Non-cooperative Games Elon Kohlberg February 2, 2015

Matching Pennies HT H1, -1-1, 1 T 1, -1 No Nash Equilibrium…

LR T 1, 00, 3 B 0, 12, 0

Mixed strategies are randomizations over the pure strategies – the rows or the columns 1-r L rRrR t T1, 00, 3 1-t B0, 12, 0 Player 1 chooses T with probability t Player 2 chooses R with probability r

LR T1, 00, 3 B0, 12, 0 Expected Payoffs to Player I: T:.4 * * 0 =.4 B:.4 * * 2 = 1.2 Best response to r=.6 is t=0, not t=.4 Try:.3 r=.6

2/3 L 1/3 R ¼ T1, 00, 3 ¾ B0, 12, 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.

StrategyPayoff

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 FootballConcert Football2, 10, 0 Concert0, 01, 2 W M

Battle of the Sexes FootballConcert Football2, 10, 0 Concert0, 01, 2 W M Mixed-strategy equilibrium: M: 2/3 F; 1/3 C W: 1/3 F; 2/3 C

Battle of the Sexes FootballConcert Football4, 10, 0 Concert0, 01, 4 W M

Battle of the Sexes FootballConcert Football4, 10, 0 Concert0, 01, 4 W M Mixed-strategy equilibrium: M: 4/5 F; 1/5 C W: 1/5 F; 4/5 C

SwerveForward Swerve6, 64, 7 Forward7, 40, 0 Chicken

SwerveForward Swerve6, 64, 7 Forward7, 40, 0 Mixed-Strategy Equilibrium S=.8 F=.2 (Makes sense: Pure strategy equilibria require breaking the symmetry…) Probability of collision: 4% Chicken

SwerveForward Swerve6, 62, 12 Forward12, 20, 0 Chicken

SwerveForward Swerve6, 62, 12 Forward12, 20, 0 Chicken Mixed-Strategy Equilibrium S=.25 F=.75 Probability of collision: 56%

Stag Hunt StagHare Stag5, 50, 3 Hare3, 03, 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 CN C2, 2, 22, 3, 2 N3, 2, 20, 0, -1 CN C2, 2, 3-1, 0, 0 N0, -1, 00, 0, 0 II I I III C N

Pollution Game CN C2, 2, 22, 3, 2 N3, 2, 20, 0, -1 Pure- strategy Nash Equilibria (i)No one cleans (ii)Two out of three clean CN C2, 2, 3-1, 0, 0 N0, -1, 00, 0, 0 II I I III C N

Pollution Game CN C2, 2, 22, 3, 2 N3, 2, 20, 0, -1 Mixed- strategy Nash Equilibria (iii) One cleans for sure; other two clean with probability 2/3. (iv) Two symmetric equilibria. CN C2, 2, 3-1, 0, 0 N0, -1, 00, 0, 0 II I I III C N