1. Chapter 12 & Module E Decision Theory & Game Theory

2. Decision Making F A decision is made for a future action. F A decision making process is a process of “selection” : Selecting one from many options (alternatives) as the decision.

3. Decision Theory F Decision theory deals with following type of decision making problems: –The outcome for an decision alternative is not certain, which is affected by some factors that are not controlled by the decision maker. –Example: Selecting a stock for investment.

4. Components of Decision Making (D.M.) F Decision alternatives - for managers to choose from. F States of nature - that may actually occur in the future regardless of the decision. F Payoff - outcome of a decision alternative under a state of nature. The components are given in Payoff Tables.

5. A Decision Table States of Nature Investment EconomyEconomy decision good bad alternatives 0.6 0.4 Apartment$ 50,000$ 30,000 Office 100,000 - 40,000 Warehouse 30,000 10,000 F

6. Criterion:Expected Payoff F Select the alternative that has the largest expected value of payoffs. F Expected payoff of an alternative: n=number of states of nature P i =probability of the i-th state of nature V i =payoff of the alternative under the i-th state of nature

7. Example Decision Alt’s Econ Good 0.6 Econ Bad 0.4Expected payoff Apartment 50,00030,000 Office 100,000-40,000 Warehouse 30,00010,000

8. Expected Value of Perfect Information (EVPI) F It is a measure of the value of additional information on states of nature. F It tells up to how much you would pay for additional information.

9. An Example If a consulting firm offers to provide “perfect information about the future with $5,000, would you take the offer? States of Nature Investment EconomyEconomy decision good bad alternatives 0.6 0.4 Apartment$ 50,000$ 30,000 Office 100,000 - 40,000 Warehouse 30,000 10,000

10. Calculating EVPI F EVPI F = EV w PI – EV w/o PI = (Exp. payoff with perfect information) – (Exp. payoff without perfect information)

11. Expected payoff with Perfect Information F EV w PI where n=number of states of nature h i =highest payoff of i-th state of nature P i =probability of i-th state of nature

12. Example for Expected payoff with Perfect Information States of Nature Investment EconomyEconomy decision good bad alternatives 0.6 0.4 Apartment$ 50,000$ 30,000 Office 100,000 - 40,000 Warehouse 30,000 10,000 h i 100,000 30,000 Expected payoff with perfect information = 100,000*0.6+30,000*0.4 = 72,000

13. Expected payoff without Perfect Information F Expected payoff of the best alternative selected without using additional information. i.e., EV w/o PI = Max Exp. Payoff

14. Example for Expected payoff without Perfect Information Decision Alt’s Econ Good 0.6 Econ Bad 0.4Expected payoff Apartment 50,00030,000 42,000 *Office 100,000-40,000 *44,000 Warehouse 30,00010,000 22,000

15. Expected Value of Perfect Information (EVPI) in above Example F EVPI = EV w PI – EV w/o PI = 72,000 - 44,000 = $28,000

16. EVPI is a Benchmark in Bargain F EVPI is the maximum $ amount the decision maker would pay to purchase perfect information.

17. Value of Imperfect Information Expected value of imperfect information = (discounted EV w PI) – EV w/o PI = (EVwPI * (% of perfection)) – EV w/o PI

18. Game Theory F Game theory is for decision making with two decision makers of conflicting interests in competition. F In decision theory: Human vs. God. F In game theory: Human vs. Human.

19. Two-Person Zero-Sum Game F Two decision makers’ benefits are completely opposite i.e., one person’s gain is another person’s loss F Payoff/penalty table (zero-sum table): –shows “offensive” strategies (in rows) versus “defensive” strategies (in columns); –gives the gain of row player (loss of column player), of each possible strategy encounter.

20. Example 1 (payoff/penalty table) Athlete Manager’s Strategies Strategies (Column Strategies) (row strat.) A B C 1$50,000$35,000$30,000 2$60,000$40,000$20,000

21. Two-Person Constant-Sum Game F For any strategy encounter, the row player’s payoff and the column player’s payoff add up to a constant C. F It can be converted to a two-person zero- sum game by subtracting half of the constant (i.e. 0.5C) from each payoff.

22. Example 2 (2-person, constant- sum) During the 8-9pm time slot, two broadcasting networks are vying for an audience of 100 million viewers, who would watch either of the two networks.

23. Payoffs of nw1 for the constant- sum of 100(million) Network 2 Network 1westernSoap Comedy western 35 1560 soap 45 5850 comedy 38 1470

24. An equivalent zero-sum table Network 2 Network 1westernSoap Comedy western -15-3510 soap - 5 8 0 comedy -12-3620

25. Equilibrium Point F In a two-person zero-sum game, if there is a payoff value P such that P=max{row minimums} =min{column maximums} then P is called the equilibrium point, or saddle point, of the game.

26. Example 3 (equilibrium point) Athlete Manager’s Strategies Strategies (Column Strategies) (row strat.) A B C 1$50,000$35,000$30,000 2$60,000$40,000$20,000

27. Game with an Equilibrium Point: Pure Strategy F The equilibrium point is the only rational outcome of this game; and its corresponding strategies for the two sides are their best choices, called pure strategy. F The value at the equilibrium point is called the value of the game. F At the equilibrium point, neither side can benefit from a unilateral change in strategy.

28. Pure Strategy of Example 3 Athlete Manager’s Strategies Strategies (Column Strategies) (row strat.) A B C 1$50,000$35,000$30,000 2$60,000$40,000$20,000

29. Example 4 (2-person, 0-sum) Row Players Column Player Strategies Strategies 1 2 3 1 4 4 10 2 2 3 1 3 6 5 7

30. Mixed Strategy F If a game does not have an equilibrium, the best strategy would be a mixed strategy.

31. Game without an Equilibrium Point F A player may benefit from unilateral change for any pure strategy. Therefore, the game would get into a loop. F To break loop, a mixed strategy is applied.

32. Example: Company ICompany II Strategies Strategies BC 2 84 3 17

33. Mixed Strategy F A mixed strategy for a player is a set of probabilities each for an alternative of the player. F The expected payoff of row player (or the expected loss of column player) is called the value of the game.

34. Example: Company ICompany II Strategies Strategies BC 2 84 3 17 Let mixed strategy for company I be {0.6, 0.4}; and for Company II be {0.3, 0.7}.

35. Equilibrium Mixed Strategy F An equilibrium mixed strategy makes expected values of any player’s individual strategies identical. F Every game contains one equilibrium mixed strategy. F The equilibrium mixed strategy is the best strategy.

36. How to Find Equilibrium Mixed Strategy F By linear programming (as introduced in book) F By QM for Windows, – we use this approach.

37. Both Are Better Off at Equilibrium F At equilibrium, both players are better off, compared to maximin strategy for row player and minimax strategy for column player. F No player would benefit from unilaterally changing the strategy.

38. A Care-Free Strategy F The row player’s expected gain remains constant as far as he stays with his mixed strategy (no matter what strategy the column player uses). F The column player’s expected loss remains constant as far as he stays with his mixed strategy (no matter what strategy the row player uses).

39. Unilateral Change from Equilibrium by Column Player probability 0.10.9 B C 0.6Strat 2 8 4 0.4Strat 3 1 7

40. Unilateral Change from Equilibrium by Column Player probability 1.0 0 B C 0.6Strat 2 8 4 0.4Strat 3 1 7

41. Unilateral Change from Equilibrium by Row Player probability 0.30.7 B C 0.2Strat 2 8 4 0.8Strat 3 1 7

42. A Double-Secure Strategy F At the equilibrium, the expected gain or loss will not change unless both players give up their equilibrium strategies. –Note: Expected gain of row player is always equal to expected loss of column player, even not at the equilibrium, since 0-sum)

43. Both Leave Their Equilibrium Strategies probability 0.80.2 B C 0.5Strat 2 8 4 0.5Strat 3 1 7

44. Both Leave Their Equilibrium Strategies probability 0 1 B C 0.2Strat 2 8 4 0.8Strat 3 1 7

45. Penalty for Leaving Equilibrium F It is equilibrium because it discourages any unilateral change. F If a player unilaterally leaves the equilibrium strategy, then –his expected gain or loss would not change, and –once the change is identified by the competitor, the competitor can easily beat the non-equilibrium strategy.

46. Find the Equilibrium Mixed Strategy F Method 1: As on p.573-574 of our text book. The method is limited to 2X2 payoff tables. F Method 2: Linear programming. A general method. F Method we use: Software QM.

47. Implementation of a Mixed Strategy F Applied in the situations where the mixed strategy would be used many times. F Randomly select a strategy each time according to the probabilities in the strategy. F If you had good information about the payoff table, you could figure out not only your best strategy, but also the best strategy of your competitor (!).

48. Dominating Strategy vs. Dominated Strategy F For row strategies A and B: If A has a better (larger) payoff than B for any column strategy, then B is dominated by A. F For column strategies X and Y: if X has a better (smaller) payoff than Y for any row strategy, then Y is dominated by X. F A dominated decision can be removed from the payoff table to simplify the problem.

49. Example: Company ICompany II Strategies Strategies A BC 1 9 72 2 11 84 3 417

50. Find the Optimal Mixed Strategy in 2X2 Table F Suppose row player has two strategies, 1 and 2, and column player has two strategies, A and B.

51. For row player: F Let p be probability of selecting row strategy 1. Then the probability of selecting row strategy 2 is (1-p). F Represent E A and E B by p, where E A (E B ) is the expected payoff of the row player if the column player chose column strategy A (B). F Set E A = E B, and solve p from the equation.

52. For column player: F Let p be probability of selecting column strategy A. Then the probability of selecting column strategy B is (1-p). F Represent E 1 and E 2 by p, where E 1 (E 2 ) is the expected payoff of the row player if the column player chose column strategy A (B). F Set E 1 = E 2, and solve p from the equation.