1 1 Lesson overview BA 592 Lesson I.6 Simultaneous Move Problems Chapter 4 Simultaneous Move Games with Pure Strategies … Lesson I.5 Simultaneous Move Theory Lesson I.6 Simultaneous Move Problems Each Example Game Introduces some Game Theory Problems Example 1: Pure CoordinationExample 1: Pure Coordination Example 2: AssuranceExample 2: Assurance Example 3: Battle of the SexesExample 3: Battle of the Sexes Example 4: ChickenExample 4: Chicken Example 5: No Equilibrium in Pure StrategiesExample 5: No Equilibrium in Pure Strategies Practice ExamplesPractice Examples Lesson I.7 Simultaneous Move Applications

2 2 BA 592 Lesson I.6 Simultaneous Move Problems Coordination Games have multiple Nash equilibria even after any dominated strategies are eliminated. Such games are hard problems to solve with game theory. Example 1: Pure Coordination

3 3 BA 592 Lesson I.6 Simultaneous Move Problems Coordination Games can be solved if the players can communicate and can agree on one Nash equilibrium. By definition of Nash equilibrium, the agreement is self enforcing: each side has no reason to break the agreement if they believe the other side will keep the agreement. Coordination games can be solved even if agreements are impossible. All that is required is the convergence (focusing) of beliefs about other players’ strategies on a focal point. Specifically, first recognize that players are, in fact, playing with all humanity, past and present, in one large game from the beginning of time. Hence, the game currently considered is only a subgame. In particular, players may have historic actions and outcomes to focus their expectations about the strategies of other players on a focal point. Example 1: Pure Coordination

4 4 BA 592 Lesson I.6 Simultaneous Move Problems Pure Coordination Games are those coordination games with equal payoffs for each Nash equilibrium. Agreements on one Nash equilibrium are simple in pure coordination games since no player cares which equilibrium is selected. Example 1: Pure Coordination

5 5 BA 592 Lesson I.6 Simultaneous Move Problems Harry and Sally meet when she gives him a ride to New York after they both graduate from the University of Chicago. They agree to meet at 7:00 at Joe's Shanghai Chinese Food Restaurant in New York. At 6:45, both remember that Joe has two restaurants, one in the Flatiron District and one in the Theater District. Define the normal form for this Pure Coordination Game, then predict an equilibrium if Harry and Sally cannot communicate further to agree on the particular restaurant. Example 1: Pure Coordination

6 6 BA 592 Lesson I.6 Simultaneous Move Problems If the Flatiron District and the Theater District are equally distant and equally desireable, then here is a normal form consistent with the data: Example 1: Pure Coordination

7 7 BA 592 Lesson I.6 Simultaneous Move Problems There are no dominate or dominated strategies, and there are two Nash equilibria. Harry and Sally should think about which of the two districts would naturally come to mind. If, say, they had previously discussed the theater, then they should choose the restaurant in the theater district. Example 1: Pure Coordination

8 8 BA 592 Lesson I.6 Simultaneous Move Problems Assurance Games are those coordination games where one of the Nash equilibria is preferred by all players. Thus, each player would select the jointly-preferred equilibrium strategy if they could be assured all other players will do likewise. Agreements on one Nash equilibrium are simple in pure coordination games since each player prefers the same equilibrium. If agreements cannot be communicated, the preferred equilibrium can be a natural focal point. Example 2: Assurance

9 9 BA 592 Lesson I.6 Simultaneous Move Problems Philosopher Jean-Jacques Rousseau described two individuals going out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, he must have the cooperation of his partner in order to succeed. An individual can get a hare by himself, but a hare is worth less than his share of a stag. This is taken to be an important analogy for social cooperation. Define a normal form for this Stag Hunt Game, then predict an equilibrium. Example 2: Assurance

10 BA 592 Lesson I.6 Simultaneous Move Problems Here is a normal form consistent with the data: On the one hand, the preferred outcome is, by definition, a natural focal point. On the other hand, players may have a mutual history of watching Bugs Bunny, which could focus their expectations about the Hare strategy. Example 2: Assurance

11 BA 592 Lesson I.6 Simultaneous Move Problems Another example of successful cooperation in a “stag hunt” is the hunting practice of orcas (known as carousel feeding). Orcas cooperatively corral large schools of fish to the surface and stun them by hitting them with their tails. Since this requires that the fish have no way to escape, it requires the cooperation of many orcas. Example 2: Assurance

12 BA 592 Lesson I.6 Simultaneous Move Problems Battle of the Sexes Games are those coordination games where one of the Nash equilibria is preferred by one player and the other equilibrium by the other players, and where all equilibria involve the players choosing the same strategy. In particular, each player would select their preferred-equilibrium strategy if they could be assured the other player will choose the same equilibrium. Example 3: Battle of the Sexes

13 BA 592 Lesson I.6 Simultaneous Move Problems Agreements on one Nash equilibrium are complicated in Battle of the Sexes Games since each player prefers a different equilibrium, so any agreement could be rejected as unfair. If agreements are impossible, finding a focal point is also more complicated because there is no jointly-preferred equilibrium to focus beliefs. Reputation becomes important: if players have a mutual history of one player dominating or playing tough, players could focus their expectations on the equilibrium that most benefits that player. Another solution is a player strategically committing to his preferred-equilibrium strategy, or strategically eliminating some alternative strategies. Example 3: Battle of the Sexes

14 BA 592 Lesson I.6 Simultaneous Move Problems A couple agreed to meet this evening, but cannot recall if they will be attending the opera or a football game. The husband would most of all like to go to the football game. The wife would like to go to the opera. Both would prefer to go to the same place rather than different ones. If they cannot communicate, where should they go? Define a normal form for this Battle of the Sexes Game, then predict an equilibrium. Example 3: Battle of the Sexes

15 BA 592 Lesson I.6 Simultaneous Move Problems Here is a normal form consistent with the data: There are two Nash equilibria, either of which can be obtained by agreement. If no such agreement is possible or acceptable, then the Football equilibrium can be a focal point if the husband has a reputation for toughness, or the Opera equilibrium if the wife has a reputation for toughness. Or, the husband can commit to the Football equilibrium by strategically eliminating his Opera strategy by breaking his glasses, and letting his wife know. Example 3: Battle of the Sexes

16 BA 592 Lesson I.6 Simultaneous Move Problems Chicken Games are the same as Battle of the Sexes Games except all equilibria involve the players choosing different strategies. (Some call such games anti-coordination games.) In particular, each player would select their preferred-equilibrium strategy if they could be assured the other player will choose the same equilibrium. Example 4: Chicken

17 BA 592 Lesson I.6 Simultaneous Move Problems Solving Chicken Games has the same complications and possibilities as solving Battle of the Sexes Games: Agreements on one Nash equilibrium are complicated since each player prefers a different equilibrium, and finding a focal point is complicated because there is no jointly-preferred equilibrium to focus beliefs. Reputation for toughness or strategic commitment can possibly solve Chicken games. Example 4: Chicken

18 BA 592 Lesson I.6 Simultaneous Move Problems Chicken is an influential model of conflict for two players. The principle of the game is that while each player prefers not to yield to the other, the outcome where neither player yields is the worst possible one for both players. The name "Chicken" has its origins in a game in which two drivers drive towards each other on a collision course: one must swerve, or both may die in the crash, but if one driver swerves and the other does not, the one who swerved will be called a “chicken”. The game has also been used to describe the mutual assured destruction of nuclear warfare. Define a normal form for this Chicken Game for Speed Racer and Racer X, then predict an equilibrium. Example 4: Chicken

19 BA 592 Lesson I.6 Simultaneous Move Problems Here is a normal form consistent with the data: There are two Nash equilibria, either of which can be obtained by agreement. If no such agreement is possible or acceptable, then Straight-Swerve can be a focal point if the Speed has a reputation for toughness, or Swerve-Straight if Racer has a reputation for toughness. Or, Speed can commit to the Straight-Swerve equilibrium by strategically eliminating his Swerve strategy by tying his steering wheel, and letting Racer X know. Example 4: Chicken

20 BA 592 Lesson I.6 Simultaneous Move Problems Strategic Uncertainty persists in those games that have no Nash equilibrium in pure strategies. Example 5: No Equilibrium in Pure Strategies

21 BA 592 Lesson I.6 Simultaneous Move Problems 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 after the kicker has committed himself. Define a normal form for this Soccer Game, then try to predict an equilibrium. Example 5: No Equilibrium in Pure Strategies

22 BA 592 Lesson I.6 Simultaneous Move Problems There is no Nash equilibrium! If the Kicker is known to kick Left, the Goalie guards Left. But if the Goalie is known to guard Left, the Kicker kicks Right. But if the Kicker is known to kick Right, the Goalie guards Right. But if the Goalie is known to guard Right, the Kicker kicks Left. An so on. So strategic uncertainty persists about kicking and guarding. Example 4: Chicken Here is a normal form consistent with the data, with payoffs in probability of scoring:

23 End of Lesson I.6 BA 592 Game Theory BA 592 Lesson I.6 Simultaneous Move Problems