Game Theory and the Nash Equilibrium Part 2

1. Game Theoryand the Nash Equilibrium Part 2 Eponine Lupo

2. Agenda • Questions from last time • 3 player games • Games larger than 2x2—rock, paper, scissors • Review/explain Nash Equilibrium • Nash Equilibrium in R • Instability of NE—move towards pure strategy • Prisoner’s Dilemma, Battle of the Sexes, 3rd Game • Application to Life

3. 3-Player Game 2 L R 2 L R L L 1 1 R R Strategy Profile: {R,L,L} is the Solution to this Game L R 3

4. Rock, Paper, Scissors Player 2 R P S R Player 1 P S • No pure strategy NE • Only mixed NE is {(1/3,1/3,1/3),(1/3,1/3,1/3)}

5. 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

6. Mixed Strategy NE • 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 • Take for example a modified game of Rock, Paper, Scissors where player 1 cannot ever play “Scissors” What now is the Nash Equilibrium? Put another way, how are Player 1 and Player 2 going to play?

7. Mixed Strategy NE • Once Player 1’s strategy of S is taken away, Player 2’s strategy R is iteratively dominated by strategy P. Player 2 R P S R Player 1 P

8. Mixed Strategy NE Player 2 q 1-q P S • Player 1 wants to have a mixed strategy (p, 1-p) such that Player 2 has no advantage playing either pure strategy P or S. • u2((p, 1-p),P)=u2((p, 1-p),S) • 1p+0(1-p) = (-1)p+1(1-p) • 1p = -2p+1 • 3p = 1 • p=1/3 p R Player 1 1-p P • Now the game has been cut down from a 3x3 to 2x2 game • There are still no pure strategy NE • From here we can determine the mixed strategy NE S1 = (1/3 , 2/3)

9. Mixed Strategy NE Player 2 q 1-q P S • Likewise, Player 2 wants to have a mixed strategy (q, 1-q)such that Player 1 has no advantage playing either pure strategy R or P. • u1(R,(q, 1-q))=u1(P,(q, 1-q)) • -1q+1(1-q) = 0q+(-1)(1-q) • -2q+1 = q-1 • 3q = 2 • q=2/3 p R Player 1 1-p P S2 = (2/3 , 1/3)

10. Mixed Strategy NE • Therefore the mixed strategy: • Player 1: (1/3Rock , 2/3Paper) • Player 2: (2/3Paper , 1/3Scissors) is the only one that cannot be “exploited” by either player. • The values of p and q are such that if Player 1 changes p, his payoff will not change but Player 2’s payoff may be affected • Thus, it is a Mixed Strategy Nash Equilibrium.

11. Nash Equilibrium in R • The Nash Equilibrium is a very unstable point • If you do not begin exactly at the NE, you cannot stochastically find the NE • Theoretically you will “shoot off” to a pure strategy: (0,0) (0,1) (1,0) or (1,1) • (similar for n players) • Consider the following: • 2 players randomly choose values for p and q • Knowing player 2’s mixed strategy (q, 1-q), player 1 adjusts his mixed strategy of (p,1-p) in order to maximize his payoffs • With player 1’s new mixed strategy in mind, player 2 will adjust his mixed strategy in order to maximize his payoffs • This see-saw continues until both players can no longer change their strategies to increase their payoffs

12. Nash Equilibrium in R • Unfortunately, I was unable to find a way to discover a mixed strategy NE in R for any number of players • Is my code wrong? • Is there simply no way to find the NE in R? • I don’t know

13. Implications • In life, we react to other people’s choices in order to increase our utility or happiness • Ignoring a younger sibling who is irritating • Accepting an invitation to go to a baseball game • Once we react, the other person reacts to our reaction and life goes on • One stage games are rare in life • Very rarely are we in a “NE” for any aspect of our lives • There is almost always a choice that can better our current utility