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

Original Hats Puzzle Todd Ebert (1998) n players 50-50 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: ???

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,…

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

Set Up 3 p(L)ayers and B(ookie) 50-50 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

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

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

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: >

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?

? 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

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

More Questions NE in randomised strategies ? Group interest?

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 ( )]

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

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

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

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

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.

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?

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

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

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

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… )]

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

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?

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

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

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, …

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

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

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

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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