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

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Presentation on theme: "1 1 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict ReadingsReadings Baye “Mixed Strategies” (see the index) Dixit Chapter 7."— Presentation transcript:

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

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

3 3 3 Part III Overview Game 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 Overview

4 4 4 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 Overview

5 5 5 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 Overview

6 6 6Overview 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. 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). 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). 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).

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

8 8 8 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict 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. Example 1: Unpredictable Actions

9 9 9 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict 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 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. Example 1: Unpredictable Actions

10 10 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict 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. Example 1: Unpredictable Actions

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

12 12 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict 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. Example 1: Unpredictable Actions

13 13 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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 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. Example 1: Unpredictable Actions

14 14 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict 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. Example 1: Unpredictable Actions

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

16 16 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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). Example 2: Mixing given Perfect Conflict

17 17 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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 Example 2: Mixing given Perfect Conflict

18 18 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 2: Mixing given Perfect Conflict

19 19 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 2: Mixing given Perfect Conflict

20 20 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 2: Mixing given Perfect Conflict

21 21 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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 = Example 2: Mixing given Perfect Conflict

22 22 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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 = Example 2: Mixing given Perfect Conflict

23 23 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 2: Mixing given Perfect Conflict

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

25 25 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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). Example 3: Mixing given Major Conflict

26 26 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 3: Mixing given Major Conflict

27 27 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 3: Mixing given Major Conflict

28 28 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 3: Mixing given Major Conflict

29 29 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict To predict actions or recommend actions, since the game has simultaneous moves and is repeated, seek a solution in four steps: 1)Eliminate dominated actions. That does not help here since there are no dominated 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). Example 3: Mixing given Major Conflict

30 30 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 3: Mixing given Major Conflict

31 31 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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 p = p, or p = 70/150 = Example 3: Mixing given Major Conflict

32 32 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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 = Example 3: Mixing given Major Conflict

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

34 34 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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). Example 4: Mixing given Minor Conflict

35 35 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 4: Mixing given Minor Conflict

36 36 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 4: Mixing given Minor Conflict

37 37 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 4: Mixing given Minor Conflict

38 38 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 4: Mixing given Minor Conflict

39 39 BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict To predict actions or recommend actions, since the game has simultaneous moves and is repeated, seek a solution in four steps: 1)Eliminate dominated actions. That does not help here since there are no dominated 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). Example 4: Mixing given Minor Conflict

40 40 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 4: Mixing given Minor Conflict

41 41 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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 = p, or p = 110/210 = The expected payoff for Toshiba (whatever its strategy) is thus 100p + 0(1-p) = 0p + 110(1-p) = 52.4 Example 4: Mixing given Minor Conflict

42 42 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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 = The expected payoff for Blu-ray Association (whatever its strategy) is thus 110q + 0(1-q) = 0q + 100(1-q) = 52.4 Example 4: Mixing given Minor Conflict

43 43 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 4: Mixing given Minor Conflict

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

45 45 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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, 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. Example 4: Mixing given Minor Conflict

46 46 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 4: Mixing given Minor Conflict

47 47 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 4: Mixing given Minor Conflict

48 48 BA 445 Lesson C.1 Strategic Uncertainty when Interests 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. Example 4: Mixing given Minor Conflict

49 49 Review Questions BA 445 Lesson C.1 Strategic Uncertainty when Interests Conflict 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.

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


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