Readings

1. Readings • Readings • Baye “Mixed Strategies” (see the index) • Dixit Chapter 7 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

2. Overview • Overview BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

3. Overview Part III OverviewGame Theory in Part III completes Game Theory in Part II for those games where information is strategically revealed or withheld. In many games, a player may not know all the information that is pertinent for the choice that he has to make at every point in the game. His uncertainty may be over variables that are either internal or external to the game. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

4. Overview On the one hand, a player may be potentially uncertain about what moves the other player is making at the same time he makes his own move; we call that strategic uncertainty. All the simultaneous move games in Part II were simple enough that that uncertainty was resolved by eliminating dominated strategies. Part III’s Lesson 1 considers games where strategic uncertainty remains because uncertainty is not resolved by eliminating dominated strategies, and because players’ interests conflict (as in sports) so players conceal information about their own moves. Lesson 2 considers games where players easily reveal information about their own moves because players’ interests align (as in setting a standard industry format). BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

5. Overview On the other hand, a player may be potentially uncertain about external circumstances, such as whether a job candidate is smart and industrious or whether a product is reliable. Part III’s Lessons 3 and 4 considers games where players’ interests are aligned enough that they want to reveal information but they conflict enough that it is costly to credibly reveal information. For example, a job candidate cannot simply state “I am smart and industrious”, but he must signal that by mastering hard classes in college. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

6. Overview BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

7. Example 1: Unpredictable Actions Example 1: Unpredictable Actions BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

8. Example 1: Unpredictable Actions Overview Unpredictable Actions in games of simultaneous moves occur when there is not a unique dominance solution and not a unique Nash equilibrium. Any one-shot unpredictable action is acceptable. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

9. Example 1: Unpredictable Actions Comment: Bob Gustavson, professor of health science and men's soccer coach at John Brown University in Siloam Springs, Arkansas, says “When you consider that a ball can be struck anywhere from 60-80 miles per hour, there's not a whole lot of time for the goalkeeper to react”. Gustavson says skillful goalies use cues from the kicker. They look at where the kicker's plant foot is pointing and the posture during the kick. Some even study tapes of opponents. But most of all they take a guess — jump left or right at the same time the kicker is committing himself to kicking left or right. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

10. Example 1: Unpredictable Actions Question: Consider a penalty kick in soccer. The goalie either jumps left or right at the same time that the kicker either kicks left or right. The kicker’s payoffs are the probability of him scoring, and the goalie’s payoffs are the probability of the kicker not scoring. Those actions and payoffs define a normal form for this Penalty Kick Game. Try to predict strategies or recommend strategies. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

11. Example 1: Unpredictable Actions • Answer: To predict actions or • recommend actions, since • the game has simultaneous • moves, seek a solution in • three steps: • Eliminate dominated actions. That does not help here since there are no dominated actions. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

12. Example 1: Unpredictable Actions • 2) Eliminate actions that are • not rationalizeable. That • does not help here since • each action is rationalizeable • (each action is a best response to some action of the other player). • On the one hand, it is rational to kick left if the Kicker believes the Goalie jumps right. On the other hand, it is rational for the Kicker to kick right if he believes the Goalie jumps left. • Likewise, it is rational for the Goalie to jump left if the Goalie believes the Kicker kicks left, and it is rational for the Goalie to jump right if the Goalie believes the Kicker kicks right. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

13. Example 1: Unpredictable Actions 3) Look for a Nash equilibrium in pure strategies (that is an action for each player in which each player’s action is a best response to the known action by the other player). That does not help here since there is no Nash equilibrium. If the Kicker were known to kick Left, the Goalie guards Left. But if the Goalie were known to guard Left, the Kicker kicks Right. But if the Kicker were known to kick Right, the Goalie guards Right. But if the Goalie were known to guard Right, the Kicker kicks Left. So there is no Nash equilibrium. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

14. Example 1: Unpredictable Actions Since there are no dominance solutions and there are no Nash equilibria for this game of simultaneous moves, actions are unpredictable, and game theory has no recommendation; either action is acceptable. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

15. Example 2: Mixing given Perfect Conflict Example 2: Mixing given Perfect Conflict BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

16. Example 2: Mixing given Perfect Conflict Overview Mixing given Perfect Conflict in repeated games is better than a systematic choice of current actions, which makes future actions predictable and exploitable by opponents (whose interests conflict). BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

17. Example 2: Mixing given Perfect Conflict Comment: Example 1’s choices for the goalie were jump left or jump right. Call those actions because, in Example 2, strategies are going to be more complicated; they will be probabilities for taking specific actions --- say, jump left with probability 0.24 and jump right with probability 0.76 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

18. Example 2: Mixing given Perfect Conflict Question: Consider the normal form below for the Penalty Kick Game in soccer, where the goalie either jumps left or right at the same time that the kicker either kicks left or right. Predict strategies or recommend strategies if this game is repeated throughout the careers of the kicker and the goalie. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

19. Example 2: Mixing given Perfect Conflict Answer: If the game were not repeated, then since there are no dominance solutions and there are no Nash equilibria (in pure strategies) for this game of simultaneous moves, actions are unpredictable, and game theory has no recommendation; either action is acceptable. For example, the Kicker could kick Left and the Goalie could jump Left. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

20. Example 2: Mixing given Perfect Conflict But since the game is repeated, actions need to become unpredictable because predictable actions can be exploited. For example, see how predicting actions helps the Goalie. If the Kicker chooses Left predictably, the Goalie can choose Left and keep the Kicker at payoff .1 and the Goalie at .9; and if the Kicker chooses Right predictably, the Goalie can choose Right and keep the Kicker at payoff .3 and the Goalie at .7. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

21. Example 2: Mixing given Perfect Conflict The Nash equilibrium strategy for the Kicker is the mixed strategy for which the Goalie would not benefit if he could predict the Kicker’s mixed strategy. Suppose the Goalie predicts p and (1-p) are the probabilities the Kicker chooses Left or Right. The Goalie expects .9p + .6(1-p) from playing Left, and .2p + .7(1-p) from Right. The Goalie does not benefit if those payoffs equal, .9p + .6(1-p) = .2p + .7(1-p), or .6 + .3p = .7 - .5p, or p = 1/8 = 0.125 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

22. Example 2: Mixing given Perfect Conflict The Nash equilibrium strategy for the Goalie is the mixed strategy for which the Kicker would not benefit if he could predict the Goalie’s mixed strategy. Suppose the Kicker predicts q and (1-q) are the probabilities the Goalie chooses Left or Right. The Kicker expects .1q + .8(1-q) from playing Left, and .4q + .3(1-q) from Right. The Kicker does not benefit if those payoffs equal, .1q + .8(1-q) = .4q + .3(1-q), or .8 - .7q = .3 + .1q, or q = 5/8 = 0.625 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

23. Example 2: Mixing given Perfect Conflict Comment: Randomizing actions adds strategies (called mixed strategies) that solve some games that have no dominance solution or Nash Equilibrium (in pure strategies, where all probability is on one particular action). For example, in the Penalty Kick Game, there was no Nash equilibrium with pure strategies, and there were multiple rationalizable pure strategies. It turns out that most games have at least one Nash equilibrium in mixed strategies. In fact, the Penalty Kick Game has a unique Nash equilibrium in mixed strategies. While any of the rationalizable strategies would be reasonable if the game were played once, if instead the game were repeated, then strategies in the unique Nash equilibrium are the only way to play that guarantees the other player cannot gain even if they used your history to correctly predict your strategy. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

24. Example 3: Mixing given Major Conflict Example 3: Mixing given Major Conflict BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

25. Example 3: Mixing given Major Conflict Overview Mixing given Major Conflict in repeated games is still better than a systematic choice of current actions, which makes future actions predictable and exploitable by opponents (whose interests conflict). BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

26. Example 3: Mixing given Major Conflict Comment: Employers are in conflict with (selfish, amoral) workers, who want to steal or shirk (not work, or steal time). However, the Work-Shirk Game is not one of total conflict (it is not like the Penalty Kick Game) because monitoring workers costs the employer but does not help the worker. Because of the conflict, the other player exploiting your systematic choice of strategy is to your disadvantage, and so there is reason to follow mixed strategies in such games. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

27. Example 3: Mixing given Major Conflict Question: Consider the Work-Shirk Game for an employee and an employer. Suppose that at the same time the employee chooses to either work or shirk (not work) and the employer chooses to either monitor the employee or not monitor the employee. Suppose if the employee chooses to work, he loses $100 of happiness from the effort of working, but he yields $400 to his employer. Suppose the employer can monitor the employee at a cost of $80. Finally, if the employee chooses to not work and the employer chooses to monitor, then the employee is not paid, but in every other case (“work” or “not monitor”), then the employee is paid $150. Predict strategies or recommend strategies if this game is repeated daily. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

28. Example 3: Mixing given Major Conflict Answer: First, complete the normal form below for the Work-Shirk Game. For example, if the employee chooses to work and the employer chooses to monitor, then the employee loses $100 of happiness from the effort of working but is paid $150, and the employer gains $400 from his employee but pays $80 for monitoring and pays $150 to his employee. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

29. Example 3: Mixing given Major Conflict • To predict actions or • recommend actions, since • the game has simultaneous • moves and is repeated, seek • a solution in four steps: • Eliminate dominated actions. That does not help here since there are no dominated actions. • Eliminate actions that are not rationalizeable. That does not help here since each action is rationalizeable (each action is a best response to some action of the other player). BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

30. Example 3: Mixing given Major Conflict 3) Look for a Nash equilibrium in pure strategies (that is an action for each player in which each player’s action is a best response to the known action by the other player). That does not help here since there is no Nash equilibrium. If the Employee were known to Work, the Employer Trusts. But if the Employer were known to Trust, the Employee Shirks. But if the Employee were known to Shirk, the Employer Monitors. But if the Employer were known to Monitor, the Employee Works. So there is no Nash equilibrium in pure strategies. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

31. Example 3: Mixing given Major Conflict 4) Look for a Nash equilibrium in mixed strategies (that is probabilities for each player in which the other player’s expected values are equal for both of his actions; in that sense, the other player cannot exploit his knowledge of the first player’s probabilities). The Nash equilibrium strategy for the Employee is the mixed strategy for which the Employer would not benefit if he could predict the Employee’s mixed strategy. Suppose the Employer predicts p and (1-p) are the probabilities the Employee chooses Work or Shirk. The Employer expects 170p - 80(1-p) from playing Monitor, and 250p - 150(1-p) from Trust. The Employer does not benefit if those payoffs equal, 170p - 80(1-p) = 250p - 150(1-p), or -80 + 250p = -150 + 400p, or p = 70/150 = 0.467 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

32. Example 3: Mixing given Major Conflict The Nash equilibrium strategy for the Employer is the mixed strategy for which the Employee would not benefit if he could predict the Employer’s mixed strategy. Suppose the Employee predicts q and (1-q) are the probabilities the Employer chooses Monitor or Trust. The Employee expects 50q + 50(1-q) from playing Work, and 0q + 150(1-q) from Shirk. The Employee does not benefit if those payoffs equal, 50q + 50(1-q) = 0q + 150(1-q), or 50 = 150 – 150q, or q = 100/150 = 0.667 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

33. Example 4: Mixing given Minor Conflict Example 4: Mixing given Minor Conflict BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

34. Example 4: Mixing given Minor Conflict Overview Mixing given Minor Conflict in games with unpredictable actions is worse than a systematic choice that makes your actions predictable to your allies (whose interests align). BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

35. Example 4: Mixing given Minor Conflict Comment 1: Blu-ray Disc is designed to supersede the standard DVD format. The disc has the same physical dimensions as standard DVDs and CDs. The name Blu-ray Disc derives from the blue-violet laser used to read the disc. Blu-ray Disc was developed by the Blu-ray Disc Association, a group representing makers of consumer electronics, computer hardware, and motion pictures. During the format war over high-definition optical discs, Blu-ray competed with the HD DVD format. Toshiba, the main company supporting HD DVD, conceded in February 2008, and the format war ended. In late 2009, Toshiba released its own Blu-ray Disc player. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

36. Example 4: Mixing given Minor Conflict Comment 2: The format war over high-definition optical discs has The Blu-ray Disc Association in some conflict with Toshiba since each group has gained expertise and lower costs in producing a particular format and, so, each would gain if their format were universally adopted. However, the Format War game is not one of total conflict (it is not like the Penalty Kick Game) or even of major conflict (like the Work-Shirk Game) because both players loose most if neither format is universally adopted. Because conflict is less important than cooperation, the other player exploiting your systematic choice of strategy is to your advantage because you both want a universal format. So there is less reason to follow mixed strategies in such games. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

37. Example 4: Mixing given Minor Conflict Question: Consider the Format Game for The Blu-ray Disc Association and Toshiba. Suppose that at the same time each player either adopts the Blu-ray format or the HD format. Suppose if both adopt the same format, then both gain $100 million from customers that value the convenience of having a universal format. Suppose if they both adopt the Blu-ray format, then The Blu-ray Disc Association gains an extra $10 million since their expertise with that format gives them lower production costs. Finally, suppose if they both adopt the HD format, then Toshiba gains an extra $10 million since their expertise with that format gives them lower production costs. Predict strategies or recommend strategies if this game is repeated yearly. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

38. Example 4: Mixing given Minor Conflict Answer: First, complete the normal form below for the Format Game. For example, if The Blu-ray Disc Association and Toshiba both adopt HD, then both gain $100 million from customers that value the convenience of having a universal format, and Toshiba gains an extra $10 million since their expertise with the HD format gives them lower production costs. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

39. Example 4: Mixing given Minor Conflict • To predict actions or • recommend actions, since • the game has simultaneous • moves and is repeated, seek • a solution in four steps: • Eliminate dominated actions. That does not help here since there are no dominated actions. • Eliminate actions that are not rationalizeable. That does not help here since each action is rationalizeable (each action is a best response to some action of the other player). BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

40. Example 4: Mixing given Minor Conflict 3) Look for a Nash equilibrium in pure strategies (that is an action for each player in which each player’s action is a best response to the known action by the other player). There are two. On the one hand, both players choose Blu-ray; on the other hand, both players choose HD. If this were an exam question, you would only be responsible for identifying one solution to this problem, so identifying either of those Nash equilibria in pure strategies is sufficient for full credit. 4) However, for a complete analysis, note there is also a Nash equilibrium in mixed strategies. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

41. Example 4: Mixing given Minor Conflict The Nash equilibrium mixed strategy for Blu-ray Association is the mixed strategy for which Toshiba would not benefit if they could predict Blu-ray Association’s mixed strategy. Suppose Toshiba predicts p and (1-p) are the probabilities Blu-ray Association chooses Blu-ray or HD. Toshiba expects 100p + 0(1-p) from playing Blu-ray, and 0p + 110(1-p) from HD. Toshiba does not benefit if those payoffs equal, 100p + 0(1-p) = 0p + 110(1-p), or 100p = 110 - 110p, or p = 110/210 = 0.524 The expected payoff for Toshiba (whatever its strategy) is thus 100p + 0(1-p) = 0p + 110(1-p) = 52.4 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

42. Example 4: Mixing given Minor Conflict The Nash equilibrium mixed strategy for Toshiba is the mixed strategy for which Blu-ray Association would not benefit if they could predict Toshiba’s mixed strategy. Suppose Blu-ray Association predicts q and (1-q) are the probabilities Toshiba chooses Blu-ray or HD. Blu-ray Association expects 110q + 0(1-q) from playing Blu-ray, and 0q + 100(1-q) from HD. Blu-ray Association does not benefit if those payoffs equal, 110q + 0(1-q) = 0q + 100(1-q), or 110q = 100 – 100q, or q = 100/210 = 0.476 The expected payoff for Blu-ray Association (whatever its strategy) is thus 110q + 0(1-q) = 0q + 100(1-q) = 52.4 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

43. Example 4: Mixing given Minor Conflict Comment: The expected payoff of 52.4 for each player in the mixed strategy Nash equilibrium is less than if both players had agreed to one format or the other. That is a general lesson in games with only minor conflict of interest. The players are better off resolving the strategic uncertainty. The remaining lessons take up the problem of revealing information. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

44. Example 4: Mixing given Minor Conflict • Comment: Here is an ethical application of the game theory of mixed strategies given minor conflict. Consider the tragedy of Kitty Genovese. … BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

45. Example 4: Mixing given Minor Conflict • Kitty Genovese  was a New York City woman who was stabbed to death near her home in the Kew Gardens neighborhood of the borough of Queens in New York City, on March 13, 1964. Although neighbors heard her scream when she was first attacked, no one called the police. The attacker then returned and murdered her. • The circumstances of her murder and the lack of reaction of neighbors prompted investigation into the social psychological phenomenon that has become known as the bystander effect or “Genovese syndrome”and especially diffusion of responsibility. --- But game theory offers an alternative interpretation. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

46. Example 4: Mixing given Minor Conflict • Suppose the cost of each neighbor phoning the police is $1. It might seem that each neighbor valued saving Kitty at less than $1 since none of them called. But that is not necessarily so. Here is what might have happened: • The value each person places on the life of the victim could have been $1,048,577 (that is, each person would pay up to $1,048,577 of their own money if that would save Kitty). • Suppose there were 21 neighbors that heard Kitty, and each believed with probability 1/2 that any one of the other neighbors would call the police. That means that each neighbor Ned believes that there is probability ½ that any one neighbor would not call, and so there is probability (1/2)20 that none of them would call. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

47. Example 4: Mixing given Minor Conflict • Should neighbor Ned call? It is a gamble. It costs $1 to call. The payoff is that with probability (1/2)20 you are the only one to call, and so you save a life, which profits you $1,048,576 (which is $1,048,577 minus the dollar for the call). • The expected value of that gamble is (1/2)20$1,048,576, which equals $1. So, there is no net expected gain and no expected loss from making that call. • Thus, although it was a less than 1 out of a million chance, each neighbor might have not called, even though each values Kitty at more than a million dollars. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

48. Example 4: Mixing given Minor Conflict • Likewise, we may value saving someone’s life at a million dollars, yet refrain from playing the lottery for $1 with the idea that, if you win, the winnings will save that live. • Kitty Genovese’s death could have been more like a one-in-a-million freak accident, rather than the lack of concern among neighbors. Her neighbors may have valued here deeply but still behaved optimally, and should have no regrets that no one of them actually called the police. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

49. Review Questions • Review Questions • You should try to answer some of the review questions (see the online syllabus) before the next class. • You will not turn in your answers, but students may request to discuss their answers to begin the next class. • Your upcoming cumulative Final Exam will contain some similar questions, so you should eventually consider every review question before taking your exams. BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict

50. BA 445 Managerial Economics End of Lesson C.1 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict