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The five pirates

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Five perfectly rational and selfish pirates (A, B, C, D, E, in order of rank) find 100 gold coins. They must decide how to distribute them according to this rule: the highest ranking pirate proposes a distribution.

Then EVERYONE votes (including him). If at least 50% vote in favor, the proposal is accepted.

If not, the proposer is thrown into the sea and the next one proposes. Everyone: (1) prefers to live rather than die, (2) wants to maximize their gold, (3) enjoys watching others die if they don't care.

What distribution should pirate A propose to survive and maximize his profit?

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Hints

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With 2 pirates (D,E), D approves his own proposal with his vote: (D,E)=(100,0).
With 3 (C,D,E), C needs 2 votes.
A proposes: A=98, B=0, C=1, D=0, E=1

Solution

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Answer: A proposes: A=98, B=0, C=1, D=0, E=1
Backward induction (perfect rationality):
With 2 pirates (D,E), D approves his own proposal with his vote:

$$ (D,E)=(100,0). $$

With 3 (C,D,E), C needs 2 votes. Buy the cheapest vote (E, which would expect 0 if C drops):

$$ (C,D,E)=(99,0,1). $$

With 4 (B,C,D,E), B needs 2 votes. Buy from D (which would expect 0 if B falls):

$$ (B,C,D,E)=(99,0,1,0). $$

With 5 (A,B,C,D,E), A needs 3 votes.
If A falls, scenario 4 leaves:

B wait 99,
C wait 0,
D wait 1,
E waits 0.

Therefore, A buys the two cheapest votes: C and E with 1 coin each.

$$ (A,B,C,D,E)=(98,0,1,0,1). $$

Votes in favor: A, C and E.

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