Review Vocabulary

Backward induction A reasoning process in which players, working backward from the last possible moves in a game, anticipate each other’s rational choices. (p. 650)

Chicken A two-person variable-sum game in which each player has two strategies: to swerve to avoid a collision, or not to swerve and cause a collision if the opponent has not swerved. Neither player has a dominant strategy; the compromise outcome, in which both players swerve, is not a Nash equilibrium, but the two outcomes in which one player swerves and the other does not are Nash equilibria. (pp. p. 641 p. 642)

Constant-sum game A game in which the sum of payoffs to the players at each outcome is a constant, which can be converted to a zero-sum game by an appropriate change in the payoffs to the players that does not alter the strategic nature of the game. (p. 635)

Dominant strategy A strategy that is sometimes better and never worse for a player than every other strategy, whatever strategies the other players choose. (p. 640)

Dominated strategy A strategy that is sometimes worse and never better for a player than some other strategy, whatever strategies the other players choose. (p. 644)

Expected value If each of the possible payoffs, , occurs with respective probabilities , then the expected value , also called the expected payoff, is

where and . (p. 632)

Fair game A zero-sum game is fair when the (expected) value of the game, obtained by using optimal strategies (pure or mixed), is zero. (p. 633)

Free rider A free rider is someone who benefits from resources, goods, services, or a policy without paying the cost for the benefit. (p. 653)

Game tree A symbolic tree, based on the rules of play in a game, in which the vertices, or nodes, of the tree represent choice points, and the branches represent alternative courses of action that the players can select. (p. 647)

Maximin In a two-person zero-sum game, the largest of the minimum payoffs in each row of a payoff matrix. (p. 624)

655

Maximin strategy In a two-person zero-sum game, the pure strategy of the row player corresponding to the maximin in a payoff matrix. (p. 624)

Mechanism design The act of designing a game to achieve a particular outcome as a Nash equilibrium. (p. 646)

Minimax In a two-person zero-sum game, the smallest of the maximum payoffs in each column of a payoff matrix. (p. 625)

Minimax strategy In a two-person zero-sum game, the pure strategy of the column player corresponding to the minimax in a payoff matrix. (p. 625)

Minimax theorem The fundamental theorem for two- person constant-sum games, stating that there always exist optimal pure or mixed strategies that enable the two players to guarantee the value of the game. (p. 637)

Mixed strategy A strategy that involves the random choice of pure strategies, according to particular probabilities. A mixed strategy of a player is optimal if it guarantees the value of the game. (p. 631)

Nash equilibrium Strategies associated with an outcome such that no player can benefit by choosing a different strategy, given that the other players do not depart from their strategies. (p. 640)

Ordinal game A game in which the players rank the outcomes from best to worst. (p. 639)

Partial-conflict game A variable-sum game in which both players can benefit by cooperation but may have strong incentives not to cooperate. (p. 637)

Payoff matrix A rectangular array of numbers. In a two- person game, the rows and columns correspond to the strategies of the two players, and the numerical entries give the payoffs to the players when these strategies are selected. (p. 623)

Prisoners’ Dilemma A two-person variable-sum game in which each player has two strategies, cooperate or defect. Defect dominates cooperate for both players, even though the mutual-defection outcome, which is the unique Nash equilibrium in the game, is worse for both players than the mutual-cooperation outcome. (pp. p. 638 p. 640)

Pure strategy A course of action a player can choose in a game that does not involve randomized choices. (p. 631)

Rational choice A choice that leads to a preferred outcome. (p. 623)

Saddlepoint In a two-person constant-sum game, the payoff that results when a row minimum and a column maximum are the same, which is the value of the game. The saddlepoint has the shape of a saddle-shaped surface and is also a Nash equilibrium. (p. 626)

Strategy One of the courses of action that a player can choose in a game; strategies are mixed or pure, depending on whether they are selected in a randomized fashion (mixed) or not (pure). (p. 623)

Total-conflict game A zero-sum or constant-sum game, in which what one player wins the other player loses. (p. 623)

Value The best outcome that both players can guarantee in a two-person zero-sum game. If there is a saddlepoint, that is the value. Otherwise, the value is the expected payoff resulting when the players choose their optimal mixed strategies. (pp. p. 626 p. 633)

Variable-sum game A game in which the sum of the payoffs to the players at the different outcomes varies. (p. 637)

Vickrey auction An auction in which bidders independently submit sealed bids for an item and the winner is the bidder who bids the highest, but he or she pays only the amount of the second-highest bid. (p. 643)

Zero-sum game A constant-sum game in which the payoff to one player is the negative of the payoff to the other player, so the sum of the payoffs to the players at each outcome is zero. (p. 624)