What Is Game Theory? Working Out What to Do When Others Are Deciding Too

The formal apparatus is small and each term does real work:

• •Players, who choose; strategies, the complete plans available to each; and payoffs, the value each player places on each combination of choices
• •A dominant strategy, which is best regardless of what others do, and which makes analysis trivial when one exists and usually does not
• •A Nash equilibrium, a combination of strategies in which no player can do better by changing theirs alone, which John Nash proved exists for every finite game when mixed strategies are allowed, a result that won a Nobel Prize in 1994
• •Mixed strategies, in which a player randomises between options, which is the right answer whenever being predictable is exploitable, as in penalty kicks and in poker
• •Zero-sum games, where one player's gain is exactly another's loss, against the far more common non-zero-sum situations where cooperation can make both better off
• •Sequential games, analysed by backward induction from the final move, against simultaneous ones
• •Complete and incomplete information, the latter covering situations where players do not know each other's payoffs, which is most of real life

A small number of simple structures explain a remarkable range of situations. The prisoner's dilemma has two players each better off defecting whatever the other does, and both worse off than if both had cooperated, which models arms races, overfishing, price wars and climate negotiation. The stag hunt has cooperation as the better outcome and defection as the safer one, which models coordination problems where trust is the issue. Chicken has two players who must decide whether to yield, with the worst outcome if neither does, which models brinkmanship and standards wars. The tragedy of the commons generalises the prisoner's dilemma to many players sharing a resource. The ultimatum game, in which one player proposes a split and the other can accept or reject for both, is the standard demonstration that people reject unfair offers at cost to themselves, which pure self-interest does not predict and which holds across cultures with variation in the threshold.

The prisoner's dilemma appears to prove cooperation impossible among self-interested players, and the way out is repetition. If the same players meet again, defection carries a future cost, and strategies that condition on past behaviour become viable. Robert Axelrod ran computer tournaments in 1980 inviting submitted strategies to play each other repeatedly, and the winner was tit for tat, submitted by Anatol Rapoport and consisting of four lines: cooperate first, then copy whatever the other player did last. Axelrod's analysis of why it did well identified four properties, being nice in never defecting first, retaliatory so that exploitation is punished, forgiving so that a single defection does not lock in mutual punishment, and clear enough that the other player can work out what it is doing. Later work showed it is vulnerable to noise, where a mistaken defection triggers endless retaliation, which more forgiving variants handle better. The general finding, that cooperation emerges when interactions are repeated and the future matters enough, underlies work on reciprocity in biology, trade and law.

The applications are wider than the economics department. Auction design is the clearest success, with theorists designing the spectrum auctions that governments use to allocate radio frequencies, raising billions more than earlier methods by getting the incentives right, work recognised by the Nobel Prize in 2020. Matching markets assign medical graduates to hospitals, children to schools and kidneys to recipients using algorithms that are stable in the game-theoretic sense, meaning no pair would prefer to abandon their assignment for each other, which won a Nobel Prize in 2012 and has been implemented in real systems that save lives. Biology uses evolutionary game theory, in which strategies are inherited behaviours and payoffs are reproductive success, to explain animal conflict, cooperation and signalling, and the evolutionarily stable strategy concept developed by John Maynard Smith is the biological analogue of an equilibrium. Nuclear deterrence was analysed in these terms throughout the Cold War, with mutual assured destruction an attempt to engineer a particular equilibrium deliberately.

The limitations are real and acknowledged within the field. The standard models assume players know their own payoffs, which people frequently do not, and that they can compute the equilibrium, which requires reasoning that experiments show people do not perform beyond a couple of steps. Many games have multiple equilibria with no principled way to choose among them, which is the equilibrium selection problem and which Thomas Schelling addressed with the idea of a focal point, an option that stands out for reasons outside the mathematics. Behavioural experiments consistently find deviations from predicted play, with people cooperating more than theory expects in one-shot games and punishing unfairness at their own expense. The honest assessment is that the framework is a tool for structuring a situation and identifying its incentives rather than a predictor of behaviour, and that its greatest practical successes have come from designing institutions so that the incentives point the right way rather than from forecasting what people will do in institutions that already exist.

Game theory analyses decisions where the outcome depends on what others choose while they are reasoning about you, and its central concept is an equilibrium from which no player can improve by changing strategy alone, which Nash proved always exists when randomised strategies are allowed. A few simple structures including the prisoner's dilemma and the stag hunt model arms races, shared resources and coordination. Repetition makes cooperation viable, as Axelrod's tournaments showed, and the field's strongest practical results are in designing auctions and matching systems rather than in predicting behaviour.