Question

Consider a two-player game with strategy sets S1 = {α1, . . . , αm} and S2 = {β1, . . . , βn}. What is a Nash equilibrium for the game?

1You are being asked for a definition.

Answer 1

Nash Equilibrium Nash equilibrium is a strategy profile (a collection of strategies, one for each player) such that each strategy is a best response (maximizes payoff) to all the other strategies An outcome a ∗ = (a ∗ 1 , ..., a ∗ n ) is a Nash equilibrium if for each player i ui(a ∗ i , a ∗ −i ) ≥ ui(ai , a ∗ −i ) for all ai ∈ Ai Nash equilibrium is self-enforcing: no player has an incentive to deviate unilaterally One way to find Nash equilibrium is to first find the best response correspondence for each player ◮ Best response correspondence gives the set of payoff maximizing strategies for each strategy profile of the other players ... and then find where they “intersect”

Nash Equilibrium L H L 7, 7 7, 1 H 1, 7 13, 13 Set of Nash equilibria = {(L, L),(H, H)}

Best Response Correspondence The best response correspondence of player i is given by Bi(a−i) = {ai ∈ Ai : ui(ai , a−i) ≥ ui(bi , a−i) for all bi ∈ Ai}. Bi(a−i) is a set and may not be a singleton In the effort game L H L 7, 7 7, 1 H 1, 7 13, 13 B1(L) = {L} B1(H) = {H} B2(L) = {L} B2(H) = {H}