• Each player has a utility function up: S → R.

We start by giving a few precise definitions. Definition 1. A n-player finite game 〈[n], (Sp)p∈[n], (up)p∈[n]〉 is formally defined as follows. • We have a set of n players [n] = {1, 2, . . . , n}. • Each player p ∈ [n] has a finite set of strategies or actions Sp. • The set S = ∏ p∈[n] Sp is the set of strategy profiles. • Each player can choose a distribution of strategies x ̃p ∈ ∆p = {xp ∈ Rp + |∑sp∈Sp xp(sp) = 1}. • Each player has a utility function up : S → R. Here, given a mixed strategy profile x ̃ ∈ ∆ = ∏p∈[n] ∆p , the payoff of player p is given by the expected utility up(x ̃) = Es∼x ̃ [up(s)] . Given a strategy profile s ∈ S, we also adopt the notation sp to denote player p’s strategy and s−p to denote everyone else’s strategies. Similarly, given x ̃ ∈ ∆, we let x ̃p denote the distribution of strategies for player p and let x ̃−p denote the distribution of strategies for everyone else. Next, we give a few definitions for types of Nash equilibria.