The Pirate Game, and Why Society Needs “Madmen”

I took a game theory course in college. One of the classic problems in it is the pirate game. Five pirates — call them A through E, oldest to youngest — divide a hundred gold coins. The most senior proposes a split, everyone votes, and a majority passes it. If it fails, the proposer goes over the side and the next one proposes. All five are brilliant, perfectly rational, greedy, and fond of their own lives. Question: what should A propose?

The standard answer is 97, 0, 1, 2, 0 (or 97, 0, 1, 0, 2). The man at the top walks off with ninety-seven, the rest divide three, and two of them get nothing at all. The method is backward induction: work out what happens with only two pirates left, then back up to three, four, five. Every step fits flush. Logically it is airtight.

My first reaction was that it’s wrong. In the real world, anybody who proposed that split would have the other four flip the table on him — jump him, beat him to death, and sort out the money afterward. If a theory produces a conclusion that far from the facts, something is wrong with the theory.

I didn’t chase it down at the time. It took me a while to understand that the pirate game was never a model meant to describe reality. It’s a puzzle — one Ian Stewart set in Scientific American in 1999 to show how sharp and how counterintuitive backward induction is. The more absurd the conclusion, the more the sharpness of the tool stands out. Once I saw that, I let it go. It’s the same move as “ignore air resistance” in physics class: it’s showing off a tool, not describing pirates.

But one question stayed with me. Why do real people flip the table? Why would the pirate in line for one coin rather take nothing at all, just to put the man at the top in the water? What is that apparently idiotic behavior actually doing inside the game?

I thought about it for years. When it finally clicked, I found it far more interesting than the puzzle itself, and it explains a lot: why a curse like “when will this sun expire? We will all perish with you” has to exist, why society needs madmen, and why the decent people are always the ones who can’t get out of the trap.

I. The Ability to Flip the Table Is the Load-Bearing Beam Under Every Fair Split

Start with where the pirate game is actually fragile. Most people assume it’s the “everyone is rational” assumption. It isn’t. The load-bearing assumption is that rationality is common knowledge — not just that everyone is rational, but that A knows B is rational, and knows that B knows C is rational, and knows that B knows that C knows D is rational, layer on layer all the way down. Aumann proved in 1995 that given this common knowledge, the backward-induction solution necessarily holds. Which means the converse: one flicker of doubt anywhere in that chain and the whole derivation comes apart.

That is a much harsher requirement than “somebody is irrational.” It isn’t enough for all five pirates to be rational — if A merely suspects that E might not be, the answer changes.

So let’s put a madman aboard and see what happens.

Say the youngest, E, is a madman: he votes no on anything under twenty coins, consequences be damned, and everyone knows it. Work backward. When it’s B’s turn to propose, B needs two more votes. C can’t be bought — C cleans up in the next round. D is satisfied with one coin. That leaves E, and E wants twenty, so B pays it. Crazy pays. But when it’s A’s turn, A also needs two more votes, and one coin for C plus two for D covers it. A simply routes around E. A well-behaved E still had a fifty-fifty shot at two coins; a madman E is guaranteed nothing.

So known madness is a conditional chip. It’s worth something only when you are the pivotal vote and nobody can replace you; otherwise people walk around you. Life works the same way. The guy with the temper either gets catered to or gets cut out of the game, and there is rarely a middle state.

The interesting case is the second one: nobody knows who the madman is, only that every person carries some probability of flipping the table.

Now A’s bet is wildly asymmetric. What he saves is a few coins; what he loses is his life. Even at a 5% chance of madness per pirate, an offer of 97, 0, 1, 2, 0 puts him over the side if either of his two bought votes flips — call it close to one chance in ten of dying. What does an A who is even slightly attached to living do? He gives away a great deal more and buys an extra vote or two as insurance. The number 97 collapses on the spot. And it spreads up the chain: B is just as afraid of dying, so B can’t get 98 in the next round either, which changes the price A has to pay for votes, and so on. The whole solution slides from an extreme split toward something close to an even one.

That is what actually happens in the world. The pirate game is a multi-round ultimatum game at heart, and the ultimatum game has been slapping the rationality assumption in the face since the first experiment in 1982: proposers offer forty to fifty percent on average, and offers below twenty percent get rejected all the time. People would rather have nothing than let you have something.

Notice how cheap the self-destruct move is in the pirate game: one coin. You give up one coin and you get to throw the man holding ninety-seven into the sea. The puzzle assumes nobody is willing to spend that coin, which is a far more absurd assumption than “everyone is rational.” Real people will spend the coin, and they will spend a great deal more than one.

Which raises the question: is the impulse to take revenge when you know you’ll lose by it a human defect, or a design?

I only worked it out when I read Robert Frank’s Passions Within Reason. His starting point is something the rational-actor model cannot explain: the commitment problem. A purely rational person cannot do three things. His threats aren’t credible — when the moment comes to pay for punishing you, he runs the numbers and drops it. His promises aren’t credible — only an idiot passes up a free advantage. His loyalty isn’t credible — he leaves when something better shows up. And because everyone knows he’s rational, he can’t frighten anyone, and nobody dares trust him or commit to him for the long haul. Rationality has become a liability.

Emotion is the solution to that problem. Anger makes you retaliate when you know the trade is a loss. Guilt keeps you from cheating when nobody is watching. Love keeps you from walking out when a better option appears. The crucial part is that emotion hijacks your rational calculation at the exact moment of decision — you can still negotiate with a man who is “rationally angry,” you cannot negotiate with a man who is actually furious, and only the second one is frightening. Irrationality isn’t a side effect of the device. It is the function.

Frank calls this a commitment device. Three conditions make it work. It has to really fire: somebody has to have actually flipped a table. It has to be unfakeable: flushing, shaking, dilated pupils lie outside voluntary control, which is what makes them credible signals. And it has to be visible: anger nobody sees deters nobody. The evolutionary logic is frequency-dependent. A population of nothing but honest cooperators gets invaded by cheats; inside a population of nothing but cheats, honest types who can recognize one another do better. So the stable state contains both.

Twenty years of experiments since have largely borne this out. Fehr and Gächter’s public-goods experiment in Nature in 2002: people will pay out of their own pocket to punish free riders, and it is precisely this money-losing punishment that holds cooperation together. A 2004 paper in Science put people in a scanner and watched the striatum’s reward circuitry light up while they punished defectors — revenge genuinely feels good, which is what the commitment device looks like at the neural level. Nowak and colleagues ran the ultimatum game through evolutionary simulation in 2000 and found that as long as there is a reputation mechanism, an apparently idiotic strategy like “death before dishonor” grows on its own and stays stable.

So the fact that people are frequently unreasonable is not a bug. It’s a feature. A population of nothing but rational, well-behaved people gets exploited to death by 97, 0, 1, 2, 0.

This is the game-theoretic content of “when will this sun expire? We will all perish with you.” When the people of Xia hurled that at their last tyrant king, they had nothing in their hands but one thing: the ability to make sure he couldn’t go on living either. Laozi’s “when the people do not fear death, what is the use of threatening them with it?” is the other face of the same coin — once the people at the bottom stop caring whether they live, the deterrent at the top drops to zero, and everybody’s bargaining position gets reshuffled.

The ability to go down together is the load-bearing beam under every fair distribution. While it is there, nobody gets 97. Once it is genuinely gone, 97, 0, 1, 2, 0 stops being a puzzle and becomes the world.

II. The Madman Is a Public Good

Push the logic one step further and you arrive somewhere slightly counterintuitive: society needs madmen.

Go back to the ship where nobody knows who the madman is. If five out of a hundred are madmen and A doesn’t know which five, he has to raise his offer to everybody. The other ninety-five rational pirates did nothing whatsoever and got a raise. And the madman? On the day he actually goes off, he loses — he forfeits the coins he could have had, sometimes his life along with them. He pays the cost and everyone collects the benefit. Economics has a name for that: a public good.

The madman is a public good. And an exceptionally cheap one: a little madness changes the behavior of every rational person in the room. Schelling called it “the rationality of irrationality” in The Strategy of Conflict in 1960 — being believed when you say you’ll flip the table is itself the hardest currency there is. In the eighties, Kreps, Wilson and that crowd proved it mathematically: give the game the tiniest probability of a “crazy type” and rational players change strategy across the board, and will even act crazy on purpose to build up the reputation. Nixon’s madman theory and the Cold War doomsday machine are the same move.

The logic isn’t confined to the ship.

Nisbett and Cohen studied the culture of honor in the American South. Herding societies, where property is easy to take and the state won’t protect it, grow a culture in which an insult must be answered with violence, and Scots-Irish settlers carried that package into the South. In their experiments, Southern students who were insulted in a hallway showed spikes in cortisol and testosterone and refused to yield the right of way. The interesting part is that the South is simultaneously the most polite place in America — sir, ma’am, please, thank you. That is not a coincidence. Politeness and violence are two faces of one system: when anyone might go off, everyone is courteous. This is exactly what Heinlein meant in 1942: “an armed society is a polite society.”

I followed that road for a while myself. Are Americans so careful about civility because the man across from you might actually pull a gun? Do decent people in China get pushed around so hard because they have nothing in their hands? Is American liberty rooted in the right to bear arms, which hands the little guy the ability to take you down with him and keeps his betters from pushing too far?

The hypothesis has serious ancestors. The civic-republican tradition from Machiavelli to Harrington always treated bearing arms as the mark of a free man — slaves and serfs do not carry weapons, and a republic relies on a citizen militia rather than mercenaries. The anthropologist Boehm argued that human egalitarianism, as against chimpanzee hierarchy, exists precisely because projectile weapons let any weak individual kill a strong one from a distance, which is what makes a coalition of the weak suppressing the would-be alpha possible at all. China ran the opposite program. The First Emperor collected every weapon in the realm and cast them into twelve bronze statues; the Yuan banned bows and blades for their Chinese subjects; Toyotomi Hideyoshi ordered Japan’s Sword Hunt in 1588. The East Asian tradition is to strip civilian arms down to nothing.

But I eventually found three places where the hypothesis breaks.

First, person against person is not person against institution. A gun is an equalizer one on one; against an institution it is nothing at all. American labor history is the best test available. Homestead in 1892, Ludlow in 1914, ten thousand armed miners at Blair Mountain in 1921 — the armed decent people lost every single time to company gunmen and government troops. What American workers eventually won, they won by organizing, striking, and voting, not by shooting. In the other direction, the right to bear arms didn’t stop Jim Crow either; an armed majority using guns to hold down a minority is the normal case. Guns do not flow to decent people. They flow to whoever is already strong and whoever is most willing to use them. In Yemen, Somalia, and the favelas of Rio everybody has a gun, and what grows there isn’t civility. It’s warlords.

Second, Japan has been thoroughly disarmed for more than four hundred years since the Sword Hunt, is polite to the point of pathology, and has a homicide rate around one twentieth of America’s. Scandinavia, Germany and Canada are also disarmed, courteous, egalitarian societies. Guns are not a necessary condition for civility.

Third, and this is the one that matters: the madman’s function is not violence, it’s deterrence — and what makes deterrence work is not firepower, it’s credibility and precision.

Which brings us to the other form the madman takes. The man who spends a year in court with his building’s property manager over a five-yuan parking fee is a madman in the economic sense: he burned a thousand yuan worth of time chasing five. But everyone in the complex pays fewer junk fees because of him. The man who turns a department’s ten years of cooked books into a thirty-thousand-word complaint. The man who says “there’s a problem with this plan” in the meeting knowing exactly what it will cost him later. The man who files a complaint with the aviation regulator every time a flight is delayed. Every one of them is a madman, and every one of them is a public good. Me, writing articles trashing cloud vendors, probably counts too.

The madmen a society needs are almost never the ones who pull a gun. They’re the ones who refuse to let it go even when letting it go is the better deal. Their function is exactly the function of E on the ship: making sure A can never be certain he gets to keep 97.

And the madman has a second, less obvious function. He is the smoke alarm.

The sociologist Granovetter has a threshold model: everyone carries a private number for how many other people have to move before he moves, and the distribution of those numbers decides whether something avalanches. The low-threshold people go first. Their existence is what lets the people upstairs read where the red line is before the avalanche arrives. A society that clears out everyone at the bottom of that distribution looks quieter, and has also ripped out its own alarm.

III. The Decent Man’s Trap: The Ladder Got Pulled Away

So who are the decent people?

Correct one misconception first: the decent man is not an unconditional cooperator. Unconditional cooperators went extinct a long time ago, because they get exploited to death. Behavioral economics experiments keep finding that roughly half of any population are conditional cooperators — I cooperate if you cooperate, and I turn on you if you screw me. The pure pushover barely exists. What we call a decent man is just a conditional cooperator with a very high threshold and a very long fuse. It isn’t that he won’t turn. It’s that it takes a long time.

So why is he always the one stuck in the trap?

The obvious answer is “because his threshold is high.” That’s wrong. A high threshold is a virtue in itself — people who can absorb a lot and don’t blow the relationship up at the first provocation are the ballast of any organization. The problem is not the man. It’s the ladder.

A healthy society hands people a whole ladder of escalation: complain, file a grievance, go public, sue, strike, vote, move away. Every rung is a small-scale table-flip — cheap, reversible, and aimed at the right target. One of Axelrod’s conditions for why tit-for-tat wins is that it’s provocable: when you get burned you hit back immediately and proportionally, instead of banking it. As long as every rung at the bottom of the ladder is still there, nothing accumulates to the point of detonation.

Anyone who builds systems knows this logic cold. A well-designed system has a whole set of graceful degradation paths: rate limiting, circuit breakers, load shedding, bulkheads, retries, each layer absorbing part of the pressure. A system with no degradation path at all has exactly one failure mode — total collapse. The decent man’s trap is a system with no degradation path.

Tear out the bottom of the ladder and only the top rung is left. The fuse gets longer, the explosion goes to full yield, and it lands off target.

Lin Chong, in the classic Chinese outlaw novel Water Margin, is exactly this man. Drill master of the emperor’s eight hundred thousand guards, a model son of a respectable family, a model decent man. A powerful official’s son harasses his wife: he swallows it. He’s framed for carrying a blade into a forbidden hall: he swallows it. He’s sentenced to exile: he swallows it. He’s nearly murdered on the road, and when the monk Lu Zhishen moves to kill the two guards escorting him, Lin Chong is the one who stops him: he swallows it. He swallows all of it right up to the night in the snowstorm at the mountain shrine, when he overhears three men outside discussing how to burn him alive — and finally understands that swallowing it all the way to the bottom also ends in death. Then he stops swallowing.

Chen Sheng, the conscript whose mutiny brought down the Qin, gave the most precise game-theoretic statement of this mechanism ever written: “Desertion is death. Rebellion is death. If we die either way, why not die for a kingdom?” Qin law held that missing your deadline was punishable by beheading, full stop, which flattened the cost of obeying the rules and the cost of revolt into the same number. Once the gradient disappears, deterrence goes to zero instantly. That is not a man going mad. That is madness done rationally. Machiavelli said the same thing from the ruler’s side of the table: treat men well or destroy them utterly, never leave a half-dead enemy standing, because a lightly wounded man takes revenge.

The “off target” part deserves its own paragraph. The anger module evolved for groups of a few dozen people: the man who screwed you is standing right there, and a fist settles it. Drop it into a modern society and the thing that screws you is usually a system — a company, a process, a policy — with no face on it. Revenge has nowhere to land, so it lands on a random passerby or on a symbol. The first is the worst kind of explosion: it changes nothing, and it pushes the public toward supporting tighter control. The second occasionally works, at an insane price.

That thing in New York at the end of 2024 — a health insurance CEO shot dead in the street — I’m not offering a moral verdict, only looking at the mechanism. It happened in the one area where America’s conventional ladder is worst: appealing a denied claim almost never works, and the regulator almost never works. Then the public reaction made the entire industry flinch. Insurers pulled executive photos off their sites overnight and hired security; one of them withdrew a brand-new policy capping anesthesia coverage by duration, officially unrelated, announced the day after the shooting. That is a madman filling in at the rung where the ladder has snapped. It is perfectly consistent with the logic of section one, and it also shows how expensive and how uncontrollable that kind of substitution is, and why a normal society would rather build a few more rungs.

Then there’s the signaling layer. Frank is explicit: anger has to be seen to deter. If a society also clamps down on the expression of anger — you can’t curse, you can’t publish, you can’t assemble — the commitment device’s signaling function is dead. The people upstairs can’t read where the red line is, and they cross it without knowing they did. The economist Kuran calls this preference falsification: everyone is sitting on a fire, nobody knows how big anybody else’s fire is, so the system looks immovable right up until the day the whole thing comes down. Eastern Europe in 1989 worked exactly like that. Afterward everyone said they never saw it coming.

IV. The Yeomanry Exits

There’s a phrase going around the Chinese internet lately: liangjiazi tuichang, the sons of good families leaving the field. The people using it are more right than they know.

The liangjiazi of the Han dynasty’s six frontier commanderies were free farmer-soldiers with land, horses and weapons — a yeomanry. Han’s great generals, Zhao Chongguo and Li Guang among them, came out of that class, and it carried the dynasty’s military. Then land consolidation bankrupted the free farmers into vagrants, private retainers and paid recruits took over, and the dynasty started downhill. The yeomanry leaving the field is not a metaphor in Chinese history. It has happened several times over — and what left, every time, was the same class: armed free men.

Exit is another form of flipping the table. Hirschman wrote a book called Exit, Voice, and Loyalty: when you’re unhappy with an organization, you either speak up or you walk. Voice is climbing the ladder. Exit is refusing to play. Lying flat, skipping marriage and children, emigrating — all exit. It is more dignified than flipping the table and it does no less damage to the system. A looks around the ship and finds that fewer and fewer pirates are willing to vote at all.

When both voice and exit are blocked, only the third option is left: Lin Chong’s night in the snow.

At this point I can restate the gun hypothesis from earlier.

Draw a two-by-two. One axis is the conventional options: courts, press, unions, ballots. The other is the nuclear option: guns. America has both. Japan and Germany have only the conventional ones and run perfectly well. Yemen and Somalia have only the nuclear one and get warlords. And a decent man with neither has nothing left but exit or detonation.

When decent people get pushed around hard, the missing variable is not the gun in the hand. It is never the neighbor doing the pushing — person-on-person violence rates are in fact vanishingly low — it’s institutions, and institutions are not afraid of guns. They are afraid of exposure, lawsuits, and votes. What protects decent people is not the nuclear option, it’s the conventional ones. The gun is the most expensive rung on this ladder, and a society that has to reach for it often is telling you the other rungs are already broken.

One layer deeper. A modern state stopped needing to confiscate swords a long time ago. A few political scientists at Harvard studied what a censorship apparatus actually deletes and found it isn’t posts criticizing the government — curse them as hard as you like and it stays up — it’s anything calling for collective action. Same principle as the pirate game. Resistance is a coordination game: my being angry accomplishes nothing, I have to know that you are angry too, and I have to know that you know that I know. The load-bearing assumption in the pirate game was common knowledge from beginning to end, and the madman’s deterrent works the same way. He is only frightening when other people know he’s a madman and he knows they know.

The Sword Hunt took away capability. The firewall takes away common knowledge. The second is cheaper, and more complete.

Coda

Writing this, I find I’ve come the whole way back around on the pirate game.

It is obviously not a model of reality. But it is good for one thing: it describes with precision what a world looks like when everybody knows that nobody will flip the table. In that world the man at the top takes 97, two men get nothing, and this is an equilibrium — nobody has any incentive to deviate.

How far a society sits from that world does not depend on how many rational people it contains. It depends on whether A can be sure. As long as A can’t be sure there is no madman on the ship, he has to give ground. As long as every low rung of the ladder is still in place, the madman never has to actually go mad — refusing to let it go is enough. As long as anger can be seen, the red line doesn’t get crossed in silence.

So don’t roll your eyes at the people around you who make a stink over things that aren’t worth it. They are the load-bearing beam, the smoke alarm, the public good. A normal society doesn’t need everyone to be a madman. It only needs to guarantee one thing:

That A can never be sure.