvy Ernan and Dale Stahl ldquoAn Empirical Model of Equilibrium Selection inSymmetric Normal-Form Gamesrdquo University of Texas Department of Eco-nomics Working Paper 1998

Hedden Trey and Jun Zhang ldquoWhat Do you Think I Think you Think StrategicReasoning in Matrix Gamesrdquo Cognition LXXXV (2002) 1ndash36

Ho Teck-Hua Colin Camerer and Keith Weigelt ldquoIterated Dominance andIterated Best-response in p-Beauty Contestsrdquo American Economic ReviewLXXXVIII (1998) 947ndash969

Ho Teck-Hua and Keith Weigelt ldquoTask Complexity Equilibrium Selection andLearning An Experimental Studyrdquo Management Science XVII (1996)659ndash679

Kahneman Daniel ldquoExperimental Economics A Psychological Perspectiverdquo in RTietz W Albers and R Selten eds Bounded Rational Behavior in Experi-mental Games and Markets (New York Springer-Verlag 1988) pp 11ndash18

mdashmdash ldquoMaps of Bounded Rationality Psychology for Behavioral Economicsrdquo Ameri-can Economic Review XCIII (2003) 1449ndash1475

Keynes John Maynard The General Theory of Interest Employment and Money(London Macmillan 1936)

Kovalchik Stephanie Colin F Camerer David M Grether Charles R Plott andJohn M Allman ldquoAging and Decision Making A Broad Comparative Study ofDecision Behavior in Neurologically Healthy Elderly and Young IndividualsrdquoJournal of Economic Behavior and Organization forthcoming

Lieberman Bernhardt ldquoExperimental Studies of Conflict in Some Two-Personand Three-Person Gamesrdquo in J H Criswell H Solomon and P Suppes edsMathematical Models in Small Group Processes (New York Stanford CAStanford University Press 1962) pp 203ndash220

Livio Mario The Golden Ratio (New York Broadway Books 2002)McKelvey Richard D and Thomas R Palfrey ldquoQuantal Response Equilibria for

Normal-form Gamesrdquo Games and Economic Behavior VII (1995) 6ndash38McKelvey Richard D and Thomas R Palfrey ldquoQuantal Response Equilibria for

Extensive Form Gamesrdquo Experimental Economics I (1998) 9ndash41Messick David M ldquoInterdependent Decision Strategies in Zero-Sum Games A

Computer-Controlled Studyrdquo Behavioral Science XII (1967) 33ndash48Milgrom Paul and Nancy Stokey ldquoInformation Trade and Common Knowledgerdquo

Journal of Economic Theory XXVI (1982) 17ndash27Mookerjee Dilip and Barry Sopher ldquoLearning and Decision Costs in Experimen-

tal Constant-Sum Gamesrdquo Games and Economic Behavior XIX (1997)97ndash132

Nagel Rosemarie ldquoUnraveling in Guessing Games An Experimental StudyrdquoAmerican Economic Review LXXXV (1995) 1313ndash1326

Ochs Jack ldquoGames with Unique Mixed Strategy Equilibria An ExperimentalStudyrdquo Games and Economic Behavior X (1995) 202ndash217

OrsquoNeill Barry ldquoNonmetric Test of the Minimax Theory of Two-person Zero-SumGamesrdquo Proceedings of the National Academy of Sciences LXXXIV (1987)2106ndash2109

Rapoport Amnon and Wilfred Amaldoss ldquoMixed Strategies and Iterative Elimi-nation of Strongly Dominated Strategies An Experimental Investigation ofStates of Knowledgerdquo Journal of Economic Behavior and Organization XLII(2000) 483ndash521

Rapoport Amnon and Richard B Boebel ldquoMixed Strategies in Strictly Competi-tive Games A Further Test of the Minimax Hypothesisrdquo Games and Eco-nomic Behavior IV (1992) 261ndash283

Rapoport Amnon and Darryl A Seale ldquoCoordination Success in NoncooperativeLarge Group Market Entry Gamesrdquo Experimental Business Research RamiZwick and Amnon Rapoport eds (Dordrecht The Netherlands Kluwer Aca-demic Publishers 2002)

Rosenthal Robert W ldquoA Bounded Rationality Approach to the Study of Non-Cooperative Gamesrdquo International Journal of Game Theory XVIII (1989)273ndash292

Rubinstein Ariel ldquoInstinctive and Cognitive Reasoning Response Times StudyrdquoTel-Aviv University and New York University Working Paper 2003 httparielrubinsteintauacilpapersResponsepdf


Schelling Thomas The Strategy of Conflict (Cambridge MA Harvard UniversityPress 1960)

Selten Reinhard ldquoFeatures of Experimentally Observed Bounded RationalityrdquoEuropean Economic Review XLII (1998) 413ndash436

Sonsino Doron Ido Erev and Sharon Gilat ldquoOn the Likelihood of RepeatedZero-Sum Betting by Adaptive (Human) Agentsrdquo Technion Israel Institute ofTechnology 2002

Sovik Ylva ldquoImpossible Bets An Experimental Studyrdquo University of Oslo De-partment of Economics 2000

Stahl Dale O ldquoIs Step-j Thinking an Arbitrary Modeling Restriction or a Fact ofHuman Naturerdquo Journal of Economic Behavior and Organization XXXVII(1998) 33ndash51

Stahl Dale O and Paul Wilson ldquoOn Playersrsquo Models of Other Players Theoryand Experimental Evidencerdquo Games and Economic Behavior X (1995)213ndash254

Tang Fang-Fang ldquoAnticipatory Learning in Two-Person Games Some Experi-mental Resultsrdquo Journal of Economic Behavior and Organization XLIV(2001) 221ndash232

Van Huyck John Ray Battalio and Richard Beil ldquoTacit Cooperation GamesStrategic Uncertainty and Coordination Failurerdquo American Economic Re-view LXXX (1990) 234ndash248

Weizsacker Georg ldquoIgnoring the Rationality of Others Evidence from Experi-mental Normal-Form Gamesrdquo Games and Economic Behavior XLIV (2003)145ndash171


contest in which people guess which faces others will judge to bethe most beautiful ldquoIt is not the case of choosing those which tothe best of onersquos judgment are really the prettiest nor even thosewhich average opinion genuinely thinks the prettiest We havereached the third degree where we devote our intelligences toanticipating what average opinion expects the average opinion tobe And there are some I believe who practice the fourth fifthand higher degreesrdquo [p 156]

The essence of Keynesrsquos observation is captured in theldquobeauty contestrdquo game in which players are asked to pick num-bers from 0 to 100 and the player whose number is closest to 2frasl3of the average wins a prize Equilibrium theory predicts eachcontestant will reason as follows ldquoEven if all the other playersguess 100 I should guess no more than 2frasl3 times 100 or 67Assuming that the other contestants reason similarly however Ishould guess no more than 45 rdquo and so on finally concludingthat the only rational and consistent choice for all the players iszero

When the beauty contest game is played in experimentalsettings the group average is typically between 20 and 35 Ap-parently some players are not able to reason their way to theequilibrium value or they assume that others are unlikely to doso If the game is played multiple times with the same group theaverage moves close to 0

There are other games where players are surprisingly closeto equilibrium however Consider a stylized business entry gamein which n players decide simultaneously whether to enter amarket with known demand d where d n The payoffs aresuch that players prefer to enter if a total of d or fewer playersenter and they prefer to stay out otherwise Equilibrium theorypredicts the total amount of entry will be very close to the demandd and this is exactly what happens even in one-shot experimentswhere the usual forces that might lead to equilibration (such aslearning) have not yet had a chance to operate

In this paper we propose an alternative to equilibrium the-orymdasha cognitive hierarchy modelmdashthat explains empirical behav-ior in both of these games In cognitive hierarchy theories eachplayer believes he understands the game better than the otherplayers Specifically CH models posit decision rules that reflectan iterated process of strategic thinking (see Binmore [1988])The iteration formalizes Seltenrsquos [1998 p 421] intuition that ldquothenatural way of looking at game situations is not based on









IIA Decision Rules






k1 f(l )







j byi(s i



j)) yenj1mi

i(s ij s i



j Ek(i(s ij)))































k1 f(h)yenh0k f(h)














Row choice

Group outcome

L H

L x mdashH 0 1



























TA

BL

EII

DA

TA

AN

DC

HE

ST

IMA

TE

SO

F

INV

AR

IOU

Sp

-BE

AU

TY

CO

NT

ES

TG

AM

ES

Su

bjec

tpo

olor

gam

eS

ourc

eaG

rou

psi

zeS

ampl

esi

zeN

ash

equ

ilrsquom

Pre

drsquon

erro

r

Dat

aF

itof

CH

mod

elB

oots

trap

ped

90

ci

Mea

nS

tdde

vM

ode

M

ean

Err

orS

tdde

vM

ode

p

11

HC

W(9

8)7

6920

047

915

21

237

150

010

151

6

05

280

165

(00

05

)p

1

3H

CW

(98)

771

200

500

150

025

915

00

0015

04

05

294

195

(00

01

)H

igh

$C

HW

714

7211

061

08

455

490

594

1

63

861

(34

49

)M

ale

CH

W7

1772

144

576

97

543

7057

60

15

558

(10

43

)F

emal

eC

HW

746

7216

355

712

156

240

557

00

93

58(1

64

9)

Low

$C

HW

749

7217

254

811

954

200

547

0

111

156

(07

38

)7

(Mea

n

18)

Nag

el(9

8)7

3442

7

549

57

748

020

494

0

126

448

(00

10

)P

CC

CH

C(n

ew)

224

0

542

542

292

500

0049

5

47

295

0(0

00

1)

p

09

HC

W(9

8)7

670

49

449

424

350

010

495

00

277

45(0

11

5)

PC

CC

HC

(new

)3

240

47

547

529

050

010

475

00

286

26(0

10

8)

Cal

tech

boar

dC

amer

er73

730

42

642

623

433

050

431

04

243

34(0

10

9)

p

07

HC

W(9

8)7

690

38

938

924

735

100

388

0

219

835

(05

16

)C

EO

sC

amer

er20

200

37

937

918

833

100

377

0

120

234

(03

18

)G

erm

anst

ude

nts

Nag

el(9

5)14

ndash16

660

37

237

220

025

110

369

0

219

434

(07

15

)80

yrol

dsK

oval

chik

3333

0

370

370

175

271

1036

9

01

194

34(0

61

7)

U

S

hig

hsc

hoo

lC

amer

er20

ndash32

520

32

532

518

633

160

327

02

163

34(1

12

2)

Eco

nP

hD

sC

amer

er16

160

27

427

418

7N

A2

3027

50

013

121

(14

35

)1

2m

ean

Nag

el(9

8)15

ndash17

480

26

726

719

925

150

265

0

219

125

(11

19

)P

ortf

olio

mgr

sC

amer

er26

260

24

324

316

122

280

244

01

114

26(2

03

7)

Cal

tech

stu

den

tsC

amer

er17

ndash25

420

23

023

011

135

300

230

01

106

24(2

73

8)

New

spap

erN

agel

(98)

3696

14

60

2728

7884

0

230

230

202

13

0023

00

010

624

(30

31

)C

alte

chC

HC

(new

)2

240

21

721

729

90

080

222

06

316

0(4

01

4)

Cal

tech

CH

C(n

ew)

324

0

215

215

257

01

8021

50

118

626

(11

31

)G

ame

theo

rist

sN

agel

(98)

27ndash5

413

60

19

119

121

80

370

191

00

92

16(2

84

7)

Mea

n1

30M

edia

n1

61

aH

CW

(98)

isH

oC

amer

er

Wei

gelt

AE

R98

C

HC

are

new

data

from

Cam

erer

H

oan

dC

hon

gC

HW

isC

amer

er

Ho

Wei

gelt

(un

publ

ish

ed)

Kov

alch

ikis

data

repo

rted

byK

oval

chik

etal

[i

npr

ess]











IVB How Regular Is







Data setStahl and




Median 186 101 189 177 071Common 154 082 173 148 073







Data setStahl and













(Common )


(Common )







Data setStahl and





(Common ) 00416 00335 00237 00215 00046



IVE Two Examples




L R Data Nash CH









L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107


































































IIA Decision Rules






k1 f(l )







j byi(s i



j)) yenj1mi

i(s ij s i



j Ek(i(s ij)))































k1 f(h)yenh0k f(h)














Row choice

Group outcome

L H

L x mdashH 0 1



























TA

BL

EII

DA

TA

AN

DC

HE

ST

IMA

TE

SO

F

INV

AR

IOU

Sp

-BE

AU

TY

CO

NT

ES

TG

AM

ES

Su

bjec

tpo

olor

gam

eS

ourc

eaG

rou

psi

zeS

ampl

esi

zeN

ash

equ

ilrsquom

Pre

drsquon

erro

r

Dat

aF

itof

CH

mod

elB

oots

trap

ped

90

ci

Mea

nS

tdde

vM

ode

M

ean

Err

orS

tdde

vM

ode

p

11

HC

W(9

8)7

6920

047

915

21

237

150

010

151

6

05

280

165

(00

05

)p

1

3H

CW

(98)

771

200

500

150

025

915

00

0015

04

05

294

195

(00

01

)H

igh

$C

HW

714

7211

061

08

455

490

594

1

63

861

(34

49

)M

ale

CH

W7

1772

144

576

97

543

7057

60

15

558

(10

43

)F

emal

eC

HW

746

7216

355

712

156

240

557

00

93

58(1

64

9)

Low

$C

HW

749

7217

254

811

954

200

547

0

111

156

(07

38

)7

(Mea

n

18)

Nag

el(9

8)7

3442

7

549

57

748

020

494

0

126

448

(00

10

)P

CC

CH

C(n

ew)

224

0

542

542

292

500

0049

5

47

295

0(0

00

1)

p

09

HC

W(9

8)7

670

49

449

424

350

010

495

00

277

45(0

11

5)

PC

CC

HC

(new

)3

240

47

547

529

050

010

475

00

286

26(0

10

8)

Cal

tech

boar

dC

amer

er73

730

42

642

623

433

050

431

04

243

34(0

10

9)

p

07

HC

W(9

8)7

690

38

938

924

735

100

388

0

219

835

(05

16

)C

EO

sC

amer

er20

200

37

937

918

833

100

377

0

120

234

(03

18

)G

erm

anst

ude

nts

Nag

el(9

5)14

ndash16

660

37

237

220

025

110

369

0

219

434

(07

15

)80

yrol

dsK

oval

chik

3333

0

370

370

175

271

1036

9

01

194

34(0

61

7)

U

S

hig

hsc

hoo

lC

amer

er20

ndash32

520

32

532

518

633

160

327

02

163

34(1

12

2)

Eco

nP

hD

sC

amer

er16

160

27

427

418

7N

A2

3027

50

013

121

(14

35

)1

2m

ean

Nag

el(9

8)15

ndash17

480

26

726

719

925

150

265

0

219

125

(11

19

)P

ortf

olio

mgr

sC

amer

er26

260

24

324

316

122

280

244

01

114

26(2

03

7)

Cal

tech

stu

den

tsC

amer

er17

ndash25

420

23

023

011

135

300

230

01

106

24(2

73

8)

New

spap

erN

agel

(98)

3696

14

60

2728

7884

0

230

230

202

13

0023

00

010

624

(30

31

)C

alte

chC

HC

(new

)2

240

21

721

729

90

080

222

06

316

0(4

01

4)

Cal

tech

CH

C(n

ew)

324

0

215

215

257

01

8021

50

118

626

(11

31

)G

ame

theo

rist

sN

agel

(98)

27ndash5

413

60

19

119

121

80

370

191

00

92

16(2

84

7)

Mea

n1

30M

edia

n1

61

aH

CW

(98)

isH

oC

amer

er

Wei

gelt

AE

R98

C

HC

are

new

data

from

Cam

erer

H

oan

dC

hon

gC

HW

isC

amer

er

Ho

Wei

gelt

(un

publ

ish

ed)

Kov

alch

ikis

data

repo

rted

byK

oval

chik

etal

[i

npr

ess]











IVB How Regular Is







Data setStahl and




Median 186 101 189 177 071Common 154 082 173 148 073







Data setStahl and













(Common )


(Common )







Data setStahl and





(Common ) 00416 00335 00237 00215 00046



IVE Two Examples




L R Data Nash CH









L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107






























































j byi(s i



j)) yenj1mi

i(s ij s i



j Ek(i(s ij)))































k1 f(h)yenh0k f(h)














Row choice

Group outcome

L H

L x mdashH 0 1



























TA

BL

EII

DA

TA

AN

DC

HE

ST

IMA

TE

SO

F

INV

AR

IOU

Sp

-BE

AU

TY

CO

NT

ES

TG

AM

ES

Su

bjec

tpo

olor

gam

eS

ourc

eaG

rou

psi

zeS

ampl

esi

zeN

ash

equ

ilrsquom

Pre

drsquon

erro

r

Dat

aF

itof

CH

mod

elB

oots

trap

ped

90

ci

Mea

nS

tdde

vM

ode

M

ean

Err

orS

tdde

vM

ode

p

11

HC

W(9

8)7

6920

047

915

21

237

150

010

151

6

05

280

165

(00

05

)p

1

3H

CW

(98)

771

200

500

150

025

915

00

0015

04

05

294

195

(00

01

)H

igh

$C

HW

714

7211

061

08

455

490

594

1

63

861

(34

49

)M

ale

CH

W7

1772

144

576

97

543

7057

60

15

558

(10

43

)F

emal

eC

HW

746

7216

355

712

156

240

557

00

93

58(1

64

9)

Low

$C

HW

749

7217

254

811

954

200

547

0

111

156

(07

38

)7

(Mea

n

18)

Nag

el(9

8)7

3442

7

549

57

748

020

494

0

126

448

(00

10

)P

CC

CH

C(n

ew)

224

0

542

542

292

500

0049

5

47

295

0(0

00

1)

p

09

HC

W(9

8)7

670

49

449

424

350

010

495

00

277

45(0

11

5)

PC

CC

HC

(new

)3

240

47

547

529

050

010

475

00

286

26(0

10

8)

Cal

tech

boar

dC

amer

er73

730

42

642

623

433

050

431

04

243

34(0

10

9)

p

07

HC

W(9

8)7

690

38

938

924

735

100

388

0

219

835

(05

16

)C

EO

sC

amer

er20

200

37

937

918

833

100

377

0

120

234

(03

18

)G

erm

anst

ude

nts

Nag

el(9

5)14

ndash16

660

37

237

220

025

110

369

0

219

434

(07

15

)80

yrol

dsK

oval

chik

3333

0

370

370

175

271

1036

9

01

194

34(0

61

7)

U

S

hig

hsc

hoo

lC

amer

er20

ndash32

520

32

532

518

633

160

327

02

163

34(1

12

2)

Eco

nP

hD

sC

amer

er16

160

27

427

418

7N

A2

3027

50

013

121

(14

35

)1

2m

ean

Nag

el(9

8)15

ndash17

480

26

726

719

925

150

265

0

219

125

(11

19

)P

ortf

olio

mgr

sC

amer

er26

260

24

324

316

122

280

244

01

114

26(2

03

7)

Cal

tech

stu

den

tsC

amer

er17

ndash25

420

23

023

011

135

300

230

01

106

24(2

73

8)

New

spap

erN

agel

(98)

3696

14

60

2728

7884

0

230

230

202

13

0023

00

010

624

(30

31

)C

alte

chC

HC

(new

)2

240

21

721

729

90

080

222

06

316

0(4

01

4)

Cal

tech

CH

C(n

ew)

324

0

215

215

257

01

8021

50

118

626

(11

31

)G

ame

theo

rist

sN

agel

(98)

27ndash5

413

60

19

119

121

80

370

191

00

92

16(2

84

7)

Mea

n1

30M

edia

n1

61

aH

CW

(98)

isH

oC

amer

er

Wei

gelt

AE

R98

C

HC

are

new

data

from

Cam

erer

H

oan

dC

hon

gC

HW

isC

amer

er

Ho

Wei

gelt

(un

publ

ish

ed)

Kov

alch

ikis

data

repo

rted

byK

oval

chik

etal

[i

npr

ess]











IVB How Regular Is







Data setStahl and




Median 186 101 189 177 071Common 154 082 173 148 073







Data setStahl and













(Common )


(Common )







Data setStahl and





(Common ) 00416 00335 00237 00215 00046



IVE Two Examples




L R Data Nash CH









L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107



















































































k1 f(h)yenh0k f(h)














Row choice

Group outcome

L H

L x mdashH 0 1



























TA

BL

EII

DA

TA

AN

DC

HE

ST

IMA

TE

SO

F

INV

AR

IOU

Sp

-BE

AU

TY

CO

NT

ES

TG

AM

ES

Su

bjec

tpo

olor

gam

eS

ourc

eaG

rou

psi

zeS

ampl

esi

zeN

ash

equ

ilrsquom

Pre

drsquon

erro

r

Dat

aF

itof

CH

mod

elB

oots

trap

ped

90

ci

Mea

nS

tdde

vM

ode

M

ean

Err

orS

tdde

vM

ode

p

11

HC

W(9

8)7

6920

047

915

21

237

150

010

151

6

05

280

165

(00

05

)p

1

3H

CW

(98)

771

200

500

150

025

915

00

0015

04

05

294

195

(00

01

)H

igh

$C

HW

714

7211

061

08

455

490

594

1

63

861

(34

49

)M

ale

CH

W7

1772

144

576

97

543

7057

60

15

558

(10

43

)F

emal

eC

HW

746

7216

355

712

156

240

557

00

93

58(1

64

9)

Low

$C

HW

749

7217

254

811

954

200

547

0

111

156

(07

38

)7

(Mea

n

18)

Nag

el(9

8)7

3442

7

549

57

748

020

494

0

126

448

(00

10

)P

CC

CH

C(n

ew)

224

0

542

542

292

500

0049

5

47

295

0(0

00

1)

p

09

HC

W(9

8)7

670

49

449

424

350

010

495

00

277

45(0

11

5)

PC

CC

HC

(new

)3

240

47

547

529

050

010

475

00

286

26(0

10

8)

Cal

tech

boar

dC

amer

er73

730

42

642

623

433

050

431

04

243

34(0

10

9)

p

07

HC

W(9

8)7

690

38

938

924

735

100

388

0

219

835

(05

16

)C

EO

sC

amer

er20

200

37

937

918

833

100

377

0

120

234

(03

18

)G

erm

anst

ude

nts

Nag

el(9

5)14

ndash16

660

37

237

220

025

110

369

0

219

434

(07

15

)80

yrol

dsK

oval

chik

3333

0

370

370

175

271

1036

9

01

194

34(0

61

7)

U

S

hig

hsc

hoo

lC

amer

er20

ndash32

520

32

532

518

633

160

327

02

163

34(1

12

2)

Eco

nP

hD

sC

amer

er16

160

27

427

418

7N

A2

3027

50

013

121

(14

35

)1

2m

ean

Nag

el(9

8)15

ndash17

480

26

726

719

925

150

265

0

219

125

(11

19

)P

ortf

olio

mgr

sC

amer

er26

260

24

324

316

122

280

244

01

114

26(2

03

7)

Cal

tech

stu

den

tsC

amer

er17

ndash25

420

23

023

011

135

300

230

01

106

24(2

73

8)

New

spap

erN

agel

(98)

3696

14

60

2728

7884

0

230

230

202

13

0023

00

010

624

(30

31

)C

alte

chC

HC

(new

)2

240

21

721

729

90

080

222

06

316

0(4

01

4)

Cal

tech

CH

C(n

ew)

324

0

215

215

257

01

8021

50

118

626

(11

31

)G

ame

theo

rist

sN

agel

(98)

27ndash5

413

60

19

119

121

80

370

191

00

92

16(2

84

7)

Mea

n1

30M

edia

n1

61

aH

CW

(98)

isH

oC

amer

er

Wei

gelt

AE

R98

C

HC

are

new

data

from

Cam

erer

H

oan

dC

hon

gC

HW

isC

amer

er

Ho

Wei

gelt

(un

publ

ish

ed)

Kov

alch

ikis

data

repo

rted

byK

oval

chik

etal

[i

npr

ess]











IVB How Regular Is







Data setStahl and




Median 186 101 189 177 071Common 154 082 173 148 073







Data setStahl and













(Common )


(Common )







Data setStahl and





(Common ) 00416 00335 00237 00215 00046



IVE Two Examples




L R Data Nash CH









L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107





































































Row choice

Group outcome

L H

L x mdashH 0 1



























TA

BL

EII

DA

TA

AN

DC

HE

ST

IMA

TE

SO

F

INV

AR

IOU

Sp

-BE

AU

TY

CO

NT

ES

TG

AM

ES

Su

bjec

tpo

olor

gam

eS

ourc

eaG

rou

psi

zeS

ampl

esi

zeN

ash

equ

ilrsquom

Pre

drsquon

erro

r

Dat

aF

itof

CH

mod

elB

oots

trap

ped

90

ci

Mea

nS

tdde

vM

ode

M

ean

Err

orS

tdde

vM

ode

p

11

HC

W(9

8)7

6920

047

915

21

237

150

010

151

6

05

280

165

(00

05

)p

1

3H

CW

(98)

771

200

500

150

025

915

00

0015

04

05

294

195

(00

01

)H

igh

$C

HW

714

7211

061

08

455

490

594

1

63

861

(34

49

)M

ale

CH

W7

1772

144

576

97

543

7057

60

15

558

(10

43

)F

emal

eC

HW

746

7216

355

712

156

240

557

00

93

58(1

64

9)

Low

$C

HW

749

7217

254

811

954

200

547

0

111

156

(07

38

)7

(Mea

n

18)

Nag

el(9

8)7

3442

7

549

57

748

020

494

0

126

448

(00

10

)P

CC

CH

C(n

ew)

224

0

542

542

292

500

0049

5

47

295

0(0

00

1)

p

09

HC

W(9

8)7

670

49

449

424

350

010

495

00

277

45(0

11

5)

PC

CC

HC

(new

)3

240

47

547

529

050

010

475

00

286

26(0

10

8)

Cal

tech

boar

dC

amer

er73

730

42

642

623

433

050

431

04

243

34(0

10

9)

p

07

HC

W(9

8)7

690

38

938

924

735

100

388

0

219

835

(05

16

)C

EO

sC

amer

er20

200

37

937

918

833

100

377

0

120

234

(03

18

)G

erm

anst

ude

nts

Nag

el(9

5)14

ndash16

660

37

237

220

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110

369

0

219

434

(07

15

)80

yrol

dsK

oval

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3333

0

370

370

175

271

1036

9

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194

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hoo

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er20

ndash32

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532

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633

160

327

02

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160

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427

418

7N

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013

121

(14

35

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2m

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Nag

el(9

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480

26

726

719

925

150

265

0

219

125

(11

19

)P

ortf

olio

mgr

sC

amer

er26

260

24

324

316

122

280

244

01

114

26(2

03

7)

Cal

tech

stu

den

tsC

amer

er17

ndash25

420

23

023

011

135

300

230

01

106

24(2

73

8)

New

spap

erN

agel

(98)

3696

14

60

2728

7884

0

230

230

202

13

0023

00

010

624

(30

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alte

chC

HC

(new

)2

240

21

721

729

90

080

222

06

316

0(4

01

4)

Cal

tech

CH

C(n

ew)

324

0

215

215

257

01

8021

50

118

626

(11

31

)G

ame

theo

rist

sN

agel

(98)

27ndash5

413

60

19

119

121

80

370

191

00

92

16(2

84

7)

Mea

n1

30M

edia

n1

61

aH

CW

(98)

isH

oC

amer

er

Wei

gelt

AE

R98

C

HC

are

new

data

from

Cam

erer

H

oan

dC

hon

gC

HW

isC

amer

er

Ho

Wei

gelt

(un

publ

ish

ed)

Kov

alch

ikis

data

repo

rted

byK

oval

chik

etal

[i

npr

ess]











IVB How Regular Is







Data setStahl and




Median 186 101 189 177 071Common 154 082 173 148 073







Data setStahl and













(Common )


(Common )







Data setStahl and





(Common ) 00416 00335 00237 00215 00046



IVE Two Examples




L R Data Nash CH









L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107




















































































TA

BL

EII

DA

TA

AN

DC

HE

ST

IMA

TE

SO

F

INV

AR

IOU

Sp

-BE

AU

TY

CO

NT

ES

TG

AM

ES

Su

bjec

tpo

olor

gam

eS

ourc

eaG

rou

psi

zeS

ampl

esi

zeN

ash

equ

ilrsquom

Pre

drsquon

erro

r

Dat

aF

itof

CH

mod

elB

oots

trap

ped

90

ci

Mea

nS

tdde

vM

ode

M

ean

Err

orS

tdde

vM

ode

p

11

HC

W(9

8)7

6920

047

915

21

237

150

010

151

6

05

280

165

(00

05

)p

1

3H

CW

(98)

771

200

500

150

025

915

00

0015

04

05

294

195

(00

01

)H

igh

$C

HW

714

7211

061

08

455

490

594

1

63

861

(34

49

)M

ale

CH

W7

1772

144

576

97

543

7057

60

15

558

(10

43

)F

emal

eC

HW

746

7216

355

712

156

240

557

00

93

58(1

64

9)

Low

$C

HW

749

7217

254

811

954

200

547

0

111

156

(07

38

)7

(Mea

n

18)

Nag

el(9

8)7

3442

7

549

57

748

020

494

0

126

448

(00

10

)P

CC

CH

C(n

ew)

224

0

542

542

292

500

0049

5

47

295

0(0

00

1)

p

09

HC

W(9

8)7

670

49

449

424

350

010

495

00

277

45(0

11

5)

PC

CC

HC

(new

)3

240

47

547

529

050

010

475

00

286

26(0

10

8)

Cal

tech

boar

dC

amer

er73

730

42

642

623

433

050

431

04

243

34(0

10

9)

p

07

HC

W(9

8)7

690

38

938

924

735

100

388

0

219

835

(05

16

)C

EO

sC

amer

er20

200

37

937

918

833

100

377

0

120

234

(03

18

)G

erm

anst

ude

nts

Nag

el(9

5)14

ndash16

660

37

237

220

025

110

369

0

219

434

(07

15

)80

yrol

dsK

oval

chik

3333

0

370

370

175

271

1036

9

01

194

34(0

61

7)

U

S

hig

hsc

hoo

lC

amer

er20

ndash32

520

32

532

518

633

160

327

02

163

34(1

12

2)

Eco

nP

hD

sC

amer

er16

160

27

427

418

7N

A2

3027

50

013

121

(14

35

)1

2m

ean

Nag

el(9

8)15

ndash17

480

26

726

719

925

150

265

0

219

125

(11

19

)P

ortf

olio

mgr

sC

amer

er26

260

24

324

316

122

280

244

01

114

26(2

03

7)

Cal

tech

stu

den

tsC

amer

er17

ndash25

420

23

023

011

135

300

230

01

106

24(2

73

8)

New

spap

erN

agel

(98)

3696

14

60

2728

7884

0

230

230

202

13

0023

00

010

624

(30

31

)C

alte

chC

HC

(new

)2

240

21

721

729

90

080

222

06

316

0(4

01

4)

Cal

tech

CH

C(n

ew)

324

0

215

215

257

01

8021

50

118

626

(11

31

)G

ame

theo

rist

sN

agel

(98)

27ndash5

413

60

19

119

121

80

370

191

00

92

16(2

84

7)

Mea

n1

30M

edia

n1

61

aH

CW

(98)

isH

oC

amer

er

Wei

gelt

AE

R98

C

HC

are

new

data

from

Cam

erer

H

oan

dC

hon

gC

HW

isC

amer

er

Ho

Wei

gelt

(un

publ

ish

ed)

Kov

alch

ikis

data

repo

rted

byK

oval

chik

etal

[i

npr

ess]











IVB How Regular Is







Data setStahl and




Median 186 101 189 177 071Common 154 082 173 148 073







Data setStahl and













(Common )


(Common )







Data setStahl and





(Common ) 00416 00335 00237 00215 00046



IVE Two Examples




L R Data Nash CH









L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107




































































IVB How Regular Is







Data setStahl and




Median 186 101 189 177 071Common 154 082 173 148 073







Data setStahl and













(Common )


(Common )







Data setStahl and





(Common ) 00416 00335 00237 00215 00046



IVE Two Examples




L R Data Nash CH









L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107





























































Data setStahl and




Median 186 101 189 177 071Common 154 082 173 148 073







Data setStahl and













(Common )


(Common )







Data setStahl and





(Common ) 00416 00335 00237 00215 00046



IVE Two Examples




L R Data Nash CH









L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107
































































Data setStahl and













(Common )


(Common )







Data setStahl and





(Common ) 00416 00335 00237 00215 00046



IVE Two Examples




L R Data Nash CH









L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107




























































(Common )


(Common )







Data setStahl and





(Common ) 00416 00335 00237 00215 00046



IVE Two Examples




L R Data Nash CH









L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107
































































Data setStahl and





(Common ) 00416 00335 00237 00215 00046



IVE Two Examples




L R Data Nash CH









L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107




























































IVE Two Examples




L R Data Nash CH









L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107

































































L C R Data Nash CH

T 0 0 2 2 1 1 33 17 25M 2 2 0 0 1 1 29 17 25B 1 1 1 1 0 0 28 67 50Data 13 04 83Nash 17 17 67CH 08 08 84
















Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107









































































Data setStahl and











VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107





























































VIA Speculation















VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107






































































VIB Money Illusion












Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107


































































Nashprediction







VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107




























































VII CONCLUSION















We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107




































































We have

e0d 12 d

Ekd yenj0

k f j e jd

yenj0k f j

yenj0

k f j e jd

F j where F j j0

k

f j

In general for k 1




1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107




























































1d f0

Fk

1 2 f0 fk

Fk

1 2 f0 fk 1

Fk

1 2 f0 fk 1 fk

Fk


Fk


1 2 f0 f1

Fk

1 2 f0 f1 f2

Fk


Fk













REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107


































































REFERENCES








Data set

Stahl andWilson

Cooper andVan Huyck




Common 139 167 074 087 153 213 130 178 042 107



























































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