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Pirate game enters the scene: five pirates stand on the deck of a ship, salt spray in their faces, staring at a chest containing one hundred gleaming gold coins. They have just plundered the treasure together, risking life and limb in equal measure. Now comes the hard part: dividing the spoils.
The rules are simple, almost absurdly so. The most senior pirate proposes a distribution. Everyone votes, including the proposer. If at least half the crew accepts, the proposal stands. If not, the proposer walks the plank, and the next pirate in line gets to make a proposal. The process repeats until someone’s plan succeeds or the pirates run out of proposers.
You might expect a fair split. After all, they shared the danger and the work. Perhaps twenty coins each seems reasonable, even generous. Some might argue for a bonus to the senior pirates who presumably led the expedition. But game theory, that cold and merciless branch of mathematics, suggests something entirely different. According to pure logic, the most senior pirate should claim ninety eight coins for himself and toss one coin each to two carefully selected crew members. The other two pirates get nothing.
This conclusion seems outrageous. It violates every intuition about fairness, cooperation, and reward for effort. Yet the math is airtight. Understanding why requires walking through the logical chain that leads to this seemingly absurd result.
Working Backwards Through Death
The key to solving the pirate game lies in thinking backwards from the end. Game theorists call this backward induction, though the concept itself is straightforward enough. Imagine what happens if the voting keeps failing until only two pirates remain.
Pirate Five, the most junior member, faces Pirate Four, who now gets to propose. What will Four suggest? The answer is brutally simple: one hundred coins for himself, zero for Five. Why would Five accept this insult? He would not. But his vote does not matter. Four needs only half the votes to win, which means he needs one vote. His own vote suffices. Five gets nothing and can do nothing about it.
This knowledge shapes everything that happens before. Move back one step. Three pirates remain: Three, Four, and Five. Pirate Three makes a proposal.
Five, the junior pirate, understands his situation perfectly. If Three’s proposal fails, Four will propose next, and Five will get nothing. This means Five will accept any offer from Three, no matter how small. Even a single coin beats the guaranteed zero that awaits him in the next round. Three knows this too.
What about Four? He wants Three’s proposal to fail because he knows he can claim everything if he gets to propose. Four will vote no regardless of what Three offers him. So Three should not waste any coins trying to buy Four’s vote. The rational move is to offer Five a single coin, keep ninety nine for himself, and let Four rage impotently. With his own vote and Five’s vote, Three gets the majority he needs.
Now move back another step. Four pirates remain, and Pirate Two must propose.
Two faces a more complex situation. He needs two votes beyond his own. Four and Five both have reasons to help him. If Two’s proposal fails, Three will propose next. We already know Three’s plan: keep ninety nine coins, give one to Five, and nothing to Four. Both Four and Five would prefer getting something from Two over getting what Three will offer.
Two can therefore buy Four’s vote with a single coin. He does not need to buy Five’s vote at all. Why? Because Five gets one coin from Three’s plan. Two can match that offer at no additional cost. So Two proposes: ninety nine coins for himself, one for Four, zero for everyone else. Four votes yes because one coin beats zero. Two votes for his own proposal. Five and Three vote no, but their opposition means nothing. The proposal passes.
Finally, we arrive at the beginning. Five pirates stand on deck, and the most senior, Pirate One, must make the first proposal.
One needs three votes total: his own plus two others. He faces a similar calculation to Two. If One’s proposal fails, Two will propose next, giving ninety nine coins to himself and one to Four. Three and Five get nothing from Two.
This creates an opportunity. One can buy the votes of Three and Five with minimal investment. Each needs only one coin to prefer One’s proposal over Two’s plan. So One proposes: ninety eight coins for himself, zero for Two, one for Three, zero for Four, one for Five.
Three and Five vote yes because one coin beats zero. Two and Four vote no because they would do better if Two proposed. But One has his majority. The proposal passes.
The Moral Dimension
Real human beings would likely reject this solution. Studies of the ultimatum game, a simpler cousin of the pirate puzzle, show that people routinely reject offers they consider unfair, even when accepting would leave them better off than rejecting. We seem to have a built in sense of fairness that sometimes overrides pure self interest.
Pirates in reality might also consider revenge, reputation, and future interactions. The model assumes this is the only game these pirates will ever play together. But real pirates worked together for extended periods. A pirate who grabbed ninety eight percent of the loot might find himself short of allies on the next voyage, or worse, facing a mutiny.
The puzzle also assumes perfect rationality and perfect information. Every pirate must understand the full chain of reasoning, believe that every other pirate understands it, believe that every other pirate believes that every other pirate understands it, and so on into infinite regress. One pirate who miscalculates or acts emotionally collapses the entire structure.
Yet these objections miss the point. The pirate game is not meant to describe how real pirates behave. It is designed to isolate a specific phenomenon: what happens when rational actors face a sequential decision with complete information and no future interaction? The answer, it turns out, is something that looks deeply unfair.
The Broader View
The pirate game belongs to a family of puzzles that demonstrate how individual rationality can produce collective outcomes that seem absurd or unjust. The tragedy of the commons shows how rational resource use leads to depletion. The prisoner’s dilemma shows how rational self protection leads to mutual harm. The pirate game shows how rational voting leads to extreme inequality.
These puzzles matter because they pop up in disguised form throughout economic and political life. Legislative voting often involves sequential proposals where later voters understand what earlier voters will do. Business negotiations frequently require thinking several moves ahead about how others will respond. Even social dynamics within groups can mirror the pirate game’s structure when people jockey for position and resources.
The puzzle also illuminates why institutions matter. Real societies do not leave distributions entirely to the logic of sequential bargaining. We create norms, laws, and procedures that override the pure mathematics of power. We tax and redistribute. We establish rights that cannot be voted away. We build systems that constrain how proposals can be made and how voting must proceed.
These institutions are not irrational. They represent a different kind of rationality, one that plays a longer game. If we expect to interact with others repeatedly, building trust and maintaining fairness becomes rational in a way the one shot pirate game cannot capture. The senior pirate who takes ninety eight coins might win the battle but lose the war.
Variations and Extensions
Changing the puzzle’s parameters reveals how sensitive the outcome is to specific assumptions. Suppose the vote requires a two thirds majority instead of a simple majority. Suddenly the senior pirate must buy more votes, which reduces his take. The exact distribution changes but the backward induction logic remains.
Or suppose pirates value something besides gold and survival. If throwing someone overboard provides satisfaction, or if shame from getting nothing carries psychological cost, the calculations shift. The puzzle’s counterintuitive result depends on pirates caring only about survival first, gold second, and bloodthirstiness third as a tiebreaker.
Adding uncertainty changes everything. If pirates are not sure how others will vote, or if they doubt each other’s rationality, risk aversion enters the calculation. A senior pirate might offer more than the minimum necessary to ensure passage, trading potential profit for security.
The most interesting variation asks what happens with a different number of pirates. With six pirates, the most senior needs three votes and can buy them for one coin each, keeping ninety seven. With seven pirates, he needs four votes and keeps ninety six. The pattern continues. As the crew grows, the senior pirate’s share shrinks, but he still captures the vast majority of the treasure.
The Limits of Game Theory
Real human behavior involves emotions, norms, bounded rationality, and concern for fairness that the pirate game strips away.
The pirate game reveals how powerful pure logic can be in shaping outcomes. It shows what happens when moral intuitions clash with mathematical reasoning. Understanding the game’s cold logic helps us recognize when similar structures appear in real life, and when we might want to intervene to prevent the logic from playing out.
The puzzle also demonstrates the strange interplay between power and vulnerability. The senior pirate seems most at risk because he proposes first, facing the possibility of walking the plank if his proposal fails. Yet this very vulnerability, combined with everyone’s knowledge of what happens next, gives him enormous power. He can dictate terms precisely because everyone understands the alternative.
This paradox appears throughout bargaining situations. Sometimes the player who moves first has an advantage because they can frame the negotiation. Sometimes the player who moves last has an advantage because they can respond to what others do. The pirate game shows that with complete information and backward induction, moving first can be incredibly powerful.
Perhaps the real lesson is not about pirates at all but about the importance of the rules we choose to live by. We can create systems where pure backward induction logic determines outcomes, or we can create systems that constrain that logic in service of other values. The choice is ours.
The math just shows us what happens after we choose.
