A population game is defined by a set of strategies S = {1, ..., n} and a continuously differentiable payoff function F : Rn + Rn. The function Fi : Rn + R, the ith component of the function F, is the payoff function for strategy i. In what follows, we take the set of strategies S as fixed and identify a population game with its payoff function F. The population game F is played by a unit-mass population of infinitesimal agents. Each agent chooses a strategy from the set S. (That is, agents always choose pure strategies.) The distribution of strategies in the population is therefore described by a population state, which is an element of the simplex X = {x Rn + : jS xj = 1}. When the population state is x, agents playing strategy i receive payoff Fi(x).

Provide a definition of Nash equilibrium for population games. A population game F is a potential game if it admits a potential function: a twice continuously differentiable function f : Rn+ R that satisfies f/xi (x) = Fi(x) for all i S and x X. State the Kuhn-Tucker first-order necessary conditions for maximizing the potential function f on the simplex X. Show that population state x X satisfies the conditions you stated in part (ii) (for some appropriate choices of the Lagrange multipliers) if and only if x is a Nash equilibrium of the population game F. Also, provide a game-theoretic interpretation of the Lagrange multipliers.