Transcript Game Theory - Meet the Faculty

Lesson overview
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 Coordination
• Example 2: Assurance
• Example 3: Battle of the Sexes
• Example 4: Chicken
• Example 5: No Equilibrium in Pure Strategies
• Practice Examples
Lesson I.7 Simultaneous Move Applications
BA 592 Lesson I.6 Simultaneous Move Problems
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Example 1: Pure Coordination
Coordination Games have multiple Nash equilibria even after any
dominated strategies are eliminated. Such games are hard
problems to solve with game theory.
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Example 1: Pure Coordination
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.
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Example 1: Pure Coordination
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.
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Example 1: Pure Coordination
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.
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Example 1: Pure Coordination
If the Flatiron District and the Theater District are equally distant
and equally desireable, then here is a normal form consistent with
the data:
Sally
Harry
Flatiron
Theater
Flatiron
1,1
0,0
BA 592 Lesson I.6 Simultaneous Move Problems
Theater
0,0
1,1
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Example 1: Pure Coordination
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.
Sally
Harry
Flatiron
Theater
Flatiron
1,1
0,0
BA 592 Lesson I.6 Simultaneous Move Problems
Theater
0,0
1,1
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Example 2: Assurance
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.
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Example 2: Assurance
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.
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Example 2: Assurance
Here is a normal form consistent with the data:
Hunter 2
Hunter 1
Stag
Hare
Stag
2,2
1,0
Hare
0,1
1,1
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.
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Example 2: Assurance
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.
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Example 3: Battle of the Sexes
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.
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Example 3: Battle of the Sexes
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.
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Example 3: Battle of the Sexes
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.
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Example 3: Battle of the Sexes
Here is a normal form consistent with the data:
Wife
Football
Football
3,2
Husband
Opera
0,0
Opera
0,0
2,3
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.
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Example 4: Chicken
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.
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Example 4: Chicken
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.
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Example 4: Chicken
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.
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Example 4: Chicken
Here is a normal form consistent with the data:
Racer X
Speed
Straight
Swerve
Straight
0,0
1,3
Swerve
3,1
2,2
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.
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Example 5: No Equilibrium in Pure Strategies
Strategic Uncertainty persists in those games that have no Nash
equilibrium in pure strategies.
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Example 5: No Equilibrium in Pure Strategies
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 after the kicker has committed himself.
Define a normal form for this Soccer Game, then try to predict an
equilibrium.
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Example 4: Chicken
Here is a normal form consistent with the data, with payoffs in
probability of scoring:
Goalie
Kicker
Left
Right
Left
.1,.9
.7,.3
Right
.8,.2
.3,.7
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.
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BA 592
Game Theory
End of Lesson I.6
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