Non-cooperative Games

Elon KohlbergFebruary 2, 2015

Matching Pennies

H T

H 1, -1 -1, 1

T -1, 1 1, -1

No Nash Equilibrium…

L R

T

1, 0 0, 3

B

0, 1 2, 0

Mixed strategies are randomizations over the pure strategies – the rows or the columns

1-rL

rR

t T 1, 0 0, 3

1-t B 0, 1 2, 0

Player 1 chooses T with probability tPlayer 2 chooses R with probability r

L R

T 1, 0 0, 3

B 0, 1 2, 0

Expected Payoffs to Player I:

T: .4 * 1 + .6 * 0 = .4

B: .4 * 0 + .6 * 2 = 1.2

Best response to r=.6 is t=0, not t=.4

Try: .3 r=.6

2/3L

1/3R

¼ T 1, 0 0, 3

¾ B 0, 1 2, 0

Expected payoff to Player

I:

T: 2/3

B: 2/3

T=1/4 is a best response

Try: t=1/4 r=1/3

Expected payoff to Player

II:

L: 3/4

R: 3/4

R=1/3 is a best response

Indifference Principle:

In a Nash equilibrium, all strategies that are assigned positive probability must have the same expected payoff.

Proof If not, switch some weight from the lower expected payoff to the higher.

Indifference Principle:

In a Nash equilibrium, all strategies that are assigned positive probability must have the same expected payoff.

Q. Is this a sufficient condition for Nash equilibrium?

No. One must also verify that no strategy assigned zero probability yields a higher payoff.

Indifference Principle:

In a Nash equilibrium, all strategies that are assigned positive probability must have the same expected payoff.

Q. Is this a sufficient condition for Nash equilibrium?

No. One must also verify that no strategy assigned zero probability yields a higher payoff.

Q. Are the two conditions sufficient for Nash equilibrium?

Yes.

Strategy Payoff

1 5.1

2 7.3

3 7.3

4 2.3

5 7.1

6 7.3

7 5.8

Equations, based on the indifference principle:

t > 0 and 1-t > 0 1-r = 2 r r = 1/3

r > 0 and 1-r > 0 1-t = 3t t = 1/4

Graphical Analysis

Consider the best response graphs:

As r goes from 0 to 1, draw the best response, t.

As t goes from 0 to 1, draw the best response, r.

Intersection?

Rationales for Mixed-strategy Equilibrium:

1. Self-enforcing beliefs

2. Frequency in a population

3. Intentional randomization

Battle of the Sexes

Football Concert

Football 2, 1 0, 0

Concert 0, 0 1, 2

WM

Battle of the Sexes

Football Concert

Football 2, 1 0, 0

Concert 0, 0 1, 2

WM

Mixed-strategy equilibrium:

M: 2/3 F; 1/3 C

W: 1/3 F; 2/3 C

Battle of the Sexes

Football Concert

Football 4, 1 0, 0

Concert 0, 0 1, 4

WM

Battle of the Sexes

Football Concert

Football 4, 1 0, 0

Concert 0, 0 1, 4

WM

Mixed-strategy equilibrium:

M: 4/5 F; 1/5 C

W: 1/5 F; 4/5 C

Swerve Forward

Swerve 6, 6 4, 7

Forward 7, 4 0, 0

Chicken

Swerve Forward

Swerve 6, 6 4, 7

Forward 7, 4 0, 0

Mixed-Strategy Equilibrium

S=.8 F=.2(Makes sense: Pure strategy equilibria require breaking the symmetry…)

Probability of collision: 4%

Chicken

Swerve Forward

Swerve 6, 6 2, 12

Forward 12, 2 0, 0

Chicken

Swerve Forward

Swerve 6, 6 2, 12

Forward 12, 2 0, 0

Chicken

Mixed-Strategy Equilibrium S=.25 F=.75

Probability of collision: 56%

Stag Hunt

Stag Hare

Stag 5, 5 0, 3

Hare 3, 0 3, 3

The norm could be to go after the stag.The norm could be to go after the hare.

Q. What is the meaning of the mixed-strategy equilibrium?

Q. What probability does it assign to Stag?

Q. How would this probability change if the value of a stag were 10 rather than 5?

Pollution Game

C N

C 2, 2, 2 2, 3, 2

N 3, 2, 2 0, 0, -1

C N

C 2, 2, 3 -1, 0, 0

N 0, -1, 0 0, 0, 0

III

III

III

C N

Pollution Game

C N

C 2, 2, 2 2, 3, 2

N 3, 2, 2 0, 0, -1

Pure- strategy Nash Equilibria(i) No one cleans (ii) Two out of three clean

C N

C 2, 2, 3 -1, 0, 0

N 0, -1, 0 0, 0, 0

III

III

III

C N

Pollution Game

C N

C 2, 2, 2 2, 3, 2

N 3, 2, 2 0, 0, -1

Mixed- strategy Nash Equilibria(iii) One cleans for sure; other two clean with probability 2/3.(iv) Two symmetric equilibria.

C N

C 2, 2, 3 -1, 0, 0

N 0, -1, 0 0, 0, 0

III

III

III

C N