Pirate game

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The Pirate Game is a simple mathematical game. It illustrates how, if assumptions conforming to a homo economicus model of human behaviour hold, outcomes may be surprising.

Dividing up the gold could prove tricky

The Game

There are five pirates, A, B, C, D and E. They find 100 gold coins. They must decide how to distribute them.

The Pirates have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.

The Pirate world's rules of distribution are thus: that the most senior pirate should propose a distribution of coins. The pirates should then vote on whether to accept this distribution; the proposer is able to vote, and has the casting vote in the event of a tie [since this is the right of the proposer]. If the proposed allocation is approved by vote, it happens. If not, the proposer is thrown overboard on the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again.

The Result

It should be expected intuitively that Pirate A will propose that the allocation shall be 20, 20, 20, 20, 20. However, this is not the theoretical result.

In the game theoretic analysis, Pirate A takes 98, Pirate B 0, Pirate C 1, Pirate D 0, and Pirate E 1.

This is apparent if we work backwards: if all except D and E have been thrown overboard, D proposes 100 for himself and 0 for E. He has the casting vote, and so this is the allocation.

If there are three left (C, D and E) C knows that D will offer E 0 in the next round; therefore, C has to offer E 1 coin in this round to make E vote with him, and get his allocation through. Therefore, when only three are left the allocation is C:99, D:0, E:1.

When B Makes his decision, he knows this; he must therefore make sure that he is not thrown overboard. He does this by offering 1 to D. Because he has the casting vote, the support only by D is sufficient. Thus he proposes B:99, C:0, D:1, E:0.

A, as a rational agent, knows that this is the allocation of coins if he is thrown overboard. He therfore offers A:98, B:0, C:1, D:0, E:1.

Hence, the allocation gives the most to A but will nevertheless be accepted is:

A: 98 coins B: 0 coins C: 1 coin D: 0 coins E: 1 coin

References

Rasmussen, Eric: Games and Information, 2004

[ v]·[ d]·[ e]	Topics in game theory
Definitions	Normal form game · Extensive form game · Cooperative game · Information set · Preference
Equilibrium concepts	Nash equilibrium · Subgame perfection · Bayes-Nash · Trembling hand · Correlated equilibrium · Sequential equilibrium · Quasi-perfect equilibrium · Evolutionarily stable strategy
Strategies	Dominant strategies · Mixed strategy · Grim trigger · Tit for Tat
Classes of games	Symmetric game · Perfect information · Dynamic game · Repeated game · Signaling game · Cheap talk · Zero-sum game · Mechanism design
Games	Prisoner's dilemma · Chicken · Stag hunt · Ultimatum game · Coordination game · Matching pennies · Minority game · Rock, Paper, Scissors · Pirate game · Dictator game
Theorems	Minimax theorem · Purification theorems · Folk theorem · Revelation principle · Bishop-Cannings theorem
Related topics	Mathematics · Economics · Behavioral economics · Evolutionary biology · Evolutionary game theory · Population genetics · Behavioral ecology · Adaptive dynamics · List of game theorists

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