Game Theory∗ Theodore L. Turocy Texas A&M University Bernhard von Stengel London School of Economics CDAM Research Report LSE-CDAM-2001-09 October 8, 2001 Contents 1 What is game theory? 4 2 Definitions of games 6 3 Dominance 8 4 Nash equilibrium 12 5 Mixed strategies 17 6 Extensive games with perfect information 22 7 Extensive games with imperfect information 29 8 Zero-sum games and computation 33 9 Bidding in auctions 34 10 Further reading 38 ∗This is the draft of an introductory survey of game theory, prepared for the Encyclopedia of Information Systems, Academic Press, to appear in 2002. 1 Glossary Backward induction Backward induction is a technique to solve a game of perfect information. It first consid￾ers the moves that are the last in the game, and determines the best move for the player in each case. Then, taking these as given future actions, it proceeds backwards in time, again determining the best move for the respective player, until the beginning of the game is reached. Common knowledge A fact is common knowledge if all players know it, and know that they all know it, and so on. The structure of the game is often assumed to be common knowledge among the players. Dominating strategy A strategy dominates another strategy of a player if it always gives a better payoff to that player, regardless of what the other players are doing. It weakly dominates the other strategy if it is always at least as good. Extensive game An extensive game (or extensive form game) describes with a tree how a game is played. It depicts the order in which players make moves, and the information each player has at each decision point. Game A game is a formal description of a strategic situation. Game theory Game theory is the formal study of decision-making where several players must make choices that potentially affect the interests of the other players. 2 Mixed strategy A mixed strategy is an active randomization, with given probabilities, that determines the player’s decision. As a special case, a mixed strategy can be the deterministic choice of one of the given pure strategies. Nash equilibrium A Nash equilibrium, also called strategic equilibrium, is a list of strategies, one for each player, which has the property that no player can unilaterally change his strategy and get a better payoff. Payoff A payoff is a number, also called utility, that reflects the desirability of an outcome to a player, for whatever reason. When the outcome is random, payoffs are usually weighted with their probabilities. The expected payoff incorporates the player’s attitude towards risk. Perfect information A game has perfect information when at any point in time only one player makes a move, and knows all the actions that have been made until then. Player A player is an agent who makes decisions in a game. Rationality A player is said to be rational if he seeks to play in a manner which maximizes his own payoff. It is often assumed that the rationality of all players is common knowledge. Strategic form A game in strategic form, also called normal form, is a compact representation of a game in which players simultaneously choose their strategies. The resulting payoffs are pre￾sented in a table with a cell for each strategy combination. 3