Dutch Books, Group-Decision Making, the Tragedy of the Commons and Strategic Jury Voting Luc Bovens...
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Transcript of Dutch Books, Group-Decision Making, the Tragedy of the Commons and Strategic Jury Voting Luc Bovens...
![Page 1: Dutch Books, Group-Decision Making, the Tragedy of the Commons and Strategic Jury Voting Luc Bovens (LSE) Wlodek Rabinowicz (Lund U.)](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cba5503460f94981754/html5/thumbnails/1.jpg)
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 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: ???
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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) • 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
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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 <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>>
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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
• <1,1,1>?
• <0,0,0>?
• <p,p,p> for 0 < p < 1?
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<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
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<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
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More Questions
• NE in randomised strategies <p,p,p>?• 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[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>)]
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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
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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…
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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>)]
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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>)]
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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
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…
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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…
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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…
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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[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>)]
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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
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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: 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
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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(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
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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, …
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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
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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
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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
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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!
• Prunan(AcqGuilty) > Prmaj(AcqGuilty)– Obvious
• Prunan(ConvInn) > Prmaj(ConvInn)– Note punan < pmaj