The Beautiful Mind—John Forbes Nash, Jr.—His works in Game Theory, geometry, etc. have provided insight into the factors that govern chance and events in our daily lives. Chapter 12 Game Theory 12.1 Introduction to Game Theory 504 12.2 The Matrix Game 505 12.3 Strictly Determined Game: The Saddle Point 507 12.4 Games with Mixed Strategies 511 12.5 Reducing Matrix Games to Systems of Equations 518 Critical Thinking and Basic Exercise 521 Summary of Important Concepts 522 Review Test 523 References 523 The Joy of Finite Mathematics Copyright © 2016 Elsevier Inc. All rights reserved. 503 504 Chapter 12 Game Theory JOHN VON NEUMANN (1903-1957) John von Neumann, one of the foremost twentieth-century mathematicians, was born in Budapest, Hungary on December 28, 1903. He was a general scientific prodigy, in the mode of some of his great predecessors. He received his formal education at the University of Berlin, the Technical Institute in Zurich, and the University of Budapest, where he earned his Ph.D. in mathematics in 1926 at the age of twenty. Von Neumann taught at the University of Hamburg from 1926 until 1929. He left Germany in 1930, just before the Second World War, taking refuge in this country, where he accepted a professorship in mathematical physics at Princeton University. He became professor of mathematics at the Institute for Advanced Study at Princeton, New Jersey, when it was founded in 1933, retaining this position until his death in 1957. Von Neumann was one of the founders of game theory. In 1944 he collaborated with Oskar Morgenstern on a book, Theory of Games and Economic Behavior. This important and initial comprehensive treatment of game theory offered a new approach to the study of economic behavior through the use of game-theoretic methods. The development of the first electronic computers at the institute was directed by von Neumann. He initiated the concept of stored programs in the computer—that programs and data should be treated similarly. This was one of the major break- throughs in computer development. Among the computers for which von Neumann is credited are the MANIAC, NORC, and ORDVAC. Von Neumann served on the U.S. Atomic Energy Commission and consulted on the Atomic Bomb Project at Los Alamos. In 1956, he received the $50,000 Enrico Fermi award for his outstanding work on computer theory, design, and construction. 12.1 INTRODUCTION TO GAME THEORY Game theory is a relatively new branch of mathematics; it is the study of the rational behavior of people in conflict situations. The conflict may be between individuals involved in a game of chance, between teams engaged in an athletic contest, between nations engaged in war, or between firms engaged in compe- tition for a share of the market for a certain product. In our study of game theory, we shall refer to each contestant as a player and restrict our investiga- tions to games employing two players. In general, however, we need not restrict ourselves to conflicts generated by two players only; such games can involve any number of players. 12.2 The Matrix Game 505 The two players involved in a game have precisely opposite interests. Each Reality is broken: why games make us better and player has a specified number of actions from which to choose. However, the how this changes the world. action taken by one of the players at a particular stage of the game is not known Jane McGonigal to the other player. Each of the players has a certain objective in mind, and he attempts to choose his actions so as to achieve this aim. Definition 12.1.1 Game Theory Game theory is concerned with the analysis of human behavior in conflict situations. In our brief introduction to the subject, we have considered only games between two persons (players); however, they may involve any number of players, teams, companies and so forth. The initial study of game theory was conducted independently by John von Neumann and Emil Borel during the 1920s. After World War II, von Neumann and Oskar Morgenstern formulated the subject area as an independent branch of mathematics. The close relationship between systems of equations and game theory will become apparent as we proceed with our study. 12.2 THE MATRIX GAME All reality is a game. Consider a modified version of the game “matching pennies.” Player P puts a 1 I. Banks coin either heads up, H, or tails up, T. A second Player P2 without knowing P1’s choice calls either “heads” or “tails.” Player P1 will pay Player P2 $5 if P1 shows H and P2 chooses H; Player P1 will pay Player P2 $3 if both have chosen T. However, if Player P2 guesses incorrectly, he must pay P1 $4 (that is, if P1 shows H and P2 guesses T,orifP1 shows T and P2 guesses H Player P2 must pay P1 $4). This game is a two-person game since there are two players involved. It can be displayed in the form of a matrix as follows: P2 HT H 254 P 1 T 4 23 The positive entries of the matrix denote the gains of Player P1, and his losses to Player P2 are the negative entries of the array. We designate P1, the role player and P2 the column player. Player P1’s two choices are each associated with a row of the matrix. Player P2’s choices are each associated with a column of the matrix. Thus, if P1 chooses row one and P2 chooses column one, then P1 pays P2 $5; instead, if P2 selects column two, then P2 pays P1 $4. Definition 12.2.1 Game Theory/Payoff Matrix The states of the game are given in the form of a matrix called the game matrix or payoff matrix. This game is a zero-sum game because whatever is lost or gained by P1 is gained or lost by P2. The matrix representation of the game is called the matrix game. Each entry of the matrix game represents the payoff to either player P1 or P2. Thus a matrix game is also referred to as the payoff matrix. 506 Chapter 12 Game Theory Definition 12.2.2 Two-person Zero Game A two-person zero-sum game is a game played by two opponents with opposing interest and such that the payoff to one player is equal to the loss of the other. The problem facing each player is what choice to make so that it will be in his best interest. That is, should P1 select row one or row two? Should P2, not knowing P1’s choice, select column one or column two? Before we proceed to discuss the method for choosing optimally, we shall summarize the basic concept and terminology constituting a two-person zero-sum game: (a) Two players, P1 and P2, are engaged in a conflict of interest. For example, in the game of “matching pennies,” P1 wants to maximize his winnings while P2 wants to minimize his losses. (b) Each player has at his disposal a set of instructions regarding the action to take in each conceivable position of the game. We shall refer to these instructions as strategies. (In the game given above each player has two strategies H and T.) (c) Associated with each strategy employed by P1, there is a certain payoff. The set of all strategies will result in an array of payoffs known as the matrix game or payoff matrix. For instance, in “matching pennies,” the payoff matrix was Strategieszfflffl}|fflffl{ of P2 HT H 254 Strategies of fP 1 T 4 23 (d) The objective of P1 is to utilize his strategies so as to maximize his win- nings; P2’s objective is to select his strategies so as to minimize his losses. (e) Such games are called zero-sum games or strictly competitive games, since the sum of the amounts won by the two players is always zero. Note: The winnings of one player are equal to the losses of the other. Like all mathematics, game theory is a tautology Example 12.2.1 Payoff Matrix whose conclusions are true because they are contained in the premises. Consider a two-person zero-sum game, the payoff matrix of which is given by R.A. Epstein P2 γ γ γ 1 2 3 β 2 1 213 P1 β 2 2 6 45 Here, there are two strategies available to Player P1; namely β1 and β2, and three strategies, γ1, γ2, γ3, available to Player P2. The entries in the matrix denote winnings of Player P1. Recall that negative entries mean that Player P1 will pay P2. In the game P1 chooses one of his two strategies, β1 or β2, and simultaneously, Player P2, without knowing P1’s choice selects γ1, γ2, γ3. The intersection of the row corresponding to P1’s choice and the column corresponding to P2’s selection gives the payoff of this play in the game. For example, if P1 chooses strategy β2 and P2 selects γ3, then P1 wins $5. If P1 chooses β1 and P2 selects γ1, then P1 receives $-2; that is, he must pay Player P2 $2. Here, we have the question of importance: What strategy should P1 choose to max- imize his winnings? At the same time, what should P2’s choice be to minimize his losses? The selection of strategies will be the focus of the remaining discussions in this chapter. 12.3 Strictly Determined Game: The Saddle Point 507 12.3 STRICTLY DETERMINED GAME: THE SADDLE POINT Let us consider a two-person game, the payoff matrix of which is given by Games lubricate the body and mind.