Game Theory with Applications to Finance and Marketing

about these pure strategies. Let x be the prob. that player 1 uses Uand y the prob. that player 2 uses L. We must have1 · x + (−1) · (1 − x) = 2 · x + (−2) · (1 − x) ⇒ x = 1 2 ;0 · y + (−1) · (1 − y) = 2 · y + (−2) · (1 − y) ⇒ y = 1 3 .These mixed strategies are said to be totally mixed, in the sense thatthey assign strictly positive prob.’s to each and every pure strategy.Here we have only one mixed strategy NE. If we have two totallymixed NE’s, then we naturally have a continuum of mixed strategyNE’s (why?). From now on, we denote the set of totally mixed strategiesof player i by Σ 0 i . Observe that Σ 0 i is simply the interior of Σ i ,when Σ i is endowed with the usual Euclidean topology.19. (Matching Pennies) Note that some games do not have pure strategyNash equilibrium:player 1/player 2 H TH 1,-1 -1,1T -1,1 1,-120. Theorem 2 (Nash, 1950): Every finite game in normal form has amixed strategy equilibrium.Theorem 2 is actually a special version of the following more generaltheorem (see Fudenberg and Tirole’s Game Theory, Theorem 1.2).Theorem 2 ′ : (Debreu-Glicksberg-Fan) Consider a strategic gameG = {I, S = (S 1 , S 2 , · · · , S I ), (u i : S → R; i ∈ I)}with I being a finite set (the game has a finite number of players). If forall i ∈ I, S i is a nonempty compact convex subset of some Euclideanspace, and if for all i ∈ I, u i is continuous in s and quasi-concave in s iwhen given s −i , then G has an NE in pure strategy.21. Suppose that r : Σ → Σ ′ is a correspondence (a multi-valued function),where Σ and Σ ′ are some subsets of R n . If r(σ) ≠ ∅ for all σ ∈ Σ, then11