1. Nash equilibrium and Best Response Functions

2. Best response functions and Nash Equilibrium
The best response function for any player i, is a function that maps the list of actions by other players into the list of actions that are best responses to what the others did.
Sometimes there is only one best response.
A Nash equilibrium is a list of actions by the players such that each player’s action is a best response to the actions of the other players.

3. Using Best response functions to find Nash equilibria
With two-person games in which there is a discrete number of strategies, there is a handy way to use the payoff tables.
The method of stars.

4. Example: The Stag Hunt Player 2 Stag Hare Stag 2 , 2 0 , 1 1 , 0 1 , 1
2 , 2
0 , 1
1 , 0
1 , 1
Player 1
Hare
B1(Stag)= {Stag} B1(Hare)={Hare}
B2(Stag)={Stag} B2(Hare)={Hare}

5. Example: The Stag Hunt Player 2 Stag Hare Stag 2 , 2 0 , 1 1 , 0 1 , 1
2 , 2
0 , 1
1 , 0
1 , 1
Player 1
Hare
B1(Stag)= {Stag} B1(Hare)={Hare}
B2(Stag)={Stag} B2(Hare)={Hare}

6. B1(Stag)= {Stag} from making a best response to each of
Player 2
Stag
Hare
Stag
2 *, 2
0 , 1
1 , 0
1 , 1
Player 1
Hare
Put a star next to the payoff (s) to Player 1
from making a best response to each of
Player 2’s actions.

7. B1(Hare)= {Hare} Player 2 Stag Hare Stag 2 *, 2 0 , 1 1 , 0 1* , 1
2 *, 2
0 , 1
1 , 0
1* , 1
Player 1
Hare

8. B2(Stag)= {Stag} from making a best response to each of
Player 2
Stag
Hare
Stag
2 *, 2 *
0 , 1
1 , 0
1 , 1*
Player 1
Hare
Now put a star next to the payoff (s) to Player 2
from making a best response to each of
Player ’s actions.

9. B2(Hare)= {Hare} Player 2 Stag Hare Stag 2 *, 2 * 0 , 1 1 , 0 1* , 1*
2 *, 2 *
0 , 1
1 , 0
1* , 1*
Player 1
Hare
Where are the Nash equilibria?

10. Nash Equilibria Found
The boxes with two stars in them are Nash equilibria.
The Stag Hunt has two Nash equilibria
In one equilibrium, both players faithfully play their part in the stag hunt.
In the other equilibrium, both players chase after hares.
Note that one equilibrium is better for both than the other, but both are Nash equilibria.

11. Example: Battle of Bismarck Sea
Imamura
Sail North
Sail South
Fly North
2 , -2
2 , -2
1 , -1
3 , -3
Kenney
Fly South
BK(Sail North)= {Fly North} BK(Sail South)={Sail South}
BI(Fly North)={Sail North, Sail South} BI(Sail South)={Sail North}
Note that Imamura has two best responses to Fly North By Kenney. (The set of best responses has two members.)

12. Starring Kenney’s Best responses
Imamura
Sail North
Sail South
Fly North
2 * , -2
2 , -2
1 , -1
3* , -3
Kenney
Fly South
BK(Sail North)= {Fly North}
BK(Sail South)={Fly South}

13. Starring Imamura’s Best responses
Sail North
Sail South
Fly North
2 * , -2*
2 , -2*
1 , -1*
3* , -3
Kenney
Fly South
BI(Fly North)= {Sail North, Sail South }
BI(Fly South)={Sail North}
Where are the Nash equilibria?

14. Best response in continuous games
Two person games where a strategy is choice of a real number.
Player 1 chooses x1, player 2 chooses x2.
Payoffs are given by functions F1(x1,x2) and F2(x1,x2).
Best response for player 1 to action x2 is the action x1 that maximizes F1(x1,x2) . Similarly for player 2.

15. Example
Two partners work together. Payoff to Partner 1 is F1(x1,x2)=ax1+bx1x2-(1/2)x12
Payoff to Partner 2 is F2(x1,x2)= ax2+bx1x2-(1/2)x22
Best response functions are found by setting derivatives with respect to own action equal to zero.
R1(x2)=a+bx2 and R2(x1)=a+bx1

16. Nash equilibrium
In Nash equilibrium, each is doing the best response to the other’s action. Thus to find Nash equilibrium, we solve the simultaneous equations:
x1 = R1(x2)=a+bx2 and x2 =R2(x1)=a+bx1
What is the solution?
How does this look in a graph?
What happens if the parameters a and b differ for the two players?

17. Many players-two strategies
Farmers and bandits
Inventors and mimics
Congested roads