An Experimental Study of Jury Voting Behavior

Abstract

This chapter uses experimental analysis to test the Feddersen and Pesendorfer (American Political Science Review 92(1):23–35, 1998) theoretical results regarding the Condorcet jury theorem. Under the assumption that jurors will vote strategically (rather than sincerely based on private information), Feddersen and Pesendorfer derive the surprising conclusion that a unanimity rule makes the conviction of innocent defendants more likely, as compared with majority rule voting. Previous experimental work largely supported these theoretical predictions regarding strategic individual behavior, but failed to find support for the conclusions about the relative merits of unanimity and majority rule procedures in terms of group decisions. We extend this literature with an experiment in which the cost of convicting an innocent defendant is specified to be more severe than the cost of acquitting a guilty defendant. This payoff asymmetry results in a higher threshold of reasonable doubt than the 0.5 level used in earlier studies. We find very little evidence of the strategic voting predicted by theory (even for our asymmetric payoff structure) and no difference between the use of unanimity and majority rules. Overall, it was very difficult for the juries in our experiment to achieve a conviction, and no incorrect convictions occurred. Our experimental results suggest that the standard risk neutrality assumption can lead to misleading conclusions. We argue that a high cost associated with convicting the innocent can interact with risk aversion to produce an even higher threshold of reasonable doubt than would result from risk neutrality, which tends to neutralize the negative effects of strategic voting under a unanimity rule.

Appendices

Appendix 1: Derivations of Nash Equilibria and Model Predictions

Our parameters

Probability the signal matches the true state: $p=0.75$

Threshold of reasonable doubt: $q=0.667$

Number of jurors: $n=12$

Number of guilty votes required to convict under majority rule: $\widehat{k}=10$

Note that while here our threshold of reasonable doubt is 0.667, and we added $4 to all of the payouts in the experiments, these computations remain valid.

• A. Nash equilibrium probability a juror will vote guilty after observing an innocent signal under unanimity voting from Feddersen and Pesendorfer (1998, p. 26, Equation 3)

  $$ \begin{array}{c}\hfill \sigma (i)=\frac{{\left(\frac{\left(1-q\right)\left(1-p\right)}{qp}\right)}^{1/(n-1)}p-\left(1-p\right)}{p-{\left(\frac{\left(1-q\right)\left(1-p\right)}{qp}\right)}^{1/(n-1)}\left(1-p\right)}\hfill \\ {}\hfill \sigma (i)=\frac{{\left(\frac{\left(1-2/3\right)\left(1-3/4\right)}{2/3\times 3/4}\right)}^{1/(12-1)}\left(3/4\right)-\left(1-3/4\right)}{3/4-{\left(\frac{\left(1-2/3\right)\left(1-3/4\right)}{2/3\times 3/4}\right)}^{1/(12-1)}\left(1-3/4\right)}\hfill \\ {}\hfill \sigma (i)=0.720389\hfill \end{array} $$

• B. Nash equilibrium probability a juror will vote guilty after observing an innocent signal under majority rule voting from Feddersen and Pesendorfer (1998, pp. 33–34, Appendix B)

  $$ \begin{array}{c}\hfill \sigma \left(i,\widehat{k}\right)=\frac{p\left(1+f\right)-1}{p-f\left(1-p\right)}\hfill \\ {}\hfill \sigma \left(i,\widehat{k}\right)=\frac{0.75\left(1+0.641859\right)-1}{0.75-0.641859(0.25)}\hfill \\ {}\hfill \sigma (i)=0.392502\hfill \end{array}, $$

  where f is determined by

  $$ \begin{array}{c}\hfill f={\left(\frac{1-q}{q}{\left(\frac{1-p}{p}\right)}^{n-\widehat{k}+1}\right)}^{1/(k-1)}\hfill \\ {}\hfill f={\left(\frac{1-0.667}{0.667}{\left(\frac{1.75}{0.75}\right)}^{12-10+1}\right)}^{1/(10-1)}\hfill \\ {}\hfill f=0.641859\hfill \end{array} $$

• C. Probability an innocent defendant will be convicted under unanimity voting from Feddersen and Pesendorfer (1998, p. 26)

  $$ \begin{array}{c}\hfill {l}_i\left(p,q,n\right)=\frac{\left(2p-1\right){\left(\frac{\left(1-q\right)\left(1-p\right)}{qp}\right)}^{1/(n-1)}}{p-\left(1-p\right)\left({\left(\frac{\left(1-q\right)\left(1-p\right)}{qp}\right)}^{1/(n-1)}\right)}\hfill \\ {}\hfill {l}_i\left(0.75,0.667,12\right)=\frac{\left(2(0.75)-1\right){\left(\frac{\left(1-0.667\right)\left(1-0.75\right)}{(0.667)(0.75)}\right)}^{1/(12-1)}}{0.75-\left(1-0.75\right)\left({\left(\frac{\left(1-0.667\right)\left(1-0.75\right)}{(0.667)(0.75)}\right)}^{1/(12-1)}\right)}\hfill \\ {}\hfill {l}_i\left(0.75,0.667,12\right)=0.790312\hfill \end{array} $$

• D. Probability a guilty defendant will be acquitted under unanimity voting from Feddersen and Pesendorfer (1998, p. 26)

  $$ \begin{array}{c}\hfill {l}_o\left(p,q,n\right)=1-\left(\frac{\left(2p-1\right)}{p-\left(1-p\right)\left({\left(\frac{\left(1-q\right)\left(1-p\right)}{qp}\right)}^{1/(12-1)}\right)}\right)\hfill \\ {}\hfill {l}_o\left(0.75,0.667,12\right)=1-\frac{\left(2(0.75)-1\right)}{0.75-\left(1-0.75\right)\left({\left(\frac{\left(1-0.667\right)\left(1-0.75\right)}{(0.667)(0.75)}\right)}^{1/(12-1)}\right)}\hfill \\ {}\hfill {l}_o\left(0.75,0.667,12\right)=0.069895\hfill \end{array} $$

• E. Probability an innocent defendant will be convicted under majority rule voting from Feddersen and Pesendorfer (1998, p. 30)

  $$ {l}_I\left(\widehat{k}\right)={\displaystyle \sum_{j=\widehat{k}}^n\left(\begin{array}{c}\hfill n\hfill \\ {}\hfill j\hfill \end{array}\right){\left({\gamma}_I\left(\widehat{k}\right)\right)}^j{\left(1-{\gamma}_I\left(\widehat{k}\right)\right)}^{n-j}}, $$

  where

  $$ {\gamma}_I\left(\widehat{k}\right)=\left(1-p\right)\sigma \left(g,\widehat{k}\right)+p\sigma \left(i,\widehat{k}\right) $$

  Our parameters and above give

  $$ \begin{array}{c}\hfill {\gamma}_I\left(\widehat{k}\right)=\left(1-0.75\right)(1)+(0.75)(0.392502)\hfill \\ {}\hfill {\gamma}_I\left(\widehat{k}\right)=0.5443765\hfill \end{array} $$

  Since for us

  $$ \begin{array}{c}\hfill \widehat{k}=10\hfill \\ {}\hfill R=12\hfill \end{array} $$

  we need only to find the sum

  $$ {\displaystyle \sum_{j=10}^{12}\left(\begin{array}{c}\hfill n\hfill \\ {}\hfill j\hfill \end{array}\right)\kern0.5em {\left({\gamma}_I\left(\widehat{k}\right)\right)}^j{\left(1-{\gamma}_I\left(\widehat{k}\right)\right)}^{n-j}} $$

  Thus,

  $$ \begin{array}{c}\hfill {l}_I(10)=\left(\begin{array}{c}\hfill 12\hfill \\ {}\hfill 10\hfill \end{array}\right){(0.5443765)}^{10}{\left(1-0.5443765\right)}^2 \\ \qquad\quad+\left(\begin{array}{c}\hfill 12\hfill \\ {}\hfill 11\hfill \end{array}\right){(0.5443765)}^{11}{\left(1-0.5443765\right)}^1\hfill \\ {}\hfill \kern1em \qquad+\left(\begin{array}{c}\hfill 12\hfill \\ {}\hfill 12\hfill \end{array}\right){(0.5443765)}^{12}{\left(1-0.5443765\right)}^0\hfill \\ {}\hfill {l}_I(10)=0.038795\hfill \end{array} $$

• F. Probability a guilty defendant will be acquitted under majority rule voting from Feddersen and Pesendorfer (1998, p. 31)

  $$ {l}_G\left(\widehat{k}\right)=1-{\displaystyle \sum_{j=\widehat{k}}^n\left(\begin{array}{c}\hfill n\hfill \\ {}\hfill j\hfill \end{array}\right){\left({\gamma}_G\left(\widehat{k}\right)\right)}^j{\left(1-{\gamma}_G\left(\widehat{k}\right)\right)}^{n-j}}, $$

  where

  $$ {\gamma}_G\left(\widehat{k}\right)=p\sigma \left(g,\widehat{k}\right)+\left(1-p\right)\;\sigma \left(i,\widehat{k}\right) $$

  Our parameters and above give

  $$ \begin{array}{c}\hfill {\gamma}_G\left(\widehat{k}\right)=(0.75)(1)+\left(1-0.75\right)(0.392502)\hfill \\ {}\hfill {\gamma}_G\left(\widehat{k}\right)=0.8481255\hfill \end{array} $$

  Thus, we have

  $$ \begin{array}{c}\hfill {l}_G(10)=1-\left(\begin{array}{c}\hfill 12\hfill \\ {}\hfill 10\hfill \end{array}\right){(0.8481255)}^{10}{\left(1-0.8481255\right)}^2\\ \qquad\qquad-\left(\begin{array}{c}\hfill 12\hfill \\ {}\hfill 11\hfill \end{array}\right){(0.8481255)}^{11}{\left(1-0.8481255\right)}^1\hfill \\ {}\hfill \kern1em \qquad-\left(\begin{array}{c}\hfill 12\hfill \\ {}\hfill 12\hfill \end{array}\right){(0.8481255)}^{12}{\left(1-0.8481255\right)}^0\hfill \\ {}\hfill {l}_G(10)=0.72935\hfill \end{array} $$

Appendix 2: Instructions for Unanimity Treatment

This is an experiment in the economics of decision-making. Various agencies have provided funds for the experiment. Your earnings will depend partly on your decisions and partly on chance. If you are careful and make good decisions, you may earn a considerable amount of money, which will be paid to you, in cash, at the end of the experiment today. The experiment will consist of three parts. We will throw a six-sided die at the end of the session. If the result of the die throw is 1, 2 or 3, you will be paid 1/2 of your cumulative earnings for Part I and all of your earnings for Part IIIa of the experiment. If the result of the die throw is 4, 5, or 6, you will be paid 1/2 of your cumulative earnings for Part II and all of your earnings for Part IIIb of the experiment. In addition, you will be paid $6 for showing up today.

We will begin by reading these instructions out loud. Please follow along. If you have any questions as we are reading, raise your hand and your question will be answered for everyone.

Before beginning, we will choose one of you to assist us in the experiment today. This person, who will be called the monitor, will help us by throwing dice and drawing colored balls from a container. The monitor will also observe procedures to ensure that the instructions are followed. The monitor will be paid the average of what all participants earn. We will now assign each of you a number from 1 to 13, and we will throw a 20-sided die to select the monitor.

In this experiment, you will be asked to predict from which randomly chosen cup a ball was drawn. We will begin by having the monitor roll a six-sided die behind a screen at the front of the room. If the roll of the die yields a 1, 2, or 3, we will draw from the blue cup, which contains three blue balls and one red ball. If the roll of the die yields a 4, 5, or 6, we will draw from the red cup, which contains three red balls and one blue ball. Therefore it is equally likely that either cup will be selected. Since the monitor will roll the six-sided die behind a screen, you will not see the result of the die throw or know which cup is being used for the draws.

Blue cup	Red cup
Used if the die roll is 1, 2, or 3	Used if the die roll is 4, 5, or 6
Contents: 3 blue balls and 1 red ball	Contents: 3 red balls and 1 blue ball

2.1 Private Draws

Once a cup is determined by the roll of the die, we will empty the contents of that cup into an unmarked container. (The container is always the same, regardless of which cup is being used.) Then we will approach each of you and draw a ball from the container. The result of this draw will be your private information and should not be shared with other participants. After each draw, we will return the ball to the container before making the next private draw so the contents of the container are always the same when we make a private draw. Each person will have one private draw, with the ball being replaced after each draw.

2.2 The Voting Process

After each person has seen a private draw, we will begin the voting process. We will approach each of you to ask for your vote: Vote “B” if you think the blue cup was emptied into the unmarked container or “R” if you think the red cup was emptied into the unmarked container. After everyone has voted, we will announce the total number of “R” and “B” votes and the monitor will announce the color of the cup that was actually emptied into the unmarked container.

2.3 Your Payoff

Your money payoff for the period depends on the cup that was actually used and the “group decision. The group decision is red if at least 10 of the 12 people vote “R.” Otherwise, if three or more people vote “B,” the group decision is blue.

Your dollar payoffs are summarized in the table below. Any correct decision earns each member of the group a $4 payoff. A correct group decision is one that matches the cup actually used. If the group decision is blue and the red cup was actually used, each member of the group earns a $2 payoff. Finally, there are no money payoffs if the group decision is red and the cup used for the draws was actually blue.

	Cup used is blue	Cup used is red
Group decision is blue	Your payoff is $4	Your payoff is $2
Group decision is red	Your payoff is $0	Your payoff is $4

2.4 Decision Sheet

This part of the experiment will consist of ten periods. The results for each period will be recorded on a separate row on the decision sheet that follows. The period numbers are listed on the left side of each row. Next to the period number is a blank that should be used to record the draw (blue or red) that you see when we come to your desk. Write b (for blue) or r (for red) in column (0) at the time the draw is made. Column 1 contains spaces to record your vote and the total number of blue and red votes, which will be announced at the end of each period. Once you see your draw, you should write your vote (B or R) in the column labeled “your vote.” At the end of each period, the monitor will announce the group decision and the color of the cup that was actually used. Record the group decision (blue or red) in column (2) and the color of the cup actually used for the draws in column (3). Recall that if at least ten of the twelve people voted “R,” and the red cup was actually used, then each of you earns $4. However, if at least ten of the twelve people voted “R” and the blue cup was actually used, then each of you earns nothing. If fewer than ten people vote “R,” the group decision is blue, and you each earn $4 if the blue cup was actually used or $2 if the red cup was actually used. Notice that the payoff for an incorrect group decision of blue is higher than the payoff for an incorrect group decision of red. You should record your earnings for the period in column (4) and keep track of your cumulative earnings for all periods in column (5).

Before we begin the periods that determine your earnings, we will go through one practice period. In this practice period, the monitor will throw the die that determines which cup will be used, and you will each see a draw from that cup. However, unlike in the periods that determine your earnings, you will observe the throw of the die, your draw will not be private, and you will not be asked to vote in this practice period.

At this time the monitor will throw the die that determines which cup is to be used. Remember that the blue cup is used if the throw is 1, 2, or 3, and the red cup is used if the throw is 4, 5, or 6. Now we will come to the desk of each person and show them a private draw from the unmarked container. If this were not a practice period, this person would record the color of the ball (b or r) in column (0). Recall that each person will have one private draw with the ball being replaced after each draw. After each person has seen a private draw, each person should record a vote, B or R, in the column labeled “your vote.” Then we will come to each desk and tally the total number of B and R votes.

Are there any questions before we begin the periods that determine your earnings? Please do not talk with anyone during the experiment. We will insist that everyone remain silent until the end of the last period. If we observe you communicating with anyone else during the experiment, we will pay you your cumulative earnings at that point and ask you to leave without completing the experiment.

At this time, the monitor will throw the die that determines which cup is to be used. Remember that the blue cup is used if the throw is 1, 2, or 3, and the red cup is used if the throw is 4, 5, or 6. Now we will bring the container to each person’s desk and draw a ball from the unmarked container. After you see a private draw, record the color of the ball (b or r) in column (0), and then we will return the ball to the unmarked container before approaching the next person.

2.5 Part II (Was Distributed After All Ten Periods of Part I Were Completed)

The group decision will be determined differently for the remaining periods of the experiment. The group decision is red if all of the 12 people vote “R.” Otherwise, if one or more people vote “B,” the group decision is blue. Notice that we require a unanimous vote to make red the group decision, but if any one person votes “B,” then the group decision is blue. The contents of the cups and the payoffs will remain the same as summarized on your new Decision Sheet.