Dutch Books, Group-Decision Making, the Tragedy of the

Commons and Strategic Jury Voting

Luc Bovens (LSE)

Wlodek Rabinowicz (Lund U.)

Original Hats PuzzleTodd 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 <a: Sc; b: Sc> not a solution?• Because Alice could increase her payoff by

unilaterally deviating from <Sc, Sc>: – Ua(<St,Sc>) > Ua(<Sc,Sc>)

• In a Nash equilibrium, none of the parties is able to increase her payoff by unilateral deviation

• NE-Solution: <a: <1/3,1/3,1/3>; b: <1/3,1/3,1/3>>

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

• <1,1,1>?

• <0,0,0>?

• <p,p,p> for 0 < p < 1?

<1,1,1>?

• E[Ua(<1,1,1>)] =

E[Ua(<1,1,1>)|D)P(D) + E[Ua(<1,1,1>)|S)P(S)

(-2) × ½ + 2 × 1/3 × ½ = -2/3

• E[Ua(<1,1,1>)] = -2/3 < 0 = E[Ua(<0,1,1>)]

• <1,1,1> is not a Nash equilibrium

<0, 0, 0>

• E[Ua(<0,0,0>)] = 0

• E[Ua(<1,0,0>)] =

E[Ua(<1,0,0>)|D)P(D) + E[Ua(<1,0,0>)|S)P(S)

(-2) (1/2) + (+2)(1/2) = 0

• E[Ua(<0,0,0>)] = 0 = 0 = E[Ua(<1,0,0>)]

• <0,0,0> is a Nash equilibrium

More Questions

• NE in randomised strategies <p,p,p>?• 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[Ua(<1,0,0>)] = 1/2 > 0 = E[Ua(<0,0,0>)]

• E[Ua(<0,1,1>)] = 0 > -1/6 = E[Ua(<1,1,1>)]

Self-Interest; Sweetened Pie;Ex-Post Evaluation

0.2 0.4 0.6 0.8 1-0.1

0

0.1

0.2

0.3

0.4

0.5

E[Ua(<1,p,p>)]

E[Ua(<p,p,p>)]E[Ua(<0,p,p>)]

½(3-Sqrt[3]) =.63…

p

Group-Interest; Sweetened Pie;Ex-Post Evaluation

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

E[Ug(<1,p,p>)]

E[Ug(<p,p,p>)]E[Ug(<0,p,p>)]

½(2-Sqrt[2]) =.29…

Self-Interest; Unsweetened Pie;Ex-Post Evaluation

0.2 0.4 0.6 0.8 1

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

E[Ua(<1,p,p>)]

E[Ua(<0,p,p>)]

E[Ua(<p,p,p>)]

Group-Interest; Unsweetened Pie;Ex-Post Evaluation

0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

E[Ug(<1,p,p>)]

E[Ug(<0,p,p>)]

E[Ug(<p,p,p>)]

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

0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1E[Ua(<1,p,p>)]

E[Ua(<0,p,p>)]E[Ua(<p,p,p>)]

½(3-Sqrt[3]) =.63…

Group-Interest & Tragedy

0.2 0.4 0.6 0.8 1

-1

-0.5

0.5

1

E[Ug(<1,p,p>)]

E[Ug(<0,p,p>)]

E[Ug(<p,p,p>)]½(2-Sqrt[2]) =.29…

Group-Interest, Hats and Ex Ante Evaluation

0.2 0.4 0.6 0.8 1

-0.2

-0.1

0.1

0.2 E[Ug(<1,p,p>)]

E[Ug(<p,p,p>)]

E[Ug(<0,p,p>)]

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[Ug(<.29,.29,.29>)] =.41… < 1 = E[Ug(<1,0,0>)]

• Hats:– E[Ug(<.29,.29,.29>)] =.10… < 1/4 E[Ug(<1,0,0>)]

Maximisation and Nash Equilibrium

• For the group-interest and ex-ante evaluation, the following coincide: – Nash Equilibrium:

• Solve E[Ug(<1,p,p>)] = E[Ug(<0,p,p>)] for 0 < p < 1

– Maximum • Let f[p] = E[Ug(<p,p,p>)]

• 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: A Sell bet Acquit

Indiv action:α Step forward to sell bet

Vote innocent

Decision Proc A iff some α A iff some α

Situation: S Same colours Innocence

# individuals:m 3 Jury size

Signal: s Detecting 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(α1,…, αm)• 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[Ug(<p,p,p>)] is maximal.

• Jury voting: choose <p,q> so that E[Ug(<<p,q>,…, <p,q>>)] is maximal.

• For these values of <p,q>, 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

00.2

0.40.6

0.810

0.2

0.4

0.6

0.8

1

-0.275-0.25

-0.225-0.2

-0.175

00.2

0.40.6

0.81

p

q

Jury Size and Majority Vote

5 10 15 20

0.2

0.4

0.6

0.8

1

Pr(AcqGuilty)

Pr(ConvInnoc)

Pr(VoteInn|InnSign) = p

Pr(VoteGuilty|GuiltySign) = q

m

Unanimity

5 10 15 20

0.2

0.4

0.6

0.8

1

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!

• Prunan(AcqGuilty) > Prmaj(AcqGuilty)– Obvious

• Prunan(ConvInn) > Prmaj(ConvInn)– Note punan < pmaj