1 1 Lesson overview BA 592 Lesson I.6 Simultaneous Move Problems Chapter 4 Simultaneous Move Games...

1 1

Lesson overviewLesson overview

BA 592 Lesson I.6 Simultaneous Move Problems

Chapter 4 Simultaneous Move Games with Pure Strategies …Chapter 4 Simultaneous Move Games with Pure Strategies …

Lesson I.5 Simultaneous Move TheoryLesson I.5 Simultaneous Move Theory

Lesson I.6 Simultaneous Move ProblemsLesson I.6 Simultaneous Move Problems

Each Example Game Introduces some Game Theory ProblemsEach 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 ApplicationsLesson I.7 Simultaneous Move Applications

2 2BA 592 Lesson I.6 Simultaneous Move Problems

Coordination Games Coordination Games havehave multiple Nash equilibria even after any multiple Nash equilibria even after any dominated strategies are eliminated. Such games are hard dominated strategies are eliminated. Such games are hard problems to solve with game theory. problems to solve with game theory.

Example 1: Pure CoordinationExample 1: Pure Coordination


Coordination Games Coordination Games can be solved if the players can can be solved if the players can communicate and can agree on one Nash equilibrium. By communicate and can agree on one Nash equilibrium. By definition of Nash equilibrium, the definition of Nash equilibrium, the agreement is self enforcing: agreement is self enforcing: each side has no reason to break the agreement if they believe the each side has no reason to break the agreement if they believe the other side will keep the agreement. other side will keep the agreement.

Coordination games can be solved even if agreements are Coordination games can be solved even if agreements are impossible. All that is required is the convergence (focusing) of impossible. All that is required is the convergence (focusing) of beliefs about other players’ strategies on a focal point. beliefs about other players’ strategies on a focal point. Specifically, first recognize that players are, in fact, playing with Specifically, first recognize that players are, in fact, playing with all humanity, past and present, in one large game from the all humanity, past and present, in one large game from the beginning of time. Hence, the game currently considered is only beginning of time. Hence, the game currently considered is only a subgame. In particular, players may have historic actions and a subgame. In particular, players may have historic actions and outcomes to focus their expectations about the strategies of other outcomes to focus their expectations about the strategies of other players on a players on a focal pointfocal point. .



Pure Coordination Games Pure Coordination Games are those coordination games with are those coordination games with equal payoffs for each Nash equilibrium. equal payoffs for each Nash equilibrium.

Agreements on one Nash equilibrium are simple in pure Agreements on one Nash equilibrium are simple in pure coordination games since no player cares which equilibrium is coordination games since no player cares which equilibrium is selected. selected.



Harry and Sally Harry and Sally meet when she gives him a ride to New York meet when she gives him a ride to New York after they both graduate from the University of Chicago. They after they both graduate from the University of Chicago. They agree to meet at 7:00 at Joe's Shanghai Chinese Food Restaurantagree to meet at 7:00 at Joe's Shanghai Chinese Food Restaurantin New York. At 6:45, both remember that Joe has two in New York. At 6:45, both remember that Joe has two restaurants, one in the Flatiron District and one in the Theater restaurants, one in the Flatiron District and one in the Theater District.District.

Define the normal form for this Define the normal form for this Pure Coordination GamePure Coordination Game, then , then predict an equilibrium predict an equilibrium if Harry and Sally cannot communicate if Harry and Sally cannot communicate further to agree on the particular restaurantfurther to agree on the particular restaurant..



If the Flatiron District and the Theater District are equally distant If the Flatiron District and the Theater District are equally distant and equally desireable, then here is a and equally desireable, then here is a normal form consistent with normal form consistent with the data:the data:


Flatiron TheaterFlatiron 1,1 0,0Theater 0,0 1,1

Sally

Harry

7 7

Flatiron TheaterFlatiron 1,1 0,0Theater 0,0 1,1

Sally

Harry


There are no dominate or dominated strategies, and there are two There are no dominate or dominated strategies, and there are two Nash equilibria. Harry and Sally should think about which of the Nash equilibria. Harry and Sally should think about which of the two districts would naturally come to mind. If, say, they had two districts would naturally come to mind. If, say, they had previously discussed the theater, then they should choose the previously discussed the theater, then they should choose the restaurant in the theater district.restaurant in the theater district.



Assurance Games Assurance Games are those coordination games where one of the are those coordination games where one of the Nash equilibria is preferred by all players. Thus, each player Nash equilibria is preferred by all players. Thus, each player would select the jointly-preferred equilibrium strategy if they would select the jointly-preferred equilibrium strategy if they could be could be assuredassured all other players will do likewise. all other players will do likewise.

Agreements on one Nash equilibrium are simple in pure Agreements on one Nash equilibrium are simple in pure coordination games since each player prefers the same coordination games since each player prefers the same equilibrium. equilibrium.

If agreements cannot be communicated, the preferred equilibrium If agreements cannot be communicated, the preferred equilibrium can be a natural focal point. can be a natural focal point.

Example 2: AssuranceExample 2: Assurance


Philosopher Jean-Jacques Rousseau Philosopher Jean-Jacques Rousseau described two individuals described two individuals going out on a hunt. Each can individually choose to hunt a stag going out on a hunt. Each can individually choose to hunt a stag or hunt a hare. Each player must choose an action without or hunt a hare. Each player must choose an action without knowing the choice of the other. If an individual hunts a stag, he 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 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 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 his share of a stag. This is taken to be an important analogy for social cooperation.social cooperation.

Define a normal form for this Define a normal form for this Stag Hunt GameStag Hunt Game, then predict an , then predict an equilibrium.equilibrium.



Here is a Here is a normal form consistent with the data:normal form consistent with the data:

Stag HareStag 2,2 0,1Hare 1,0 1,1

Hunter 2

Hunter 1

On the one hand, the preferred outcome is, by definition, a On the one hand, the preferred outcome is, by definition, a natural focal point. On the other hand, players may have a natural focal point. On the other hand, players may have a mutual history of watching Bugs Bunny, which could focus their mutual history of watching Bugs Bunny, which could focus their expectations about the Hare strategy.expectations about the Hare strategy.



Another example of successful cooperation in a “stag hunt” is the Another example of successful cooperation in a “stag hunt” is the hunting practice of orcas (known as carousel feeding). Orcas hunting practice of orcas (known as carousel feeding). Orcas cooperatively corral large schools of fish to the surface and stun cooperatively corral large schools of fish to the surface and stun them by hitting them with their tails. Since this requires that the them by hitting them with their tails. Since this requires that the fish have no way to escape, it requires the cooperation of many fish have no way to escape, it requires the cooperation of many orcas.orcas.



Battle of the Sexes Games Battle of the Sexes Games are those coordination games where are those coordination games where one of the Nash equilibria is preferred by one player and the one of the Nash equilibria is preferred by one player and the other equilibrium by the other players, and where all equilibria other equilibrium by the other players, and where all equilibria involve the players choosing the same strategy. In particular, involve the players choosing the same strategy. In particular, each player would select their preferred-equilibrium strategy if each player would select their preferred-equilibrium strategy if they could be assured the other player will choose the same they could be assured the other player will choose the same equilibrium. equilibrium.

Example 3: Battle of the SexesExample 3: Battle of the Sexes


AgreementsAgreements on one Nash equilibrium are complicated in Battle of on one Nash equilibrium are complicated in Battle of the Sexes Games since each player prefers a different the Sexes Games since each player prefers a different equilibrium, so any agreement could be rejected as unfair. equilibrium, so any agreement could be rejected as unfair.

If agreements are impossible, finding a focal point is also more If agreements are impossible, finding a focal point is also more complicated because there is no jointly-preferred equilibrium to complicated because there is no jointly-preferred equilibrium to focus beliefs. focus beliefs. ReputationReputation becomes important: if players have a becomes important: if players have a mutual history of one player dominating or playing tough, mutual history of one player dominating or playing tough, players could focus their expectations on the equilibrium that players could focus their expectations on the equilibrium that most benefits that player.most benefits that player.

Another solution is a player Another solution is a player strategically committingstrategically committing to his to his preferred-equilibrium strategy, or strategically eliminating some preferred-equilibrium strategy, or strategically eliminating some alternative strategies.alternative strategies.



A coupleA couple agreed to meet this evening, but cannot recall if they agreed to meet this evening, but cannot recall if they will be attending the opera or a football game. The husband 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 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 like to go to the opera. Both would prefer to go to the same place rather than different ones. If they cannot communicate, where rather than different ones. If they cannot communicate, where should they go?should they go?

Define a normal form for this Define a normal form for this Battle of the Sexes GameBattle of the Sexes Game, then , then predict an equilibrium.predict an equilibrium.




Football OperaFootball 3,2 0,0

Opera 0,0 2,3

Wife

Husband

There are two Nash equilibria, either of which can be obtained by There are two Nash equilibria, either of which can be obtained by agreement. If no such agreement is possible or acceptable, then agreement. If no such agreement is possible or acceptable, then the Football equilibrium can be a focal point if the husband has a the Football equilibrium can be a focal point if the husband has a reputation for toughness, or the Opera equilibrium if the wife has reputation for toughness, or the Opera equilibrium if the wife has a reputation for toughness. Or, the husband can commit to the a reputation for toughness. Or, the husband can commit to the Football equilibrium by strategically eliminating his Opera Football equilibrium by strategically eliminating his Opera strategy by breaking his glasses, and letting his wife know. strategy by breaking his glasses, and letting his wife know.



Chicken Games Chicken Games are the same as Battle of the Sexes Games except are the same as Battle of the Sexes Games except all equilibria involve the players choosing different strategies. all equilibria involve the players choosing different strategies. (Some call such games anti-coordination games.) In particular, (Some call such games anti-coordination games.) In particular, each player would select their preferred-equilibrium strategy if each player would select their preferred-equilibrium strategy if they could be assured the other player will choose the same they could be assured the other player will choose the same equilibrium. equilibrium.

Example 4: ChickenExample 4: Chicken


Solving Chicken Games has the same complications and Solving Chicken Games has the same complications and possibilities as solving Battle of the Sexes Games: Agreements possibilities as solving Battle of the Sexes Games: Agreements on one Nash equilibrium are complicated since each player on one Nash equilibrium are complicated since each player prefers a different equilibrium, and finding a focal point is prefers a different equilibrium, and finding a focal point is complicated because there is no jointly-preferred equilibrium to complicated because there is no jointly-preferred equilibrium to focus beliefs. focus beliefs. ReputationReputation for toughness or for toughness or strategic commitmentstrategic commitment can possibly solve Chicken games.can possibly solve Chicken games.



ChickenChicken is an influential model of conflict for two players. The is an influential model of conflict for two players. The principle of the game is that while each player prefers not to yield 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 to the other, the outcome where neither player yields is the worst possible one for both players. The name "Chicken" has its possible one for both players. The name "Chicken" has its origins in a game in which two drivers drive towards each other 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 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 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 who swerved will be called a “chicken”. The game has also been used to describe the mutual assured destruction of nuclear used to describe the mutual assured destruction of nuclear warfare.warfare.

Define a normal form for this Define a normal form for this Chicken Game Chicken Game for Speed Racer for Speed Racer and Racer X, then predict an equilibrium.and Racer X, then predict an equilibrium.




Straight SwerveStraight 0,0 3,1Swerve 1,3 2,2

Racer X

Speed

There are two Nash equilibria, either of which can be obtained by There are two Nash equilibria, either of which can be obtained by agreement. If no such agreement is possible or acceptable, then agreement. If no such agreement is possible or acceptable, then Straight-Swerve can be a focal point if the Speed has a reputation Straight-Swerve can be a focal point if the Speed has a reputation for toughness, or Swerve-Straight if Racer has a reputation for for toughness, or Swerve-Straight if Racer has a reputation for toughness. Or, Speed can commit to the Straight-Swerve toughness. Or, Speed can commit to the Straight-Swerve equilibrium by strategically eliminating his Swerve strategy by equilibrium by strategically eliminating his Swerve strategy by tying his steering wheel, and letting Racer X know. tying his steering wheel, and letting Racer X know.



Strategic Uncertainty Strategic Uncertainty persists in those games that have no Nash persists in those games that have no Nash equilibrium in pure strategies.equilibrium in pure strategies.

Example 5: No Equilibrium in Pure StrategiesExample 5: No Equilibrium in Pure Strategies


Bob Gustavson,Bob Gustavson, professor of health science and men's soccer professor of health science and men's soccer coach at John Brown University in Siloam Springs, Arkansas, coach at John Brown University in Siloam Springs, Arkansas, says “When you consider that a ball can be struck anywhere from 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 60-80 miles per hour, there's not a whole lot of time for the goalkeeper to react”. Gustavson says skillful goalies use cues goalkeeper to react”. Gustavson says skillful goalies use cues from the kicker. They look at where the kicker's plant foot is from the kicker. They look at where the kicker's plant foot is pointing and the posture during the kick. Some even study tapes pointing and the posture during the kick. Some even study tapes of opponents. But most of all of opponents. But most of all they take a guess — jump left or they take a guess — jump left or right after the kicker has committed himselfright after the kicker has committed himself..

Define a normal form for this Define a normal form for this Soccer GameSoccer Game, then try to predict an , then try to predict an equilibrium.equilibrium.

Example 5: No Equilibrium in Pure StrategiesExample 5: No Equilibrium in Pure Strategies


Left RightLeft .1,.9 .8,.2

Right .7,.3 .3,.7

Goalie

Kicker

There is no Nash equilibrium! If the Kicker is known to kick 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 Goalie guards Left. But if the Goalie is known to guard Left, the Kicker kicks Right. But if the Kicker is known to kick 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 Right, the Goalie guards Right. But if the Goalie is known to guard Right, the Kicker kicks Left. An so on.guard Right, the Kicker kicks Left. An so on.

So strategic uncertainty persists about kicking and guarding. So strategic uncertainty persists about kicking and guarding.


Here is a Here is a normal form consistent with the data, with payoffs in normal form consistent with the data, with payoffs in probability of scoring:probability of scoring:

23 23

End of Lesson I.6End of Lesson I.6

BA 592 Game BA 592 Game TheoryTheory


1 1 Lesson overview BA 592 Lesson I.6 Simultaneous Move Problems Chapter 4 Simultaneous Move Games...

Documents

Transcript of 1 1 Lesson overview BA 592 Lesson I.6 Simultaneous Move Problems Chapter 4 Simultaneous Move Games...