Nash Equilibrium Econ 171 Suggested Viewing A Student s Suggestion Video game theory lecture Open Yale Economics Ben Pollack s Game Theory Lectures http oyc yale edu economics game theory Monte Hall Problem Let s Draw a Game Tree HW Problem 2 11 Perfect Recall Clicker Question 1 Strategy A2 Stategy B2 Strategy C2 Strategy A1 Strategy B1 Player 1 Strategy C1 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 B1 and Player 2 plays C2 a Nash equilibrium A Yes B No Clicker Question 2 Strategy A2 Stategy B2 Strategy C2 Strategy A1 Strategy B1 Player 1 Strategy C1 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 A1 and Player 2 plays B2 a Nash equilibrium A Yes B No Clicker Question 3 Strategy A2 Stategy B2 Strategy C2 Strategy A1 Strategy B1 Player 1 Strategy C1 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 Definition A strategy profile is a Nash Equilibrium if each player s strategy maximizes his payoff given the strategies used by the other players 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 Battle of Sexes Bob Movie A Movie B 2 1 0 0 0 0 1 2 Movie A Alice Movie B BRA A A BRA B B BRB A A BRB B B Prisoners Dilemma Game Player 2 Cooperate P L A Cooperate y E R 1 Defect Defect 10 10 0 11 11 0 1 1 Best Responses and Nash Equilibria for this game BR2 a z BR2 b w x z BR2 c y BR2 d y z BR1 w b BR1 x b BR1 y b BR1 z a d Find Nash equilibria for these games Chicken Pure coordination Driving Game How many Nash equilibria There might be just one There might be more than one There might not be any Rock Paper Scissors Where is Nash equilibrium The Romney game Textbook example 3 candidates Winner takes all Strategy positive effort 1 for self or negative effort 2 for smeared candidate Starting point Candidate 1 has initial score 2 candidates 2 and 3 have initial scores 0 Nash equilibria All positive Candidates 2 and 3 both smear 1 Other Nash equilibria 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 Hunt Hunter 3 does Hare Hunter 3 does Stag Hunter 2 Hunter 2 Stag Stag Hare 0 0 1 0 1 1 2 2 2 0 1 0 Stag Hunter 1 Hare Stag 1 0 0 1 1 0 Hare Find the Nash equilibria Hunter 1 Hare 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 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 All three outcomes A B and C are Nash equilibria