Game Theory

You know the rules, and so do I.…

We know the game and we’re gonna play it.

—RICK ASTLEY

Game theory covers an incredibly broad spectrum of scenarios of cooperation and competition, but the field began with those resembling heads‑up poker: two‑person contests where one player’s gain is another player’s loss. In adversarial situations, nearly every choice directly or indirectly affects other people, and these effects determine how the conflict turns out. As John Nash proved in 1951, every two‑player game has at least one equilibrium—a set of strategies where neither player can unilaterally improve their own outcome. In rock‑paper‑scissors, for example, the equilibrium tells us, perhaps unexcitingly, to choose completely at random, each gesture a third of the time. Once both players adopt that mix, neither has an incentive to switch. Nash’s discovery earned him a Nobel Prize, and the Nash equilibrium now underpins vast swaths of economic and social theory.

But as computer scientists quickly point out, knowing an equilibrium exists is not the same as being able to find it or to get there. Game theory “predicts the agents’ equilibrium behavior typically with no regard to the ways in which such a state will be reached,” observes Christos Papadimitriou. And indeed, by the early 2000s it was proved that simply finding a Nash equilibrium is computationally intractable—PPAD‑complete—making it, in a sense, as hard as the halting problem that Turing wrestled with. As Papadimitriou puts it, “If an equilibrium concept is not efficiently computable, much of its credibility as a prediction of the behavior of rational agents is lost.” MIT’s Scott Aaronson agrees: if we rely on the existence of equilibria to justify market policies, then the intractability of finding those equilibria should temper that reliance. Or, as eBay’s former research director Kamal Jain said, “If your laptop cannot find it, neither can the market.”

The core challenge is recursion. Keynes described successful investing as “anticipating the anticipations of others,” and in poker the same dynamic is called “leveling”—thinking about what your opponent thinks you think. As poker pro Vanessa Rousso advises, you want to “play one level above your opponent.” But endlessly cycling into the opponent’s head is exhausting and often futile. The Nash equilibrium offers a way out: step out of the recursive rabbit hole and go directly to the best strategy, assuming rational play. Yet the universe of games is vast, and many are not so straightforward. Algorithmic game theory therefore asks not just which strategies are theoretically ideal, but how machines—and people—actually compute them.

Dominant Strategies, for Better or Worse

Even when we can reach an equilibrium, its stability does not guarantee that it is good. Nowhere is that better illustrated than in game theory’s most famous game: the prisoner’s dilemma. Two accomplices are held separately, each deciding whether to stay silent (“cooperate”) or inform (“defect”). If both stay silent, they walk away with half a million dollars each. If one defects and the other stays silent, the defector gets the full million while the silent one serves ten years. If both defect, they split the sentence: five years apiece.

No matter what your partner does, defecting is better for you personally. If they stay silent, you get the whole prize; if they defect, you at least avoid the harsh full sentence. Defection is therefore not merely an equilibrium but a dominant strategy—the best response to every possible move by the opponent. Yet when both follow this logic, both spend five years in prison, a dramatically worse outcome for everyone. The equilibrium of rational individual play is not the outcome that is best for the group.

Algorithmic game theory quantifies this gap with a measure called the “price of anarchy.” It captures how much worse a decentralized, selfish outcome is compared to perfect coordination. In the prisoner’s dilemma, the price is effectively infinite: the gap can be made arbitrarily large by increasing the stakes. But other games are far less dire. When Tim Roughgarden and Éva Tardos studied selfish routing in traffic—whether cars on roads or packets on the Internet—they proved the price of anarchy is at most 4/3. A free‑for‑all is only about 33% worse than a perfectly managed system. That may explain why the Internet works as well as it does without a central routing authority, though it also tempers hopes that self‑driving cars will bring a congestion utopia. The low price of anarchy cuts both ways: uncoordinated commutes are already close to optimal.

The Tragedy of the Commons

In 1968, ecologist Garrett Hardin scaled the two‑player prisoner’s dilemma into a multi‑player scenario. Picture a village commons where every herder can graze livestock. Adding one more animal benefits the individual herder with only a tiny, diffuse harm to everyone else. Yet when all herders follow this logic, the commons is devastated and no grass remains for anyone—a tragedy. This lens is now central to understanding pollution, climate change, and resource depletion. As Carnegie Mellon game theorist Avrim Blum notes, when leaded gasoline was a few cents cheaper, each driver’s personal health cost seemed minimal, yet the collective effect was enormous. The same dynamics push corporations and nations to be a little more reckless than their peers for a competitive edge, even though the shared equilibrium is a ravaged Earth, all for no net economic gain.

Tragedies of the commons arise even with a clean conscience. Consider company vacation policy. When an employer offers unlimited vacation, workers want to take as much as possible but also slightly less than their colleagues, to appear more loyal and promotable. Everyone looks to everyone else for a baseline and then undercuts it. The Nash equilibrium of this game is zero vacation—a race to the bottom. As Mathias Meyer, CEO of Travis CI, writes, “People will hesitate to take a vacation as they don’t want to seem like that person who’s taking the most vacation days.” The same logic plays out at the national level: the United States works some of the longest hours in the developed world, and the average worker takes only half of granted vacation days.

Between firms, the race is just as ruinous. Two shopkeepers in a small town each would rather close on Sunday, but if one stays open and the other doesn’t, the open shop gains all the customers. The equilibrium is for both to work seven days a week, exhausted and no better off competitively. During the 2014 holiday season, retailer after retailer pushed opening times earlier to avoid losing share—Macy’s, Target, Kmart each opened sooner, culminating in Kmart opening at 6:00 a.m. on Thanksgiving and staying open for forty‑two hours straight. A similar dynamic drives arms races, whether in weaponry, college amenities, or conspicuous consumption: each participant tries to gain a slight edge, but collectively everyone spends more for the same relative standing. As The Onion satirized, razor blade manufacturers can end up in a pointless spiral.

In all these situations, the stability of the bad equilibrium is the curse. Individually rational players cannot shift it from within. But that doesn’t mean it can’t be fixed—it means the solution must come from outside.

Mechanism Design: Change the Game

If the rules of the game force a bad strategy, perhaps we shouldn’t try to change strategies; perhaps we should change the game. This is the insight of “mechanism design,” sometimes called reverse game theory. While game theory asks what behavior emerges under given rules, mechanism design asks what rules will produce the behavior we want.

Return you and your accomplice to the jail cell, but now you belong to a crime syndicate. The Godfather has made it clear that informants sleep with the fishes. The payoffs are worsened—death for defection—yet by altering the incentives, the equilibrium shifts; both prisoners cooperate and walk away half a million dollars richer, minus a tithe. Worsening every outcome made everyone better off.

For the shopkeepers, a verbal truce is unstable; a legally binding contract that, say, requires a shop that opens on Sunday to pay its proceeds to the other shop changes the equilibrium in the same way. Mechanism design explains why Phil Libin’s offer of $1,000 for taking a vacation at Evernote was likely misguided. Increasing the prize money in the prisoner’s dilemma doesn’t break the bad equilibrium—it just raises the stakes. The problem isn’t that vacations aren’t attractive; it’s that the game’s logic forces a zero‑vacation bottom. What Libin needed wasn’t a carrot but a stick: making a certain amount of vacation compulsory shifts the bottom upward. Several companies, like Panera Bread, have done exactly that.

A league commissioner is a designer of the same kind. Without structured seasons and off‑hours, professional basketball would degenerate into a sleepless, haggard arms race of ever‑shifting practices, much like war. Even Wall Street, where microseconds count, closes every day at 4:00 p.m. sharp so that brokers can sleep without being ambushed by a runaway competitive equilibrium. The stock market is more a sport than a war.

Scaling up, mechanism design provides a powerful argument for government regulation. Mandated minimum vacations, limits on shop hours, and environmental laws are all ways to change the game so that the equilibrium moves toward the common good. Religion can play a similar role: “Remember the Sabbath day” solves the shopkeepers’ problem whether enforced by an all‑powerful deity or by the social pressure of a community. Omniscience and omnipotence offer an even stronger guarantee that bad actions have dire consequences, making God the ultimate mechanism designer.

Persuasion principles can also act as informal mechanism design. Robert Cialdini’s six levers—reciprocity, commitment, liking, social proof, scarcity, and authority—subtly reshape the psychological payoffs. Casinos exploit all of them: free drinks trigger reciprocity, chips replace cash to lower the pain of committing, personalized service builds liking, big‑win displays offer social proof, limited‑time offers create scarcity, and dealer advice invokes authority. The house always wins.

Framing, too, can change the game without altering its formal structure. An Israeli daycare introduced a fine for late pickups, intending to deter tardiness. Instead, lateness increased: the fine reframed the situation from a social obligation to a market transaction, and parents felt they had purchased the right to be late. Once social norms are eroded, they are difficult to rebuild. The ultimatum game offers another example of how fairness perceptions alter behavior: one person proposes a split of a sum of money and the other can accept or reject. Purely rational responders should accept any offer, but offers below about 30% are consistently rejected across cultures—people will sacrifice their own gain to punish what feels unfair. Distinguishing between distributive justice (equal shares) and procedural justice (fair processes) helps explain why people can agree on the need for fairness yet disagree violently on what it requires.

Mechanism Design by Evolution

The redwoods of California illustrate nature’s version of the tragedy. They are tall only because they compete for sunlight; as Richard Dawkins observes, the forest canopy is an aerial meadow that wastes enormous energy on wood “stilts” instead of photosynthesis. If the trees could agree to stay short, the whole ecosystem would thrive. But such coordination seems impossible without an outside authority.

Yet cooperation does appear in nature, and often it rides on the back of emotions that individuals cannot entirely control. Consider two seemingly unrelated acts: a man writes a vindictive online review of a shoddy vacuum, gaining nothing but satisfaction; a woman tackles a purse‑snatcher, risking injury to return $40 to a stranger. Both actions are individually irrational but socially valuable. Anger and outrage override rational self‑interest and make it known that you will retaliate, even at a cost. As economist Robert Frank explains, “If people expect us to respond irrationally to the theft of our property, we will seldom need to, because it will not be in their interests to steal it.” Emotion is mechanism design written into the species—an involuntary contract that needs no outside enforcement.

Love does the same for marriage. The commitment problem is this: why not always be ready to leave for a better option? Knowing the other party is equally ready to leave would prevent the long‑term investments that make the relationship worthwhile. Love, like organized crime, changes the payoffs. The emotional bond makes leaving so painful that the equilibrium shifts to staying together. As Frank puts it, “The worry that people will leave relationships because it may later become rational for them to do so is largely erased if it is not rational assessment that binds them in the first place.” Being able to fall involuntarily in love makes you a more trustworthy partner; your capacity for heartbreak is the very thing that makes you a reliable accomplice. Happiness, as Robert Frank suggests, is the lock.

Information Cascades: The Tragic Rationality of Bubbles

Learning from others is often rational: a busy restaurant is probably good, an empty concert hall a warning. But when private information is weak and public action dominates, herds can stampede off a cliff with no one to blame.

Auctions make the mechanics painfully clear. In a sealed‑bid first‑price auction, bidders write down a single number; the winner pays their own bid. To avoid overpaying, each bidder must guess what others will bid, leading to a recursive tangle of bid‑shading. English (ascending) and Dutch (descending) auctions add public bidding, which can merge private uncertainty with crowd behavior. Imagine oil companies bidding for drilling rights based on unreliable geological surveys. The first company, with a promising survey, bids high. The second, with mixed data, takes the high bid as confirmation and bids even higher. The third disregards its poor survey because two others seem confident, and so on. An information cascade forms: each subsequent bidder rationally infers from the actions of predecessors, but the aggregate “information” detaches from reality. As economist David Hirshleifer explains, once someone ignores their own signal and follows the herd, their action adds no new information to the public pool, and the pool stops growing. The cascade can send valuations skyrocketing on nothing.

A dramatic real‑world cascade occurred in April 2011, when two Amazon sellers using algorithmic pricing saw their bots ratchet a developmental biology textbook to over $23 million. Neither algorithm set an upper limit. Likewise, a deadly skiing accident in Washington State stemmed from a cascade: every member of the party felt uneasy about the snow conditions but stayed silent, assuming others knew better. In the 2010 flash crash, stocks momentarily traded for $100,000 or for a penny, as high‑speed trading algorithms amplified one another’s moves without being anchored to fundamental value. The fault, again, lies less with the players than with the game.

Information cascades offer a rational theory of bubbles, fads, and herd behavior. They remind us to be wary when public information swamps private judgment—when we know more about what people are doing than why they’re doing it. We should hesitate to override our own doubts, and if we go along, we might find a way to voice our reluctance, so others don’t mistake our action for genuine confidence. Stick‑to‑your‑convictions types will be wrong more often than herd followers, but they provide a positive externality: their behavior remains genuinely informative. There may come a time when one stubbornly independent person saves the entire herd from disaster.

To Thine Own Self Compute

Algorithmic game theory reveals that the burden of strategizing is itself a cost. Mechanism design can address not only the outcomes but also the mental recursion required. The Vickrey auction demonstrates this elegantly. In a Vickrey (second‑price sealed‑bid) auction, the highest bidder wins but pays only the second‑highest bid. This simple twist makes honesty a dominant strategy: bidding your true value is always optimal. Bidding more risks overpaying; bidding less risks losing without saving any money, because your bid only determines whether you win, not what you pay. The auction shades your bid for you. Paul Milgrom calls it “one of the best things you can see.”

The Vickrey auction is strategy‑proof—you need not think about what others are doing. As Noam Nisan says, “You would like to get some kind of rules of society where it’s not worthwhile to lie, and then people won’t lie so much, right? … Vickrey shows, here is the way to do that.” The implications go far beyond auctions. Roger Myerson’s revelation principle proves that any game that requires strategic masking of the truth can be transformed into a game that requires only simple honesty. Imagine you have an agent playing the game on your behalf; you straightforwardly tell them your true desires. The principle says we can embed the agent’s strategic optimization directly into the rules, making truthfulness the dominant strategy. This is the mechanism designer’s holy grail. The FCC’s spectrum auctions, designed with such principles, have raised over $44.8 billion while encouraging truthful bidding.

When faced with intractable recursions and bad equilibria, you have multiple levers. If you cannot change the game through formal rules, you can still exercise control over which games you play. Many conflicts are best avoided entirely. Mutually assured destruction (MAD) is the extreme example: when both sides possess enough destructive power to annihilate each other, the only winning move is not to play, as the film WarGames dramatized. Deterrence works on a similar logic—a credible threat that shifts the adversary’s calculus. Criminal justice relies on deterrence; business giants use their scale to price out competitors. However, threats must be credible; if your red line is crossed and you don’t act, you lose all deterrent power.

Containment strategies aim to prevent a problem from spreading after it appears—stop the bleeding first, solve the root cause later. Quarantining malicious software, using honeypots to divert attackers, or containing a reputational crisis all follow this pattern. In a war of attrition, the side that outlasts the other wins, but sometimes it is rational to lose a battle to win the war—a sacrifice play or hollow victory. For the smaller player in a lopsided struggle, guerrilla tactics change the game by avoiding head‑on confrontation. The American Revolution started with guerrilla methods; Dollar Shave Club’s viral video was a modern business equivalent. The key is to avoid fighting the last war: while the dominant player keeps perfecting old strategies, a nimble competitor can punch above its weight with innovation.

Every game ends. Having a credible exit strategy is essential, whether it’s an IPO, a treaty clause, or a plan to step away gracefully from a long‑term commitment without burning bridges. Burning bridges or scorched‑earth tactics offer only short‑term satisfaction and usually escalate the damage. If no good exit appears, you might attempt a Hail Mary pass—a desperate final effort—or even burn the boats to remove the temptation of retreat, as Hernán Cortés did.

The guiding thread is to seek out games where honesty is the dominant strategy, to be skeptical of situations where public information drowns out private judgment, and to remember that changing the rules is almost always more effective than redoubling effort. Don’t hate the player—change the game. Or, as the philosophy distilled from all of this advises: try to compute less about others, and just be yourself.