# 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.Recent research in political science addresses from a theoretical point of view the question of how individuals behave in juries. The classic Condorcet jury theorem deals with a model in which all jurors have identical preferences (they want to convict the guilty and acquit the innocent) but differ in their probability of making a correct decision. The theorem states that if the jury decision is made by majority rule, then the probability of a correct decision by the group is higher than that of any individual, and the probability of a correct decision goes to one as the size of the jury becomes very large.

Austen-Smith and Banks (1996) challenge the foundations of the Condorcet jury theorem by questioning whether any game theoretic basis can be given for the type of behavior assumed by the theorem. They demonstrate that if individuals start from a common prior about guilt of the defendant and then obtain private information, it is generally not a Nash equilibrium to vote sincerely (i.e., based only on one's private information). Subsequent work by Wit (1996) and McLennan (1996) reestablished that the conclusions of the theorem still hold if individuals vote strategically according to a symmetric mixed strategy equilibrium of the game.

Feddersen and Pesendorfer (1998) analyze the Nash equilibrium of the Condorcet jury theorem, with particular attention to the effect of the decision rule. They compare the unanimous decision rule with rules that require only a majority or supermajority to convict. They conclude that the unanimous rule results in probabilities of convicting an innocent defendant that are higher than those for majority rule and that do not go to zero as the number of jurors goes to infinity.

The Feddersen and Pesendorfer model assumes that the jury decision is a simultaneous move game, in which all jurors vote without any communication beforehand. As Coughlan (2000) shows, if it is possible to have a straw poll before the vote, then there exist equilibria in which voters reveal their information in the straw poll and vote optimally in the actual vote based on the pooled information from the straw vote. This type of behavior would eliminate the unattractive aspects of unanimity, because decisions under majority rule should be identical to those under unanimity.

There has been relatively little experimental work to investigate the implications of these theories. The one exception is work by Ladha, Miller, and Oppenheimer (1996), who run experiments in which a jury makes a sequence of decisions under majority rule. They find some evidence of strategic behavior and that asymmetric Nash equilibria are sometimes played.

We report here on experiments run to test the Feddersen-Pesendorfer and Coughlan predictions. Our experiments thus consider groups faced with a decision problem isomorphic to the one modeled in those articles. We address the following questions: whether players vote strategically, whether unanimity rule leads to more incorrect convictions than majority rule, how this varies with group size, and whether the effects of a straw poll are the same as those predicted by Coughlan (2000).

As described in detail below, our experiments were conducted using an abstract group decision situation that avoided any direct reference to legal terms like "guilt" or "innocence." Specifically, the experimental implementation of the Feddersen-Pesendorfer jury game was done using colored jars and balls. For each group, the true state of the world (which we analogize here to either "guilty" or "innocent") was determined by randomly assigning the group a red jar (guilty) or a blue jar (innocent) with equal probability. The red jar contained predominantly red balls (guilty signals) and the blue jar contained predominantly blue balls (innocent signals). Before any voting occurred, each subject blindly selected one ball (with replacement) from her group's (randomly assigned) jar. The subject then observed the color of her selected ball, but did not observe the predominant color of the jar from which it was drawn, or the color of the balls selected from the same jar by the other members of the group. Then each gr oup member privately voted for either a red jar (analogous to voting to "convict") or a blue jar (analogous to voting to "acquit"). The votes were then aggregated to form a group decision of red (convict) or blue (acquit) according to the specified voting rule, majority or unanimity. If the group decision matched the color of the group's assigned jar (analogous to convict if guilty, or acquit if innocent), then everyone in the group received a small token payment plus a large prize. If the group decision was incorrect, everyone in the group received only the small token payment. All of this was carefully explained in the instructions, which were read aloud (see Appendix A). We consider the effects of three treatment variables: voting rule (majority or unanimity), group size, and effect of a straw poll. In each treatment we consider individual and group behavior.

In the data, we find clear evidence of strategic behavior that is roughly consistent with the Feddersen-Pesendorfer game theoretic model. That is, a large percentage (between 30% and 50%) of subjects vote red (convict) even when they draw a blue ball (innocent signal). The percentage increases with the group size, as is predicted. Although individual behavior is explained well by the game theoretic model, there are some discrepancies between data and theory at the group level. For example, in our experiments there were fewer decisions analogous to incorrect convictions under unanimity rule than under majority rule, while Nash theory predicts the opposite. In the case of no straw poll, we are able to explain most of these discrepancies, using the logit version of quantal response equilibrium (QRE).

THE CONDORCET JURY MODEL

The Condorcet jury model applies to a general class of group decision problems in which members have a common interest but hold different beliefs about the true state of the world. By a common interest is meant that if the state of the world were common knowledge, then all group members would agree about which decision to make. The differences in beliefs create an information aggregation problem, which creates potential obstacles for the group to reach a consensus and make the "correct" decision. This class of decision problems has application to many settings, including juries in criminal and civil trials, corporate strategy decisions by boards of directors or partners, hiring and tenure decisions by faculty committees, examinations judged subjectively by committees, and so forth. The first of these, trial juries, was the subject of an article by Feddersen and Pesendorfer (1998), which is the main motivation for our work.

Several caveats are in order. First, the model we study applies to a broad class of settings, and it does not capture all the interesting institutional details of a single setting, such as a trial jury. Second, the model is an approximation, and it leaves out some contextual components that may affect behavior in specific applications. Third, it is meant to be simple, so that we can isolate and study certain phenomena of theoretical interest. Fourth, it is flexible, in the sense that one can analyze the model using a wide range of different assumptions about the degree of sophistication or rationality of group members. In this sense, although it is unquestionably a formal theoretical model, it allows one to explore the implications of bounded rationality as well as the implications of rational choice within the confines of the same model. It thus provides a very nice framework for conducting experiments to compare rational choice and bounded rationality.

Model Structure and Notation

We consider a game with a set N = {1, 2,..., n} of n players (e.g., jurors), and let 1 [leq] k [leq] n represent the number of votes needed for a positive decision (e.g., conviction). The game begins by nature choosing a state of the world in [Omega] = {G, I}, [1] with probability s and 1 -- s, respectively. The players do not observe the state that is selected, but each obtains some private information about the state. If the true state is G, then each player observes an independent Bernoulli random variable, which is g with probability p and i with probability 1 -- p. If the true state is I, then each juror observes an independent Bernoulli random variable, which is i with probability p and g with probability 1 -- p. After observing their private information, players vote for one of two actions in X = {C, A}. [2] If k or more players vote for C, then the group decision is C; otherwise the decision is A. The utility u: X x [Omega] [mapsto] [Re] of each player is defined by u(A, I) = u(C, G) = 0, U(C, I) = -q , and U(A, G) = q - 1,where 0 [less than] q [less than] 1.

In all our experiments, we used the values s = .5, q = .5, and p = .7. We will be concerned with two different voting rules. For the majority rule case, k is the least integer greater than n/2; for the unanimity case, k = n.

We will distinguish between two kinds of behavior, which we call naive and strategic. Naive voters ignore the group strategy aspect of the decision problem and simply vote as if they were the only voter. In the jury setting, that means if they receive a guilty signal they vote to convict, and if they receive an innocent signal they vote to acquit. This is the kind of behavior assumed by Condorcet.

The second kind of behavior is strategic. Contemporary game theorists, including some political scientists (e.g., Austen-Smith and Banks 1996) argue that we should assume individuals behave strategically rather than naively. They also prove formally that naive and strategic behavior can have dramatically different logical implications in the Condorcet jury model. In particular, strategic behavior by jurors is modeled using game theory, which predicts that under certain specific circumstances it is optimal for jurors to vote against their signal. Our predictions about strategic behavior in our experiments are given by the choice probabilities at symmetric Nash equilibria.

We also consider statistical versions of both types of behavior. Since we observe all kinds of choices in our experiment, it is necessary to introduce an error component to individual choices. For the case of naive behavior, we do this by assuming that naive subjects vote with their signal with some fixed probability, [gamma], and make an error with probability 1 -- [gamma]. The probability of correct choice is assumed to be independent of the signal received and the same across all treatments. This becomes a free parameter of the naive behavior model, which allows us to fit the data to the model by standard estimation methods. For the case of strategic behavior, we incorporate the error structure into the equilibrium concept by using QRE. The QRE model (explained in more detail below) assumes that players may deviate with some probability from best responses, and the probability of deviation depends on the expected payoff difference between the best response and the deviation. We use a logit parameterizatio n of QRE that includes a free parameter, [lambda], which determines payoff responsiveness. Higher values of [lambda] in the strategic model correspond approximately to higher values of [gamma] in the naive model.

Strategic Behavior: Nash Equilibrium

To characterize equilibria, a strategy for a voter is a function [sigma]: {g, i} [mapsto] [10, 1], taking signals into account in the probability of voting for conviction. There are trivial equilibria to the above game in which voters ignore their information. Of special interest, however, are symmetric "informative" equilibria to the game. Symmetric equilibria require that players with the same signal adopt the same (mixed) strategy. Informative equilibria are those in which the players do not ignore their information.

As shown by Feddersen and Pesendorfer (1998), for the case of unanimity, the unique symmetric informative equilibrium requires that [sigma](g) = 1 and that the posterior probability of the defendant actually being guilty conditional on player i receiving an innocent signal and all other players voting guilty must be equal to q. That is,

q = [frac{(1 - p)[[g.sup.n-1].sub.G]}{(1 - p)[[g.sup.n-1].sub.G] + [pg.sup.n-1].sub.I]}],

where

[g.sub.G] = p[sigma](g) + (1 -p)[sigma](i) (1)

and

[g.sub.I] = (1 - p)[sigma](g) + p[sigma](i) (2)

are the probabilities that an individual votes to convict when the defendant is guilty or innocent, respectively. Using [sigma](g) = 1, this implies that

[sigma](i) = [frac{[D.sub.n]p - (1 - p)}{p - [D.sub.n](1 - p)}],

where

[D.sub.n] = [lgroup][frac{(1 - q)(1 - p)}{qp}][rgroup].sup.[frac{1}{n - 1}]]

Also, the probability of an incorrect jury decision to convict the innocent (Pr[C\I] = [([g.sub.I]).sup.n]) or acquit the guilty (Pr[A\G] = 1 - [([g.sub.G]).sup.n]) is determined from the above formula. Table 1 gives the equilibrium values of [sigma](i), Pr[A\G], and Pr[C\I], respectively, for unanimity rule for certain values of n.

For majority rule, the formulas are slightly more complicated, but the symmetric equilibrium is simpler, namely, [sigma](g) = 1.0 and [sigma](i) = 0.0, regardless of the size of n. [3] We can then compute the corresponding probabilities of convicting an innocent defendant: (pr[C\I] = [[[Sigma].sub.k][greater than]n/2] (n k) [(1 - p).sup.k][p.sup.n-k] and of acquittinG a guilty defendant (Pr[A\G] = [[[Sigma].sub.k][greater than]n/2] These numbers also are reported in Table 1 for the parameter values used in our experiment.

In the case of a straw poll, the game expands so that the players have two votes. A strategy is now a specification of how to vote in the straw poll, as a function of the player's signal, and then a specification of how to vote in the final vote, as a function of the signal and the observed outcome of the straw poll. Let [[sigma].sub.0] : {g, i} [rightarrow] [0, 1] be the probability of voting for conviction on the straw poll. Define [N.sub.0] = N [cup] {0} to be the possible outcomes of the poll. Then let [[sigma].sub.1] : {g, i} X [N.sub.0]. In the case of a straw poll, Coughlan (2000) shows that, for the parameters used here, there is a fully informative equilibrium in which all players reveal their signals on the straw vote, that is, [[sigma].sub.0](i) = 0, and [[sigma].sub.0](g) = 1. They then vote based on the majority outcome of the straw poll in the final vote, that is, [[sigma].sub.1] (s, M([[sigma].sub.0])) = M ([[sigma].sub.0]) for all signals s [epsilon] {i, g}, where M([[sigma].sub.0]) is zero, one, or some appropriate mixing probability according to whether the majority outcome of the straw poll was to convict, acquit, or a tie.

Strategic Behavior: Quantal Response Equilibrium

The above solutions all assume no error. As a general way to incorporate decision error, McKelvey and Palfrey (1995, 1998) propose QRE, which is a statistical version of Nash equilibrium. [4] The basic idea is that it is unreasonable to expect individuals always to behave in perfect accord with rationality and always choose best responses to the other players. Instead, they choose better responses more often than worse responses. QRE is not a deterministic model but instead specifies a probability distribution, [[sigma].sup.*](.), over actions. The probabilities are ordered by the expected payoffs of the actions, EU(.), according to some specific function, called a quantal response function, which is just the statistical version of a best response function. [5] Actions with higher expected payoff will be played more frequently, and actions with lower expected payoffs will be played less frequently. So, for any individual i, and any pair of actions available to i, let us say a and b, [[[sigma].sup.*].sub.i](a) [greater than] [[[sigma].sup.*].sub.i](b) if and only if [EU.sub.i](a) [greater than] [EU.sub.i](b). The "E" of QRE stands for equilibrium, in the sense that the expected payoffs, EU(.), are themselves derived from the equilibrium probabilities, [[sigma].sup.*](.). One can think of an iterative process in which a given profile of choice probabilities for all the players' results determines a profile of expected payoffs for every action, which in turn (via the quantal response function) generates a new profile of choice probabilities. A QRE is just a fixed point of this iterative process. QRE retains the rational expectations flavor of a Nash equilibrium but relaxes the assumption that players choose optimal responses.

There are several ways to justify a formal model with the above properties. The idea that individuals choose stochastically rather than deterministically has been proposed for a long time, for example, to motivate reinforcement learning and discrete choice econometrics. Alternatively, one can "rationalize" stochastic choice if players have stochastic utility functions. Harsanyi (1973) proposes a model in this vein: The game payoff matrix is viewed as just an approximation of the utilities of the player over outcomes in the game, and each player's actual utility varies around these means according to some statistical distribution. In a QRE, for every action an individual might choose, there is a privately observed payoff disturbance for that action, and one then looks at the Bayesian equilibrium to the corresponding game of private information. This is equivalent to smoothing the best response curves of the players and then looking at a fixed point of these smoothed response functions, which is exactly what QRE does.

We focus on a particularly tractable form of QRE, called logit QRE, in which the quantal response functions are logit curves. That is, for any pair of actions a and b, we let In [[sigma](a)/[sigma](b)] = [lambda] . [EU(a) - EU(b)], where [lambda] is a response parameter.

In our game players have different information and in some cases make a sequence of decisions. Therefore, we turn to the extensive form of the game; we represent profiles of action probabilities as behavior strategies and apply ORE to the "agent" form of this game. [6] Formally, let p = ([p.sub.1], ..., [p.sub.n]) be a completely mixed profile of behavior strategies, where [p.sub.i] = {[p.sub.ijk]} and [p.sub.ijk] is the probability that player i, with signal j [epsilon] {i, g}, votes for k [epsilon] {c, a}. Let [[bar{u}].sub.ijk](p) denote the expected utility to player i from taking action k with signal j, given p. Then [p.sup.*] is a logit equilibrium if and only if, for all i, j, k,

[[p.sup.*].sub.ijk] = [frac{[e.sup.-[lambda][[bar{u}].sub.ijk]([p.sup.*])]}{[[sum].sub.l][e .sup.-[lambda][[bar{u}].sub.ijl]([p.sup.*])]}],

where again [lambda] [greater than] 0 is a free parameter that determines the slope of players' logit response curves.

As we vary [lambda] from 0 to [infty], we can map out a family of QREs that correspond to different levels of rationality (or, more precisely, payoff responsiveness). When [lambda] = 0, response curves are completely flat, so all strategies are used with equal probability (pure error, or zero rationality). When [lambda] approaches [infty], logit response curves converge to standard best response curves, so players use only optimal strategies (no error, or perfect rationality). This family of QREs has several interesting properties, which are described by McKelvey and Palfrey (1995, 1998). For example, if we consider a convergent sequence of logit equilibria for a sequence of [lambda] values converging to [infty], the limit point must be a Nash equilibrium (or sequential equilibrium for extensive form games) of the underlying game. In this sense, Nash equilibrium is just a very special boundary case of QRE, which corresponds to perfect rationality.

We also consider an alternate model of errors combined with strategic behavior, called the noisy Nash model (NNM), which also looks at statistical variation around the Nash equilibrium but differs from QRE in two ways. First, it does not incorporate the rational expectations assumption of QRE. The NNM model assumes that individuals follow Nash behavior with some fixed probability [gamma] (to be estimated) and choose randomly with probability 1 - [gamma]. [7] Like QRE, in the limiting case when [gamma] approaches 1, the prediction approaches Nash equilibrium. Second, for intermediate values of the error term, the NNM will often differ from QRE. The difference is twofold. (1) NNM assigns the same probability of deviating from Nash equilibrium (1 - [gamma]), regardless of the expected utility loss from such a deviation. (2) NNM is not an equilibrium model. Recall that QRE is defined as a fixed point in terms of choice probabilities and logit responses. That is, each player's errors (deviations from Nash play) af fect the expected payoffs of all the other players and hence will indirectly affect all other players' logit responses. In contrast, under NNM, there is no such feedback, so that one player's deviation from Nash play has no indirect effect on any other player's deviation from Nash play. See Fey, McKelvey, and Palfrey (1996), McKelvey and Palfrey (1998), McKelvey, Palfrey, and Weber (n.d.), for further discussion of the differences between QRE and NNM. Similar to QRE, we can map out a family of NNM predictions by varying the free parameter [gamma] from 0 to 1.[8] When [gamma] = 0, all strategies are used with equal probability (pure error, or zero rationality). When [gamma] approaches 1, predictions of NNM converge to Nash equilibrium.

In figures 1 and 2, the symmetric portion of the logit QRE correspondence for jury sizes n = 3 and n = 6 and both the majority and unanimity voting rules are displayed as the thick solid curves.[9] Each graph is on the unit square of mixed behavior strategies of a representative player. The horizontal and vertical axes of each graph correspond to the probability of voting to convict, given innocent and guilty signals, respectively. At the center of each unit square is the "pure error" logit equilibrium that corresponds to [lambda] = 0. As [lambda] increases, the equilibrium curves converge to the symmetric Nash equilibrium, which is on the upper boundary of the unit square (the upper left corner, in the case of majority rule).

In a similar fashion, one constructs the correspondences defined by the NNM model and the naive (nonstrategic) model by varying [gamma] between 0 and 1. Referring to figures 1 and 2, the NNM correspondence is simply the line segment that connects the center of the probability square (pure error) to the symmetric Nash equilibrium. The naive model correspondence is the dashed line segment that connects the center to the upper left corner of the strategy space. This vertex corresponds to "honest" voting.

In Appendix B we fully characterize and compute the symmetric quantal response equilibrium correspondence for the three-person unanimous jury game with a straw vote. The characterization of the majority rule QRE correspondence is similar and is not included in the appendix. Our efforts to compute the symmetric quantal response equilibrium correspondence for majority rule juries with a straw vote were unsuccessful. Also, we found that the six-person jury game with a straw vote is too complex to compute the QRE correspondence, using our numerical methods.

EXPERIMENTAL DESIGN

We conducted four experiments with undergraduate and graduate students at the California Institute of Technology. Each experiment had twelve subjects (plus one subject who was used as a monitor). Each experiment had four sessions. Between sessions, two treatment variables were altered: the decision rule (majority or unanimity) and the straw poll (taken or not). The treatment variables were manipulated according to the design in Table 2, which gives the particulars of each experiment. In each session, subjects participated in a sequence of fifteen "matches." [10] In each match the subjects were randomly divided into groups of size n, where n was one of the treatment variables, and a game similar to that described in the previous section was conducted. In Table 2, the values of each treatment variable are given per session. For example, U/N (15) means 15 matches with the unanimity decision rule and no straw poll. All matches in the same experiment were run with the same number of subjects.

Subjects were not told that the experiment was intended to represent a jury decision. The states of the world were called the red jar and blue jar instead of guilty and innocent, and rather than a choice between convict or acquit, the subjects were instructed to guess whether the true jar was red or blue. The complete instructions are given in Appendix A.

Briefly, each match proceeded as follows. Subjects were told that there were two jars, a red one and a blue one. The red jar contained seven red balls and three blue; the blue jar contained seven blue balls and three red. One of the jars was selected for each group, as determined by the roll of a die by the monitor (a subject chosen at random from the group at the beginning of the experiment). [11] The subjects were not told which jar had been selected, but they were each allowed to choose one ball at random from the jar. [12] They then voted for either the red jar or the blue jar. Two decision rules were investigated: majority rule and unanimity. The decision rule, which was explained to them before the session, was used to determine the group decision, and payoffs were based on whether the group decision was correct or incorrect. Each subject received fifty cents if the group decision was correct, five cents if incorrect. Subjects were paid in cash at the end of the experiment, plus a "show up fee" of $5.

RESULTS

Jury Behavior without Deliberation

The first two columns of Table 3 show the realizations of the voter strategies in the case of no straw poll. As discussed in the previous section, our experiment was run with colored jars and balls to represent states of the world and evidence. We frequently interpret the results of the experiment, however, in terms of the underlying jury problem that the experiment represents. Thus, a red jar represents guilt, and a blue jar represents innocence. A red ball is evidence for guilt and a blue ball is evidence for innocence. A vote for the red jar represents a vote to convict and a vote for the blue jar is a vote to acquit. The columns of Table 3 and of subsequent tables are labeled to reflect this correspondence.

The data support our predictions for both the majority rule treatment and the unanimity treatment. Under majority rule, the subjects voted the same direction as their signals more than 94% of the time; the only exception corresponds to the analogous case of innocent signals in a six-member jury, in which case the subjects erred 21% of the time. This seems like a surprising result but can be explained by the fact that the Nash equilibrium in the six-person majority rule experiments is weak. In the symmetric equilibrium of that game, jurors who receive innocent (blue) signals are indifferent between voting to acquit and voting to convict. In the QRE, this indifference leads to a prediction that players with innocent (blue) signals will vote against their signal more frequently than players with guilty (red) signals, for every positive value of [lambda].

In the case of unanimity, jurors with a red (guilty) signal tend to vote with the signal--strongly so for groups of three, less strongly for groups of six. However, when they got a blue (innocent) signal, 36% of subjects voted red (to convict) in the three-person groups, 48% did so in the six-person groups. For the three-person groups, this is very close to the Nash predicted value of .314. For the six-person groups, the rate is significantly below the predicted value of .651.

Since the Nash equilibrium of the game requires that in some cases a pure strategy is adopted, any observations in which subjects do not follow that strategy are enough to reject statistically the Nash equilibrium as a model of behavior. Thus, any game theoretic framework that explains the data must incorporate a model of the error source. Earlier, we proposed three alternatives that incorporate error: naive (nonstrategic), logit QRE, and NNM.

Table 4 gives the results of estimating the free parameters in these three models. Because of the symmetry of the game, all three make identical predictions for the three-person majority rule case. Also, the aggregate choice frequencies from all the nondeliberation data are superimposed (large dots) in figures 1 and 2. The first (and perhaps most important) thing to observe is that the naive model does very poorly. In other words, voters are behaving as if they understand the strategic subtleties of the decision problem. For the NNM model we estimate [gamma] to be in the 0.90 to 0.93 range for three-person groups. Since (1 - [gamma]) is the probability of choosing randomly, this corresponds to an error rate of less than 5%. For six-person groups, we estimate [gamma] to be in the 0.75 to 0.78 range, which corresponds to an error rate slightly greater than 10%. For this size group, the QRE fits significantly better than the NNM under both majority and unanimity rules. In the case of three-person groups under u nanimity, the fits of QRE and NNM are almost identical, with no significant difference between the two. The naive model is rejected in favor of the QRE for all treatments in which the two models make distinct predictions.

Group Behavior with Deliberation

As is evident from Table 3, in the case of a straw poll, the final vote cannot be predicted very well using the equilibrium of the Feddersen and Pesendorfer model. On average, subjects voted against a red (guilty) signal about 15% of the time, independent of group size or voting rule. With a blue (innocent) signal, between 16% and 37% vote to convict against their signal depending on the treatment. This is a higher rate of voting against signal than for those who obtain a red (guilty) signal, but we do not get differences as large as those between majority rule and unanimity as in the case of no straw poll.

Table 5 presents the results of the straw poll sessions based on the Coughlan equilibrium, which predicts that subjects will reveal their signal in the straw poll and then base their vote in the final round on the majority outcome of the straw pa11. For the most part, the subjects do use the straw poll to reveal their signal. More than 90% in every cell (except 89.7% in one cell) revealed their signal in the straw poll. Of some interest is the additional finding that voting against innocent signals occurs with about twice the frequency of voting against guilty signals in all four treatments.

In the final vote, when the outcome of the straw poll is not a tie, under all treatments players vote with the public signal at least 83% of the time. One might expect the proportion to be higher. In an equilibrium of the Coughlan type, individuals in the final vote should ignore their own signal and only pay attention to the public signal. Table 6 gives the result of a probit analysis of the final vote by the individual's private signal and the publicly available information. For public information, we use the number of other individuals who voted red (to convict) in the straw poll plus the information of the individual (1 for a red [guilty] signal, 0 for a blue [innocent] signal). If voters are following the Coughlan equilibrium, then they should base their vote on the public information and ignore their private signal. The probit analysis indicates that this is not the case. The individual's private information has a significant effect on the final vote for all combinations of treatment variables.

We believe there are several reasons the straw poll does not work exactly as predicted by Coughlan (2000). The simplest explanation is based on the results of the treatment without a straw poll. Even in the simplest group--three-person majority rule--some players fail to vote sincerely. This source of error causes a small effect if there is no straw poll, and the QRE discussion in the last section showed how these small effects can be accounted for using an equilibrium model with errors. With a straw poll, the relatively small effects of errors in the initial voting stage become compounded in the second stage, since the noisy behavior in the straw poll means that individuals are not sure how to interpret the poll results. If they believe there is some likelihood that others will not perfectly follow the first-stage equilibrium, then they should give some weight to their own private information in the second stage. This has a snowball effect on straw poll behavior, because they know that others will be uncert ain how to interpret the straw vote. Since QRE is an equilibrium model, it can capture this effect of compounding errors.

Appendix B presents a QRE analysis of the straw vote game for the three-person unanimity treatment, including a table that presents the maximum likelihood estimates of [lambda] and [gamma]. Our computational algorithm failed to converge in its computation of the QRE correspondence for the majority rule game or the six-person games. The main findings are summarized as follows, and a more detailed account is contained in Appendix B.

There are several important features of the QRE correspondence. First, for higher values of [lambda], there are multiple symmetric equilibria that correspond to the various Nash equilibria. Second, the graph is not well behaved, as it contains two points of bifurcation. Third, one of the branches corresponds to the informative equilibrium studied by Coughlan (2000), and it is the component of the QRE correspondence that most closely matches the data. Fourth, this component has the feature that voters condition their final vote on their own signal as well as the vote outcome in the final round. In particular, in the jury setting, for any fixed number of votes to convict in the straw vote, the probability of voting to convict in the final stage is higher if one observes a guilty signal than if one observes an innocent signal. Fifth, the probability of voting to convict in the final vote is monotonic in the number of straw votes to convict.

These features of the QRE are consistent with the data and with the simple probit analysis of Table 6. Yet, although the main qualitative predictions of QRE are found in the data, the quantitative fit is less successful. In fact, the maximum likelihood fit of the NNM is better than the fit of the QRE model. Further details and discussion are in Appendix B.

Group Decision Accuracy

Table 7 summarizes the accuracy of the final decision as a function of the experimental treatment variables. We compare the actual data with the Nash equilibrium predictions of error rates, given in Table 1. First, in the experimental data with no straw poll, under unanimity the probability of voting incorrectly when the true state is blue (to convict an innocent defendant) decreases from .190 to .029 in our data as the group becomes larger (this difference is significant at the .05 level using a difference of proportion test, with t = 2.223). The Nash theory predicts the opposite (from .14 to .19). Furthermore, in the blue (innocent) state, the error rate for six-person unanimous groups is lower than the error rate for majority groups (.03 vs. .30). This difference (significant at the .01 level, t = 2.924) is also counter to the Nash theory. As for majority rule groups, error rates decline with larger size in the true red (guilty) state and increase with larger size in the blue (innocent) state. This is exac tly the opposite of what is predicted in the Nash equilibrium.

We view the contradictions with the aggregate theoretical predictions as surprising, especially because the individual choice frequencies are not very different from the theoretical levels. We interpret this to mean that the accuracy of group decisions is not a robust phenomenon. That is, small changes in individual choice behavior can result in large changes in the probability of an erroneous group judgment. This is especially true for unanimity rule, where our data suggest that a small amount of individual decision error can produce a much larger number of "acquittals" than is predicted by the Nash equilibrium.

As evidence in support of this claim, we also computed expected group accuracy under the QRE model. To compute these values, we used the maximum likelihood QRE estimates of [sigma](i) and [sigma](g) from Table 4 and substituted into equations 1 and 2 to get values of [g.sub.G] and [g.sub.I]. Then the probabilities P(C\I) and P(A\G) were computed by the corresponding binomial formulas used in Table 1. We see that the accuracy implications of the QRE estimates match the data better than the Nash predictions. All the above discrepancies between the data and the Nash predictions except one are resolved by the QRE predictions. In particular, the QRE predicts that, under unanimity, the probability of convicting an innocent defendant should decrease (from .19 to .07) as the size of the jury increases. This is consistent with the data.

The experiments with a straw poll suggest that the poll increases the accuracy of judgments in the guilty state but has essentially no effect in the innocent state. In a jury setting, it appears that the probability of convicting the innocent does not increase with deliberation, but the probability of acquitting the guilty declines substantially. In this case we can compute the Nash predictions. With a straw vote, the fully informed Nash equilibrium predicts that both majority and unanimous juries of the same size should have identical accuracy. This is true since the equilibrium strategy is to reveal individual signals in the straw poll, and then vote on the final ballot based on whether the number of reported guilty signals exceeds [frac{n}{2}].

Group Accuracy in the QRE

As pointed out earlier, Feddersen and Pesendorfer (1998) imply that large unanimous juries will convict innocent defendants with fairly high probability, whereas the likelihood of such errors in large majority juries will vanish. Even in juries of six, the probability of convicting the innocent is predicted to be higher with unanimity rule than with majority rule. This did not happen in our experiment. We found in both the three- and six-person groups that errors of this kind are more prevalent with majority than with unanimity rules.

We now show that our empirical findings are consistent with the logit equilibrium. Specifically, for any logit parameter of [lambda] [less than] [infty], the probability of conviction goes to zero in large unanimous juries. In contrast, for any [lambda] [greater than] 0, the probability of a majority-rule error in the symmetric informative logit QRE goes to 0 as jury size increases, regardless of the state of the world. We first show that the probability of an individual innocent vote in a logit QRE is bounded below by an expression that is independent of n.

THEOREM 1. Fix [lambda] [less than] [infty]. For every [delta] [greater than] 0, there exists N ([delta], [lambda]), such that for all n [greater than] N ([delta], [lambda]) the probability of acquittal in any logit QRE is greater than 1 - [delta], regardless of whether the defendant is innocent, or guilty.

Proof. Writing [[p.sup.*].sub.iga] and [[p.sup.*].sub.igc] for the probability that an individual votes to acquit or convict, respectively, with a guilty signal, it follows that in any logit QRE,

[[p.sup.*].sub.iga] = [frac{1}{1 + [e.sup.[lambda][[[bar{u}].sub.igc]([p.sup.*])-[[bar{u}].sub.iga]([p.s up.*])]]}] [greater than] [frac{1}{1 + [e.sup.[lambda]]}],

since [[bar{u}].sub.igc]([p.sup.*]) - [[bar{u}].sub.iga]([p.sup.*]) [less than] 1. A similar argument establishes that [[p.sup.*].sub.iia] [greater than] 1/1 + [e.sup.[lambda]].

The theorem now follows immediately, because the lower bound on both [p.sub.iga] and [p.sub.iia] is independent of n.

Q.E.D.

THEOREM 2. Fix [lambda] [less than] [infty], and consider the logit QRE of the unanimous jury game with a straw poll. For every [delta] [greater than] 0, there exists N ([delta], [lambda]), such that for all n [greater than] N([delta], [lambda]) the probability of acquittal is greater than 1 - [delta], regardless of whether the defendant is innocent or guilty.

Proof. The argument from the previous theorem applies to the final stage of the straw poll in exactly the same way as it applies to the case with no straw poll. Therefore, the lower bound on [p.sub.iga] and [p.sub.iia] identified in the previous theorem is the same and is independent of n. The result follows immediately.

Q.E.D.

We see that (logit) QRE behavior is entirely consistent with the traditional jurisprudential theory that argues for unanimous juries as a protection against conviction of the innocent. The reason is that the Nash equilibrium conviction/acquittal probabilities are not robust to decision errors. In order for the probability of convicting the innocent to increase, the probability of voting to acquit must go to zero extremely fast in the number of jurors (on the order of 1/n), since the probability of conviction is equal to [(1 - [[p.sup.*].sub.ia]).sup.n]. When the probability of voting to acquit goes to 0 any slower than this, defendants will always be acquitted by large juries. That is, unanimous juries will become completely uninformative.

In our unanimity rule data, we found two interesting results that are relevant to the jury setting. First, both with and without straw polls, the innocent are wrongly convicted less frequently by large juries than small juries. Second, both with and without straw polls, the probability of acquitting the guilty increases with n. Both of these findings are the opposite of what Nash theory predicts, but they are consistent with the logit QRE, which predicts that unanimous juries will become more heavily biased toward acquittal as their size increases. Under majority rule this acquittal bias does not occur, but at the nontrivial cost of roughly 50% higher wrongful conviction rates of innocent defendants.

Individual Behavior, Heterogeneity, and Asymmetric Equilibria

The theoretical work underlying this experiment focuses entirely on symmetric informative equilibria. As was noted in passing, there are many equilibria in these voting games, all of which satisfy standard refinement criteria, such as perfection, properness, and stability. We now investigate whether our data can be interpreted as evidence either for uninformative equilibria or for asymmetric equilibria.

There is clear evidence against uninformative equilibria. The unique uninformative equilibrium for the unanimity game has all players always voting to acquit, independent of their actual signal. In contrast, Table 7 clearly showed that voting is very informative in the unanimity games, with and without straw polls. We conclude that our data do not provide evidence of uninformative equilibria.

The issue of asymmetric equilibria is both more subtle and more problematic. A necessary condition for asymmetric equilibria is some evidence of heterogeneity in the observed decision rules of different players within the same treatment. We find strong evidence for heterogeneity in our data. As a simple way to categorize behavior, we divide players into three strategy types, based on how analogous jurors would vote when they receive an innocent signal. [13] Strategy type 1 always votes sincerely, that is, [sigma](i) = 0. We call these honest voters "sincere." Strategy type 2 mixes when an innocent signal is received, that is, [sigma](i) [greater than] 0. We call these voters "mixers." Strategy type 3 always votes to convict, independent of the signal, that is, [sigma](i) = 1. We call these voters "convicters." [14] To implement this classification, we use frequency of choices.

Table 8 presents the breakdown of strategy types by session. Overall, 23% of the voters are always sincere, 58% mix, and 19% always vote to convict.

The numbers in parentheses indicate the number of guilty votes and the total number of innocent signals observed by all voters classified in that cell. For example, in session 1, in the unanimity/no-deliberation treatment, four subjects always voted sincerely when they received an innocent signal. They received a total of 31 innocent signals and voted to acquit in every instance. In that same session, seven subjects mixed, that is, they neither always voted to acquit nor always voted to convict when they received an innocent signal. They received a total of 57 innocent signals and voted to convict 24 times.

To illustrate the range of possible asymmetric equilibria in jury games, consider the simplest case of three-person juror unanimity juries with no deliberation. In this case there is an equilibrium with two convicter types and one sincere voter. To see that this is an equilibrium, first look at the convicters. Either of them is pivotal if and only if the sincere player votes to convict, which happens only if she receives a guilty signal. Thus the best response is to vote to convict with a guilty signal and to vote either to convict or to acquit (indifferent) with an innocent signal. For the sincere voter, since the other two players are always voting to convict, conditioning on being pivotal is uninformative, so sincere behavior is strictly optimal for either signal. Using similar reasoning, one can easily show that for any probability p [epsilon] [0, 1], there is an equilibrium in which one sincere voter, one punitive, and one mixer will vote to convict with a innocent signal with probability p (and always vote to convict with a guilty signal). An even wider range of asymmetric equilibria exist with six-person juries. Ladha (1998) has shown that the pure strategy asymmetric Nash equilibria maximize the accuracy of information aggregation in jury games.

We have not fully characterized the asymmetric equilibria, but the point is clear. There are lots of equilibria, so the empirical restrictions of equilibrium behavior are limited. But we point out that in all these equilibria for three-person juries, the aggregate probability of a vote to convict, given an innocent signal, is above one in three.

The issue of asymmetric equilibria is further complicated if one takes into account the possibility that subjects may make errors. In Figure 3 we plot the full QRE correspondence for the three-person unanimity game under the assumption that iteratively dominated strategies are not played (i.e., a player who receives a guilty signal always votes guilty). Even under this assumption, it is evident from the figure that the full QRE correspondence is quite complicated. First, it bifurcates in several places. This is not a generic feature of the QRE correspondence, and it happens here because the game is symmetric and hence is not generic. In generic games, as shown by McKelvey and Palfrey (1995, 1998), the principal branch of the QRE correspondence selects a unique equilibrium (i.e., there is no bifurcation). For nongeneric games, such as these jury games, such a selection is no longer possible. In this case, the forks or bifurcations in the QRE correspondence are places where the equilibrium correspondence branc hes to connect to the asymmetric equilibria. Thus, if we drop the focus on asymmetric equilibria, the QRE gives no guidance as to which asymmetric equilibrium should be selected. Second, if subjects make errors, then in addition to the limiting points of the correspondence, one predicts that other points on the correspondence could occur. If there is a failure to coordinate on any equilibrium, then matters are further complicated.

Figure 3 deals only with the three-person jury. For the case of a six-person jury, matters become even more complicated and there are too many parameters to solve. Figure 4 plots the symmetric part of the QRE correspondence for a six-person jury using the unanimity rule. This graph is the same as in Figure 2 but is viewed from a different projection into the coordinate axes. In this view, we see that even the principal branch of the symmetric part of the QRE correspondence is not monotonic in [lambda]. Thus, if one follows the principal branch of the QRE starting at [lambda] = 0, one reaches a point (at about [lambda] = 15) where the curve bends backward for a while. In this region, the symmetric branch of the QRE is multiple valued. For the six-person case, we have not been able to compute the full QRE correspondence that would include the asymmetric QRE. But since there is no unique selection of a symmetric QRE for some values of [lambda], this may even increase the probability of a lack of coordination as well as the tendency toward asymmetric behavior.

LEARNING

The game we implemented in the laboratory is complicated. We asked subjects to make choices based on limited partial information in an uncertain environment with asymmetric information. They were not guided by any clues that would make it easy to connect the group decision problem they were solving to real-life situations that they have experienced. This was intentional, since we did not wish to bias their decisions or distort their induced preferences in ways that are difficult to predict or measure. Furthermore, in order to make the most of this information, they needed to anticipate how other subjects in the room would make decisions based on their (different) partial information.

Given the complexity of the task, we expected to find trends in the data. For example, the concept of conditioning a decision on the event that one's vote is pivotal is a very subtle notion, yet that is the main idea underlying the equilibrium of these games. It is conceivable that in early rounds subjects are less aware of this equilibrium effect and experience is required before it affects their decision making. The difficulty of the task was one factor that motivated the repeated trials feature of our design.

For the majority rule experiments, the aggregate data are a close match to the equilibrium prediction that voters will be sincere. This is not surprising, since the simple intuitive rule of voting sincerely corresponds to the optimal decision rule when one conditions on being pivotal. In six-person groups with majority rule, we found significantly more deviation from sincere voting, but this can be explained by the fact that voting sincerely is only a weak equilibrium in a group that size. That is, with a blue (innocent) signal, a voter is indifferent between voting red (guilty) and voting blue (innocent).

To test whether the observed phenomenon of strategic voting was partly the result of learning, we focus on the decisions by voters with blue (innocent) signals in the unanimity treatments with no straw poll. This was the only case in our experiment in which players are predicted to vote against their signal in the Nash equilibrium. For these voters, we compared the frequency of voting against their blue signal (to convict) for the first five rounds of data and the last five rounds of data. In all four sessions, the frequency of voting red (guilty) with a blue (innocent) signal in the last five rounds was higher than in the first five rounds and was closer to the Nash equilibrium prediction.

As expected, the time trend data from other voter types in other treatments do not show much of a pattern. Breaking the data down by session, treatment, and signal (a total of 28 cases), we found that in 61% of the cases the decision frequencies moved closer to the Nash ec1uilibrium or remained constant in the later periods, [15] whereas in 54% of the cases the decision frequencies were farther from the Nash equilibrium in the later periods.

CONCLUSION

Our experiments reveal a definite tendency to vote strategically in the case of a unanimous rule and no straw vote. In three-person groups more than one-third of the time players with blue (innocent) signals voted against this signal, which is very close to the Nash equilibrium prediction. In six-person groups, this frequency increased significantly, also as predicted by the theory, but the increase was not as great as Nash equilibrium predicts.

Individual behavior is explained well by the game theoretic model, but at the level of group decisions, there are numerous discrepancies. In particular, contrary to Feddersen and Pesendorfer (1998), in our experiments there were fewer decisions analogous to incorrect conviction under unanimity rule than under majority rule, and larger juries may convict fewer innocent defendants than smaller juries under unanimity. We simultaneously accounted for the individual and group data by using quantal response equilibrium to model the error.

Our study suggests that during deliberation, which we implemented experimentally with nonbinding straw polls, voters tend to reveal their signals but not to the full extent predicted by the theory. Most of them then use the public information to make their final vote. Thus, the introduction of straw polls largely (but not completely) eliminates strategic voting by juries in the case of the unanimity rule. Surprisingly, straw polls may lead to more strategic voting under the majority rule. The fact that there is a residual of strategic revelation even with straw polls results in some departure from the "sincere" equilibrium path of the final stage. As expected, it appears that straw polls increase the frequency with which juries make correct decisions in the case of a guilty defendant.

Serena Guarnaschelli is a Ph.D candidate, Richard D. MeKelvey is Professor of Political Science, and Thomas R. Palfrey is Professor of Economics and Political Science, Division of the Humanities and Social Sciences, California Institute of Technology, M/C 228-77, Pasadena, CA 91125.

Support from the National Science Foundation (Grant #SBR9617854) is gratefully acknowledged. We thank Tara Butterfield for research assistance, John Patty for help in running the experiments, and Tim Reed and Charles Smith for writing the computer program for the experiments. We also thank Tim Feddersen, Susanne Lohmann, Krishna Ladha, the audiences at several academic conferences and seminars, three referees, and the Editor for their comments.

(1.) Read "G" = Guilty, "I" = Innocent.

(2.) Read "C" = Convict, "A" = Acquit.

(3.) This is true for both the case of odd and even n, although it is somewhat more difficult to prove for even n, since the rule is no longer symmetric (n/2 votes to acquit suffice to acquit, but more than n/2 votes to convict are necessary to convict).

(4.) See Rosenthal (1989), Chen, Friedman, and Thisse (1977) and Zauner (1999) for alternative formulations of similar ideas. The QRE approach has been used in various applied areas, such as the travelers' dilemma (Capra et al, 1999), all-pay auctions and public goods games (Anderson, Goeree, and Holt 1998a, 1998b), and international conflict (Signorino 1999).

(5.) We use a logit specification for the quantal response function, which is explained below.

(6.) See MeKelvey and Palfrey 1998 for details, where this version is referred to as the AQRE (for agent QRE). We use the notation QRE to refer to this version as well.

(7.) That is, they vote to convict or to acquit with equal probability.

(8.) As with QRE, we limit attention to the NNM corresponding to the symmetric mixed strategy Nash equilibrium.

(9.) Later we present and discuss the asymmetric components of this correspondence. The large dots in the figures are explained in the data analysis section.

(10.) The last three sessions of experiment CJ1 were truncated to ten matches each, due to one particularly slow subject.

(11.) The die was rolled once for each group in each match, so in each match, different groups could have different states.

(12.) This was accomplished by placing the balls in random order on their computer screen, with the colors hidden. Subjects then used the mouse to select one ball and reveal its color. To convince them that this procedure was conducted honestly, before the experiment we generated the order of the samples for each match, each group, each possible state, and each subject. The samples in the experiment were generated according to this list. Subjects recorded which ball they selected in each match, and they were free to peruse the list after the match to verify that the correct number of balls of each color were present, and that the ball they selected was of the correct color.

(13.) Since nearly all jurors always vote to convict with a guilty signal, we do not break down the individual strategy types based on that. There are no voters who always vote to acquit.

(15.) In all but one of the cases in which there was no change between the first and last periods, all the early data are exactly at the Nash equilibrium.

(16.) The use of francs (or some other form of artificial currency) is common in experimental work, and is done primarily to allow for calibration of total payout via adjusting the exchange rate rather than adjusting the payments on the payoff tables viewed by the subject. This allows numbers in the payoff tables to be in whole numbers and avoids the use of decimals.

(17.) A more interesting alternative is to estimate aversion of QRE that incorporates a second "tremble" term, similar to NNM, but that (unlike NNM) is accounted for in the computation of equilibrium. This would effectively bound the QRE a fixed distance from 0 and 1. We have not pursued this due to issues of computational feasibility.

REFERENCES

Anderson, Simon P., Jacob K. Goeree, and Charles A. Holt. 1998a. "Rent Seeking with Bounded Rationality: An Analysis of the All Pay Auction." Journal of Political Economy 106 (August): 828 -53.

Anderson, Simon P., Jacob K. Goeree, and Charles A. Holt. 1998b. "A Theoretical Analysis of Altruism and Decision Error in Public Goods Games." Journal of Public Economics 70 (November): 297-323.

Austen-Smith, David, and Jeffrey S. Banks. 1996. "Information Aggregation, Rationality, and the Condorcet Jury Theorem." American Political Science Review 90 (March): 34-45.

Capra, Monica, Jacob K. Goeree, Rosario Gomez, and Charles A. Holt. 1999. "Anomalous Behavior in a Traveler's Dilemma?" American Economic Review 89 (June): 678-90.

Chen, Hsiao-Chi, James W. Friedman, and Jaques Thisse. 1997. "Boundedly Rational Nash Equilibrium: A Probabilistic Choice Approach." Games and Economic Behavior 18 (January): 32-54.

Coughlan, Peter. 2000. "In Defense of Unanimous Jury Verdicts: Mistrials, Communication, and Strategic Voting," American Political Science Review 94 (June): 375-93.

Feddersen, Timothy, and Wolfgang Pesendorfer. 1998. "Convicting the Innocent: The Inferiority of Unanimous Jury Verdicts under Strategic Voting." American Political Science Review 92 (March): 23-36.

Fey, Mark, Richard D. McKelvey, and Thomas R. Palfrey. 1996. "An Experimental Study of the Constant Sum Centipede Game." International Journal of Game Theory 25 (3): 269-87.

Harsanyi, John. 1973. "Games with Randomly Disturbed Payoffs." International Journal of Game Theory 2 (1): 1-23.

Ladha, Krishna. 1998. "Collective Hypothesis Testing by Bayesian Actors under Majority-Rule Voting." University of Chicago. Typescript.

Ladha, Krishna, Gary Miller, and Joseph Oppenheimer. 1996. "Information Aggregation by Majority Rule: Theory and Experiments." Washington University. Typescript.

McKelvey, Richard D., and Thomas R. Palfrey. 1995. "Quantal Response Equilibria for Normal Form Games." Games and Economic Behavior 10 (July): 6-38.

MeKelvey, Richard D., and Thomas R. Palfrey. 1998. "Quantal Response Equilibria for Extensive Form Games." Experimental Economics 1 (1): 9-41.

McKelvey, Richard D., Thomas R. Palfrey, and Roberto Weber. N.d. "The Effects of Payoff Magnitude and Heterogeneity on Behavior in 2X2 Games with Unique Mixed Strategy Equilibria." Journal of Economic Behavior and Organization. Forthcoming.

McLennan, Andrew. 1998. "Consequences of the Condorcet Jury Theorem for Beneficial Information Aggregation by Rational Agents." American Political Science Review 92 (June): 413-8.

Rosenthal, Robert. 1989. "A Bounded-Rationality Approach to the Study of Noncooperative Games." International Journal of Game Theory 18 (3): 273-92.

Signorino, Curtis S. 1999. "Strategic Interaction and the Statistical Analysis of International Conflict." American Political Science Review 93 (June): 279-98.

Wit, Jorgen. 1998. "Rational Choice and the Condorcet Jury Theorem." Games and Economic Behavior 22 (February): 364-76.

Zauner, Klaus. 1999. "A Payoff Uncertainty Explanation of Results in Experimental Centipede Games." Games and Economic Behavior 26 (January): 157-85.

APPENDIX A: EXPERIMENT INSTRUCTIONS

Decision-Making Experiment (Condorcet Jury)

This is an experiment in group decision making, and you will be paid for your participation in cash, at the end of the experiment. Different subjects may earn different amounts. What you earn depends partly on your decisions, partly on the decisions of others, and partly on chance.

The entire experiment will take place through computer terminals, and all interaction between you will take place through the computers. It is important that you not talk or in any way try to communicate with other subjects during the experiments.

We will start with a brief instruction period. During the instruction period, you will be given a complete description of the experiment and will be shown how to use the computers. If you have any questions during the instruction period, raise your hand, and your question will be answered so everyone can hear. If any difficulties arise after the experiment has begun, raise your hand, and an experimenter will come and assist you.

In this experiment, one subject will act as a monitor. The monitor will be paid a fixed amount ($20.00) for the experiment. The monitor will assist in running the experiment by generating random numbers for use in the experiment, and serving as someone who can check that the experiment is run correctly. We will select the monitor by having you each select an envelope. Open your envelope, and read the card inside to determine if you are the monitor.

[PASS OUT ENVELOPES]

Will the monitor please go to the master computer, and we will now pass out record sheets to the subjects in the experiments.

[PASS OUT RECORD SHEETS]

Please put your name at the top of the record sheet. The experiment you are participating in is broken down into 4 separate sessions. Each session will consist of 15 matches. At the end of the last match of the last session, you will be paid the total amount you have accumulated during the course of the 4 sessions. Everyone will be paid in private, and you are under no obligation to tell others how much you earned. Your earnings are denominated in francs. [16] Your dollar earnings are determined by multiplying your earning in francs by a conversion rate. In this experiment, the conversion rate is .005, meaning that 100 francs is worth 50 cents.

We will now begin the computer instructions for the first session. We will teach you about the experiment and how to use the computer by going through a short practice match. During the instruction session, do not hit any keys until you are told to do so, and when you are told to enter information, type exactly what you are told to type. You are not paid for the practice match.

[START JURY PROGRAM]

When the computer prompts you for your name, type your full name. Then hit the ENTER key, and confirm by typing Y.

First Screen

You now see the first screen of the experiment. At the top center of the screen, you see your subject number. Please record that on your record sheet now. This session consists of 15 matches. In each match, you will be matched with another group of subjects. Your group number and the number of subjects you are matched with is on the top right side of your screen. In this session, each of you is matched with two other subjects, in groups of size 3.

There are two jars, which we call the red jar and the blue jar. The red jar contains 7 red balls and 3 blue balls. The blue jar contains 7 blue balls and 3 red balls. At the beginning of the match, we will randomly choose one of the two jars for your group. This will be done by the monitor, who will roll a die to determine the color. If an odd number is rolled, the jar will be blue; if an even number is rolled, the jar will be red. Different groups may have different colors. You will not be told which jar has been chosen. You may now press a key to advance to the next screen.

Second Screen

To help you determine which jar has been selected for your group, each member of your group will be allowed to select one ball, at random, from the jar. This is done on the computer by randomly ordering the balls in the jar for each subject, but leaving the colors hidden. You can then use the mouse to click on one of ten balls. When you select a ball, its color will be revealed to you and displayed on your screen, and you must record this information on your record sheet. This is called your "Sample." You are not told the colors of the sample balls drawn by the other members of your group. Note that the balls are numbered. This is just so that you can remember and record which ball you selected.

Each subject in your group selects a ball on their own, and only sees their own sample ball. The balls are ordered differently for each subject. However, all the members of your group have the same number of red and blue balls. That is, if the true color of the jar for your group were red, then all members of your group are drawing their sample balls from a list containing 7 red and 3 blue balls. If the true color of your group's jar is blue, then all members of your group would be drawing their sample balls from a list containing 3 red and 7 blue balls. Please select a ball now, and record its color and number on your record sheet. When you have finished recording, press a key to continue.

Third Screen

After everyone has drawn the sample, you will be asked to vote for either a red jar or a blue jar.

[READ EITHER A OR B]

A Unanimity instructions: If (and only if) everyone in your group votes red, your group decision is red. Otherwise, your group decision is blue. That is, if anyone in your group votes blue, the group decision is blue.

B Majority rule instructions: If a majority (two or more) of your group votes red, then the group decision is red. Otherwise, your group decision is blue. That is, your group's decision will be whichever color receives more votes.

The table on this screen tells you your payoffs. If your group's decision is the same as the true color, then the decision is correct, and each member of your group earns 100 francs. If your group decision is different from the true color of the jar, then the group decision is incorrect, and each member of the group earns 10 francs. You may vote by selecting, with your mouse, either the red or the blue jar and then confirming. In the actual experiment, you may vote however you like. In the practice match, please vote red if you are on the left side of the room, and blue if you are on the right side of the room. Please vote now.

Fourth Screen

After everyone has voted, the vote is tallied, and you and the others in your group will be told what the result of the vote was, and what the true color of the jar for your group was. Please hit a key to proceed.

Fifth Screen

After all decisions have been made, you are given a summary of the decisions up to now. Please record this information on your record sheet. This marks the end of the match. After you have recorded all of the information, please hit a key to continue.

Sixth Screen--Display Overhead

We will now illustrate in more detail how the sample is drawn. Your computer screen now shows the balls you drew your sample from with all of the colors revealed. The ball that you drew is outlined with a white box.

Prior to the experiment, all of the lists of sample balls for each subject, and each match, and each possible true jar color have been generated by a random number generator. We have a printout of this information that will be available to the monitor. You now see on the overhead projector screen a copy of the first page of this data, which is for the practice match. Find your subject number and the true color of your group's jar. Compare the display on your computer screen to that on the overhead projector. The color of the balls should be identical. Please look at the overhead projector screen. Note also that the ordering of the balls for all subjects is different, even if they are drawing from the same jar.

During the actual experiment, you will not be shown this screen. However, your samples will be done in exactly the same way. We have printed out the similar sheet for all of the matches to follow. If you record the number of the ball that you drew each match, then at the end of the experiment, if you wish, you may ask the monitor to check your sample against this sheet to verify that the ball you selected was the correct color.

After a match is over, and everyone has recorded the information about their match, you will be randomly rematched into another group consisting of different subjects. A new jar will be randomly drawn for your group, you will get a new sample from a jar of the group color, you will vote, and a group decision will be determined as described above. After we have finished 15 matches in this manner, the first session of the experiment will be over, and we will give you instructions for the next session.

Are there any questions before we begin with the actual experiment?

[EXPERIMENTER TAKES QUESTIONS]

We will now begin with the actual experiment. If there are any problems from this point on, raise your hand, and an experimenter will come and assist you.

[START EXPERIMENT]

Rules for Experiment Session #2

This experiment session will also last for 15 matches. The rules are the same as in the first session, with one exception.

[READ A OR B]

A Recall that in the previous session, your group decision was red if and only if all three members of your group voted red. Otherwise your group decision was blue. In this session there will be a slight change in the voting rule. Now, if a majority (two or more) of your group votes red, then the group decision is red. Otherwise, your group decision is blue. That is, your group's decision will be whichever color receives more votes.

B Recall that in the previous session, your group decision was red if two or more members of your group voted red. Otherwise your group decision was blue. In this session there will be a slight change in the voting rule. Now, if (and only if) everyone in your group votes red, your group decision is red. Otherwise, your group decision is blue. That is, if anyone in your group votes blue, the group decision is blue.

Are there any questions?

Rules for Experiment Session #3

This experiment session will also last for 15 matches. The rules are the same as experiment session #2, with one exception. In this experiment, each group will conduct one preliminary vote before making the final vote. In the preliminary vote, you may vote for either the red jar or the blue jar. You do not receive any earnings for the preliminary vote. The number of blue and red votes in the preliminary vote will be revealed to you and the other members of your group. After you have been told this preliminary vote outcome, your group will conduct the final vote, in the same manner as in experiment session #2. We will conduct a practice session to illustrate how the preliminary vote is conducted.

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Rules for Experiment Session #4

This experiment session will also last for 15 matches. The rules are the same as experiment session #1, with one exception. As in the previous experiment, each group will conduct one preliminary vote before making the final vote. You do not receive any earnings for the preliminary vote. The Outcome of this preliminary vote will be revealed to you and the other members of your group, including the exact number of red votes and blue votes. You are not paid anything for this outcome. You must record this information on your information and record sheet. Then your group will conduct the final vote, in the same manner as experiment session #1. That is,

[READ A OR B]

A Unanimity instructions: If (and only if) everyone in your group votes red, your group decision is red. Otherwise, your group decision is blue. That is, if anyone in your group votes blue, the group decision is blue.

B Majority rule instructions: If a majority (two or more) of your group votes red, then the group decision is red. Otherwise, your group decision is blue. That is, your group's decision will be whichever color receives more votes.

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APPENDIX B: SOLUTION FOR THE CASE N = 3, UNANIMITY, WITH STRAW POLL

Notation and Strategies

We use the notation of Feddersen and Pesendorfer (1998) and Coughlan (2000) in this appendix. To translate to the experiment, read guilty as "red," innocent as "blue," convict as "vote red," and acquit as "vote blue."

Denote by:

* s = {G, I} the signal (either guilty or innocent);

* p = the probability of receiving the correct signal;

* [v.sub.t] = {[C.sub.t], [A.sub.t]} the vote (either convict or acquit) in the straw poll (t = 0) and in the decisive final vote (t = 1);

* i = {0, 1, 2} the number of other convicting votes in the straw poll;

* j = {0, 1, 2} the number of other matching signals.

Payoffs are defined as follows:

* 0 for making the right decision (that is, convicting the guilty and acquitting the innocent);

* q for acquitting a guilty defendant (where q [epsilon] [0, 1]);

* q - 1 for convicting an innocent defendant.

Each agent's strategy consists of a pair of probabilities, [[sigma].sub.0], [[sigma].sub.1], that specify the likelihood of voting to convict in the straw poll (as a function of one own's signal) and in the final vote (as a function of one's own signal, one's own vote in the straw poll, and others' vote in the straw poll):

[[sigma].sub.0](S) = [frac{[e.sup.[lambda][EU.sub.SP]([C.sub.0], s)]}{[e.sup.[lambda][EU.sub.SP]([C.sub.0], s)] + [e.sup.[lambda][EU.sub.SP]([A.sub.0], s)]}], (3)

[[sigma].sub.1](s, [v.sub.0], i) = [frac{[e.sup.[lambda][EU.sub.FV]([C.sub.1], s, [v.sub.0], i)]}{[e.sup.[lambda][EU.sub.FV]([C.sub.1], s, [v.sub.0], i)] + [e.sup.[lambda][EU.sub.FV]([A.sub.1], s, [v.sub.0], i)]}], (4)

where [EU.sub.SP] ([EU.sub.FV]) is the expected utility for casting a given vote in the straw poll (final vote).

Expected Utility for the Straw Poll

[EU.sub.sp]([v.sub.0], s) =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [pi](j) is the (conditional) probability of other j matching signals:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that is, in a more compact form,

[pi](j) = (2 - \j - 1\)[[p.sup.j+1][(1-p).sup.2-j] + [p.sup.2-j][(1 - p).sup.j+1]].

[mu](i, j) is the (conditional) probability that i other jurors will vote to convict in the straw poll when the number of their matching signals is j:

[mu](i, j) =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [EU.sub.FIN] is the weighted combination of the expected utilities from voting either to convict or acquit in the final vote, where weights are given by the strategy for the final vote:

[EU.sub.FIN](s, [v.sub.0], i) = [[sigma].sub.1](s, [v.sub.0], i)[EU.sub.FV]([C.sub.1], s, [v.sub.0], i)

+ [1 - [[sigma].sub.1](s, [v.sub.0], i)][EU.sub.FV]([A.sub.1], s, [v.sub.0], i).

Expected Utility for the Final Vote

In order to make the analysis clearer, consider first the case of a guilty signal (for now, s = G). By the end of the straw poll, all jurors can observe all the votes (so that they know j), but they still do not know what the true state is (either guilty or innocent) and what signals the other jurors received (where the number of other guilty signals can be 0, 1, or 2). This leads to 2 . 3 = 6 information sets. The conditional probability of being in any of these sets is given by:

[hat{[beta]}](i, k) = [frac{[beta](i, k)}{[[[sum].sup.6].sub.l=1] [beta](i, l)}],

where

[beta](i, k) =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII)

For an intuition about these coefficients, notice that we ordered the information sets so that the defendant is guilty in the first three of them (k = 1, 2, 3), innocent otherwise. So [beta]([hat{i}], k) are the conditional probabilities for the signal being correct (as in this case we are taking s = G) for k = 1, 2, 3; incorrect otherwise.

For the case of s = I, we can define similar coefficients:

[hat{[varepsilon]}](i, k) = [frac{[varepsilon](i,k)}{[[[sum].sup.6].sub.l=1][varepsilon](i,l)}],

where

[varepsilon](i, k) =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Notice that the unanimity rule in the final vote implies that, whenever a juror votes to acquit, independently of what the other jurors voted, the payoffs are determined (0 if the true state is innocent, q - 1 otherwise). So:

[EU.sub.FV]([A.sub.1], s, [v.sub.0], i) =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The formulas for [EU.sub.FV] when the final vote is to convict are more complicated. Consider the case in which the juror receives a guilty signal and votes to convict in the straw poll:

[EU.sub.FV]([C.sub.1], G, [C.sub.0], i) = (q - 1) [[[sum].sup.3].sub.k=1] [hat{[beta]}](i,k)[1 - [theta](i, k)]

- q [[[sum].sup.3].sub.k=1] [hat{[beta]}] (i, k + 3)[[theta](i, k)].

To understand this formula, notice that the final payoff is 0 only if the true state is guilty and everyone is voting to convict. The payoff is q - 1 (as shown in the first addend) if the true state is guilty (so that the sum is over k = 1, 2, 3) but someone votes to acquit (with probability as expressed in 1 - [theta]). The second addend represents the other case, when the true state is innocent, but everyone votes to convict. The [theta]s are defined as follows:

[theta](i, k) =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where x = 1.5 -\k - 1.5\ and w = 1.5 - \k - 2.5\.

The case for a guilty signal and a straw poll vote to acquit ([EU.sub.FV]([C.sub.1], G, [A.sub.0], i)) is similar, except that everyone else observes one vote to convict less than in the previous case. The formulas are as above, but for subtracting an observed vote to convict in the [theta]s; for example, [[sigma].sub.1](G, [C.sub.0], 1) becomes [[sigma].sub.1](G, [C.sub.0], 0).

The case for an innocent signal is similar, except that it involves the [hat{[varepsilon]}] coefficients, and the exponents in the formulas are switched. More precisely, consider the case of a straw poll vote to convict:

[EU.sub.FV]([C.sub.1], I, [C.sub.0], i) = (q - 1) [[[sum].sup.3].sub.k=1] [hat{[varepsilon]}](i, k + 3)[1 - [eta](i,k)]

-q [[[sum].sup.3].sub.k=1] [hat{[varepsilon]}](i, k)[[eta](i, k)],

where

[eta](i, k) =

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using the above analysis, we constructed a numerical algorithm to compute the logit QRE correspondence. The logit equilibrium correspondence has multiple solutions, due to the multiplicity of Nash equilibria in the voting game with a straw poll. There is exactly one of these that converges to the fully informative equilibrium identified by Coughlan (2000). That is, for this branch of the equilibrium correspondence, as [lambda] becomes large, voters vote informatively in the straw poll (i.e., vote the color of their signal), and, in the final vote, they vote according to the majority outcome in the straw poll.

The key feature of the QRE correspondence is that, for all positive values of [lambda], voter strategies in the final vote stage are influenced by their private signal as well as the outcome of the straw poll, in contrast to the full communication equilibrium. Moreover, this conditioning follows a very logical pattern. For any combination of how the other jurors vote in the straw poll, the QRE predicts that a voter is more likely to vote to acquit if they received an innocent signal than if they had received a guilty signal. For example, consider the QRE choice probabilities at our estimated value of [lambda]. If both other jurors voted to acquit in the straw poll, a voter with an innocent signal is predicted to vote to acquit approximately 75% of the time, while a voter with a guilty signal is predicted to vote to acquit slightly less than half the time. Similar comparisons hold for the cases in which one other voter voted to acquit in the straw poll and in which neither other voter voted to acquit. These q ualitative comparisons are all borne out in the data.

We then fit the data to the logit QRE solutions using maximum likelihood analysis. Because there are multiple QRE solutions, this required searching over all branches of the QRE correspondence and over all values of lambda. Perhaps not surprisingly, the branch that best fits the data is the one that converges to the full information equilibrium identified by Coughlan (2000). The estimated value of [lambda] is 38.34. We also fit the NNM model to the data in a similar fashion. Table B-1 shows the parameter estimates.

While the QRE picks up the qualitative features of conditional voting in the second stage, it does not provide a very good fit quantitatively. In fact, the NNM estimates provide a somewhat better fit to the data than the QRE, even though NNM does not make the right qualitative predictions.

The reason the quantitative fit of QRE is not as tight as the NNM fit is not entirely clear. QRE predictions of choice probabilities vary substantially across information sets. For some of these sets, the QRE choice probabilities converge to 0 or 1 very quickly (especially in the second stage); for others they converge very slowly (especially in the first stage). Yet, the data in the first stage have less noise than the data in the second stage! Therefore, we cannot get a good fit either for the first stage or for the second stage with a single value of X.

There is a second possible explanation. In the QRE, the frequency of voting informatively should be much higher with a guilty signal than with an innocent signal. This reflects the fact that voters who receive innocent signals are less likely to be pivotal in the second stage, so their vote in the first stage is less likely to make a difference. Consequently, the QRE choice probabilities conditional on an innocent signal converge to the informative Nash equilibrium more slowly than the QRE choice probabilities conditional on a guilty signal. The data support this qualitative prediction, but the difference is not statistically significant (.067 for innocent signals, .046 for guilty signals).

Based on this post hoc analysis, several fixes may improve the QRE fit, such as estimating a separate value for [lambda] in the first and second stages, for different signals. Since these fixes are ad hoc, we have not pursued them. [17]

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Author: | GUARNASCHELLI, SERENA; McKELVEY, RICHARD D.; PALFREY, THOMAS R. |
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Publication: | American Political Science Review |

Geographic Code: | 1USA |

Date: | Jun 1, 2000 |

Words: | 14230 |

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