An Experimental Study of The Jury Voting Model with Ambiguous - - PowerPoint PPT Presentation

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

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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.

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Jury Theorem

◮ Old insight: If

(i) each group member’s information is positively correlated with the true state

• f 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).

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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).

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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).

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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.

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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,

• r 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.

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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.

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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
• f 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.

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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).

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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.

1To generate ambiguity in the laboratory setting, we adopted the method of Stecher et al

(2011).

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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.

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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

Stage 1

Bet 3

All subjects are to be paid according to the decisions they made in three bets at the end of the experiment. 18 / 35

Revealed Updating Rules

Bet 3

◮ Rational jurors update their prior beliefs after receiving new information as

follows:3 → The standard Bayesian updating; or → The Maximum Likelihood Updating (MLU) (Gilboa and Schmeidler, 1993);

• r

→ The Full Bayesian Updating (FBU) (Pires, 2002). Bet 3 White Black Indifferent Do Not Bet Bet 1, Bet 2 White, White or Yellow Bayes’ Rule

• thers
• thers
• thers

Black, Black or Yellow

• thers

Bayes’ Rule

• thers
• thers

White, Black or Yellow FBU MLU

• thers
• thers

Black, White or Yellow FBU MLU

• thers
• thers

Indifferent, Indifferent FBU MLU Bayes’ Rule

• thers

Stage 2 Graphs 3In our experiment, whenever subjects’ updating behaviour would not conform to either of these categories, we will deem their behaviour as inconsistent. 19 / 35

Stage 2

To emulate the jury voting scenario, Stage 2 consists of one trial round and twenty subsequent rounds of decision-making between two alternatives. → Specifically, at the beginning of each round, the computer randomly and independently selects one among two possible urns, a ‘Blue’ or a ‘Red’ one, with equal probability.

◮ Each urn contains 100 balls, either red or blue. ◮ The urn is said to be Red (Blue) if it predominantly contains Red (Blue)

balls.

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Stage 2 (Cont’d)

→ Next, the computer randomly and independently assigns each subject to a group

• f other five subjects (N = 6), tasked with guessing the right color of the urn

selected for that round. → Before each subject casts their vote about the color of the urn for the round, they receive a private information regarding the color of a randomly and independently drawn ball, with replacement, from that urn.

◮ For the ambiguous treatments only, before the computer randomly draws

the ball for each subject, a graph of 10 bar charts is shown to all subjects. Each of the bar charts contains 10,000 realisations of 21 different proportions of the two colors, comprised between 60/40 and 80/20, that is any realisation of the percentage of the predominant balls of a given color in the set p = {0.60, 0.61, . . . , 0.79, 0.80}.4

◮ We repeat a similar routine for the other treatments dealing with the 70/30

and 90/10 compositions, that is any realisation of the percentage of the predominant balls of a given color in the set p = {0.70, 0.71, . . . , 0.89, 0.90}.

4These realisations are obtained in much the same way as the ones for Stage 1. 21 / 35

Stage 2 (Cont’d)

→ Once subjects cast their vote, each vote gets next aggregated to form a specific group decision in accordance with the voting rule which applies to those subjects’ treatment.

◮ The default for the group decision is set to be ‘Blue’ if that group falls short

• f meeting the minimum number of red votes for a ‘Red’ decision to be

reached: For the (strict) majority voting rule the minimum number of red votes is four out of six; whereas for the unanimity voting rule that minimum number of red votes is six out of six.

◮ No feedback regarding either the color of the selected urn, or other subjects’

votes are provided during the experimental session.

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Stage 2: FP-U Treatment

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Stage 2: A-U Treatment

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Stage 2: A-U Treatment

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Questionnaire

Lastly, subjects are asked to complete a questionnaire after Stage 2.

◮ The questionnaire involves personality traits, locus of control and a few

demographic questions. Answers in the questionnaire are only meant to be used as control variables in our empirical analysis of the experimental data.

◮ Subjects are informed that their answers to the questionnaire do not affect the

payments they receive at the very end of the experimental sessions they participated in.

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Payments

Subjects are incentivised in taking part in the experiment and paying attention to their choices during the experiment as follows.

◮ They receive NZD10.00 as a show-up fee; ◮ NZD2.00 for each correctly placed bet, for a maximum of NZD6.00 attainable for

choices made in Stage 1; additionally,

◮ They receive NZD14.00 if their group decision in the randomly selected round

• ut of the twenty rounds they were involved in is correct;

◮ Otherwise, they receive respectively (i) NZD13.00 or (ii) NZD5.00 if their group

decision is incorrect in that randomly selected round (false negative or false positive - alias type II or type I error). Hence, the maximum and minimum payments subjects can receive in an experimental session are NZD30.00 and NZD15.00, respectively.

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The Experiment

Our experiment was conducted between July and September 2017 and April and May 2018 at DECIDE (Laboratory for Business Decision Making) based at the University of Auckland.

◮ Subjects were recruited among students at the University of Auckland using

ORSEE (Greiner, 2015).

◮ A total of 336 subjects participated in 16 experimental sessions. ◮ All sessions were computerised, using z-Tree (Fishbacher, 2007). ◮ Preceding each stage, in each of these treatments, separate instructions were

given to subjects by the experimenter.

◮ The subjects’ total rewards from this experiment consisted of the earnings from

Stage 1, Stage 2 and the show-up fee. → This resulted in an average reward per subject of NZD26.00.

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Descriptive Statistics

Experimental Realisations for p = 0.70 and p ∈ [0.60, 0.80]

N = 6 k = 4 k = 6 Number of individual decisions 480 720 Number of group decisions 80 120 All subjects Red votes with red signals 71.2% 43.9% Red votes with blue signals 8.9% 16.6% Ambiguity-averse subjects Red votes with red signals 52.9% 87.9% Red votes with blue signals 0% 14.9% SEU maximising subjects Red votes with red signals 75.2% 30.9% Red votes with blue signals 13.4% 9.0% Ambiguity-loving subjects Red votes with red signals 58.7% 47.1% Red votes with blue signals 1.3% 27.4% Wrong group outcomes 28.8% 51.7% True jar ‘Blue’ (Type I error) 2.4% 0% True jar ‘Red’ (Type II error) 57.9% 100% N = 6 k = 4 k = 6 Number of individual decisions 1,320 1,680 Number of group decisions 220 280 All subjects Red votes with red signals 60.6% 57.7% Red votes with blue signals 19.9% 17.9% Ambiguity-averse subjects Red votes with red signals 62.3% 66.7% Red votes with blue signals 20.3% 22.2% SEU maximising subjects Red votes with red signals 72.9% 58.7% Red votes with blue signals 16.1% 18.2% Ambiguity-loving subjects Red votes with red signals 50.2% 45.8% Red votes with blue signals 24.9% 13.0% Wrong group outcomes 36.8% 52.5% True jar ‘Blue’ (Type I error) 11.0% 0% True jar ‘Red’ (Type II error) 66.7% 98.7%

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Descriptive Statistics

Experimental Realisations for p = 0.80 and p ∈ [0.70, 0.90]

N = 6 k = 4 k = 6 Number of individual decisions 600 720 Number of group decisions 100 120 All subjects Red votes with red signals 59.5% 68.5% Red votes with blue signals 20.3% 24.4% Ambiguity-averse subjects Red votes with red signals 53.3% 60.7% Red votes with blue signals 0% 31.3% SEU maximising subjects Red votes with red signals 69% 65.5% Red votes with blue signals 34.2% 19% Ambiguity-loving subjects Red votes with red signals 51.6% 75.4% Red votes with blue signals 11% 30.4% Wrong group outcomes 35% 45.8% True jar ‘Blue’ (Type I error) 2% 0% True jar ‘Red’ (Type II error) 66.7% 94.8% N = 6 k = 4 k = 6 Number of individual decisions 480 720 Number of group decisions 80 120 All subjects Red votes with red signals 57.5% 46.4.5% Red votes with blue signals 26.7% 21.2% Ambiguity-averse subjects Red votes with red signals 48.3% 55.6% Red votes with blue signals 16.1% 47.7% SEU maximising subjects Red votes with red signals 62.2% 56.7% Red votes with blue signals 26.7% 20.2% Ambiguity-loving subjects Red votes with red signals 56.4% 38.1% Red votes with blue signals 20.7% 12.6% Wrong group outcomes 40% 45% True jar ‘Blue’ (Type I error) 10.5% 0% True jar ‘Red’ (Type II error) 66.7% 98.2%

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Logit Estimations That Explain Individual Decisions

Ambiguity-loving Subjects - High Precision

(1) (2) (3) (4) VARIABLES Red Vote Red Vote Red Vote Red Vote Red Sample 0.388*** 0.394*** 0.384*** 0.393*** (0.104) (0.102) (0.0967) (0.0987) Unanimity Rule 0.226 0.242* 0.281** 0.173 (0.148) (0.144) (0.131) (0.143) Ambiguity 0.132 0.145 0.0934 0.0846 (0.145) (0.140) (0.139) (0.141) Red Sample * Unanimity

• 0.0325
• 0.0405
• 0.0328
• 0.0449

(0.146) (0.146) (0.140) (0.142) Ambiguity * Unanimity

• 0.331**
• 0.368**
• 0.399**
• 0.327*

(0.157) (0.154) (0.170) (0.172) Red Sample * Ambiguity

• 0.0947
• 0.0962
• 0.0991
• 0.0928

(0.142) (0.144) (0.140) (0.138) Round

• 0.000690
• 0.000728
• 0.000668
• 0.000679

(0.00179) (0.00179) (0.00176) (0.00180) Locus of Control – Yes Yes Yes Personality Traits – – Yes Yes Self-reported Abilities – – – Yes Observations 880 880 880 880 Robust standard errors clustered at subject level in parentheses *** p<0.01, ** p<0.05, * p<0.1

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Logit Estimations That Explain Individual Decisions

Ambiguity-averse Subjects - High Precision

(1) (2) (3) (4) VARIABLES Red Vote Red Vote Red Vote Red Vote Red Sample 0.679*** 0.627*** 0.780*** 0.813*** (0.171) (0.164) (0.211) (0.209) Unanimity Rule 0.501** 0.540*** 0.738*** 0.743*** (0.214) (0.147) (0.217) (0.273) Ambiguity 0.284 0.509** 0.684*** 0.918*** (0.263) (0.220) (0.205) (0.240) Red Sample * Unanimity

• 0.395**
• 0.354**
• 0.510**
• 0.546**

(0.187) (0.172) (0.241) (0.233) Ambiguity * Unanimity

• 0.103
• 0.215
• 0.353**
• 0.163

(0.276) (0.203) (0.175) (0.234) Red Sample * Ambiguity

• 0.269*
• 0.267*
• 0.244*
• 0.270**

(0.144) (0.146) (0.147) (0.134) Round

• 0.00238
• 0.00238
• 0.00197
• 0.00194

(0.00353) (0.00358) (0.00359) (0.00361) Locus of Control – Yes Yes Yes Personality Traits – – Yes Yes Self-reported Abilities – – – Yes Observations 320 320 320 320 Robust standard errors clustered at subject level in parentheses *** p<0.01, ** p<0.05, * p<0.1

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Logit Estimations That Explain Individual Decisions

MLU Subjects - High Precision

(1) (2) (3) (4) VARIABLES Red Vote Red Vote Red Vote Red Vote Red Sample 0.419** 0.419** 0.431** 0.412** (0.166) (0.166) (0.172) (0.173) Unanimity Rule 0.263 0.263 0.295 0.303 (0.189) (0.189) (0.184) (0.200) Ambiguity 0.332* 0.330* 0.349* 0.315 (0.181) (0.184) (0.191) (0.202) Red Sample * Unanimity 0.00938 0.00864 0.00610 0.0198 (0.176) (0.174) (0.176) (0.179) Ambiguity * Unanimity

• 0.256
• 0.257
• 0.379**
• 0.372**

(0.179) (0.177) (0.170) (0.166) Red Sample * Ambiguity

• 0.111
• 0.111
• 0.126
• 0.105

(0.176) (0.176) (0.183) (0.186) Round

• 0.00104
• 0.00104
• 0.000970
• 5.48e-05

(0.00212) (0.00213) (0.00214) (0.00197) Locus of Control – Yes Yes Yes Personality Traits – – Yes Yes Self-reported Abilities – – – Yes Observations 580 580 580 580 Robust standard errors clustered at subject level in parentheses *** p<0.01, ** p<0.05, * p<0.1

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Results

The experimental data suggest the existence of a strong link between an individual’s attitude toward ambiguity and their voting behavior. → There is evidence of a reduced tendency to vote to convict (reduced probability

• f voting for red) especially for ambiguity-loving subjects.

→ There is evidence of a reduced tendency to vote to convict (reduced probability

• f voting for red) especially for subjects adhering to MLU (comprising of

subjects who are either ambiguity-averse or ambiguity-loving). → There is evidence of a reduced tendency to vote to convict (reduced probability

• f voting for red) especially for ambiguity centered around high precision levels.

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