Game Theory and the Pirate Puzzle game

I found an interesting blog about game theory, named Mind Your Decisions.

As a way of getting you in game theory mode–the mode in which nothing is as it appears–here’s a joke:

The dumbest kid in the world

A young boy enters a barber shop and the barber whispers to his customer, “This is the dumbest kid in the world. Watch while I prove it to you.”

The barber puts a dollar bill in one hand and two quarters in the other, then calls the boy over and asks, “Which do you want, son?”

The boy takes the quarters and leaves.

“What did I tell you?” said the barber. “That kid never learns!” Later, when the customer leaves, he sees the same young boy coming out of the ice cream store.

“Hey, son! May I ask you a question? Why did you take the quarters instead of the dollar bill?”

The boy licked his cone and replied, “Because the day I take the dollar, the game is over!”

The Mind Your Decisions blog uses the pirate puzzle game to make their point about fair division. The scenario and rules are listed below. Before you read the answer, think about who you think has the advantage in this scenario. Hint, it does not pay to be 2nd, but why?

Pirate Puzzle game

Three pirates (A, B, and C) arrive from a lucrative voyage with 100 pieces of gold. They will split up the money according to an ancient code dependent on their leadership rules. The pirates are organized with a strict leadership structure—pirate A is stronger than pirate B who is stronger than pirate C.

The voting process is a series of proposals with a lethal twist. Here are the rules:

1. The strongest pirate offers a split of the gold. An example would be: “0 to me, 10 to B, and 90 to C.”
2. All of the pirates, including the proposer, vote on whether to accept the split. The proposer holds the casting vote in the case of a tie.
3. If the pirates agree to the split, it happens.
4. Otherwise, the pirate who proposed the plan gets thrown overboard from the ship and perishes.
5. The next strongest pirate takes over and then offers a split of the money. The process is repeated until a proposal is accepted.

Pirates care first and foremost about living, then about getting gold. How does the game play out?

The solution

At first glance it appears that the strongest pirate will have to give most of the loot. But a closer analysis demonstrates the opposite result—the leader holds quite a bit of power.

The game can be solved by thinking ahead and reasoning backwards. All pirates will do this because they are a very smart bunch, a trait necessary for surviving on the high seas.

Looking ahead, let’s consider what would happen if pirate A is thrown overboard. What will happen between pirates B and C? It turns out that pirate B turns into a dictator. Pirate B can vote “yes” to any offer that he proposes, and even if pirate C declines, the situation is a tie and pirate B holds the casting vote. In this situation, pirate C has no voting power at all. Pirate B will take full advantage of his power and give himself all 100 pieces in the split, leaving pirate C with nothing.

But will pirate A ever get thrown overboard? Pirate A will clearly vote on his own proposal, so his entire goal reduces to buying a single vote to gain the majority.

Which pirate is easiest to buy off? Pirate C is a likely candidate because he ends up with nothing if pirate A dies. This means pirate C has a vested interest in keeping pirate A alive. If pirate A gives him any reasonable offer—in theoretical sense, even a single gold coin—pirate C would accept the plan.

And that’s what will happen. Pirate A will offer 1 gold coin to pirate C, nothing to pirate B, and take 99 coins for himself. The plan will be accepted by pirates A and C, and it will pass. Amazingly, pirate A ends up with tremendous power despite having two opponents. Luckily, the opponents dislike each other and one can be bought off.

The game illustrates the spoils can go to the strongest pirate or the one that gets to act first, if the remaining members have conflicting interests. The leader has the means to buy off weak members.

Don’t get caught up in the exact assumptions or outcomes of the game—just remember the basic lesson. In the real world, it might be necessary to buy a vote with 20 gold coins. Nonetheless, the general logic is the same. Here are some of the main insights from the game:

Lessons

  • Players should think ahead and reason backwards
  • A  leader can win by exploiting conflict among weaker members
  • Players derive worth from voting power, and some players can be bought off
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About Jorge Costales

- Cuban Exile [veni] - Raised in Miami [vidi] - American Citizen [vici]
This entry was posted in 2TG Favorites, Random Observations and tagged . Bookmark the permalink.

3 Responses to Game Theory and the Pirate Puzzle game

  1. Jason H says:

    I actually strongly disagree with the sentiment that A holds the most power–I think the inverse is true. A is trying to bargain for his life.

    We take it for granted and assume that the pirates value gold more than their lives, which isn't true. Pirate A values his life more than his gold. Pirate C should know this. It doesn't matter one way or the other to Pirate B, because if Pirate A dies, Pirate B will receive all the gold. Any settlement less than 100 gold to Pirate B will fail.

    Thus, Pirate A must bargain with Pirate C, in which case, Pirate C will hold all of the power between the two, because Pirate A's life is in the hands of Pirate C.

    Using this kind of logic, Pirate A will fail to live unless if he can appease Pirate C, so he -must- appease Pirate C. Then the question of Pirate C's sell-out point is brought into play.

    How much is enough? Because we can't discern how much is enough, it becomes a broken example.

    I think in the pirate's game we are downplaying the pirates' greed.

  2. Ted says:

    Not true, Jason H. Pirate C's top priority is getting the most possible coins. Pirate A knows that if he thrown overboard, Pirate C will get nothing. Therefore even if Pirate A offers just 1 coin to Pirate C, Pirate C will have no choice but to accept it.

    They both have an interest in keeping Pirate A alive. Pirate C would rather get 1 coin and let Pirate A live, then to let Pirate A die and get nothing. And Pirate A knows this.

    If Pirate A offers himself 99 coins and Pirate C 1 coin then dares Pirate C to vote him thrown overboard, what can Pirate C do? Either throw him overboard and get no coins from Pirate B or agree on the proposal and get 1 coin from Pirate A.

  3. Tom says:

    Ted, pirate C also knows that he will move up the seniority ladder if pirate A is thrown out. Therefoe he has a strong interest in A being voted down. Since by the rules he doesnt have to worry about his life, C will want to maximize his share *while A is still alive*. So A will not want to risk his life by offering too little which means he will offer C at least 50 coins.

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