Nash Equilibrium Note Handout-----written by Professor J. G. Powers

A group of individuals is said to be involved in a game whenever the fate of an individual in the group depends not only on its own actions, but also on the actions of the other individuals in the group.

One thing to recall when working with games is that one of the fundamental assumptions used throughout economics is that economic agents are rational. A rational consumer seeks to maximize his or her utility. A rational firm seeks to maximize its profits. If player 1 is rational, she seeks to maximize her payoff, and does not care about how player 2 does.

Perhaps the most fundamental concept in game theory is Nash Equilibrium.

Nash Equilibrium is a set of strategies (or actions) one for each player, such that no player has an incentive to deviate from its part of the strategy---no player has the incentive to take a different action:

------Each Player is doing the best that he or she can, given what the other players are doing
------The action a player takes is a best response to what the other players are doing.
------Or given what the other players are doing, no one player has the incentive to change their action unilaterally.                               Player2
                        L                 R
Player 1 T       4,9             1,5
             B       5,6             4,8

Here B is a dominant strategy for player 1, it is best no matter what player 2 is doing.
If player 2 plays L, player 1's best response is to play B 5>4 a higher payoff
if player 2 plays R, player 1's best response is to play B 4>1 a higher payoff

Looking at the payooffs, 2 knows 1 will play B, so what should 2 do? If 1 plays B, 2's bbest response is to play R since 8>6 (B,R) is the Nash Equilibrium of this game, no player has the incentive to unilaterally change its action.
                               Player2
                        L                 R
Player 1 T       4,2             6,9
             B       8,7             1,3


Here if 2 plays L, 1 should play B since 8>4
if 1 plays B, 2 should play L since 7>3
So (B,L) is a Nash Equilibrium, given what the other is doing neither player has the incentive to change their actions. Each player's action is a best response to what the other player is doing.

But we haven't finished yet, (T,R) is also a Nash Equilibrium of this game.
If 1 plays T, 2 should play R, and if 2 plays R, 1 should play T.
There is also a "mixed strategy" Nash Equilibrium for this game:

A mixed strategy involves players randomizing between certain actions. The concept of a mixed strategy equilibrium can perhaps be seen most clearly in the children's game Rock-Paper-Scissors.
                               Player1
                        R                 P             S
Player 2 R       0,0             -1,1           1,-1
             P       1,-1             0,0            -1,1

             S       -1,1             1,-1           0,0

Randomize it. The equilibrium of this game is to play each strategy with a one-third probability.

E.G.
                               Player2
                        L                 R
Player 1 T       2,7             4,5
             B       9,6             3,8


There is no N.E in pure strategies. For 1 to be willing to mix, 2 must be playing L and R in such a way (L with probablity a, R with probablity (1-a)) such that player 1 is indifferent between playing T and B---they both provide the same expected payoff.

Player1 plays T: 2a+4(1-a)
            plays B: 9a+3(1-a)   set two equations equal and solve for a
Player2 plays L: 7b+6(1-b)
            plays R: 5b+8(1-b)  set two equal to solve for b.

END