Nash equilibrium and Best Response Functions Best response

Nash equilibrium and Best Response Functions

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.

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.

Example: The Stag Hunt Player 2 Stag Player 1 Hare Stag 2 , 2 0, 1 Hare 1, 0 1 , 1 B 1(Stag)= {Stag} B 1(Hare)={Hare} B 2(Stag)={Stag} B 2(Hare)={Hare}

Example: The Stag Hunt Player 2 Stag Player 1 Hare Stag 2 , 2 0, 1 Hare 1, 0 1 , 1 B 1(Stag)= {Stag} B 1(Hare)={Hare} B 2(Stag)={Stag} B 2(Hare)={Hare}

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

B 1(Hare)= {Hare} Player 2 Stag Player 1 Hare Stag 2 *, 2 0, 1 Hare 1, 0 1* , 1

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

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

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.

Example: Battle of Bismarck Sea Imamura Sail South Sail North Kenney Fly North 2 , -2 Fly South 1 , -1 3 , -3 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. )

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

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

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

Example • Two partners work together. Payoff to Partner 1 is F 1(x 1, x 2)=ax 1+bx 1 x 2 -(1/2)x 12 • Payoff to Partner 2 is F 2(x 1, x 2)= ax 2+bx 1 x 2(1/2)x 22 • Best response functions are found by setting derivatives with respect to own action equal to zero. • R 1(x 2)=a+bx 2 and R 2(x 1)=a+bx 1

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: x 1 = R 1(x 2)=a+bx 2 and x 2 =R 2(x 1)=a+bx 1 What is the solution? How does this look in a graph? What happens if the parameters a and b differ for the two players?

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