= Game theory
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Game theory studies decisions whose consequences depend on what other decision makers choose. A bidder's gain depends on competing bids in an <auction>; a driver's travel time depends on other drivers in a <congestion game>. This strategic interdependence makes choosing well a different problem from <optimization> against a fixed environment: each participant's <best response> depends on the others' choices.
A <normal-form game> specifies the players, their available <pure strategies>, and their <payoffs> for each <strategy profile>. Larger <payoffs> mean more preferred outcomes for that player. Unless stated otherwise, a two-player table below lists $(u_1,u_2)$: the row player's <payoff> first, the column player's second. Each player maximizes their own <expected utility>; the numbers need not represent money, and comparing different players' numerical scales is unnecessary.
The central question is whether everyone's choices fit together. A <Nash equilibrium> consists of mutual <best responses>: no <unilateral deviation> improves a player's <expected payoff>. It can involve definite choices, or <mixed strategies> that deliberately randomize. <Matching pennies> shows why randomization can be essential; the <coordination game> shows how players can share an interest in coordinating while disagreeing over which outcome to choose. Such shared gains are absent from a two-player <zero-sum game>, where one player's gain is exactly the other's loss.
These ideas are useful both for analyzing incentives and for designing them. <Mechanism design> asks how changing the rules changes participants' <best responses>; <evolutionary game theory> studies how successful strategies spread. The mathematical appeal is that local comparisons of <payoffs> can reveal a globally consistent pattern of choices. Consistency alone does not imply <Pareto efficiency>, uniqueness, or that actual play will converge to a <Nash equilibrium>.
John Nash's https://pmc.ncbi.nlm.nih.gov/articles/PMC1063129/[1950 paper, Equilibrium Points in N-Person Games]{ref} established existence of a <mixed strategy> <Nash equilibrium> for every <finite game>. His https://cs.uwaterloo.ca/~klarson/teaching/F04-886/papers/nash51.pdf[1951 paper, Non-Cooperative Games, Annals of Mathematics 54, 286–295]{ref} developed the theory beyond two-player <zero-sum games>. The distinction between conflicting and shared interests is fundamental to the subject.
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