Download The Penalty Kick Zero-Sum Games Main theorem: There exists a

Economics 2020b Lecture 6:
The Penalty Kick
Zero-Sum Games
Main theorem:
There exists a number, 𝑉, such that Player 1 can
guarantee to receive at least 𝑉, and Player 2 can
guarantee to pay no more than 𝑉.
(If the game is constant-sum, where the sum of the two
players’ payoffs is always the same number, 𝑐, then:
There exists a number, 𝑉, such that Player 1 can
guarantee 𝑉 and Player 2 can guarantee 𝑉 − 𝑐.)
Proof:
In a Nash equilibrium of a zero-sum game, each player’s
equilibrium strategy guarantees the player’s
equilibrium payoff.
Conclusion:
In a zero-sum game
- The expected outcome is determined!
- In some cases, e.g., if the game is 2 × 2, the
probabilities over the pure strategies are also
determined!
The penalty kick in soccer
Player 1: Striker
Player 2: Goalie
Payoff to player 1:
probability of goal
Payoff to player 2: - probability of goal
L
R
l
.2
.9
r
.8
.5
You are an outside observer betting on the outcome:
Q. What is the probability of a goal?
Q. What is the probability of a dive to the Right?
Numerical analysis:
L
R
l
.2
.9
r
.8
.5
Graphical analysis
0
L
R
l
.2
.9
r
.8
.5
1
Varying the parameters of the game
The goalie improves her ability to block the shot when
she dives to the Right.
Game changes from
L
R
l
.2
.9
r
.8
.5
to
L
R
l
.2
.9
r
.8
.3
What do you expect?
- Probability of a goal will decrease?
- Probability of dive to the right will increase?
Graphical analysis
L
R
l
.2
.9
r
.8
.5
0
to
L
R
l
.2
.9
r
.8
.3
1