Thomas R. Palfrey

An experimental study of the centipede game

We report on an experiment in which individuals play a version of the centipede game. In this game, two players alternately get a chance 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 the larger portion of the pile, and the other player getting the smaller portion. If one views the experiment as a complete information game, all standard game theoretic equilibrium concepts predict the first mover should take the large pile on the first round. The experimental results show that this does not occur. An alternative explanation for the data can be given if we reconsider the game as a game of incomplete information in which there is some uncertainty over the payoff functions of the players. In particular, if the subjects believe there is some small likelihood that the opponent is an altruist, then in the equilibrium of this incomplete information game, players adopt mixed strategies in the early rounds of the experiment, with the probability of taking increasing as the pile gets larger. We investigate how well a version of this model explains the data observed in the centipede experiments.

Quantal Response Equilibria for Extensive Form Games

This article investigates the use of standard econometric models for quantal choice to study equilibria of extensive form games. Players make choices based on a quantal-choice model and assume other players do so as well. We define an agent quantal response equilibrium (AQRE), which applies QRE to the agent normal form of an extensive form game and imposes a statistical version of sequential rationality. We also define a parametric specification, called logit-AQRE, in which quantal-choice probabilities are given by logit response functions. AQRE makes predictions that contradict the invariance principle in systematic ways. We show that these predictions match up with some experimental findings by Schotter et al. (1994) about the play of games that differ only with respect to inessential transformations of the extensive form. The logit-AQRE also implies a unique selection from the set of sequential equilibria in generic extensive form games. We examine data from signaling game experiments by Banks et al. (1994) and Brandts and Holt (1993). We find that the logit-AQRE selection applied to these games succeeds in predicting patterns of behavior observed in these experiments, even when our prediction conflicts with more standard equilibrium refinements, such as the intuitive criterion. We also reexamine data from the McKelvey and Palfrey (1992) centipede experiment and find that the AQRE model can account for behavior that had previously been explained in terms of altruistic behavior.

Voter Participation and Strategic Uncertainty

The paradox of not voting is examined in a model where voters have uncertainty about the preferences and costs of other voters. In game-theoretic models of voter participation under complete information, equilibrium outcomes can have substantial turnout even when voting costs are relatively high. In contrast, when uncertainty about preferences and costs is present, only voters with negligible or negative net voting costs participate when the electorate is large.

A Strategic Calculus of Voting

ConclusionThere are several major insights which this game theoretic analysis has produced. First, we have shown that equilibria exist with substantial turnout even when both the majority is much larger than the minority and the costs of voting are exceptionally high. For example, in large electorates using the status quo rule, we show mixed-pure equilibria with turnout roughly equal to twice the size of the minority when the cost is nearly equal to 37% (e−1 × 100) of the reward (B). Second, with large electorates the many equilibria appear to reduce to just two types, the type just mentioned and a type with almost no turnout. Third, we have shown that turnout may rise as the costs of voting rise. This results when all members of a team “adjust” their turnout probabilities so that the probability of being pivotal increases to match the increased cost of voting. We have also shown that turnout is nearly invariant with costs in large electorates where turnout probabilities approach one or zero. Fourth, the actual split of the vote is likely to be a biased measure of the actual distribution of preferences in the electorate. Because majorities have greater incentives to free-ride, they will turn out less heavily than minorities. Elections can be relatively close, even when one alternative is supported by a substantial majority of the electorate. The probability that the majority will win does not seem to be closely related to the size of the electorate or its size relative to the minority. However, turnout is quite strongly correlated with the relative sizes of the minority and the majority.

The Relationship Between Information, Ideology, and Voting Behavior

The question of voter sophistication is important for understanding voter and candidate behavior in mass elections. We develop an index of voter information - based on perceptual data - and find that it is significantly related to ideological extremism and voting behavior. Individuals with a high level of information tend to be more extreme than those with low levels and are much more likely to vote.

Risk Averse Behavior in Generalized Matching Pennies Games

In experimental studies of behavior in 2×2 games with unique mixed strategy equilibria, observed choice frequencies are systematically different from mixed-strategy Nash predictions. This paper examines experimental results for a variety of such games, and shows that a structural econometric model which incorporates risk aversion into a quantal response equilibrium explains the data very well. Moreover, risk aversion estimates are stable across the different games and are close to those obtained from laboratory and field auction data, as well as from individual lottery choice experiments.

An Experimental Study of Jury Decision Rules

We present experimental results on groups facing a decision problem analogous to that faced by a jury. We consider three treatment variables: group size (three or six), number of votes needed for conviction (majority or unanimity), and pre-vote deliberation. We find evidence of strategic voting under the unanimity rule: A large fraction of our subjects vote for a decision analogous to conviction even when their private information indicates a state analogous to innocence. This is roughly consistent with the game theoretic predictions of Feddersen and Pesendorfer. Although individual behavior is explained well by the game theoretic model, there are discrepancies at the level of the group decision. Contrary to Feddersen and Pesendorfer, in our experiments there are fewer outcomes analogous to incorrect convictions under unanimity rule than under majority rule. In the case of no deliberation, we simultaneously account for the individual and group data using quantal response equilibrium.

The Paradox of Voter Participation? A Laboratory Study

It is widely believed that rational choice theory is grossly inconsistent with empirical observations about voter turnout. We report the results of an experiment designed to test the voter turnout predictions of the rational choice Palfrey–Rosenthal model of participation with asymmetric information. We find that the three main comparative statics predictions are observed in the data: the size effect, whereby turnout goes down in larger electorates; the competition effect, whereby turnout is higher in elections that are expected to be close; and the underdog effect, whereby voters supporting the less popular alternative have higher turnout rates. We also compare the quantitative magnitudes of turnout to the predictions of Nash equilibrium. We find that there is undervoting for small electorates and overvoting for large electorates, relative to Nash equilibrium. These deviations from Nash equilibrium are consistent with the logit version of Quantal Response Equilibrium, which provides a good fit to the data, and can also account for significant voter turnout in very large elections.

Altruism, reputation and noise in linear public goods experiments

We report the results of a public goods experiment using a design that enables us to directly measure individual response functions in voluntary contributions games and estimate error rates. In addition, following Andreoni (1988), we employ two treatments in order to measure the extent to which voluntary contribution is due to reputation effects. The partners treatment involves a fixed group of subjects playing a repeated game. The strangers treatment approximates a one-shot game by randomly changing group assignments after each play. Our data shows that essentially the only difference between the two treatments is the amount of noise in the data, with the strangers treatment being the noisier of the two. This noise manifests itself in two distinct ways. First, there is more variation of decision rules across subjects in the strangers treatment. Second, individual behavior is, on average, less consistent with a cutpoint decision rule in the strangers treatment, which produces higher estimates of individual error rates. The differences between the strangers and partners data are virtually the same as differences between data from experienced and inexperienced subjects. This suggests an explanation for the finding in Andreoni (1988) that there was greater contribution under the strangers treatment in the standard homogeneous environment in which one direction of errors (under-contribution) are censored. Our results also support his conclusion that reputation effects do not appear to play a prominent role in repeated linear public goods voluntary contribution games. Many findings from past public goods experiments are consistent with our model of random variation in a population of subjects who are, on average, neither altruistic nor spiteful.

Equilibria in Campaign Spending Games: Theory and Data

We present a formal game-theoretic model to explain the simultaneity problem that makes it difficult to obtain unbiased estimates of the effects of both incumbent and challenger spending in U.S. House elections. The model predicts a particular form of correlation between the expected closeness of the race and the level of spending by both candidates, which implies that the simultaneity problem should not be present in close races and should be progressively more severe in the range of safe races that are empirically observed. This is confirmed by comparing simple OLS regression of races that are expected to be close with races that are not, using House incumbent races spanning two decades.