1. 1 1 Slide © 2006 Thomson South-Western. All Rights Reserved. Slides prepared by JOHN LOUCKS St. Edward’s University

2. 2 2 Slide © 2006 Thomson South-Western. All Rights Reserved. Chapter 5 Utility and Game Theory n Introduction to Game Theory Pure Strategy Pure Strategy Mixed Strategies Mixed Strategies Dominated Strategies Dominated Strategies

3. 3 3 Slide © 2006 Thomson South-Western. All Rights Reserved. Introduction to Game Theory n In decision analysis, a single decision maker seeks to select an optimal alternative. n In game theory, there are two or more decision makers, called players, who compete as adversaries against each other. n It is assumed that each player has the same information and will select the strategy that provides the best possible outcome from his point of view. n Each player selects a strategy independently without knowing in advance the strategy of the other player(s). continue

4. 4 4 Slide © 2006 Thomson South-Western. All Rights Reserved. Introduction to Game Theory n The combination of the competing strategies provides the value of the game to the players. n Examples of competing players are teams, armies, companies, political candidates, and contract bidders.

5. 5 5 Slide © 2006 Thomson South-Western. All Rights Reserved. n Two-person means there are two competing players in the game. n Zero-sum means the gain (or loss) for one player is equal to the corresponding loss (or gain) for the other player. n The gain and loss balance out so that there is a zero- sum for the game. n What one player wins, the other player loses. Two-Person Zero-Sum Game

6. 6 6 Slide © 2006 Thomson South-Western. All Rights Reserved. n Competing for Vehicle Sales Suppose that there are only two vehicle dealer- ships in a small city. Each dealership is considering three strategies that are designed to take sales of new vehicles from the other dealership over a four-month period. The strategies, assumed to be the same for both dealerships, are on the next slide. Two-Person Zero-Sum Game Example

7. 7 7 Slide © 2006 Thomson South-Western. All Rights Reserved. n Strategy Choices Strategy 1: Offer a cash rebate Strategy 1: Offer a cash rebate on a new vehicle. on a new vehicle. Strategy 2: Offer free optional Strategy 2: Offer free optional equipment on a equipment on a new vehicle. new vehicle. Strategy 3: Offer a 0% loan Strategy 3: Offer a 0% loan on a new vehicle. on a new vehicle. Two-Person Zero-Sum Game Example

8. 8 8 Slide © 2006 Thomson South-Western. All Rights Reserved. 2 2 1 2 2 1 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B n Payoff Table: Number of Vehicle Sales Gained Per Week by Dealership A Gained Per Week by Dealership A (or Lost Per Week by Dealership B) (or Lost Per Week by Dealership B) -3 3 -1 3 -2 0 3 -2 0 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Two-Person Zero-Sum Game Example

9. 9 9 Slide © 2006 Thomson South-Western. All Rights Reserved. n Step 1: Identify the minimum payoff for each row (for Player A). row (for Player A). n Step 2: For Player A, select the strategy that provides the maximum of the row minimums (called the maximum of the row minimums (called the maximin). the maximin). Two-Person Zero-Sum Game Example

10. 10 Slide © 2006 Thomson South-Western. All Rights Reserved. n Identifying Maximin and Best Strategy RowMinimum 1-3-2 2 2 1 2 2 1 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B -3 3 -1 3 -2 0 3 -2 0 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Best Strategy For Player A MaximinPayoff Two-Person Zero-Sum Game Example

11. 11 Slide © 2006 Thomson South-Western. All Rights Reserved. n Step 3: Identify the maximum payoff for each column (for Player B). (for Player B). n Step 4: For Player B, select the strategy that provides the minimum of the column maximums the minimum of the column maximums (called the minimax). (called the minimax). Two-Person Zero-Sum Game Example

12. 12 Slide © 2006 Thomson South-Western. All Rights Reserved. n Identifying Minimax and Best Strategy 2 2 1 2 2 1 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B -3 3 -1 3 -2 0 3 -2 0 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Column Maximum 3 3 1 3 3 1 Best Strategy For Player B MinimaxPayoff Two-Person Zero-Sum Game Example

13. 13 Slide © 2006 Thomson South-Western. All Rights Reserved. Pure Strategy n Whenever an optimal pure strategy exists: n the maximum of the row minimums equals the minimum of the column maximums (Player A’s maximin equals Player B’s minimax) n the game is said to have a saddle point (the intersection of the optimal strategies) n the value of the saddle point is the value of the game n neither player can improve his/her outcome by changing strategies even if he/she learns in advance the opponent’s strategy

14. 14 Slide © 2006 Thomson South-Western. All Rights Reserved. RowMinimum 1-3-2 CashRebate b 1 0%Loan b 3 FreeOptions b 2 Dealership B -3 3 -1 3 -2 0 3 -2 0 Cash Rebate a 1 Free Options a 2 0% Loan a 3 Dealership A Column Maximum 3 3 1 3 3 1 Pure Strategy Example n Saddle Point and Value of the Game 2 2 1 2 2 1 SaddlePoint Value of the game is 1

15. 15 Slide © 2006 Thomson South-Western. All Rights Reserved. Pure Strategy Example n Pure Strategy Summary n Player A should choose Strategy a 1 (offer a cash rebate). n Player A can expect a gain of at least 1 vehicle sale per week. n Player B should choose Strategy b 3 (offer a 0% loan). n Player B can expect a loss of no more than 1 vehicle sale per week.

16. 16 Slide © 2006 Thomson South-Western. All Rights Reserved. Mixed Strategy n If the maximin value for Player A does not equal the minimax value for Player B, then a pure strategy is not optimal for the game. n In this case, a mixed strategy is best. n With a mixed strategy, each player employs more than one strategy. n Each player should use one strategy some of the time and other strategies the rest of the time. n The optimal solution is the relative frequencies with which each player should use his possible strategies.

17. 17 Slide © 2006 Thomson South-Western. All Rights Reserved. Mixed Strategy Example b1b1b1b1 b2b2b2b2 Player B 11 5 a1a1a2a2a1a1a2a2 Player A 4 8 4 8 n Consider the following two-person zero-sum game. The maximin does not equal the minimax. There is not an optimal pure strategy. ColumnMaximum 11 8 11 8 RowMinimum 4 5 Maximin Minimax

18. 18 Slide © 2006 Thomson South-Western. All Rights Reserved. Mixed Strategy Example p = the probability Player A selects strategy a 1 (1  p ) = the probability Player A selects strategy a 2 If Player B selects b 1 : EV = 4 p + 11(1 – p ) If Player B selects b 2 : EV = 8 p + 5(1 – p )

19. 19 Slide © 2006 Thomson South-Western. All Rights Reserved. Mixed Strategy Example 4 p + 11(1 – p ) = 8 p + 5(1 – p ) To solve for the optimal probabilities for Player A we set the two expected values equal and solve for the value of p. 4 p + 11 – 11 p = 8 p + 5 – 5 p 11 – 7 p = 5 + 3 p -10 p = -6 p =.6 Player A should select: Strategy a 1 with a.6 probability and Strategy a 1 with a.6 probability and Strategy a 2 with a.4 probability. Strategy a 2 with a.4 probability. Hence, (1  p ) =.4

20. 20 Slide © 2006 Thomson South-Western. All Rights Reserved. Mixed Strategy Example q = the probability Player B selects strategy b 1 (1  q ) = the probability Player B selects strategy b 2 If Player A selects a 1 : EV = 4 q + 8(1 – q ) If Player A selects a 2 : EV = 11 q + 5(1 – q )

21. 21 Slide © 2006 Thomson South-Western. All Rights Reserved. Mixed Strategy Example 4 q + 8(1 – q ) = 11 q + 5(1 – q ) To solve for the optimal probabilities for Player B we set the two expected values equal and solve for the value of q. 4 q + 8 – 8 q = 11 q + 5 – 5 q 8 – 4 q = 5 + 6 q -10 q = -3 q =.3 Hence, (1  q ) =.7 Player B should select: Strategy b 1 with a.3 probability and Strategy b 1 with a.3 probability and Strategy b 2 with a.7 probability. Strategy b 2 with a.7 probability.

22. 22 Slide © 2006 Thomson South-Western. All Rights Reserved. Mixed Strategy Example n Value of the Game For Player A: EV = 4 p + 11(1 – p ) = 4(.6) + 11(.4) = 6.8 For Player B: EV = 4 q + 8(1 – q ) = 4(.3) + 8(.7) = 6.8 Expected gain per game for Player A Expected loss per game for Player B

23. 23 Slide © 2006 Thomson South-Western. All Rights Reserved. Dominated Strategies Example RowMinimum -2 0-3 b1b1b1b1 b3b3b3b3 b2b2b2b2 Player B 1 0 3 1 0 3 3 4 -3 3 4 -3 a1a1a2a2a3a3a1a1a2a2a3a3 Player A ColumnMaximum 6 5 3 6 5 3 6 5 -2 6 5 -2 Suppose that the payoff table for a two-person zero- sum game is the following. Here there is no optimal pure strategy. Maximin Minimax

24. 24 Slide © 2006 Thomson South-Western. All Rights Reserved. Dominated Strategies Example b1b1b1b1 b3b3b3b3 b2b2b2b2 Player B 1 0 3 1 0 3 Player A 6 5 -2 6 5 -2 If a game larger than 2 x 2 has a mixed strategy, we first look for dominated strategies in order to reduce the size of the game. If a game larger than 2 x 2 has a mixed strategy, we first look for dominated strategies in order to reduce the size of the game. 3 4 -3 3 4 -3 a1a1a2a2a3a3a1a1a2a2a3a3 Player A’s Strategy a 3 is dominated by Player A’s Strategy a 3 is dominated by Strategy a 1, so Strategy a 3 can be eliminated.

25. 25 Slide © 2006 Thomson South-Western. All Rights Reserved. Dominated Strategies

26. 26 Slide © 2006 Thomson South-Western. All Rights Reserved. Dominated Strategy

27. 27 Slide © 2006 Thomson South-Western. All Rights Reserved. n Two-Person, Constant-Sum Games (The sum of the payoffs is a constant other than zero.) (The sum of the payoffs is a constant other than zero.) n Variable-Sum Games (The sum of the payoffs is variable.) (The sum of the payoffs is variable.) n n -Person Games (A game involves more than two players.) (A game involves more than two players.) n Cooperative Games (Players are allowed pre-play communications.) (Players are allowed pre-play communications.) n Infinite-Strategies Games (An infinite number of strategies are available for the players.) (An infinite number of strategies are available for the players.) Other Game Theory Models

28. 28 Slide © 2006 Thomson South-Western. All Rights Reserved. End of Chapter 5