An Experimental Study of The Jury Voting Model with Ambiguous Information Simona Fabrizi Steffen Lippert Addison Pan University of Auckland DECIDE Workshop – Auckland 6 July 2018 1 / 35
Small Group Decision Making ◮ Much real world negotiation and decision making takes place in small groups. ◮ Group members cast votes and determine the collective decision according to some voting rule. ◮ Groups: committee, a board, a jury, an electorate. ◮ Decisions: donor organ allocation, parliamentary decisions, pronouncing a defendant guilty or innocent. ◮ Decision rules: majority, super-majority, unanimity. 2 / 35
Small Group Decision Making Our Problem ◮ The group faces only two alternatives and there are only two states of nature. ◮ All group members agree on the optimal decision in each state. ◮ True state of nature is unknown at the time of the decision. Ex-ante, each state is equally likely. ◮ Each group member has some private information, a private “signal” about the true state of nature. 3 / 35
Jury Theorem ◮ Old insight: If (i) each group member’s information is positively correlated with the true state of nature, (ii) the information of distinct members is conditionally independent, given the state of nature, and (iii) all jury members cast their votes simultaneously and according to their private information (informative voting), then majority rule is asymptotically efficient (Condorcet Jury Theorem: Condorcet, 1785). 4 / 35
Strategic Voting and Jury Paradox ◮ Informative voting may not be a Nash equilibrium in the voting game (Strategic voting: Austen-Smith & Banks, 96). ◮ Individual vote affects the collective decision only if the vote is pivotal. ◮ If all others vote informatively, conditioning on pivotality is informative. ◮ A voter’s rational choice might be to not vote informatively. ◮ Becomes worse if the voting rule is more demanding (Jury Paradox: Feddersen & Pesendorfer, 98) ◮ With the unanimity rule, conditioning on pivotality is very informative. ◮ Unanimity voting is an inferior rule under strategic voting, especially as the size of the jury grows larger. The more demanding the hurdle for conviction, the more likely a jury will convict an innocent. ◮ Experimental evidence: Voters vote strategically and the probability of reaching a wrong decision is higher under unanimity (Guarnaschelli, McKelvey, & Palfrey 00; Goeree & Yariv, 11). 5 / 35
Strategic Voting and Jury Paradox ◮ Results use the assumption that the individuals’ private information has a reliability that is commonly known and can be precisely measured. → Common prior assumption, Bayesian updating. ◮ We propose that the reliability of the voters’ information is not precisely measured. ◮ Consequence: Payoffs when voters are not pivotal become decision-relevant (Pan, 18). 6 / 35
On the way to Ambiguity in Jury Voting Models ◮ Ellis (16): Theory model of majority voting with ambiguous jurors’ values. ◮ Fabrizi & Pan (17): Theory model of voting (with unanimous and non-unanimous voting rules) under an ambiguous signal precision with ambiguity-averse jurors. ◮ Pan, Fabrizi, & Lippert (18): Theoretical and experimental research allowing for voters with non-congruent views about the precision of their signals. → This paper: How do attitudes toward ambiguity in single-person decisions translate into behaviour in ambiguous voting situation? ◮ Kelsey and le Roux (17): Experimental study of the effect of ambiguity attitudes as identified in single-person decisions in two-player games. 7 / 35
Our Study ◮ This experimental study examines individual voting behaviour in a jury voting model with ambiguous information. ◮ We find a link between an individual’s attitude towards ambiguity and their voting behaviour in collective decision-making in ambiguous settings. ◮ Voters who are not ambiguity neutral vote less often to “convict”. ◮ The inferiority of the unanimity rule is reduced. ◮ Robust to including locus of control, personality traits, self-reported ability. 8 / 35
Quick Review: Jury Voting Model à la FP An Illustrative Example ◮ A jury consisting of six jurors, N = 6. ◮ Common prior: Probability a defendant is Guilty or Innocent is P ( G ) = P ( I ) = 1 / 2. ◮ Before casting their votes, each juror receives a private i.i.d. signal, drawn from a common distribution, g or i . ◮ Signals’ precision is P ( g | G ) = P ( i | I ) = p = 0 . 8. ◮ After receiving their signals, jurors simultaneously vote to either acquit, a , or to convict, c , the defendant. ◮ The jury verdict, Acquittal, A , or Conviction, C , is determined by a voting rule k , identifying the minimum number of votes needed for conviction. ◮ Jurors have common values, as follows: u ( A , I ) = u ( C , G ) = 0 , u ( C , I ) = − q = − 0 . 9 , u ( A , G ) = − (1 − q ) = − 0 . 1 . 9 / 35
Quick Review: Jury Voting Model à la FP Results for this Illustrative Example Absent any ambiguous information → Informative voting is the unique symmetric and responsive Nash equilibrium under the majority voting rule ( k = 4). → Jurors receiving an innocent signal no longer has a strict preference to vote to acquit under the unanimity voting rule ( k = 6). We use this example, and a variant of it, for our experimental treatments. 10 / 35
Experimental Design For our experimentation, we implement eight treatments. All eight treatments contain three stages. ◮ Two decision stages. 1. We replicate Cohen et al. (00): Ellsberg three-color urn experiment to elicit each subject’s ambiguity attitude and updating rule under ambiguity. 2. We then let subjects perform decision-tasks in which we vary the reliability of the information they receive and the voting rules by which group decisions are reached. ◮ One questionnaire stage. ◮ Demographics, self-assessed abilities, locus of control, personality traits. ◮ The first stage and the questionnaire are the same across all treatments. The differences across treatments are only featured by the set-ups of the second stage. 11 / 35
Experimental Design Stage 2 ◮ Voting rules: ◮ Majority (k=4) and Unanimity (k=6). ◮ Signal precisions: ◮ High, two versions: p = 0 . 8 and p ∈ [0 . 7 , 0 . 9] ◮ Low, two versions: p = 0 . 7 and p ∈ [0 . 6 , 0 . 8] → Four FP treatments: p = 0 . 7 or p = 0 . 8, each subjected to ‘M’-ajority rule (FP-M) or ‘U’-nanimity rule (FP-U). → Four Ambiguity treatments, p ∈ [0 . 6 , 0 . 8] or p ∈ [0 . 7 , 0 . 9], each subjected to ‘M’-ajority rule (A-M) or ‘U’-nanimity rule (A-U). 12 / 35
Stage 1 We computerize the experiment of Cohen et al. (2000) and conduct it with real cash prizes. ◮ Subjects are asked to place three consecutive bets on the colors of a randomly selected ball from a standard 3-color Ellsberg urn. ◮ Subjects are initially told that the urn contains 90 balls, of which 30 are white, and the remaining 60 are either black or yellow. ◮ The exact composition of the Ellsberg urn is then determined at random by the computer and not revealed to the subjects. 1 ◮ Next, the computer randomly selects a ball from that urn with replacement until a ‘non-yellow’ ball is selected. Subjects are not told about the color of the selected ball. 1 To generate ambiguity in the laboratory setting, we adopted the method of Stecher et al (2011). 13 / 35
Stage 1 Bets 1 & 2 Then subjects place Bets 1 and 2. Each bet has four alternatives. → In Bet 1, subjects choose: ◮ ‘White’; ◮ ‘Black’; ◮ ‘Indifferent’ between White or Black (computer places a bet on White or Black with equal probability); ◮ ‘Do Not Bet’, renouncing to the prospect of a positive earning. → In Bet 2, subjects choose: ◮ ‘White or Yellow’; ◮ ‘Black or Yellow’; ◮ ‘Indifferent’ between White or Yellow or Black or Yellow (computer places a bet on White or Yellow or Black or Yellow with equal probability) ◮ ‘Do Not Bet’, renouncing to the prospect of a positive earning. 14 / 35
Stage 1 Bet 3 ◮ After Bet 2, subjects are told that the ball selected for Bets 1 and 2 was ‘non-yellow’ and that it was placed back into the urn. ◮ Next, the computer draws another ball, the color of which is once again not revealed to the subjects. ◮ Subjects then place Bet 3, consisting of the same options as in Bet 1. 15 / 35
Stage 1 Bets 1 & 2 We adopted the method of Stecher, Shields and Dickhaut (2011) to generate 10,000 realizations of the proportion of black and yellow balls. 16 / 35
Revealed Attitudes Toward Ambiguity Bets 1 & 2 Jurors’ attitudes towards ambiguity should be consistent with one of the following expected utility models of decision-making: 2 → The Subjective Expected Utility (SEU) model; or → The Expected Utility Model with Multiple Priors ( Gilboa and Schmeidler, 1989 ), generalised by Hurwicz α -criteria ( Hurwicz, 1951 ), with the α : (1 − α ) weight mixture of Maxmin preference and Maxmax preference. Bet 1 White Black Indifferent Do Not Bet Bet 2 White or Yellow SEU α > 1 / 2 inconsistent inconsistent Black or Yellow α < 1 / 2 SEU inconsistent inconsistent Indifferent inconsistent inconsistent SEU or α = 1 / 2 inconsistent Do Not bet inconsistent inconsistent inconsistent inconsistent 2In our experiment, whenever subjects’ updating behaviour would not conform to either of these categories, we will deem their behaviour as inconsistent. 17 / 35
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