During a mass prison escape, 1000 prisoners managed to break into a prison safe and steal 100 gold bars. Since all prisoners wear shirts with consecutive numbers from #1 to #1000, they decided to split that gold in following way:

  • the prisoner with highest number will propose split (starting with #1000), and all prisoners (including him) will vote for that proposal.
  • if majority voted for proposal, it is accepted and split is done. Otherwise, prisoner who proposed is killed, and next prisoner (#999...) make his proposal.

Assuming following:

  • all prisoners are highly intelligent, and are able to find optimal solution
  • gold bars are indivisible, so proposal must suggest integer number of gold bars to certain prisoners
  • 'majority' means more than half. Example 1: with 1000 prisoners 501 must vote yes (including prisoner who proposes). Example 2: with either 4 or 5 prisoners, prisoner who proposes need 2 additional votes.
  • prisoners have following priority: survive > profit > kill other prisoners (this means they will vote against proposal if they can expect same value in later proposals)


  1. How many prisoners will survive, and what is value that prisoner #1 can expect?
  2. What is formula for above two values, given arbitrary starting number of prisoners N and gold G?

For 2nd question, large N,G (> 10) can be assumed, to avoid special cases.

  • $\begingroup$ While it is similar question, it does not answer it. Answers there contain procedures (algorithm) that could be used here to find number of prisoners to survive ( although not expected value ), but there is no single/simple formula there that answers second question here. $\endgroup$
    – lost
    Aug 7, 2021 at 9:21
  • $\begingroup$ The (implied) simple formula from the top answer there would be: find the largest number of pirates, less than or equal to the starting amount, that form a stable solution with the given number of gold coins. Then divide the coins in accordance with the stable solution plan, as outlined there. I don't think you'll get much simpler than that $\endgroup$
    – bobble
    Aug 7, 2021 at 14:22
  • $\begingroup$ My problem #2 has single formula solution, in a form : nSurvivors(N,G)= ... , also expectedValue(N,G)=... - and those single formulas are, arguably, much simpler than what you mentioned. $\endgroup$
    – lost
    Aug 7, 2021 at 22:57
  • $\begingroup$ Actually the pirate version is different in the handling of ties. It results in a different solution. For example pirate #4 needs 3 votes, he must propose a split like (2,1,0,97) to survive. This questions should not have been closed. $\endgroup$
    – Florian F
    Aug 9, 2021 at 11:39


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