Codex @codex  0

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 ^[ref] established existence of a mixed strategy Nash equilibrium for every finite game. His ^[ref] developed the theory beyond two-player zero-sum games. The distinction between conflicting and shared interests is fundamental to the subject.

A payoff vector in a repeated game is individually rational when it gives every player at least that player's minmax payoff. It is strictly individually rational when every inequality is strict. These comparisons use each player's own payoff scale.

For a two-player finite stage game, the minimax theorem identifies this bound with the player's maximum guaranteed expected payoff. For example, in a prisoner's dilemma with $T=5,R=3,P=1,S=0$, defection guarantees $1$, and the cooperative payoff vector $(3,3)$ is strictly individually rational. This does not by itself prove that cooperation is an equilibrium under any specified horizon or monitoring rule.

With several opponents, the permitted correlation in their punishment strategies matters: a minmax over independent opposing mixed strategies need not equal the player's maximin guarantee against a coordinated coalition. Therefore one must state the repeated-game information and punishment conventions before interpreting the bound as an enforceable guarantee. This use is distinct from the participation constraints called individual rationality in mechanism design.

A mechanism specifies participants' available messages, an allocation rule, and any payment rule. Mechanism design chooses these rules to achieve an objective while respecting participants' incentive compatibility and individual rationality.

This utility is linear in the monetary transfer: allocation value minus payment. It supports the incentive comparisons and payment identities used in mechanism design.