1 1 BA 210 Lesson III.5 Strategic Uncertainty when Interests ConflictOverviewOverview

2 2 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 210 Lesson III.5 Strategic Uncertainty when Interests ConflictOverview

3 3 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 5 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 6 considers games where players easily reveal information about their own moves because players’ interests align (as in setting a standard industry format). BA 210 Lesson III.5 Strategic Uncertainty when Interests ConflictOverview

4 4 Lesson III.5 Strategic Uncertainty when Interests Conflict Example 1: Unpredictable Actions Example 2: Mixing with Perfect Conflict Example 3: Mixing with Major Conflict Example 4: Mixing with Minor Conflict Summary Review Questions Lesson Overview

5 5 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict Example 1: Unpredictable Actions

6 6 BA 210 Lesson III.5 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

7 7 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict Question: Consider a penalty kick in soccer. The goalie either jumps left or right after the kicker has committed himself to kicking 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

8 8 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict Answer: To predict actions or recommend actions, since the game has simultaneous moves, first look for dominate or dominated actions. There are none. Then look for a Nash equilibrium. There is none. 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

9 9 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict Finally, look to see if any action can be eliminated because it is not rationalizable (that is, it is not a best response to some action by the other player. But all actions are rationalizable. 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

10 BA 210 Lesson III.5 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

11 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict Example 2: Mixing with Perfect Conflict

12 BA 210 Lesson III.5 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 with Perfect Conflict

13 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict Question: Consider the normal form below for the Penalty Kick Game in soccer. Predict strategies or recommend strategies if this game is repeated throughout the careers of the kicker and the goalie. Example 2: Mixing with Perfect Conflict

14 BA 210 Lesson III.5 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 with Perfect Conflict

15 BA 210 Lesson III.5 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 with Perfect Conflict

16 BA 210 Lesson III.5 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 with Perfect Conflict

17 BA 210 Lesson III.5 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 with Perfect Conflict

18 BA 210 Lesson III.5 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 with Perfect Conflict

19 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict Example 3: Mixing with Major Conflict

20 BA 210 Lesson III.5 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 with Major Conflict

21 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict Question: Consider the Work-Shirk Game for an employee and an employer. Suppose if the employee chooses to work, he looses $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 with Major Conflict

22 BA 210 Lesson III.5 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 looses $100 of happiness from the effort of working but is paid $150, and the employer gain $400 from his employer but pays $80 for monitoring and pays $150 to his employee. Example 3: Mixing with Major Conflict

23 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict To predict actions or recommend actions, since the game has simultaneous moves, first look for dominate or dominated actions. There are none. Then look for a Nash equilibrium in pure strategies. There is none. 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. Example 3: Mixing with Major Conflict

24 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict Since the game is repeated, actions need to become unpredictable because predictable actions can be exploited. 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 with Major Conflict

25 BA 210 Lesson III.5 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 with Major Conflict

26 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict Example 4: Mixing with Minor Conflict

27 BA 210 Lesson III.5 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 with Minor Conflict

28 BA 210 Lesson III.5 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 with Minor Conflict

29 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict Question: Consider the Format Game for The Blu-ray Disc Association and Toshiba. Suppose 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 with Minor Conflict

30 BA 210 Lesson III.5 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 with Minor Conflict

31 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict To predict actions or recommend actions, since the game has simultaneous moves, first look for dominate or dominated actions. There are none. Then look for a Nash equilibrium in pure strategies. There are two. On the one hand, both players choose Blu-ray; on the other than, both players choose HD. There is also a Nash equilibrium in mixed strategies. Example 4: Mixing with Minor Conflict

32 BA 210 Lesson III.5 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 with Minor Conflict

33 BA 210 Lesson III.5 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 with Minor Conflict

34 BA 210 Lesson III.5 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 with Minor Conflict

35 BA 210 Lesson III.5 Strategic Uncertainty when Interests ConflictSummarySummary

36 Review Questions BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict Review Questions  You should try to answer some of the following questions 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.

37 End of Lesson III.5 BA 210 Lesson III.5 Strategic Uncertainty when Interests Conflict BA 210 Introduction to Microeconomics