Two petrol stations face each other across a road. If both charge a fair price, both make a decent profit. If one cuts its price, it steals the customers and makes more, for a while. So each is tempted to cut. Both cut. Both now earn less than before, and neither can raise prices alone without losing everyone. Nobody was stupid. Each did the sensible thing given what the other might do.
That pattern, a trap built entirely out of sensible choices, is the first thing game theory teaches you to see. It then teaches how to get out of the trap: by repeating the game, by changing who moves first, by changing the rules. The same toolkit explains the design of radio-spectrum auctions and the matching of kidney donors.
A game has three ingredients: the players (who decides), the strategies (what each can do) and the payoffs (what each gets for every combination of choices, in whatever they care about: money, time, votes, survival). Two players with two options each fit in a small grid, the payoff matrix. One player picks the row, the other the column, and each cell holds two numbers: the row player's payoff first, the column player's second.
Two ideas solve most such grids. A dominant strategy is one that is best for you whatever the other does: if it exists, play it. When neither player has one, look for a Nash equilibrium: a pair of choices where neither player could do better by changing their own choice alone. John Nash proved in 1950 that every finite game has at least one, possibly in mixed strategies (rolling dice between options). Nash shared the 1994 economics Nobel prize with John Harsanyi and Reinhard Selten for it.
Each cell: your payoff (row player), then the column player's. Edit the numbers; the analysis updates. Shaded = a best response for that player; outlined cell = Nash equilibrium.
Without JavaScript: in the prisoner's dilemma (5, 3, 1, 0), defecting is dominant for both players, the only Nash equilibrium is both defecting for 1 each, and both cooperating would give 3 each.
Try the presets and read the verdicts. Chicken and the driving game have two pure equilibria and a mixed one at 50:50; matching pennies has no pure equilibrium at all, only the mixed one, which is why a goalkeeper and a penalty-taker must be unpredictable. In matching pennies, if the row player leans even slightly toward heads, the column player exploits it.
Merrill Flood and Melvin Dresher devised it at the RAND Corporation in 1950, and Albert Tucker gave it its story: two suspects, questioned separately. Stay silent (cooperate) and both get a light sentence. Betray the other (defect) and you walk free while he gets the worst. Both betray and both get a heavy sentence. With points instead of years, the standard payoffs are: temptation 5, mutual cooperation 3, mutual defection 1, being betrayed 0.
Whatever the other does, defecting pays you more: 5 beats 3 if she cooperates, 1 beats 0 if she defects. So both defect and both get 1, when both cooperating would have given 3 each. The equilibrium is stable and bad. That is the logic of arms races, overfishing, price wars and the two petrol stations. Now play it. Ten rounds against a computer opponent whose rule you can pick; you know the game is exactly ten rounds long.
Payoffs per round: both cooperate 3 each; you defect on a cooperator 5 to 0; both defect 1 each. Tit for tat copies your last move; the grudger cooperates until you defect once, then never again.
Against tit for tat, the best ten-round total is 32: cooperate nine times, defect on the last round when it can no longer be punished. Against always-defect, nothing beats 10. Against always-cooperate you can take 50. The right play depends on who is on the other side, which is the whole subject in one sentence.
Meet the same person again and again and the logic changes: defecting today invites revenge tomorrow. In 1980 the political scientist Robert Axelrod invited game theorists to send in programs for a repeated prisoner's dilemma, each program playing every other for 200 moves. The winner, in this tournament and in a second, larger one, was the simplest entry: tit for tat, sent by the psychologist and game theorist Anatol Rapoport. It cooperates first, then copies the opponent's last move. Axelrod's 1984 book The Evolution of Cooperation drew the lesson: successful strategies were nice (never defect first), provocable (punish defection at once), forgiving (return to cooperation) and clear (easy to read).
Tit for tat never beats the partner it faces; the best it can do in a single match is tie. It won by getting good scores out of nearly everyone. Run your own small tournament. Six kinds of players meet in a 200-round round robin; add copies of Always-defect and see whether the winner changes.
Entrants: Always cooperate, Tit for tat, Grudger, Pavlov (cooperate only if both did the same last round), Coin flip, plus the Always-defect copies. Everyone meets everyone else for 200 rounds; bars show the average points per round. Payoffs 5, 3, 1, 0. A toy field, far smaller than Axelrod's.
With no Always-defect, the nice, retaliating strategies fill the top. Add more defectors and the field gets harsher: the defectors climb to the top once enough naive cooperators and coin-flippers are there to be exploited, while their scores against each other stay at 1. Success in a repeated game depends on who else is playing. Turn on the 5% slips and tit for tat loses ground too: after one accidental defection two tit-for-tat players can lock into a spiral of mutual retaliation. Later work found more forgiving variants, such as generous tit for tat, and the Pavlov rule do better in noisy worlds. Cooperation without a referee needs a future worth protecting, which is why long-term relationships, reputations and repeated dealings support trust that one-off meetings cannot.
Not every game is a conflict. In the driving game, both of you want to pick the same side; which side hardly matters. Load "Which side to drive" above: two equilibria, both good, and nothing in the payoffs says which. Conventions (driving on the left in the UK, Japan and India, on the right in most of the rest of the world) are equilibria picked by history. The stag hunt adds risk: both hunting stag gives 4 each, but hunting hare alone is safe at 3. Two equilibria again, one better but riskier. Thomas Schelling, in The Strategy of Conflict (1960), noticed that when payoffs cannot choose, players use focal points: the obvious, salient option, like meeting "at the clock" in a station. Many real-world failures, from bank runs to standards wars, are coordination games stuck on the bad equilibrium.
When players move in turn, you solve the game backwards: start at the end, see what the last mover would do, and work back. Take a company deciding whether to enter a market held by an incumbent. The incumbent says: "If you enter, I'll start a price war." Should the entrant believe it? Only if fighting is actually best for the incumbent after entry happens. If, once the entrant is in, the war would cost the incumbent more than sharing the market, the threat is empty. A rational entrant sees that and enters. Reinhard Selten's refinement of Nash equilibrium, subgame-perfect equilibrium, rules such empty threats out.
Payoffs (entrant, incumbent): stay out (0, 10); enter and incumbent accommodates (4, 5); enter and incumbent fights (−2, your slider). The commitment option makes accommodating cost the incumbent 3 (so it pays only 2).
The slider has a threshold. With a war worth less than 5 to the incumbent, the threat is empty and the entrant walks in. Raise the war's payoff above 5, or use the commitment button, and the threat becomes credible, the entrant stays out, and the incumbent never has to fight. Commitment devices (spare factory capacity already built, a contract that punishes backing down, a reputation) have this strange property: by taking away your own options they can make you better off.
An auction is a game where the rules decide the payoffs. In a sealed-bid second-price auction, the highest bidder wins but pays only the second-highest bid. William Vickrey analysed it in 1961 (it had been used by stamp collectors since 1893). Here, bidding exactly your true value is a dominant strategy: bidding more risks paying above your value, bidding less risks losing an item you would have profited from. In a first-price auction you pay what you bid, so you shave your bid below your value, trading a bigger profit if you win against a smaller chance of winning. With four bidders whose values are equally likely to be anywhere in a range, the equilibrium bid is three-quarters of your value (in general, (n−1)/n of it).
Values for all four bidders are drawn uniformly between 0 and 100. The three rivals bid their full value in the second-price format and three-quarters of it in the first-price format. Your average profit is per auction, including those you lose.
Slide the share. In the second-price format your profit peaks at 100% of your value; in the first-price format it peaks near 74%. Also watch the seller's average revenue: it is about 60 in both formats, a first glimpse of the revenue equivalence theorem. Real auctions differ from this textbook case. When bidders are unsure of the item's true value (oil leases, spectrum) the winner tends to be the one who overestimated most, the winner's curse.
Game theory asks: given the rules, what will players do? Mechanism design turns it around: what rules would make self-interested players do what we want? Two famous cases.
Spectrum auctions. Governments own radio frequencies and need to hand them to phone companies. For decades they used beauty contests and lotteries. From July 1994 the US Federal Communications Commission instead ran a "simultaneous multiple round" auction, where many related licences are sold together over many rounds and bidders see each other's bids. Theorists including Paul Milgrom and Robert Wilson helped shape the design, and they shared the 2020 economics Nobel prize for improvements to auction theory and new auction formats. In 2000 the UK's 3G auction raised £22.5 billion from five licences.
Kidney exchange. Many people who need a kidney have a willing donor who is the wrong blood type. Selling kidneys is illegal almost everywhere, so prices cannot match people. Alvin Roth, Tayfun Sönmez and Utku Ünver designed algorithms (their 2004 paper in the Quarterly Journal of Economics) in which pair A's donor gives to pair B's patient while B's donor gives to A's: a swap, extended to longer cycles and to chains started by an altruistic donor. The first paired exchange recorded was in South Korea in 1991 and the first in the US in 2000; by 2012 the US National Kidney Registry had organised a 60-surgery chain, and by 2014 one of 70. Roth shared the 2012 economics Nobel prize with Lloyd Shapley for the theory of stable allocations and the practice of market design.
Game theory predicts what people do under the rules. Mechanism design chooses the rules.
When your best move depends on others' moves, look for dominant strategies, then for equilibria, and remember that a stable outcome can be a bad one. Repetition, commitment and well-designed rules are the three ways out, and each is a way of changing the game rather than the players.
Try this: this week, spot a situation where you and someone else are each waiting for the other to move first or to give in, and ask what the payoff grid looks like and whether it is a game you can change.
Sources. Dominance, Nash equilibrium, mixed strategies, backward induction and the second-price result are standard textbook game theory (Gibbons; Osborne; Polak's Yale lectures). The prisoner's dilemma was devised by Flood and Dresher at RAND in 1950 and named by Tucker (Wikipedia, "Prisoner's dilemma"). Nash, "Equilibrium points in n-person games," PNAS 36, 1950; the 1994 economics Nobel prize went to Nash, Harsanyi and Selten. Axelrod's tournaments (c. 1980) and 1984 book: tit for tat, by Anatol Rapoport, won both; matches were 200 moves; the death-spiral under noise and generous tit for tat are as described in Wikipedia's "Tit for tat". Schelling, The Strategy of Conflict (1960). The second-price auction was described by Vickrey in 1961 and used by stamp collectors from 1893. The FCC has held spectrum auctions since July 1994 using a simultaneous multiple round design; the UK's 2000 auction raised £22.5 billion from five 3G licences (Wikipedia, "Spectrum auction"). The 2020 prize went to Milgrom and Wilson, the 2012 prize to Roth and Shapley. Kidney exchange history from Wikipedia's "Kidney paired donation"; Roth, Sönmez & Ünver, "Kidney Exchange," QJE 2004. Every number an interactive shows was computed in node from the code on this page: ten-round best score against tit for tat 32, against always-defect 10, against always-cooperate 50; the four-bidder first-price optimum is near 0.74 (theory 0.75); the average winning price is near 60 in both formats; the toy tournament's ranking depends on the chosen field and is not Axelrod's result.