Game Theory
Lecture 2
Review
Prisoner’s Dilemma
Chicken
Matching Pennies
The Prisoner’s Dilemma
Colin and Anselm are arrested by the police for suspected delinquency. The police
have insufficient evidence for a conviction, and, having separated them, visit each of
them to offer the same deal: if one confesses to delinquency, then he goes free and the
other receives five-year sentence. If neither confess, then they are both are sentenced to
one year for the lesser charge of missing class. If each confesses, then they both receive
two year sentences. They must both make their decisions as to whether to confess or not
simultaneously, without knowledge of the other’s decision.
Anselm
N
C
Colin
N
C
-1/2, -1/2
-5, 0
0, -5
-2, -2
The possible decisions of the players are called strategies in game theory. The numbers
in each box are the utilities, called payoffs, that the players receive when the
corresponding pair of strategies is played. For example, if Anselm does not confess and
Colin does, then the outcome will be (-5,0), so Anselm will get 5 years in prison and
Colin will walk free. Players in game theory are assumed to be only interested in their
own payoffs.
Players: Anselm, Colin
Anselm’s Strategies: N, C
Anselm’s Payoffs: -5, -2, -1, 0
Colin’s Strategies: N, C
Colin’s Payoffs: -5, -2, -1, 0
Outcomes: (N, N) (N, C) (C, N) (C, C)
Nash equilibrium is (C, C)
Work through sequential game
Chicken
After years of fierce competition, Nick and Trace have agreed to have a competition
to determine who the better man is. They are going to take their cars and go to opposite
ends of Vineyards Boulevard and drive their cars straight at each other as fast as they can.
The better man will be determined by whoever has the most nerve and forces the other
driver to swerve from the road first. The one who swerves will be the “chicken.” Neither
Nick nor Trace wants to be the chicken. If they both swerve, then they will both be
chickens, but the humiliation when they both swerve is not as great as when only one
swerves. However, if neither man swerves, then the game ends in a fiery wreck. Their
payoffs are listed below.
Trace
Nick
C
S
C
0, 0
1, -1
S
-1, 1
-10, -10
Players: Nick, Trace
Nick’s Strategies: C, S
Nick’s Payoffs: -10, -1, 0, 1
Trace’s Strategies: C, S
Trace’s Payoffs: -10, -1, 0, 1
Outcomes: (C, C) (C, S) (S, C) (S, S)
Nash equilibria in pure strategies are (C, S) and (S, C)
Work through sequential game
Sequential game has first-mover advantage
Matching Pennies
Johanna and Alex love to gamble. Their favorite game is called matching pennies, in
which the each player takes a penny and chooses heads or tails without the other
knowing; if both pennies match when they are revealed, then Johanna wins; otherwise,
Alex wins.
Alex
Johanna
H
T
Players: Johanna, Alex
Johanna’s Strategies: H, T
Johanna’s Payoffs: -1, 1
Alex’s Strategies: H, T
Alex’s Payoffs: -1, 1
Outcomes: (H, H) (H, T) (T, H) (T, T)
No Nash equilibrium in pure strategies
Work through sequential game
Sequential game has second-mover advantage
H
1, -1
-1, 1
T
-1, 1
1, -1