### The Five Pirates Game

I read this game theory problem on Ravikiran’s blog and just laughed. It is amazing how otherwise intelligent people can be rather silly. People who believe the results of non-cooperative game theory are in for a rude shock when they finally encounter the real world. But first, let me describe the game:

There are five pirates (numbered 1 through 5) and 100 gold coins. Pirate 1 proposes a distribution (for example 100,0,0,0,0) and all the pirates vote. If more than 50 percent of the pirates vote down the proposal, pirate 1 is out with nothing (killed or otherwise sent off) and pirate 2 gets to decide. The question is, what should Pirate 1 propose.

As incredible as it may sound, I have read that the solution is (98,0,1,0,1). I won’t go into the logic of this solution except to say that pirates 3 and 5 are at a disadvantage because of the 50 percent rule and even numbers, so pirate 1 senses they will jump at anything and gives them as little as possible. Pirate 1 is a brilliant game theorist and is convinced that he will win 3 to 2 and get away with 98 percent of the loot.

But suppose pirates 2 through 5 are poor game theorists. They naively think that the loot ought to be evenly distributed. Therefore they vote down the “brilliant” solution of pirate 1 (he’s gone, he gets nothing) and pirate 2, not knowing anything about game theory, proposes that the loot be evenly split up. The others readily agree, since they have no way of knowing what will happen if they vote down the offer and they naively think that they should accept a “fair” solution, they agree.

This is the important point: the expected payoff for “brilliant” game theorists is zero. The expected payoff for “idiot” game theorists is 25 gold coins. Obviously it doesn’t pay to know anything about game theory does it? Ravikiran made the same point.

So here’s the question: what is wrong with traditional non-cooperative game theory? I will let my readers ponder this point for a week and then post my answer to that question next week, but if you have comments, they would be most welcome.

There are five pirates (numbered 1 through 5) and 100 gold coins. Pirate 1 proposes a distribution (for example 100,0,0,0,0) and all the pirates vote. If more than 50 percent of the pirates vote down the proposal, pirate 1 is out with nothing (killed or otherwise sent off) and pirate 2 gets to decide. The question is, what should Pirate 1 propose.

As incredible as it may sound, I have read that the solution is (98,0,1,0,1). I won’t go into the logic of this solution except to say that pirates 3 and 5 are at a disadvantage because of the 50 percent rule and even numbers, so pirate 1 senses they will jump at anything and gives them as little as possible. Pirate 1 is a brilliant game theorist and is convinced that he will win 3 to 2 and get away with 98 percent of the loot.

But suppose pirates 2 through 5 are poor game theorists. They naively think that the loot ought to be evenly distributed. Therefore they vote down the “brilliant” solution of pirate 1 (he’s gone, he gets nothing) and pirate 2, not knowing anything about game theory, proposes that the loot be evenly split up. The others readily agree, since they have no way of knowing what will happen if they vote down the offer and they naively think that they should accept a “fair” solution, they agree.

This is the important point: the expected payoff for “brilliant” game theorists is zero. The expected payoff for “idiot” game theorists is 25 gold coins. Obviously it doesn’t pay to know anything about game theory does it? Ravikiran made the same point.

So here’s the question: what is wrong with traditional non-cooperative game theory? I will let my readers ponder this point for a week and then post my answer to that question next week, but if you have comments, they would be most welcome.

## 4 Comments:

I thought ravikiran discussed a scenario where only two people were involved. Even with 5 people here, so many assumptions have been made including that the guy with the coins is brilliant while others are stupid. I think models are much more sophisticated now. Such a game is useless without the 'punisher' who can't take inequity/freeloaders.

yum yum

By Anonymous, at 12:01 PM

Most games in game theory don't lead to a very "sensible" equilibrium. The theorists say that the equilibrium is the rational strategy to use, and sometimes there are either too many assumptions or not enough information to correctly judge just what may be the best strategy.

However, game theory is somewhat useful when applied to real life management economics.

Theory of anything is confusing as hell, but application is what's important.

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