Clicker Game Choose one of the numbers below

Clicker Game: Choose one of the numbers below. You will get 1 point if your number is the closest number to 3/4 of the average of the numbers chosen by all class members, otherwise you will get 0 points… • • • A) 6 B) 5 C) 4 D) 3 E) 2

Nash Equilibrium The real John Nash Hollywood’s Version

Clicker Question-A Chicken Game Player 2 Swerve Hang Tough 0, 0 0, 1 1, 0 -10, -10 Pllayer 1 Hang Tough Does either player have a dominant strategy? A) Yes B) No

Nash Equilibrium in Chicken Game? Player 2 Swerve Hang Tough 0, 0 0, 1 1, 0 -10, -10 Pllayer 1 Hang Tough How many Nash equilibria does this game have? A) None B) Exactly one C) More than one

Definition A strategy profile is a Nash Equilibrium if each player’s strategy maximizes his payoff given the strategies used by the other players.

Clicker Question Strategy A 2 Stategy B 2 Strategy C 2 Strategy A 1 Strategy B 1 Player 1 Strategy C 1 6, 0 2, 3 1, 1 3, 2 2, 3 2, 0 4, 1 2, 8 5, 2 Is the outcome where Player 1 plays B 1 and Player 2 plays C 2 a Nash equilibrium? A) Yes B) No

Clicker Question Strategy A 2 Stategy B 2 Strategy C 2 Strategy A 1 Strategy B 1 Player 1 Strategy C 1 6, 0 2, 3 1, 1 3, 2 2, 3 2, 0 4, 1 2, 8 5, 2 Is the outcome where Player 1 plays A 1 and Player 2 plays B 2 a Nash equilibrium? A) Yes B) No

Best response mapping Best response for a player is a mapping from actions by the others to the action (or actions) that maximizes the player’s payoffs given the actions of the others. In Nash equilibrium, every player is doing the best response to what the other players are doing.

Prisoners’ Dilemma Game Player 2 Cooperate P L A Cooperate y E R 1 Defect 10, 10 0, 11 11, 0 1, 1

Battle of Sexes Movie A Alice Movie B BRA(A)=A BRA(B)=B Bob Movie B 2, 1 0, 0 1, 2 BRB(A)=A BRB(B)=B

Best Responses and Nash Equilibria for this game? BR 2(a)=z BR 2(b)={w, x, z} BR 2(c)=y BR 2(d)={y, z} BR 1(w)=b BR 1(x)=b BR 1(y)=b BR 1(z)={a, d}

Find Nash equilibria for these games • Chicken • Pure coordination (Driving Game)

How many Nash equilibria (in pure strategies)? There might be just one. There might be more than one. There might not be any.

Rock, Paper Scissors, Where is Nash equilibrium?

When is Nash equilibrium “the right answer”? 1. Players are “rational”. Each player’s strategy maximizes his payoff, given his beliefs about the strategies used by the other players. 2. Each player’s beliefs about the other players’ strategies are correct. When is 2) a reasonable assumption?

3 -Hunter Stag Hunter 3 does Hare Hunter 3 does Stag Hunter 2 Stag 2, 2, 2 0, 1, 0 Stag Hunter 1 Hare Stag Hare 1, 0, 0 1, 1, 0 Find the Nash equilibria Stag Hunter 1 Hare 0, 0, 1, 1 1, 0, 1 1, 1, 1

Weakly dominated strategies? • Nobody will use a strictly dominated strategy in Nash equilibrium. • If there is a strictly dominant strategy for all players, it is a Nash equilibrium. (example Prisoners’ Dilemma. ) • Nash equilibrium does not exclude possibility of using a weakly dominated strategy. (A voting example with unanimous preferences. )

Nash and domination Every Nash equilibrium survives the iterated elimination of strictly dominated strategies. Not every outcome that satisfies the iterated elimination of strictly dominated strategies is a Nash equilibrium.

Clicker Question Strategy A 2 Stategy B 2 Strategy C 2 Strategy A 1 Strategy B 1 Player 1 Strategy C 1 6, 0 2, 3 1, 1 3, 2 2, 3 2, 0 4, 1 2, 8 5, 2 Does this game have more than one Nash equilibrium? A) Yes B) No

Game of previous slide reduced by IDSDS Strategy B 2 Strategy C 2 Strategy A 1 3, 2 4, 1 Strategy C 1 2, 0 5, 2 Find the Nash equiibria for reduced game. These must be Nash equilibria for the original game. Note that strategy profiles C 1, B 2 and A 1, C 2 are not Eliminated by IDSDS, but are not Nash equillibria.

Clicker Question: What are the Nash Equilibria for this game? 4, 1 A) Player 1 plays a and Player 2 plays z. B) Player 1 plays d and Player 2 plays z. C) Player 1 plays b and Player 2 plays y. D) Both outcomes A) and B) are Nash equilibria. E) There are no Nash equilibria

A coordination game • You choose one of three parties to go to Party X, Party Y, Party Z You like big parties and your payoff will be the number of people who attend the same party that you do. We will play this repeatedly with clickers. After each round, you will see how many people chose each option. Then you play again.

Which Party do you choose? • A) Party Y • B) Party X • C) Party Z

Fictional play version • You do the best response given the average of previous responses. • Will this converge? • If it converges, it converges to Nash equilibrium. Why?

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