1. Eponine Lupo

2.  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, 3 rd Game  Application to Life

3. 14, 24, 328, 30, 27 30, 16, 2413, 12, 50 1 2 L R R L 16, 24, 3030, 16, 24 30, 23,1414, 24, 32 1 2 L R R L LR 3 Strategy Profile: {R,L,L} is the Solution to this Game

4. 0, 0-1, 11, -1 0, 0-1, 1 1, -10, 0 Player 2 R P S Player 1 R P S No pure strategy NE Only mixed NE is {(1/3,1/3,1/3),(1/3,1/3,1/3)}

5.  “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 4, 60, 44, 4 5, 30, 01, 7 1, 13, 52, 3 Player 2 L C R Player 1 U M D Strategy Profile: {D,C} is the Nash Equilibrium **There is no incentive for either player to deviate from this strategy profile

6.  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.  Once Player 1’s strategy of S is taken away, Player 2’s strategy R is iteratively dominated by strategy P. 0, 0-1, 11, -1 0, 0-1, 1 Player 2 R P S Player 1 R P

8.  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 -1, 11, -1 0, 0-1, 1 Player 2 q 1-q P S Player 1 p R 1-p P  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.  u 2 ((p, 1-p),P)=u 2 ((p, 1-p),S) 1p+0(1-p) = (-1)p+1(1-p) 1p = -2p+1 3p = 1 p=1/3 S 1 = (1/3, 2/3)

9. -1, 11, -1 0, 0-1, 1 Player 1 p R 1-p P  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.  u 1 (R,(q, 1-q))=u 1 (P,(q, 1-q)) -1q+1(1-q) = 0q+(-1)(1-q) -2q+1 = q-1 3q = 2 q=2/3 S 2 = (2/3, 1/3) Player 2 q 1-q P S

10.  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.  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.  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.  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