1.a payoff uncertainty explanation of results in.pdf

Games and Economic Behavior 26, 157185 (1999)Article ID game.1998.0649, available online at http://www.idealibrary.com on

A Payoff Uncertainty Explanation of Results inExperimental Centipede Games

Klaus G. Zauner

Department of Economics, University of Sydney, Sydney NSW 2006, Australia

Received August 24, 1994

This paper investigates how well a simple two-sided incomplete information gamewith a full support prior can be used to explain non-Nash equilibrium outcomesobserved in the centipede game experiments. I estimate the variance of the un-certainty about preferences in several versions of the model, select two models,and compare these models to the estimation results of the altruism model and thequantal response models. It is shown that the two selected models have a betterfit than the one-parameter altruism and quantal response models and that the es-timated variance can explain all qualitative features of these experimental results.Journal of Economic Literature Classification Numbers: C72, C19, C 44, C91. 1999Academic Press

1. INTRODUCTION

Nash equilibrium is one of the most pervasive solution concepts usedin game-theoretic models of economic phenomena. Experimental evidencesuggests that even in very simple games (for example, Brown and Rosenthal,1990; Ochs, 1995) a Nash equilibrium is rarely a good prediction for theoutcome of a strategic interaction.

*Previous versions were titled Bubbles, Speculation and a Reconsideration of the Cen-tipede Game Experiments, UC San Diego, October 1993 and A Reconsideration of theCentipede Game Experiments, UC San Diego, August 1994. I am grateful to one of thereferees who suggested the current title.

I would like to thank the two referees, Vince Crawford, Walter Heller, Bob Marks, RichardMcKelvey, Thomas Palfrey, Paul Pezanis-Christou, Joel Sobel, Garey Ramey, and audiencesat the Seventh World Congress of the Econometric Society, AGSM, Texas A & M and UCSan Diego for very helpful comments, Richard McKelvey and Thomas Palfrey for their data,and especially Max Stinchcombe for encouragement and support. All errors are mine. E-mail:[email protected].

1570899-8256/99 $30.00

Copyright 1999 by Academic PressAll rights of reproduction in any form reserved.

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

158 klaus g. zauner

This paper uses a simple two-sided incomplete information game(Harsanyi, 1973) with a full-support prior to explain systematic deviationsfrom Nash equilibrium behavior. In choosing this model this paper followsHarsanyi (1973). Classical game theory assumes that in any game 0 everyplayer has precise knowledge of the payoff function of every other player(as well as of his own). But it is more realistic to assume thateven ifeach player does have exact knowledge of his own payoff functionhe canhave at best only somewhat inexact information about the other playerspayoff functions : : : . The payoff function of every player is subject to ran-dom disturbances : : : , due to small stochastic fluctuations in his subjectiveand objective conditions (e.g., in his mood, taste, resources, social situa-tion, etc.) (Harsanyi, 1973, pp. 12). These random disturbances comefrom the strategic uncertainty (Crawford, 1997) players face in strategicsituations. This strategic uncertainty should be interpreted very loosely.For example, players can be uncertain about the rationality of players,about strategy choices, about altruism, about beliefs, etc. This strategicuncertainty is modeled as a random perturbation to each players payoffs.

To this end, each agents payoffs are independently perturbed acrossnodes with normal noise; each agent is told his own payoffs in the game0, but not the payoffs of the other agents; and then the equilibrium ofthe new game assuming that the perturbations are common knowledge isexamined. When agents play the game, they know their own preferencesthough they may be uncertain about the preferences of the other agents.This equilibrium model gives systematic deviations from the Nash equilib-rium of the underlying (unperturbed) game. Standard maximum likelihoodmethods can be used to estimate the variance of the uncertainty about pay-offs that fits the data best.

One important feature to note is that the added noise structure is un-biased in the sense that the expected value of the perturbation is 0. Thismeans that in expected value agents play the original game. This modeltherefore explains deviations from the Nash equilibrium without a system-atic change in the underlying payoffs as in the usual incomplete informationgames with finite types (for example, Kreps et al., 1982; McKelvey and Pal-frey, 1992). The perturbations used here are only a slight variation fromthe Bayesian Nash equilibrium theme (Harsanyi, 19671968). There is oneimportant difference. The usual incomplete information games with finitetypes distort the preferences of players in a way which depends on the gamethat is analyzed. Usually a different perturbation (type) is required to fitthe data and this perturbation is game-specific. Put differently, dependingon the game, different types of players are introduced which are made toplay in close similarity to the observed data. Then one computes a BayesianNash equilibrium where this type occurs with a strictly positive probability.By construction, an equilibrium of this perturbed game will involve play

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

explanation of centipede game results 159

similar to the observed data. This procedure is ad hoc since the introducedtype depends on the game that is analyzed. The model used in this paperavoids such a procedure and gives an approach to account for systematicdeviations from the Nash equilibrium without introducing game-dependenttypes.

These perturbations are applied to the centipede game experiments ofMcKelvey and Palfrey (1992). In this game, two players alternately get achance to take the larger portion of a continually escalating pile of money.As soon as one person takes, the game ends with that player getting thelarger portion of the pile, and the other player getting the smaller portion.Passing strictly decreases ones payoff if the opponent takes the larger pileon the next move. In case the opponent also passes on his next move, oneends up in the same choice situation with reversed roles of players andincreased payoffs. (In the experiments the payoffs were doubled.) In theexperiments the game ended after (at most) four or six moves.

The results of these experiments contradict standard game-theoretic wis-dom; in particular, the only Nash equilibrium of the game above involvestaking the large pile on the first occasion. In other words, the (conditional)probability of taking at the first decision node is 1 in any Nash equilibriumof the centipede game. Standard game theory would thus predict that oneshould not pass and give the opponent a chance to move since if the oppo-nent is self-interested and completely rational he would take the larger pileon his next move and one would lose out. The data show that the game-theoretic prediction of taking is rarely the outcome of the experiments. The(conditional) probability of taking at the first move is very different from1 and the probability of taking the larger pile increases as the game pro-gresses. McKelvey and Palfrey (1992) study three versions of the centipedegame (a four-move version, a high-payoff version, and a six-move version).The versions of the games and a summary of the experimental results ofthe centipede game experiments of McKelvey and Palfrey (1992) are givenin the next section.

This paper considers five versions of this Harsanyi-type model with dif-ferent normal noise structures: constant variance (basic model), variancedepending linearly on the players own payoff (OWN), variance dependinglinearly on the opponents payoff (OPP), standard deviation depending lin-early on the players own payoff (OWN2), and standard deviation depend-ing linearly on the opponents payoff (OPP2). For computational reasonsthis paper uses an agent normal form analysis. Each player observes onlythe realization of the noise term corresponding to his current take payoffand not all realizations of his noise terms (at each node) at the beginningof the game.1 This is a sensible approach given the experimental design. A

1I am very grateful to Richard McKelvey and Tom Palfrey for pointing this out to me.

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

160 klaus g. zauner

player faces strategic uncertainty at each decision node in the game, i.e.,the uncertainty is not resolved till the last move. The variance of the un-certainty about preferences is estimated for each of these versions and thetwo models that fit best are selected based on maximum likelihood.

The model with constant variance and the model where the variance de-pends linearly on the players own payoff fit the data best. These modelscan account for all patterns in the data of the centipede game experiments.For example, the probability of taking the larger pile increases as the gameprogresses. It is remarkable that only a moderate level of noise is requiredfor the probability of taking the larger pile at the first node to be very dif-ferent from 1 (the only Nash equilibrium outcome) and that such a simplemodel can account for all the qualitative features in the data. A compar-ison of the goodness of fit of these five models with the altruism model(McKelvey and Palfrey, 1990) and the quantal response models (McKelveyand Palfrey, 1995b) show that the two selected models have a better fit.

For these two models a model with time-varying variance is estimated andit is shown that the variance decreases with repeated play of the game, i.e.,players get closer to the Nash equilibrium with repeated play. Forecastsusing the estimated value of the parameter show that it would take less than100 periods for play to converge to the Nash equilibrium of the unperturbedcentipede game.

There are several other explanations for the phenomena observed in thecentipede game experiments. One explanation is altruism in the form ofincomplete information (Kreps et al., 1982). Suppose that there is a smallamount of incomplete information. There is a small number of altruistsamong the players who pass at every decision node. In this case it maybe worthwhile for a selfish player to give the opponent a chance to movefor two reasons: (1) there is a small probability that the opponent is analtruist who passes on each occasion and (2) by imitating an altruist it ispossible for a selfish player to make the opponent believe that the opponentfaces an altruist and induce the opponent to pass on the next move. Thismodel coupled with errors in the strategy choices of players are the maincomponents in McKelvey and Palfreys (1992) analysis.

Another explanation relies on the irrationality of players (Rosenthal,1981). If it is common knowledge that there is a small probability thata player passes at the last move then it pays off to pass with positive prob-ability on the second to last move. As one moves closer to the beginningof the game one can make the following observation. The probability ofpassing increases as the difference between the payoff to passing and thepayoff to taking decreases. This in turn makes the payoff difference smallerat previous steps in the game, until finally the payoff to taking is lower thanthe payoff to passing. The equilibrium here involves agents passing forsome time before starting to take. Rosenthal (1981) calls this a cascade.

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148


Still another explanation is the quantal response model of McKelvey andPalfrey (1995a, 1995b, 1996). Players make choices based on a quantalchoice model. The underlying idea of the quantal response model is theimperfect implementation of strategies: the probability of implementing aparticular strategy is increasing in the equilibrium payoff of the strategy.The expected payoffs are calculated from the equilibrium distribution ofjoint strategies. This quantal response model has a different noise structureand a different interpretation than the model of this paper. It is similarto the model used here in that it gives an internally consistent equilibriummodel that explains non-Nash behavior without a systematic change of theunderlying preferences of players. McKelvey and Palfrey (1995b) use a logitspecification of the error structure and estimate an agent normal form anda normal form version of the quantal choice model for the centipede games.

The plan of this paper is as follows: Section 2 describes the centipedegames and summarizes the data of the centipede game experiments. Sec-tion 3 presents the basic Harsanyi-type model that is estimated in Sec-tion 4. Section 5 gives the estimation results for the other four versions ofthe model. Two models are selected based on maximum likelihood. Thissection also provides a comparison of the goodness of fit between severalone-parameter models. Section 6 shows the estimation results for the modeltaking into account earlier and later plays of the game. Section 7 concludes.

2. THE CENTIPEDE GAME EXPERIMENTS

In this section a summary of McKelvey and Palfreys (1992) centipedegame experiments and results is presented. In their version of the centipedegame, two players alternately get a chance to take the larger portion ofa continually escalating pile of money. As soon as one person takes, thegame ends with that player getting the larger portion of the pile, and theother player getting the smaller portion. There are three versions of thecentipede game given in Figs. 13. All games have a similar structure. Forexample, in the four-move game in Fig. 1 player 1 can choose between

FIG. 1. Four-move centipede game.

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

PC015148

162 klaus g. zauner

FIG. 2. High-payoff centipede game.

take (going down) which ends the game with payoffs $0.40 for player1 and $0.10 for player 2 and pass (give player 2 the move). If player 1passes player 2 has the choice between take which ends the game withpayoffs $0.20 for player 1 and $0.80 for player 2 or pass (give the moveback to player 1), etc.

With these extensive-form games they ran a series of experiments in-volving students at Cal Tech and at the Pasadena City College. I refer totheir paper for more information on the experimental design. They tookgreat care to eliminate potential supergame or cooperative behavior. It wasmade common knowledge that no subject was ever matched with anothersubject more than once. Therefore one would expect that there should beno differences between earlier and later plays of the game.

All three extensive-form games have only one Nash equilibrium outcome,namely, taking the larger pile on the first occasion, i.e., going down at thefirst decision node. If the Nash equilibrium is played player 1 gets $0.40($1.60 in the high-payoff version) and player 2 gets $0.10 ($0.40 in the high-payoff version). The unique subgame perfect equilibrium involves takingthe larger pile on each occasion. McKelvey and Palfrey observed that thestudents almost never follow this game-theoretic recommendation.

Their results are summarized in Tables I and II. For more details seeMcKelvey and Palfrey (1992, pp. 808811). Table I gives the number ofgames played of each version of the centipede game together with thenumber of games ending at node i 2 I, denoted ni, where the endnodes are

FIG. 3. Six-move centipede game.


TABLE IOutcomes of the Experiments

Experiment Number of games n1 n2 n3 n4 n5 n6 n7

Four moves 281 20 100 104 43 14 High payoffs 100 15 37 32 11 5 Six moves 281 2 18 56 108 71 22 4

counted from left to right. Table II gives the implied take probabilities(the (conditional) probability that the player takes the larger pile when itis his turn to move) of each of the versions.

The data summarized in these tables point to several very interestingphenomena. The most important of these are2: First, only 7% of the four-move games, 15% of the high-payoff games, and 1% of the six-move gamesend with the Nash equilibrium outcome, i.e., end with player 1 taking thelarge pile on the first move. Second, the implied take probability increaseslater in the game. Third, the implied probability of using strictly dominatedstrategies is strictly positive (1 p4 D 0:25 in the four-move, 1 p4 D 0:31in the high-payoff, and 1 p6 D 0:15 in the six-move centipede game).Fourth, there is more taking in later plays of the game. This is surpris-ing since there is no game-theoretic reason to expect subjects to play anydifferently in earlier games than in later games because of the matchingscheme used in these experiments.

McKelvey and Palfrey (1992) explain these results with incomplete infor-mation (Harsanyi, 19671968) and errors in actions. They develop a modelusing the recent literature on signaling and reputation (Kreps and Wilson,1982; Kreps et al., 1982; Milgrom and Roberts, 1982). There is a smallnumber of altruists among the players who choose pass at every deci-sion node. It may be worthwhile for selfish players to mimic altruists. Byimitating an altruist it is possible for a selfish player to make the oppo-nent believe that the opponent faces an altruist. This induces the opponentto pass and therefore increases the selfish players payoff.3 McKelvey and

2McKelvey and Palfrey (1992, pp. 808811).3See Kreps (1990) for a lucid explanation of this reputation effect.

TABLE IIImplied Take Probabilities

Experiment p1 p2 p3 p4 p5 p6

Four moves 0.07 0.38 0.65 0.75 High payoffs 0.15 0.44 0.67 0.69 Six moves 0.01 0.06 0.21 0.53 0.73 0.85

164 klaus g. zauner

Palfrey (1992) estimate a level of altruism of about 5%. In addition, theyallow for errors in actions and errors in beliefs which are found to besignificant. They also find significant levels of learning in the sense thatsubjects learn to make fewer errors over time. Homogeneity of beliefs isrejected but rational expectations, or on-average correct beliefs, cannot berejected.

3. THE MODEL

In this paper the following Harsanyi-type model (1973) is proposed. Eachagents payoffs are independently perturbed with additive noise acrossendnodes. Before the start of the game each agent gets to know his ownpayoffs (i.e., the realizations of the noise terms corresponding to his ownpayoffs) but not the other agents payoffs (i.e., the realizations of the noiseterms corresponding to the payoffs of the other agents); then the equilib-rium of the new perturbed game is examined assuming that the originalgame and the distribution of the noise terms is common knowledge andthat agents are completely rational in the (perturbed) game. When agentsplay the perturbed game, they know their own payoffs though they may beuncertain about the payoffs of the other agents. In choosing this model Ifollow Harsanyi (1973). The introduced random disturbances come fromthe strategic uncertainty players face in a strategic situation. This modelputs all the uncertainty a player might possibly face (about players, aboutplayers strategy choices, about players rationality or irrationality, altruism,etc.) in the payoffs.

Denote player i and endnode z by i; z and let D 1; 1; : : : ; 1; 5;2; 1; : : : ; 2; 5. Denote the distribution of by . Each agent i getsto know the realization of i; but not the realization of j; for i 6Dj. After getting this information they play the game. It is assumed thatthere is common knowledge of the original game, the perturbation , thedistribution , as well as complete rationality of players. The extensive formof this perturbed game is given in Fig. 4.4 Zauner (1993) has shown that

4It is possible to write down a version of the gang-of-four (Kreps et al., 1982) model. Let0 D 0; : : : ; 0 2


FIG. 4. Perturbed four-move centipede game.

an equilibrium exists (under the assumption of continuous information) insuch incomplete information games in extensive form.

For the Harsanyi-type (1973) model in this paper the following assump-tions are made:

The prior , i.e., the distribution of , has full support (every nonemptyopen set in

166 klaus g. zauner

by a different agent with the same payoffs as the player to whom the agentbelongs. Each agents payoff is independently perturbed with additive nor-mal noise across nodes. Agents get to know the realization of the noiseterms pertaining to their own payoffs but not the realization of the noiseterms pertaining to the other agents payoffs before the extensive form isplayed. This means that each player observes only the realization of thenoise term corresponding to his current take payoff and not all realiza-tions of his noise terms (at each node) at the beginning of the game. This isa reasonable approach given how the game was played in the experiments.A player faces strategic uncertainty at each decision node in the game andthe uncertainty is not resolved until the last move.

That there is a difference to a normal form approach can be seen ina simple example. Suppose we perturb the extensive form of the followingone-player game. The player has a choice between L and R. If he choosesL the game ends. If he chooses R, he has to choose between l and r. Thepayoff at each endnode is 0.6 Now perturb the payoffs by adding indepen-dent standard normal distributions to each payoff. In the agent normalform approach we calculate the probability of choosing l at the seconddecision node by PrN0; 1 > N0; 1 D 1=2, where N0; 1 is a normaldistribution with mean 0 and variance 1. The probability of L at the firstdecision node can then be calculated as PrN0; 1 > N0; 1=2 D 1=2.On the other hand, in the normal form approach we calculate the prob-ability that strategy L is played as PrN0; 1 > maxN0; 1;N0; 1 DR11 FxFxf xdx D 1=3, where Fx is the cumulative distribution

function and f x is the probability density function of a normal distri-bution with mean 0 and variance 1. This example shows that different equi-libria can exist in the perturbed games depending on whether we take anormal form or agent normal form approach. Note that, according tothe invariance property, a solution concept should depend only on the re-duced normal form of the game (Kohlberg and Mertens, 1986, p. 1010).These perturbations are therefore able to predict deviations from the in-variance property of game-theoretic solution concepts.

If the perturbation has full support, i.e., every nonempty open set in


FIG. 5. Perturbed four-move centipede game with Gaussian noise.

across nodes) normal noise terms, where N0; 2 denotes a normally dis-tributed random variable with mean 0 and variance 2. Note also that thismodel satisfies scale invariance. Adding a constant to the payoffs of a playerwill not change the strategies and multiplying the payoffs of a player by apositive constant will only increase the variance by a multiplicative constantand leave the strategies unchanged.

Since the centipede game is a game of perfect information, i.e., infor-mation sets are singletons, it is easy to compute the take probabilities inthe perturbed game using an agent normal form approach. Start at thelast decision node and compute the probability of taking at the last node,p42. The probability p42 is the probability that the perturbed utilityfor taking at the last node is greater than the perturbed utility for pass-ing at that node. Then compute p32 which is the probability that theperturbed utility for taking at the second to last node is greater than theperturbed utility for passing at that node in a similar fashion taking into ac-count p42. Similar computations give p22 and p12. For example,p42 and p32 are computed in the following way:

p4 D P3:20CN0; 2 > 1:60CN0; 2D P3:20 1:60 > N0; 22;

p3 D P1:60CN0; 2> p40:80CN0; 2 C 1 p46:40CN0; 2

}D P1:60 p40:80 1 p46:40 > N0; 21C p24 C 1 p42}:

From the computation of the perturbed equilibrium above one sees thatthe probability of take at the last node, p4, is calculated as the proba-bility that a random draw from N0; 22 is less than some critical value.Similarly, the calculations above show that the probability of take at thesecond to last node, p3, is calculated as the probability that a random draw

168 klaus g. zauner

from N0; 21 C p24 C 1 p42 is less than some critical value. Thevariance which roughly corresponds to the uncertainty faced at a particu-lar node is larger at the last node than at the second to last node (22 vs.21 C p24 C 1 p42). Smaller variances at the initial nodes seem to in-dicate that there is less uncertainty at the earlier nodes than at the laternodes. Intuitively, decisions at the earlier nodes should be much more un-certain than decisions at the later nodes.

But, as the above calculations show, it is not enough to just look at thevariances. The (expected) payoffs (i.e., critical values used in the calcula-tions) have to be taken into account. Note that the (expected) value to tak-ing is decreasing as one moves closer to the beginning of the game whereasthe variance associated with taking remains constant. In other words, thevariance to (expected) value-to-taking ratio increases as one moves closerto the beginning of the game. Intuitively, decisions at the earlier nodes aremuch more uncertain than at the later nodes since they involve a relativelyhigher variance. Similar arguments can be made about the relationship be-tween the variance associated with passing and the (expected) value topassing.

This intuition can be strengthened by comparing the (subgame perfect)Nash equilibrium of the original game with the equilibrium of the perturbedgame. The (behavioral) strategies that are induced by the equilibrium ofthe perturbed game can be seen in Fig. 6. For the four-move centipede

FIG. 6. Implied take probabilities for the perturbed four-move game for different valuesof the variance (basic model).


game Fig. 6 plots the four take probabilities for different values of thevariance 2. (The x axis shows the variance and the y axis the equilibriumtake probabilities of the perturbed game.) Figure 7 shows a similar plotfor the six-move game. Only a very small variance is required to move awayfrom the only Nash equilibrium outcome (which involves p1 D 1). As oneshrinks the variance (moves from the right to the left in the figures) oneapproaches the unique subgame perfect equilibrium of the centipede game(p1 D p2 D p3 D p4 D 1).

Note that for most values of the variance the implied take probabili-ties increase later in the game, i.e., that p1 < p2 < p3 < p4. Put differently,the difference between the implied take probabilities of the subgame per-fect equilibrium of the unperturbed game and the implied take probabil-ities of the equilibrium of the perturbed game (for fixed variance) increasesearlier in the game. This reflects, loosely speaking, the increased strategicuncertainty at earlier nodes.

The phenomenon of increased take probabilities later in the gamecomes from the addition of independent Gaussian noise to the payoff ofthe centipede game in such a way that the standard deviation is moderaterelative to the payoff differences. The added noise guarantees that thereis a strictly positive probability that players pass at the last move. As onemoves closer to the beginning of the game the process of reducing and(later reversing) the difference between the payoff to taking and the pay-

FIG. 7. Implied take probabilities for the perturbed six-move game for different valuesof the variance (basic model).

170 klaus g. zauner

off to passing will increase the probability of passing. Note that instead ofa cascade (Rosenthal, 1981) of irrational behavior, one has the uniqueequilibrium phenomenon in a perturbed game.

Using the take probabilities, the equilibrium distribution for outcomesof the game can be computed as a function of 2 so that standard maximumlikelihood methods can then be used to estimate the variance 2 of themodel in Fig. 5 that fits the data best.

From the take probabilities it is possible to compute the probability

s2 D s12; : : : ; s52of observing each of the possible outcomes, T, PT, PPT, PPPT, : : : , where Tdenotes take and P denotes pass. The entries in the vector

s2 D s12; : : : ; s52are the expected frequencies of the different endnodes. Therefore, s12 Dp12, s22 D 1 p12p22, etc. is the likelihood of observinga particular vector n D n1; : : : ; nmC1 of outcomes given 2. The (log-)likelihood function is thus given by

L2 D5YiD1si2ni ; (1)

L2 D5XiD1ni ln si2: (2)

Similar expressions can be derived for the six-move and high-payoff cen-tipede game. Standard maximum likelihood methods can now be used toestimate the variance 2 of the model in Fig. 5 that fits the data best.

4. ESTIMATION AND RESULTS

In order to maximize the likelihood function a grid search is done overthe values of 2. The results of this estimation are given in Tables IIIand IV. Table III gives the estimated take probabilities and Table IV theexpected frequencies of the endnodes of the four-move, high-payoff, andsix-move centipede game, respectively.

Note that these perturbations can explain all patterns McKelvey and Pal-frey (1992) found in the data of their experiments.

In particular, the maximum likelihood estimation shows that:

Each terminal node has strictly positive expected frequency ( Ofi > 0 forall i 2 I).

PC015148


TABLE IIIEstimated Take Probabilities (Basic Model)

Experiment O 2 Op1 Op2 Op3 Op4 Op5 Op6 L

Four moves 0.534 0.214 0.301 0.676 0.939 418.334High payoffs 7.685 0.212 0.321 0.704 0.949 150.290Six moves 4.856 0.186 0.213 0.215 0.436 0.814 0.980 506.382Sum L 1075.01

The model predicts subjects will choose strictly dominated actions withpositive probability (1 Op4 D 0:061 in the four-move centipede game,1 Op4 D 0:051 in the high-payoff centipede game, and 1 Op6 D 0:02in the six-move centipede game).

The predicted probability of take ( Op) increases as one gets closerto the last move.

At every node there is a higher predicted probability of taking in thefour-move centipede game than in the corresponding move of the six-movecentipede game (0.214 vs. 0.186 in the first move, 0.301 vs. 0.213 in thesecond move, 0.676 vs. 0.215, etc.).

At every node there is a higher predicted probability of taking in thehigh-payoff centipede game than in the corresponding move of the six-move centipede game (0.212 vs. 0.186 in the first move, 0.321 vs. 0.213 inthe second move, etc.).

Comparing the four-move centipede game to the last four moves of thesix-move centipede game, there is a higher predicted probability of takingin the six-move centipede game than in the corresponding move of thefour-move centipede game (0.939 vs. 0.980 in the last move, 0.676 vs. 0.814in the next to last move).

Comparing the high-payoff centipede game to the last four moves ofthe six-move game, there is a higher predicted probability of taking in thesix-move centipede game, even though the payoffs in the high-payoff gamesare identical to the payoffs in the last four moves of the six-move game(0.949 vs. 0.980 in the last move, 0.704 vs. 0.814 in the next to last move,etc.).

TABLE IVExpected Frequencies of Endnodes (Basic Model)

Experiment Of1 Of2 Of3 Of4 Of5 Of6 Of7Four moves 0.214 0.237 0.372 0.167 0.010 High payoffs 0.212 0.253 0.377 0.150 0.008 Six moves 0.186 0.173 0.138 0.219 0.231 0.052 0.001

172 klaus g. zauner

The model predicts that a proportion of subjects chooses pass onevery occasion (0.01 in the four-move, 0.008 in the high-payoff, and 0.001in the six-move centipede game).

This model also gives the rest of the identified features of the data. Thereare differences between the earlier and later plays of the game in a giventreatment. Subjects played each version of the centipede game 10 times.To see whether there are significant differences between earlier and laterplays, these 10 rounds are split up into the first five rounds and the last fiverounds. Then the above model is re-estimated. In the four-move centipedegame a variance for the first five rounds of 0:8335 and a variance for the lastfive rounds of 0:362395 with a log-likelihood value of 409:925 is estimated.In the six-move centipede game a variance for the first five rounds of 6:013and a variance for the last five rounds of 3:42405 with a log-likelihood valueof 501:999 is estimated. The difference between the model with constantvariance and the model with different variances in the first five roundsand last five rounds is significant at the 1% level. Estimated variances fallbetween earlier and later play. This seems to suggest that more frequentplay gets players closer to the Nash equilibrium of the centipede gamesince as the variance, 2, approaches 0 the equilibrium of the perturbedgame approaches the subgame perfect Nash equilibrium of the centipedegame.7

This simple model can therefore account for all patterns in the data.This model provides a more parsimonious explanation of the results ofthe centipede game experiments than the reputation model of Kreps et al.(1982) used by McKelvey and Palfrey (1992) since only one parameter isused. It is remarkable that this model can capture all features of the dataof the centipede game experiments.

It is necessary to make a couple of comments about the model. First,even though this specification gives all qualitative features of the data, ithas its weaknesses. In particular, as we saw, for example, in Fig. 6 (Fig. 7) itis impossible in the four-move (six-move) game to lower the probability oftake at the first node, p1, below 0:2 (0:15). The data show that the impliedtake probability at the first node is 0:07 in the four-move and 0:01 in thesix-move game. The estimation results show that this model overestimatesthe implied take probabilities. The estimated implied take probabilitiesare 0:214 in the four-move and 0:186 in the six-move game.

Second, the perturbations used are the same regardless of payoffs. It ispossible that the uncertainty players face is larger when the payoff is larger.Since payoffs increase later in the game, such models would increase the

7A general proof of the upper-hemi-continuity of incomplete information games is con-tained in Stinchcombe and Zauner (1993) and Zauner (1993).


variances later in the game. Several such models are considered in the nextsection.

5. OTHER SPECIFICATIONS AND GOODNESS OF FIT

This section considers other specifications of the error structure and com-pares several such specifications. In the last section we saw that the esti-mated take probabilities at the first node are considerably higher thanthe implied take probabilities of the data and that the above specifica-tion of the noise structure seems to have the feature that the amount ofuncertainty faced at a particular node is unrelated to the payoff at thatnode. In some sense players have more information about payoffs at laterdecision nodes. Several models that naturally increase the variance later inthe game are considered.

One alternative to the basic model above involves a multiplicative in-stead of an additive noise structure. That is to say, the payoffs are x1C instead of xC , where x is the payoff and the noise term has a normaldistribution with mean 0. This means that the variances of the perturbationsshould depend on each players own payoff. To obtain a multiplicative noisestructure, we change (in the four-move centipede game for example) thepayoffs of player 1 to 0:40 C N0; 0:4022; 0:20 C N0; 0:2022; 1:60 CN0; 1:6022; 0:80 C N0; 0:8022; 6:40 C N0; 6:4022, and the pay-offs for player 2 to 0:10 C N0; 0:1022; 0:80 C N0; 0:8022; 0:40 CN0; 0:4022; 3:20 C N0; 3:2022; 1:60 C N0; 1:6022, where we gofrom left to right in Fig. 5. Similar changes are made to the high-payoffand six-move game. A possible interpretation of this noise structure is thatpeople face (strategic) uncertainty and that this uncertainty is bigger iftheir own payoff is larger. Table V gives the estimated take probabilitiesfor this specification. This model fits worse than the basic model since thesum of the likelihood values of the four-move, high-payoff, and six-movegames 1090:7 is smaller than that for the basic model 1075:01.

There is nothing special about squaring the payoff. Multiplication of thevariance with a players own payoff gives another specification. To make

TABLE VEstimated Take Probabilities: Variance Depends on Players Own Squared Payoff (OWN2)

Experiment O 2 Op1 Op2 Op3 Op4 Op5 Op6 LFour moves 1.108 0.144 0.268 0.354 0.665 427.100High payoffs 2.199 0.231 0.336 0.380 0.619 153.027Six moves 1.191 0.088 0.125 0.154 0.276 0.356 0.659 510.569Sum L 1090.70

PC015148

174 klaus g. zauner

this change in the four-move centipede game, player 1s payoff at the firstendnode is 0:40CN0; 0:402 and player 2s payoff is 0:10CN0; 0:102and similarly for the other endnodes and the other games. For the four-move (six-move) game Fig. 8 (Fig. 9) plots the implied take probabilitiesfor different values of the variance.

Table VI gives the estimation results for the model where the variancedepends linearly on each players own payoff. Table VI shows the estimatedtake probabilities for this model.

This model fits better than the basic model and the model with the mul-tiplicative noise structure since the log-likelihood is higher 1010:5.

Another alternative class of models makes the variance dependent onthe opponents payoff. This arises when subjects think about whether theycare about the other players well-being or not. This specification allowsplayers to care in either directions. They prefer others get larger payoffs be-cause of altruism or they prefer others get smaller payoffs because of envy.To make the variance depend on the opponents squared payoff, changethe payoffs of player 1 for the four-move centipede game in Fig. 5 to0:40 C N0; 0:1022; 0:20 C N0; 0:8022; 1:60 C N0; 0:4022; 0:80 CN0; 3:2022; 6:40 C N0; 1:6022, and the payoffs for player 2 to0:10 C N0; 0:4022; 0:80 C N0; 0:2022; 0:40 C N0; 1:6022; 3:20 CN0; 0:8022; 1:60 C N0; 6:4022 (going from the left endnode to theright endnode). Similar changes are made in the six-move and high-payoff

FIG. 8. Implied take probabilities for the perturbed four-move game for different valuesof the variance (OWN model).


FIG. 9. Implied take probabilities for the perturbed six-move game for different valuesof the variance (OWN model).

games. Table VII gives the estimation results. This model fits worse thanthe basic model.

The last model we consider is a model where the variance of the noiseterms depends linearly on the opponents payoff at that node. Thereforewe change, in the four-move centipede game for example, player 1s payoffat the first endnode to 0:40CN0; 0:102 and player 2s payoff to 0:10CN0; 0:402 and similarly for the other endnodes and the other games.Table VIII shows the estimated take probabilities for this model.

A comparison according to the log-likelihood function8 shows that themodel OWN (i.e., where the error variance is a linear function of each

8The sum of squared errors criterion, the AI criterion, or the BI criterion selects the samemodels. Note that all the models considered have one parameter. For a discussion of thesecriteria, see Amemiya (1985, p. 146).

TABLE VIEstimated Take Probabilities: Variance Depends Linearly on Players Own Payoff (OWN)


176 klaus g. zauner

TABLE VIIEstimated Take Probabilities: Variance Depends on Opponents Squared Payoff (OPP2)

Experiment O 2 Op1 Op2 Op3 Op4 Op5 Op6 L

Four moves 2.282 0.154 0.363 0.292 0.565 438.908High payoffs 4.542 0.227 0.411 0.336 0.546 154.478Six moves 1.688 0.079 0.224 0.124 0.335 0.272 0.576 539.280Sum L 1132.67

players own payoff) fits the data best (log-likelihood of 1010:5). Thesecond best model selected is the basic model from Section 3 (with a log-likelihood of 1075:01).

Table IX compares the estimation results of different one-parametermodels for the four-move game. The columns show the actual frequen-cies and the expected frequencies for the altruism model (McKelvey andPalfrey, 1990, p. 12), the normal form quantal response model (NQR)(McKelvey and Palfrey, 1995b, p. 30), the agent quantal response model(AQR) (McKelvey and Palfrey, 1995b, p. 30), the basic model (Basic),the model where the variance depends on each players own squared pay-off (OWN2), the model where the variance depends on each players ownpayoff (OWN), the model where the variance depends on the opponentssquared payoff (OPP2) and the model where the variance depends on theopponents payoff (OPP), respectively. The rows of the table show the fre-quencies at the first, second, etc. endnode. Table X shows the similar es-timation results for the six-move game. It does not include the altruismmodel (McKelvey and Palfrey, 1990) since the estimation results for thesix-move game are not available.

This comparison shows that according to the log-likelihood9 the basicmodel and the model where the variance depends on each players ownpayoff fit the centipede data best. This is true for the four-move as wellas for the six-move experiments. All models (except the model where the

9As above, other criteria (AIC, BIC, sum of squared errors) give the same results.

TABLE VIIIEstimated Take Probabilities: Variance Depends Linearly on Opponents Payoff (OPP)



TABLE IXComparison of Expected Frequencies of Endnodes in Four-Move Game for Different Models

Fr. Act. Alt. NQR AQR Basic OWN2 OWN OPP2 OPP

Of1 0.071 0.857 0.203 0.247 0.214 0.144 0.054 0.154 0.165Of2 0.356 0.122 0.210 0.235 0.237 0.230 0.289 0.307 0.243Of3 0.370 0.018 0.220 0.351 0.372 0.221 0.498 0.157 0.185Of4 0.153 0.003 0.216 0.156 0.167 0.269 0.150 0.215 0.250Of5 0.049 0.000 0.150 0.011 0.010 0.136 0.009 0.166 0.157L 1000.6 437.5 425.0 418.3 427.1 402.8 438.9 438.5

variance depends on each players own payoff) overestimate the actual fre-quency at the first node but share similar properties.10

6. LEARNING TO PLAY THE NASH EQUILIBRIUM

This section looks at a model that incorporates play in each of the 10 timeperiods t D 1; 2; : : : ; 10. It tries to answer the question whether play differssignificantly throughout the 10 periods. The question is whether players getcloser to the Nash equilibrium prediction through more frequent play.Falling estimated variances and significantly different play during these 10periods would suggest that players learn. The exact learning process isleft unmodeled.11 Falling variances over time are taken as an indicationthat the uncertainty that players face decreases. The smaller the variances,

10A two-parameter quantal response model (McKelvey and Palfrey, 1995b, p. 30) has alog-likelihood of 402.5 (four-move game) and 454.3 (six-move game).

11I do not attempt to build a theoretical model of learning in games. There is an extensiveliterature on learning in games. See, for example, Fudenberg and Kreps (1993), Fudenbergand Levine (1996), Kandori (1997), and the references therein.

TABLE XComparison of Expected Frequencies of Endnodes in Six-Move Game for Different Models

Fr. Act. NQR AQR Basic OWN2 OWN OPP2 OPP

Of1 0.007 0.136 0.237 0.186 0.088 0.002 0.079 0.090Of2 0.064 0.137 0.192 0.173 0.114 0.027 0.207 0.132Of3 0.199 0.135 0.136 0.138 0.123 0.136 0.088 0.114Of4 0.384 0.146 0.205 0.219 0.186 0.511 0.209 0.180Of5 0.253 0.162 0.192 0.231 0.174 0.290 0.113 0.149Of6 0.078 0.175 0.036 0.052 0.207 0.033 0.174 0.209Of7 0.014 0.109 0.001 0.001 0.107 0.001 0.129 0.126L 536.6 533.9 506.4 510.6 453.4 539.3 526.1

PC015148

178 klaus g. zauner

TABLE XIEstimated Take Probabilities in Four-Move Centipede Game for Time-Disaggregated Data

(Basic Model)

Period n O 2 Op1 Op2 Op3 Op4 L1 29 0.951 0.233 0.245 0.534 0.877 46.5752 29 0.835 0.228 0.250 0.564 0.892 42.8193 29 0.733 0.223 0.260 0.595 0.907 46.9764 29 0.643 0.219 0.273 0.628 0.921 44.6355 29 0.564 0.216 0.292 0.662 0.934 45.7756 29 0.495 0.213 0.315 0.696 0.946 38.2717 29 0.435 0.211 0.342 0.730 0.957 35.4598 29 0.382 0.211 0.374 0.762 0.966 41.7589 29 0.335 0.212 0.408 0.792 0.975 40.225

10 20 0.294 0.216 0.445 0.820 0.982 28.353410.846

Note: Model: 2t D expt1, OD 0:951, OD 0:1304, LD 410:846.

the better informed players are and the closer players are to the Nashequilibrium prediction of the centipede game. Note that play convergesto the unique subgame perfect equilibrium of the centipede game as thevariance converges to 0 (cf. Fig. 7).

I consider a very simple model12:

2t D expt1; (3)

where t D 1; 2; 3; : : : ; 10Note that if D 0, then the variance is constant and we have the model

analyzed in earlier sections. If > 0 then variances fall, the uncertainty ofplayers diminishes, and the probability of a wrong move goes down withrepeated play.

Tables XI and XII summarize the estimation results for the four-moveand six-move games using the basic model. Table XI (Table XII) showsthe estimated implied take probabilities for the four-move game (thesix-move game). It is now possible to test whether there is a significantdifference of play between periods. Performing a likelihood-ratio test forthe null hypothesis H0x D 0 shows that the null hypothesis for the four-move game and for the six-move game is rejected at the 1% level.13

12For models with a richer dynamic structure, see, e.g., Crawford (1995) and the referencestherein.

13The log-likelihood for the restricted model (Table III) is L D 418:334 (four-move game)and L D 506:382 (six-move game) and the log-likelihood for the unrestricted model is L D410:846 (four-move game) and L D 503:033 (six-move game).


TABLE XIIEstimated Take Probabilities in Six-Move Centipede Game for Time-Disaggregated Data

(Basic Model)

Period n O 2 Op1 Op2 Op3 Op4 Op5 Op6 L1 29 6.647 0.195 0.202 0.211 0.353 0.741 0.960 54.2642 29 6.103 0.192 0.205 0.211 0.374 0.762 0.967 50.3403 29 5.603 0.189 0.208 0.211 0.396 0.782 0.972 57.9604 29 5.144 0.187 0.211 0.213 0.420 0.801 0.977 51.6125 29 4.723 0.186 0.213 0.216 0.444 0.820 0.981 49.5876 29 4.337 0.185 0.215 0.219 0.469 0.837 0.985 48.8577 29 3.982 0.184 0.217 0.224 0.494 0.853 0.988 48.3248 29 3.656 0.184 0.218 0.230 0.519 0.867 0.991 55.2019 29 3.357 0.184 0.218 0.237 0.544 0.881 0.993 48.568

10 20 3.082 0.185 0.218 0.245 0.570 0.894 0.995 38.321503.033


Similarly, Tables XIII and XIV summarize the estimation results for thefour-move and six-move games using the model OWN, i.e., the model wherethe variance depends linearly on each players own payoff. Table XIII (Ta-ble XIV) shows the estimated implied take probabilities for the four-move game (six-move game) using the OWN model. Again, it is now possi-ble to test whether there is a significant difference of play between periods.Performing a likelihood-ratio test for the null hypothesis H0x D 0 shows

TABLE XIIIEstimated Implied Take Probabilities in Four-Move Centipede Game for Time-Disaggre-

gated Data (OWN Model)

Period n O 2 Op1 Op2 Op3 Op4 L

1 29 0.305 0.050 0.180 0.630 0.907 42.1942 29 0.282 0.050 0.197 0.657 0.916 43.1313 29 0.260 0.050 0.219 0.684 0.924 44.3194 29 0.241 0.051 0.245 0.712 0.932 44.7765 29 0.222 0.052 0.277 0.739 0.939 44.6906 29 0.206 0.055 0.315 0.765 0.946 37.7277 29 0.190 0.059 0.358 0.790 0.953 32.5618 29 0.176 0.066 0.404 0.814 0.959 39.7499 29 0.162 0.076 0.454 0.836 0.965 38.564

10 20 0.150 0.090 0.506 0.857 0.970 26.055393.766


180 klaus g. zauner

TABLE XIVEstimated Implied Take Probabilities in Six-Move Centipede Game for Time-Disaggregated

Data (OWN Model)

Period n O 2 Op1 Op2 Op3 Op4 Op5 Op6 L1 29 0.642 0.002 0.016 0.077 0.461 0.839 0.966 50.5822 29 0.608 0.002 0.018 0.087 0.496 0.853 0.969 45.5623 29 0.576 0.002 0.021 0.099 0.532 0.867 0.973 51.4944 29 0.546 0.002 0.024 0.114 0.567 0.879 0.976 43.7335 29 0.517 0.002 0.026 0.133 0.601 0.891 0.979 40.3746 29 0.489 0.002 0.029 0.155 0.634 0.902 0.982 39.3787 29 0.464 0.003 0.032 0.181 0.665 0.912 0.984 40.0698 29 0.439 0.003 0.035 0.211 0.695 0.921 0.986 48.9829 29 0.416 0.004 0.039 0.245 0.723 0.930 0.988 41.481

10 20 0.394 0.005 0.043 0.282 0.748 0.938 0.990 43.587445.240


that the null hypothesis for the four-move and for the six-move game isrejected at the 1% level.14

In both cases we see that there is a significant difference of play betweenperiods and that the variance goes down as the game is played more often.Repeated play leads to a decreased uncertainty of players and therefore tobehavior which more closely resembles the Nash equilibrium prediction forthe centipede game. Subjects seem to obtain more experience as they playthe centipede games more frequently.

Figures 1013 plot the estimated implied take probabilities for thefour-move and six-move games using the basic model and the model OWN.In addition, forecasts of the implied take probabilities using the estimatedmodels are included for periods 1150 (four-move game) and 11100 (six-move game). Figure 10 (Fig. 12) gives the estimated take probabilitiesfor the four-move game using the basic model (model OWN). Figure 11(Fig. 13) gives the estimated implied take probabilities for the six-movegame using the basic model (model OWN).

Players get close to the Nash equilibrium prediction rather quickly. Inthe four-move centipede game it would take them only 50 periods to playthe (unique subgame perfect) Nash equilibrium. In the six-move centipedegame it takes longer for play to settle at the Nash equilibrium. It would take

14The log-likelihood for the restricted model (Table VI) is L D 402:838 (four-move game)and L D 453:375 (six-move game) and the log-likelihood for the unrestricted model is L D393:766 (four-move game) and L D 445:240 (six-move game).


FIG. 10. Estimated implied take probabilities for the four-move game (basic model):estimated (rounds 110); forecasts (rounds 1150).

FIG. 11. Estimated implied take probabilities for the six-move game (basic model): esti-mated (rounds 110); forecasts (rounds 11100).

182 klaus g. zauner

FIG. 12. Estimated implied take probabilities for the four-move game (OWN model):estimated (rounds 110); forecasts (rounds 1150).

FIG. 13. Estimated implied take probabilities for the six-move game (OWN model):estimated (rounds 110); forecasts (rounds 11100).


them 80100 periods to play the Nash equilibrium. Play converges fasterto the Nash equilibrium for the model OWN than for the basic model.There is some evidence that the play of more experienced subjects wouldgive predictions closer to the Nash equilibrium as those found in theexperiments.

7. CONCLUSIONS

This paper tried to explain non-Nash equilibrium outcomes observed inthe centipede game experiments of McKelvey and Palfrey (1992) by a sim-ple model a` la Harsanyi (1973). According to this model, the doubts play-ers have in a strategic situation about aspects of the game are modeled asrandom disturbances to each players payoffs. Five different models wereconsidered that differed according to the specification of the error terms.In each of the models the variance of the uncertainty of players was esti-mated. Two models were selected according to a likelihood criterion andcompared against other one-parameter models, in particular against thealtruism model, the normal-form quantal response model, and the agentquantal response model. This comparison showed that the two models ofthis paper fit the centipede game data better than the one-parameter quan-tal response or altruism models. The quantal response model and the mod-els in this paper share important features. Each of these models tries toexplain non-Nash equilibrium outcomes without a systematic change in thepayoffs and are able to capture the possible violation of the invarianceprinciple of Kohlberg and Mertens (1986, p. 110). The invariance princi-ple states that solution concepts should only depend on the reduced nor-mal form.

In addition, it was shown that the estimated variance of the uncertaintycan capture all qualitative features of the data. It is remarkable that thesesimple one-parameter models are able to generate such a result.

For the two models selected above the question was asked whether playis significantly different throughout the 10 periods it was played. The es-timation results show that estimated variances are significantly differentalong play. The variances fall quickly with time which suggests that sub-jects learn and get close to the Nash equilibrium prediction moderatelyfast.

A possible extension of the above results is the inclusion of a hetero-geneity parameter. The application of these random disturbances to otherexperimental data like the ultimatum bargaining game and to equilibriumrefinements in experimental games (Banks et al., 1994; Brandts and Holt,1992, 1993; Camerer and Weigelt, 1988) would also be desirable.

PC015148

184 klaus g. zauner

REFERENCES

Amemiya, T. (1985). Advanced Econometrics. Cambridge, MA: Harvard Univ. Press.Banks, J., Camerer, C., and Porter, D. (1994). An Experimental Analysis of Nash Refinements

in Signaling Games, Games Econom. Behav. 6, 131.Brandts, J., and Holt, C. (1992). An Experimental Test of Equilibrium Dominance in Sig-

nalling Games, Amer. Econom. Rev. 82, 13501365.Brandts, J., and Holt, C. (1993). Adjustment Patterns and Equilibrium Selection in Experi-

mental Signaling Games, Internat. J. Game Theory 22, 279302.Brown, J., and Rosenthal, R. (1990). Testing the Minimax Hypothesis: A Re-Examination of

ONeills Game Experiment, Econometrica 58, 10651061.Camerer, C., and Weigelt, K. (1988). Experimental Tests of a Sequential Equilibrium Repu-

tation Model, Econometrica 56, 136.Crawford, V. (1995). Adaptive Dynamics in Coordination Games, Econometrica 63, 103143.Crawford, V. (1997). Theory and Experiment in the Analysis of Strategic Interaction, in

Advances in Economics and Econometrics: Theory and Applications, Seventh World Congress,(D. Kreps and K. Wallis, Eds.), Vol. 1, pp. 206242. Cambridge: Cambridge Univ. Press.

Fudenberg, D., and Kreps, D. (1993). Learning Mixed Equilibria, Games Econom. Behav. 5,320367.

Fudenberg, D., and Levine, D. (1996). Theory of Learning in Games, mimeo.Harsanyi, J. (19671968). Games with Incomplete Information Played by Bayesian Players,

Parts I, II, III, Manage. Sci. 14, 159182, 320332, 468502.Harsanyi, J. (1973). Games with Randomly Disturbed Payoffs: A New Rationale for Mixed

Strategy Equilibrium Points, Internat. J. Game Theory 2, 123.Kandori, M. (1997). Evolutionary Game Theory in Economics, in Advances in Economics

and Econometrics: Theory and Applications, Seventh World Congress (D. Kreps and K. Wallis,Eds.), Vol. 1, pp. 243277. Cambridge: Cambridge Univ. Press.

Kohlberg, E., and Mertens, J.-F. (1986). On the Strategic Stability of Equilibria, Economet-rica 54, 10031037.

Kreps, D. (1990). A Course in Microeconomic Theory. Princeton, NJ: Princeton Univ. Press.Kreps, D., and Wilson, R. (1982). Reputation and Imperfect Information, J. Econom. Theory

27, 253279.Kreps, D., Milgrom, P., Roberts, J., and Wilson, R. (1982). Rational Cooperation in the

Finitely Repeated Prisoners Dilemma, J. Econom. Theory 27, 245252.McKelvey, R., and Palfrey, T. (1990). An Experimental Study of the Centipede Game, Social

Science Working Paper 732. Pasadena, CA: California Institute of Technology.McKelvey, R., and Palfrey, T. (1992). An Experimental Study of the Centipede Game, Econo-

metrica 60, 803836.McKelvey, R., and Palfrey, T. (1995a). Quantal Response Equilibria for Normal Form

Games, Games Econom. Behav. 10, 638.McKelvey, R., and Palfrey, T. (1995b). Quantal Response Equilibria for Extensive Form

Games, Social Science Working Paper 947. Pasadena, CA: California Institute of Technol-ogy.

McKelvey, R., and Palfrey, T. (1996). A Statistical Theory of Equilibrium in Games, Japan.Econom. Rev. 47, 186209.

Milgrom, P., and Roberts, J. (1982). Predation, Reputation, and Entry Deterrence, J.Econom. Theory 27, 280312.


Ochs, J. (1995). Games with Unique Mixed Strategy Equilibria: An Experimental Study,Games Econom. Behav. 10, 174189.

Rosenthal, R. (1981). Games of Perfect Information, Predatory Pricing and the Chain-StoreParadox, J. Econom. Theory 25, 92100.

Stinchcombe, M., and Zauner, K. (1993). Inference Preserving Perturbations of ExtensiveForm Games, Working Paper. San Diego: Univ. of California.

Zauner, K. (1993). Equilibrium in Games with Incomplete Information: Generalization toUnbounded Utility Functions, mimeo. San Diego: Univ. of California.

1. INTRODUCTION2. THE CENTIPEDE GAME EXPERIMENTSFIG. 1.FIG. 2.FIG. 3.TABLE ITABLE II

3. THE MODELFIG. 4.FIG. 5.FIG. 6.FIG. 7.

4. ESTIMATION AND RESULTSTABLE IIITABLE IV

5. OTHER SPECIFICATIONS AND GOODNESS OF FITTABLE VFIG. 8.FIG. 9.TABLE VITABLE VIITABLE VIIITABLE IX

6. LEARNING TO PLAY THE NASH EQUILIBRIUMTABLE XTABLE XITABLE XIITABLE XIIITABLE XIVFIG. 10.FIG. 11.FIG. 12.FIG. 13.

7. CONCLUSIONSREFERENCES

1.a payoff uncertainty explanation of results in.pdf

Documents

selected models

centipede game experiments

article id game

classical game theory

thequantal response

nonnash equilibrium

experimental results

economic behavior