Portfolio Management puzzles, solved step by step

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  1. 066Five pirates, ranked A to E by seniority, must split 100 gold coins. The most senior pirate proposes a split and everyone votes. If at least half the votes, including his own, are in favour, the split stands; otherwise he is thrown overboard and the next pirate proposes. Pirates are perfectly rational, want to survive first and maximise coins second, and vote no when indifferent. What does A propose?Logic brainteasersHardHedge fundsQuantitative asset management

    Try it first

    How many coins does A keep?

    Show the worked solution

    A proposes 98 for himself, 0 for B, 1 for C, 0 for D and 1 for E. Solve from the end. With two pirates, D keeps all 100 because his own vote is half. With three, C buys E for 1 coin. With four, B buys D for 1. With five, A needs two votes and buys the two pirates who get nothing under B's plan, C and E, for one coin each, keeping 98.

    Where do you start?

    At the end, where there is no choice left. Think of planning a train journey with connections: you start from the time you must arrive and work back to when you must leave. A sequential game is solved backwards, because each pirate's vote depends only on what he would get if the current proposal failed. With two pirates left, D proposes 100 for himself; his own vote is half, which is enough. So E gets nothing if it ever comes to that, and E knows it.

    Solve it backwards: each proposer buys the pirates who would get nothing nextPirate APirate BPirate CPirate DPirate Esenior to junior5 pirates leftneeds 3 votes9801014 pirates leftneeds 2 votesgone990103 pirates leftneeds 2 votesgonegone99012 pirates leftneeds 1 votesgonegonegone1000readupwarda vote bought with 1 coin: this pirate gets 0 in the row belowthe proposer, who votes for himself
    Read the grid from the bottom up: each proposer keeps everything except one coin for each vote he needs, and he buys the pirates who would get nothing in the row below, so A ends with 98 and pays C and E one coin each.

    How does each step follow from the one below?

    With three pirates, C needs two votes, his own and one more. E gets 0 if C dies, so one coin buys E: C proposes [99, 0, 1]. With four, B needs two votes; under C's plan D gets 0, so B buys D for one coin: [99, 0, 1, 0]. A vote is worth exactly one coin more than that pirate's fallback, so a proposer always buys the cheapest voters, the ones left with nothing in the next round. With five, A needs three votes. Under B's plan C and E get nothing, so one coin each buys them, and A proposes [98, 0, 1, 0, 1].

    State the assumptions, because the answer rests on them. If an indifferent pirate voted yes, A could buy votes for zero coins. If the rule needed a strict majority, the counts change. Interviewers often change one rule as a follow-up to see whether you rebuild the chain or reach for a memorised answer. The buy-side lesson is about incentives: what someone will accept depends on their alternative, not on fairness.

    Where candidates lose it

    The trap is reasoning forwards from fairness, proposing an even split or generous bribes to the next in line. Without the backward chain you cannot know who is cheap to buy, and B, the obvious ally, is in fact the most expensive vote because he inherits the power if A dies.

    The second loss is skipping the stated assumptions. Say that indifferent pirates vote no and that exactly half passes; they decide whether the bribe is one coin or zero.

    What the interviewer asks next

    • What if a proposal needs a strict majority rather than half?
    • With the same rules, what happens with 200 pirates and 100 coins?
    • Where do you see the same logic, what someone accepts depends on their outside option, in a debt restructuring?