Download presentation

Presentation is loading. Please wait.

Published byAriana Verne Modified over 2 years ago

1
Non-cooperative Games Elon Kohlberg February 2, 2015

2
Matching Pennies HT H1, -1-1, 1 T 1, -1 No Nash Equilibrium…

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

4
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

5
LR T1, 00, 3 B0, 12, 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

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

7
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.

8
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.

9
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.

10
StrategyPayoff 15.1 27.3 3 42.3 57.1 67.3 75.8

11
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

12
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?

13
Rationales for Mixed-strategy Equilibrium: 1.Self-enforcing beliefs 2.Frequency in a population 3.Intentional randomization

14
Battle of the Sexes FootballConcert Football2, 10, 0 Concert0, 01, 2 W M

15
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

16
Battle of the Sexes FootballConcert Football4, 10, 0 Concert0, 01, 4 W M

17
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

18
SwerveForward Swerve6, 64, 7 Forward7, 40, 0 Chicken

19
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

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

21
SwerveForward Swerve6, 62, 12 Forward12, 20, 0 Chicken Mixed-Strategy Equilibrium S=.25 F=.75 Probability of collision: 56%

22
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?

23
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

24
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

25
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

Similar presentations

OK

Nash Equilibrium - definition A mixed-strategy profile σ * is a Nash equilibrium (NE) if for every player i we have u i (σ * i, σ * -i ) ≥ u i (s i, σ.

Nash Equilibrium - definition A mixed-strategy profile σ * is a Nash equilibrium (NE) if for every player i we have u i (σ * i, σ * -i ) ≥ u i (s i, σ.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on rocks soil and minerals Ppt on seasons in hindi Ppt on asymptotic notation of algorithms definition Ppt on motivational short stories Ppt on immigration Ppt on bond length and bond Ppt on centring point Ppt on teacher's day messages Ppt on viruses and anti viruses name Download ppt on real numbers for class 9