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The expectation of the seller's total receipts over the buyers' valuation types and any randomization in the sales mechanism. In an auction it is the sum of the bidders' expected payments; with a posted price it is price times sale probability. It does not subtract selling costs or the seller's opportunity cost of retaining the item, so it is revenue rather than profit.

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.

Mechanism design chooses allocation and payment rules while accounting for how agents' information and incentives affect their reports. Social choice theory studies collective choice from preferences, while auction design also uses monetary transfers.

A single-item auction in which the highest bidder wins and pays the highest competing bid. Under private values and quasilinear utility, bidding one's valuation is a dominant strategy: if the competing maximum is below one's value, winning at that price is best; if it is above one's value, losing is best; equality gives zero utility either way. Thus truthful bidding is an ex post Nash equilibrium.