Game Theory: Mixed Strategy Nash Equilibrium

1. Game Theory: Mixed Strategy Nash Equilibrium Margaret Banker 8 December 2010

2. Review Notation • Strategy “s”  one of the strategies in a strategy set • s1  Strategy of player 1 • s2  Strategy of player 2 • U Utility/Payoff • u1(s1, s2) Player 1’s payoff if player 1 plays strategy s1and player 2 plays strategy s2 Example Game: Example strategy of: Player 1: s1=T Player 2: s2= L Example Payoffs: u1(T,L)=3 u2(T,L)=5 Player 2 L R T Player 1 B

3. Mixed Strategy Nash Equilibrium • What is a Mixed Strategy? • What is a Nash Equilibrium? • What is a Mixed Strategy Nash Equilibrium??

4. Mixed Strategy • “The act of selecting a strategy according to a probability distribution” • A player uses a mixed strategy when: • He is indifferent between several pure strategies • Game is not solvable using pure strategies (no dominant or efficient strategies) • Keeping the opponent guessing is desirable, i.e. when the opponent can benefit from knowing the next move

5. Calculating Payoffs of Mixed Strategy - General Player 2 q (1-q) L R • s1 = (p , (1-p)) • s2 = (q , (1-q)) • Calculating Payoffs: u1(s1 , s2) = p*q*80 + p*(1-q)*0 + (1-p)*q*0 + (1-p)*(1-q)*30 u2(s1 , s2) = p*q*30 + p*(1-q)*0 + (1-p)*q*0 + (1-p)*(1-q)*80 p T Player 1 (1-p) B

6. Calculating Payoffs of Mixed Strategy - Example Player 2 1/2 1/2 L R • s1 = (1/4 , 3/4) • S2 = (1/2 , 1/2) • Calculating Payoffs: U1(s1 , s2) = ¼*½*80 + ¼*½*0 + ¾*½*0 + ¾*½*30 = (1/8)*80 + (3/8)*30 = (170/8) U2(s1 , s2) = ¼*½*30 + ¼*½*0 + ¾*½*0 + ¾*½*80 = (1/8)*30 + (3/8)*80 = (270/8) ¼ T Player 1 ¾ B

7. Nash Equilibrium • “A strategy profile is a Nash Equilibrium if and only if each player’s prescribed strategy is a best response to the strategies of others” • Equilibrium that is reached even if it is not the best joint outcome Player 2 L C R Strategy Profile: {D,C} is the Nash Equilibrium *There is no incentive for either player to deviate from this strategy profile* U Player 1 M D

8. Best Response • Deals with responding to the beliefs you hold of the other player’s strategy • Example

9. Mixed Strategy Nash Equilibrium • Example: Penalty Shots • Sometimes there is NO pure Nash Equilibrium, or there is more than one pure Nash Equilibrium • In these cases, use Mixed Strategy Nash Equilibriums to solve the games Goalie L R L Shooter R *No Pure Nash Equilibrium*

10. Mixed Strategy Nash Equilibrium • Example: Penalty Shots • If Shooter ALWAYS shoots Left (pure strategy sS = L), then Goalie will respond by anticipating Left (pure strategy sG=L) uS(L, L)=10 • If Shooter ALWAYS shoots Right (pure strategy sS=R), then Goalie will respond by anticipating Right (pure strategy sG=R) uS(R,R)=40 Goalie L R L Shooter R

11. Mixed Strategy Nash Equilibrium • Example: Penalty Shots • If Shooter adopts mixed strategy (½, ½), what pure strategy will the Goalie play? • Calculate and compare Goalie’s payoffs: uG ((½, ½), L) = ½*90 + ½*20 = 55 uG ((½, ½), R) = ½*30 + ½*60 = 45 Thus, Goalie Pwill play sG=L • Strategy Profile : ((½, ½), L) Calculate Shooter’s Payoff: Is he better off than playing a pure strategy? uS((½,½),L) = ½*10 + ½*80 = 45 Goalie L R ½ L Shooter ½ R uS(L, L) = 10 uS(R,R) = 40 uS((½,½),L) = 45

12. Mixed Strategy Nash Equilibrium • Example: Penalty Shots • Recall definition of Nash Equilibrium: • What strategy profile ((p , (1-p) (q , (1-q)) is a Mixed Strategy Nash Equilibrium? Goalie q 1-q L R “A strategy profile is a Nash Equilibrium if and only if each player’s prescribed strategy is a best response to the strategies of others” pL Shooter 1-p R

13. Mixed Strategy Nash Equilibrium “A strategy profile is a Nash Equilibrium if and only if each player’s prescribed strategy is a best response to the strategies of others” • Example: Penalty Shots • Shooter wants to have a mixed strategy (p, 1-p) such that Goalie has no advantage playing either pure strategy L or R. • uG((p, 1-p),L)=uG((p, 1-p),R) 90p+20(1-p) = 30p+60(1-p) 70p+20 = -30p+60 100p = 40 p=.40 Goalie q 1-q L R pL Shooter 1-pR sS = (.40 , .60)

14. Mixed Strategy Nash Equilibrium “A strategy profile is a Nash Equilibrium if and only if each player’s prescribed strategy is a best response to the strategies of others” • Example: Penalty Shots • Likewise, Goalie must choose mixed strategy (q, 1-q) such that Shooter is indifferent between his pure strategies, i.e. such that some mixed strategy (p, 1-p) is Shooter’s Best Response • uS(L, (q, 1-q))=uS(R, (q, 1-q)) 10q+70(1-q) = 80q+40(1-q) -60q+70 = 40q+40 100q = 30 q=.30 Goalie q 1-q L R pL Shooter 1-pR sG = (.30 , .70)

15. Result! • Therefore the mixed strategy: • Shooter: (.4L , .6R) • Shoot Left 40% of time, Shoot Right 60% of time • Goalie: (.3F , 0.7R) • Anticipate Left 30% of time, Anticipate Right 70% of time is the only one that cannot be “exploited” by either player. • Thus, it is a Mixed Strategy Nash Equilibrium.

16. Resources • Duffy, John. “Introduction to Game Theory: Elements of a Game.” University of Pittsburgh, Department of Economics.< http://www.pitt.edu/~jduffy/econ1200/Lectures.htm >. • GameTheory.net < http://www.gametheory.net> • “Glossary of Game Theory Terms.” Gametheory.net. <http://www.gametheory.net/dictionary/#I>. • Sönmez, Tayfun. “Econ 308 Lecture Notes 4.” Boston College, Department of Economics. <http://www2.bc.edu/~sonmezt/>. • Sönmez, Tayfun. “Econ 308 Lecture Notes 7.” Boston College, Department of Economics. <http://www2.bc.edu/~sonmezt/>. • Walker, Paul. “Chronology of Game Theory.” University of Canterbury: Economics and Finance. October, 2005. <http://www.econ.canterbury.ac.nz/personal_pages/paul_walker/gt/hist.htm>. • Watson, Joel. “Strategy: An Introduction to Game Theory.” University of California, San Diego.WW Norton & Company: New York. 2008.