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Dutch Books, Group-Decision Making, the Tragedy of the Commons and Strategic Jury Voting Luc Bovens (LSE) Wlodek Rabinowicz (Lund U.)

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Original Hats Puzzle Todd Ebert (1998) n players chance of white or black hat Signal: colour of other players’ hats Simultaneously guess colour of one’s own hat Prize for group iff at least one correct guess and no incorrect guesses; passes are allowed n = 3: guess opposite colour iff you see two hats of same colour => ¾ chance of win; n > 3: ???

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Dutch Books Different betting rates => Dutch books Bet 1: [P: 1; S:3] on Q; Bet 2: [P:1; S:3] on not-Q Fair bets: Pr(Q) = Price/Stake –Less than fair bet on H: [P=1; S:1.50] –More than fair bet on H: [P=1; S: 3] Kolmogorov axioms: –Pr(C) = 0; –Pr(Q) = 1-Pr(not-Q); –Pr(Q or R) = Pr(Q) + Pr(R) for mut excl events Why should my degrees of belief satisfy the Kolmogorov axioms? Suppose: Pr(Q) = 1/3; Pr(not-Q) = 1/3 => then you should be willing to sell bet 1 and bet 2 => Dutch book can be made against you Dutch book as justification for Kolmogorov axioms, intransitive preferences,…

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Our Hats Puzzle A Dutch book can be made against a group of rational players who are –not allowed to engage in pre-play communication … are self-interested => cf. Prisoners’ Dilemma … are group-interested

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Set Up 3 p(L)ayers and B(ookie) chance of black and white hat (D) Hats have different colours Pre-signal: Bet 1: B sells single bet on (D) to L: –[P=3; S=4] Post-signal: Bet 2: B buys single bet on (D) from L: –[P=2; S=4] D is true: –Bet 1: L at +1 –Bet 2: L at -2 –Total: L at -1 D is false –Bet 1: L at -3 –Bet 2: L at +2 –Total: L at -1 Either way: –L at -1 –B at +1

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What’s Wrong? Post-signal: Bet 2: B buys single bet on (D) to L: –[P=2; S=4] (↓) D is true: –Player is sure to get the bet (↑)D is false: –Player has a 1/3 chance of getting the bet D is true: –Bet 2: L at -2 D is false –Bet 2: L at +2 IGNORE BOOKIE! Analogy: bet on snow at noon tomorrow for P(S) = 1/2

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Moral Degrees of belief –Matches willingness to bet –May not match the expression of our willingness to bet our posted betting rates

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Nash Equilibrium Scissors, Stones and Paper Game Why is not a solution? Because Alice could increase her payoff by unilaterally deviating from : –U a ( ) > U a ( ) In a Nash equilibrium, none of the parties is able to increase her payoff by unilateral deviation NE-Solution: ; b: >

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NE-Solution Alice sees two hats of the same colour She needs to determine a conditional strategy for players who see two hats of the same colour ? for 0 < p < 1?

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? E[U a ( )] = E[U a ( )|D)P(D) + E[U a ( )|S)P(S) (-2) × ½ + 2 × 1/3 × ½ = -2/3 E[U a ( )] = -2/3 )] is not a Nash equilibrium

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E[U a ( )] = 0 E[U a ( )] = E[U a ( )|D)P(D) + E[U a ( )|S)P(S) (-2) (1/2) + (+2)(1/2) = 0 E[U a ( )] = 0 = 0 = E[U a ( )] is a Nash equilibrium

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More Questions NE in randomised strategies ? Group interest?

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Sweeten the Pie Sweeten the pie: –Before: B buys single bet on (D) to P: [P=2; S=4] –Now: B buys single bet on (D) to P: [P=2; S=3] E[U a ( )] = 1/2 > 0 = E[U a ( )] E[U a ( )] = 0 > -1/6 = E[U a ( )]

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Self-Interest; Sweetened Pie; Ex-Post Evaluation E[U a ( )] ½(3-Sqrt[3]) =.63… p

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Group-Interest; Sweetened Pie; Ex-Post Evaluation E[U g ( )] ½(2-Sqrt[2]) =.29…

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Self-Interest; Unsweetened Pie; Ex-Post Evaluation E[U a ( )]

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Group-Interest; Unsweetened Pie; Ex-Post Evaluation E[U g ( )]

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Tragedy of the Commons Revenue of barn-fed cow is 1 Revenue of commons-fed cow is 2/i (+ε), with i being the # of cows on the commons Three farmers, each with one cow Cost of barn-feeding or commons-feeding is 1 Standard case: –Individual rationality: 2 cows on commons leading to total utility of 3 and depletion of common –Group rationality: 1 cow on commons, leading to total utility of 4 and pay off two other farmers Cf. over-fishing, unitisation of oil fields etc.

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New Tragedy No pre-play communication: –What if the farmers need to decide independently whether to bring their cow to the commons? –What if the state cannot designate a single person who is allowed to bring her cow to the commons, but can only manipulate the farmers’ inclinations to bring their cows to the commons, say, through social advertisement?

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Self-Interest and Tragedy E[U a ( )] ½(3-Sqrt[3]) =.63…

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Group-Interest & Tragedy E[U g ( )] ½(2-Sqrt[2]) =.29…

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Group-Interest, Hats and Ex Ante Evaluation E[U g ( )] ½(2-Sqrt[2]) =.29…

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Effects of pre-play communication # of Cows –Self-interest: E[#cows] =3×.63 = 1.90 < 2 –Group-interest: E[#cows] =3×.29 =.88 < 1 Cost of no pre-play communication for group: Cows: –E[U g ( )] =.41… )] Hats: –E[U g ( )] =.10… )]

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Maximisation and Nash Equilibrium For the group-interest and ex-ante evaluation, the following coincide: –Nash Equilibrium: Solve E[U g ( )] = E[U g ( )] for 0 < p < 1 –Maximum Let f[p] = E[U g ( )] Solve f’[p] = 0 and f’’[p] < 0 for 0 < p < 1

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Strategic Voting Condorcet Jury Theorem: The chance that a majority vote is correct approaches 1 as the number of independent and partially reliable voters goes to infinity. A fortiori: …a unanimity vote … Or not? –My vote only matters if it is pivotal –But if it is pivotal, then there is a majority who voted guilty –So even if I receive an innocent signal, I still have good reason to vote guilty –… unless others reason in the same way! –What is the probability with which I should vote guilty when I receive an guilty signal and what is the probability with which I should vote innocent when I receive an innocent signal?

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Similarity of Structure Group action: ASell betAcquit Indiv action:αStep forward to sell bet Vote innocent Decision ProcA iff some α Situation: SSame coloursInnocence # individuals:m3Jury size Signal: sDetecting same colours Detecting innocence

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Parameters Utilities: U(A,S), U(not-A,S),… Priors: P(S) Reliability of signals: P(s|S); P(not-s|not-S) Decision Procedure: A = f(α 1,…, α m ) Pr(α|s=1) = p? –Sell when same signal; –Vote innoc when innoc signal Pr(not-α|s=0) = q? –Not sell when diff signal (q = 1) –Vote guilty when guilty signal

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Challenge Hats: choose p so that E[U g ( )] is maximal. Jury voting: choose so that E[U g (,…, >)] is maximal. For these values of, what is the probability of acquitting the guilty, convicting the innocent,... …for variable jury size, majority vs unanimity voting, …

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11-person Jury, Unanimity, U as a function of p and q values p q

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Jury Size and Majority Vote Pr(AcqGuilty) Pr(ConvInnoc) Pr(VoteInn|InnSign) = p Pr(VoteGuilty|GuiltySign) = q m

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Unanimity Pr(ConvInnoc) Pr(AcqGuilty) Pr(VoteInn|InnSign) = p Pr(VoteGuilty|GuiltySign) = q m

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Explananda q = 1 but p << 1 for Unanimity –Under pivotality, there is a strong signal for guilt, even if I receive an innocent signal Jury size ~ Pr(AcqGuilty) under Unanimity –Obvious Jury size ~ Pr(ConvInn) under Unanimity –Note decreasing p-values! Pr unan (AcqGuilty) > Pr maj (AcqGuilty) –Obvious Pr unan (ConvInn) > Pr maj (ConvInn) –Note p unan < p maj

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