1 What is a Nash Equilibrium?
In game theory, a Nash Equilibrium (NE) is a situation where no player could gain by changing their own strategy holding all other player’s strategies as fixed. The famous John Nash showed that there exists a Nash equilibrium, possibly in mixed strategies (those with stochastic decisions) for every finite game.
Formally, consider a game with players \(j=1, \dots, J\) and let \(\mathscr{S}_{j}\) denote the set of all possible strategies for player \(j\). Let \(s^*=(s_{j}^*, s_{-j}^*)\) be a strategy profile that is, a set consisting of one strategy for each player where \(s_{-j}^*\) denotes the \(J-1\) strategies of all the players besides \(j\). Let \(u_{j}(s_{j},s^*_{-j})\) be player \(j\)’s payoff function of the strategies. The strategy profile \(s^*\) is a Nash equilibrium if \[ u_{j}(s_{j}^*, s_{-j}^*)\geq u_{j}(s_{j},s_{-j}^*);\quad \forall s_{j}\in S_{j}. \]
A game can have more than one Nash equilibrium. Even if the equilibrium is unique, it might be weak whereby a player might be indifferent among several strategies given the other player’s choices. The equilibrium is unique and called a strict Nash equilibrium if the inequality if the inequality is strict so one strategy is the unique best response \[ u_{j}(s_{j}^*, s_{-j}^*)\geq u_{j}(s_{j},s_{-j}^*);\quad \forall s_{j}\in S_{j};~s_{j}\neq s_{j^*}. \] The strategy set \(\mathscr{S}_{j}\) can be different for different players, and its elements can be a variety of mathematical objects.
2 Pure Equilibrium vs Mixed Equilibrium
A pure strategy is a deterministic choice: player \(j\) selects a single element \(s_j \in \mathscr{S}_j\) with certainty. A mixed strategy generalizes this by allowing player \(j\) to randomize, selecting each pure strategy according to some probability distribution \(\sigma_j \in \Delta(\mathscr{S}_j)\), where \(\Delta(\mathscr{S}_j)\) is the set of all probability distributions over \(\mathscr{S}_j\). The payoff under a mixed strategy profile \(\sigma = (\sigma_1, \dots, \sigma_J)\) is then replaced by the expected payoff \[ U_j(\sigma_j, \sigma_{-j}) = \sum_{s \in \mathscr{S}} u_j(s)\prod_{k=1}^{J}\sigma_k(s_k). \] A pure strategy Nash equilibrium (PSNE) is a NE in which every player’s equilibrium strategy is a pure strategy — a degenerate mixed strategy placing probability 1 on a single action. A mixed strategy Nash equilibrium (MSNE) is one in which at least one player randomizes non-trivially. In a MSNE, a player is willing to randomize only if they are indifferent among all pure strategies in the support of their mixing distribution; each pure strategy in the support must be a best response.
Every finite game (finitely many players, each with a finite pure strategy set) has at least one Nash equilibrium, possibly in mixed strategies.
3 Strict vs Non-Strict Equilibrium
A Nash equilibrium \(s^*\) is strict if every player has a unique best response: for all \(j\) and all \(s_j \neq s_j^*\), \[ u_j(s_j^*, s_{-j}^*) > u_j(s_j, s_{-j}^*). \] It is weak (or non-strict) if at least one player is indifferent between their equilibrium strategy and some other strategy, so the inequality holds with equality for some \(s_j \neq s_j^*\). Strict Nash equilibria are in some sense more robust: a player has no incentive to deviate even if they break ties arbitrarily. Weak equilibria can be fragile, as small perturbations to payoffs may destroy them. Note that a strict NE must be a pure strategy equilibrium, since a player who mixes must be indifferent among all strategies in their support.
4 Equilibria for Coalitions
The Nash equilibrium concept assumes players deviate unilaterally; it does not account for groups of players jointly coordinating a deviation. Several refinements address this.
A strong Nash equilibrium (Aumann, 1959) requires that no coalition \(C \subseteq \{1,\dots,J\}\) can jointly deviate in a way that makes every member of \(C\) strictly better off. Formally, \(s^*\) is a strong NE if for every coalition \(C\) and every joint deviation \(\hat{s}_C \in \prod_{j \in C}\mathscr{S}_j\), there exists some \(j \in C\) such that \[ u_j(s_j^*, s_{-j}^*) \geq u_j(\hat{s}_j, \hat{s}_{-C}^*, s_{-C}^*). \] Strong NE is a very demanding concept and frequently fails to exist. A weaker notion is the coalition-proof Nash equilibrium (Bernheim, Peleg & Whinston, 1987), which only rules out self-enforcing joint deviations — those that are themselves immune to further defection by sub-coalitions.
5 Discussion of Fixed Point Theorems
Nash’s existence result is proved using fixed point arguments. Each player’s best response correspondence \(\text{BR}_j: \prod_{k \neq j}\Delta(\mathscr{S}_k) \rightrightarrows \Delta(\mathscr{S}_j)\) maps opponents’ mixed strategies to the set of the player’s optimal mixed responses. A Nash equilibrium is precisely a fixed point of the joint best response map \(\text{BR}(\sigma) = \prod_j \text{BR}_j(\sigma_{-j})\). For finite games, the strategy simplex is compact and convex, and \(\text{BR}\) has convex values and a closed graph, so [[kakutanis-fixed-point-theorem]] applies directly. In games with a continuum of pure strategies, [[brouwers-fixed-point-theorem]] is used instead. A formal treatment of these results is given in [[fixed-point-theorems]].