Since my prisoners’ hats post seems to be the most frequently read post on my blog, I thought I’d put up another riddle. This was one of my favorites that a friend told me when he was practicing for finance job interviews.
4 pirates come across 1000 gold pieces. After a fair amount of arguing, the following system is chosen for divvying up the loot:
- The pirates will draw straws. The order of straws will determine a fixed order for the remainder of the divvying process.
- The first pirate (from the straw order) will propose a distribution of the gold. This proposal is put to a vote. If a majority (greater than 50%) of the pirates agree on the proposal, they distribute the gold appropriately and they are done. Otherwise, the first pirate is killed and they move onto the next pirate.
- Assume that pirates are perfectly rational actors that vote based upon the following desired outcomes (in preferred order):
- A pirate wants to live.
- All else being equal, a pirate wants the most gold.
- If a pirate is going to live and will get the same amount of gold through two different outcomes, the pirate will vote to see more blood.
Given all of this, what does the first pirate propose, and what is the maximum amount of gold he can take?
Answers to questions I’ve received by email:
The pirates know their order before the first proposal.
Pirates vote on their own proposals.
A very nicely worded solution was emailed to me by Jack Challis some time ago:
The answer is for the first guy to give 998 to himself and 1 to #3 and 1 to #4. If either #3 or #4 vote no on this process and #1 is killed, then #2 can say I’ll take everything. #3 realizes his goose is cooked if #2 dies and votes with #2. So the original offer of 998 to #1 improves his position — he lives and gets something. Similarly for #4 instead of nothing he gets something (he always lives). So this solution is an equilibrium — nobody can make a choice which improves their position. Great problem. Two interesting generalizations: 1) What happens with 5 people? Again we work backwards. Start with when it has whittled down to three. #3 says I want everything and #4 agrees. So #2 wants to offer #4 and #5 one coin each to win their loyalty. That means #1 should offer #3 1 coin to improve his position and 2 coins to #4 . So for 5 we have 1: 997 2: 0 3: 1 4: 2 5: 0 2) Much harder question — what happens if you don’t know the ordering (1,2,3,4) ahead of time. You just know whose first and then second and third only after each round of voting. Consider when there are three players remaining. If you really do value your life at infinity, then the risk of becoming the #3 man is too great so #2 can take everything and the others will agree with him. So the expected payout from this course of action is 333 coins (1/3 prob* 1000 payout). #1 can save his life by matching the payout — he can divvy up 333 to the other 3 and keep a coin for himself.