Chapter 6 Game Theory (Module 4) 1

Learning Objectives Students will be able to: Explain the principles of zero-sum, two-person games. Analyze pure strategy games and use dominance to reduce the size of the game. Solve mixed strategy games when there is no saddle point. 2

Chapter Outline 1. Introduction 2. Language of Games 3. The Minimax Criterion 4. Pure Strategy Games 5. Mixed Strategy Games 6. Dominance 3

1. Introduction Game theory is the study of how optimal strategies are formulated in conflict. Game theory has been effectively used for: War strategies, Union negotiators, Competitive business strategies. 4

Introduction (continued) Game models are classified by: the number of players, the number of strategies employed and the sum of all payoffs. A zero sum game implies that what is gained by one player is lost for the other. 5

2. Language of Games Consider a duopoly(احتكارثنائي) competitive business market in which one company is considering advertising in hopes of luring (attracting) customers away from its competitor. The company is considering radio and/or newspaper advertisements. Let’s use game theory to determine the best strategy. Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process. 6

Example 1 Suppose there are only two lighting fixture stores, X and Y, in Urbana, Illinois. (This is called a duopoly.) The respective market shares have been stable up until now, but the situation may change. The daughter of the owner of store X has just completed her MBA and has developed two distinct advertising strategies, one using radio spots and the other newspaper ads. Upon hearing this, the owner of store Y also proceeds to prepare radio and newspaper ads. The 2 × 3 payoff matrix in Table M4.1 shows what will happen to current market shares if both stores begin advertising. By convention, payoffs are shown only for the first game player, X in this case. Y’s payoffs will just be the negative of each number. For this game, there are only two strategies being used by each player. If store Y had a third strategy, we would be dealing with a 2 × 3 payoff matrix. 7

Language of Games (continued) Below is the payoff matrix (as a percent of change in market share) for Store X. A positive number means that X wins and Y loses, while a negative number implies Y wins and X loses. Example 1: STORE X’s PAYOFFs Y’s strategy 1 (use radio) Y’s strategy 2 (use newspaper) X’s strategy 1 3 5 X’s strategy 2 1 -2 This slide introduces the difference between “natural” and “assignable” causes. The next several slides expand the discussion and introduce some of the statistical issues. 8

Language of Games (continued) Store X’s Strategy Stores Y’s Strategy Outcome (% change in market share) X1: Radio Y1: Radio X wins 3 Y loses 3 Y2: Newspaper X wins 5 Y loses 5 X2: Newspaper X wins 1 Y loses 1 X loses 2 Y wins 2 This slide introduces the difference between “natural” and “assignable” causes. The next several slides expand the discussion and introduce some of the statistical issues. 9

Language of Games (continued) A positive number in Table M4.1 means that X wins and Y loses. A negative number means that Y wins and X loses. It is obvious from the table that the game favors competitor X, since all values are positive except one. If the game had favored player Y, the values in the table would have been negative. In other words, the game in Table M4.1 is biased against Y. However, since Y must play the game, he or she will play to minimize total losses. To do this, Player Y would use the minimax criterion, our next topic. 10

The minimax criterion is used in a two-person zero-sum game. Each person should choose the strategy that minimizes the maximum loss. Note: This is identical to maximizing the minimum gains. 11

The Minimax Criterion (continued) The lower value of the game is equal to the maximum of the minimum values in the rows. The upper value of the game is equal to the minimum of the maximum values in the columns. 12

The Minimax Criterion (continued) Lower Value of the Game: Maximum of the minimums in rows STORE X’s PAYOFFs Y1 (radio) Y2 (newspaper) Minimum Gain (for X) X1 3 5 X2 1 -2 Maximum Loss (for Y) Upper Value of the Game: Minimum of the maximums in columns 13

4. Pure Strategy Games An equilibrium or saddle point condition exists if the upper and lower values are equal. This number is called the value of the game, and an equilibrium or saddle point condition exists. This is called a pure strategy because each of both players will follow only one strategy. STORE X’s PAYOFFs Y1 (radio) Y2 (newspaper) Minimum X1 3 5 X2 1 -2 Maximum Saddle point: Both upper and lower values are 3, this is called the value of the game. 14

Pure Strategy Games (continued) The value of the game is the average or expected game outcome if the game is played an infinite number of times. 15

Example 2: Saddle point Lower value Upper value STORE X’s PAYOFFs Y1 (radio) Y2 (newspaper) Minimum row number X1 10 6 X2 -12 2 Maximum column number Saddle point Lower value Upper value 16

5. Mixed Strategy Game A mixed strategy game exists when there is no saddle point. Use the expected gain or loss approach. The goal of this approach is for a player to play each strategy a particular percentage of time so that the expected value of the game does not depend upon what the opponent does. 17

Mixed Strategy Games (continued) This will only occur if the expected value of each strategy is the same.. Example: Y1 Y2 Minimum row number X1 4 2 X2 1 10 Maximum column number Lower value Upper value No saddle point! 18

Mixed Strategy Games (continued) Let P be the percentage of time that player Y chooses strategy Y1 and (1-P) be the percentage of time that he chooses strategy Y2. If player X chooses to play strategy X1, the gain of Y is 4P + 2(1-P), and if he chooses to play strategy X2, the gain of Y is 1p + 10(1-P). If these two expected values are equal, then the expected value for player Y does not depend on the strategy chosen by X. Y1 (P) Y2 (1-P) Expected Gain X1 4 2 4P + 2(1-P) X2 1 10 1P + 10(1-P) 19

Mixed Strategy Games (continued) (P) Y2 (1-P) Expected Gain X1 4 2 4P + 2(1-P) X2 1 10 1p + 10(1-P) Set these two equations equal to each other and solve for P 4P + 2(1-P) = 1P + 10(1-P) 4P – 2P – 1P + 10P = 10 – 2 P = 8/11 and 1-P = 3/11 Expected payoff = = 1P + 10(1-P) = 1(8/11) + 10(3/11) = 3.46 20

Mixed Strategy Games (continued) Performing a similar analysis for player X: Y1 (P) Y2 (1-P) X1 (Q) 4 2 X2 (1-Q) 1 10 Expected Gain 4Q + 1(1-Q) 2Q + 10(1-Q) Set these two equations equal to each other and solve for Q 4Q + 1(1-Q) = 2Q + 10(1-Q) 4Q – 1Q – 2Q + 10Q = 10 – 1 Q = 9/11 and 1-Q = 2/11 Expected payoff = 2Q + 10(1-Q) = 2(9/11) + 10(2/11) = 3.46 21

Mixed Strategy Games (continued) (P) Y2 (1-P) Expected Gain X1 (Q) 4 2 4P + 2(1-P) X2 (1-Q) 1 10 1p + 10(1-P) 4Q + 1(1-Q) 2Q + 10(1-Q) Set these two equations equal to each other and solve for P Set these two equations equal to each other and solve for Q 22

6. Dominance Dominance is a principle that can be used to reduce the size of games by eliminating strategies that would never be played. Note: A strategy can be eliminated if all its game’s outcomes are the same or worse than the corresponding outcomes of another strategy. 23

Dominance (continued) Initial game Y1 Y2 X1 4 3 X2 2 20 X3 1 X3 is a dominated strategy, it is dominated by both X1 and also X2 Game after removal of dominated strategy Y1 Y2 X1 4 3 X2 2 20 24

Dominance (continued) Initial game Y1 Y2 Y3 Y4 X1 -5 4 6 -3 X2 -2 2 -20 Y2 and Y 3 are dominated by Y1 and Y4 Game after removal of dominated strategies Y1 Y4 X1 -5 -3 X2 -2 -20 Reconsider examples 1 and 2 for dominance! 25

Dominance (continued) STORE X’s PAYOFFs Y1 (radio) Y2 (newspaper) X1 3 5 X2 1 -2 X2 is dominated by X1, Reduced matrix: STORE X’s PAYOFFs Y1 (radio) Y2 (newspaper) X1 3 5 X has only one choice (X1), then Y will choose Y1 to minimize its losses. So, the value of the game is 3. 26

Lab Exercise # 6 1. Solve Example 1 for pure strategy games using QM for Windows. 2. Solve Example 2 for pure strategy games using QM for Windows. 3. Solve the Mixed Strategy Example using QM for Windows.

Example 2 48

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Mixed Strategy Example 50

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