In 1971, Wilson proved that “almost all” finite games have an odd number of mixed Nash equilibria. Since then, several other proofs have been given, but always for mixed extensions of finite games. In this paper, we state a new oddness theorem for large classes of polynomial payoff functions and semi-algebraic sets of strategies, and we provide some applications to recent models of games on networks such that Patacchini-Zenou’s model about juvenile delinquency and conformism, Calvo ́-Armengol-Patacchini-Zenou’s model about social networks in education, Konig- Liu-Zenou’s model about R&D networks, Helsley-Zenou’s model about social networks and interactions in cities.

We model the search problem of a reward maximizing agent who encounters a finite sequence of alternatives, each of which yields an ex-ante unknown reward. The agent pays a cost for exploring each alternative and decides when to terminate the search. This setting is an adaptation of Weitzman (1979) in the framework of lists. We characterize the solution to the search problem: the optimal policy uses a specific notion of reservation rewards that is sensitive to the list structure.