Game Theory Formalism: Nash Equilibrium & Dominant Strategy

Normal-form games

A normal-form (strategic-form) game is defined by a triple G = (N, S, u): a finite set of players N, a strategy set Sᵢ for each player i, and a payoff function uᵢ: S₁ × S₂ × ... × Sₙ → ℝ giving each player's payoff for every combination of strategies. For two players with finite strategy sets, this is exactly the payoff matrix you have likely seen for the Prisoner's Dilemma.

Solving a game means answering: which strategy profile(s) (s₁*, ..., sₙ*) will rational, self-interested players actually reach? The rest of this article works through the standard toolbox, in order of increasing generality.

Dominant and dominated strategies

Strategy sᵢ' strictly dominates strategy sᵢ for player i if sᵢ' gives strictly higher payoff than sᵢ against every strategy the other players might choose:

(s₋ᵢ denotes "the strategies of all players other than i".) If a player has a strategy that strictly dominates every alternative available to them, that strategy is their dominant strategy — a rational player plays it regardless of what anyone else does. In the Prisoner's Dilemma, Defect strictly dominates Cooperate for both players, which is precisely why (Defect, Defect) is such a robust prediction even though it is collectively worse than (Cooperate, Cooperate).

Weak domination is the corresponding non-strict version: sᵢ' weakly dominates sᵢ if it is never worse and sometimes strictly better. Weakly dominated strategies are "safe to eliminate" in most practical analyses, but eliminating them can, in rare cases, remove a legitimate equilibrium — strict domination is always safe.

Iterated elimination of dominated strategies (IEDS)

Even when no single strategy dominates all others outright, you can often narrow down the solution by repeatedly removing dominated strategies:

Consider a 3×3 game where Player 1 has strategies {Top, Middle, Bottom} and Player 2 has {Left, Center, Right}. Suppose Bottom is strictly dominated by Top for Player 1 — remove Bottom. In the reduced 2×3 game, Right may now be strictly dominated by Center for Player 2 (since Bottom, which made Right attractive, is gone) — remove Right. Continue until the game shrinks to a single cell, or no further strategy can be eliminated.

Best-response functions

Player i's best-response function BRᵢ(s₋ᵢ) returns the strategy (or set of strategies, if there is a tie) that maximises i's payoff given the other players' strategies s₋ᵢ:

Best-response analysis is the single most useful tool for finding Nash equilibria by hand in a small payoff matrix: for every column (Player 2's strategy), underline Player 1's best payoff; for every row, underline Player 2's best payoff. Any cell with both payoffs underlined is a Nash equilibrium — by construction, neither player can do better by unilaterally switching.

Finding pure-strategy Nash equilibria

Putting best-response analysis into practice on the classic Battle of the Sexes game (a couple prefers different events but most prefer being together):

Underlining best responses: if Player 2 picks Opera, Player 1's best response is Opera (2 > 0) — mark it. If Player 2 picks Football, Player 1's best response is Football (1 > 0) — mark it. Doing the same for Player 2 reveals two cells where both payoffs are marked: (Opera, Opera) and (Football, Football). Both are pure-strategy Nash equilibria — this game has no unique prediction, which is exactly the coordination problem the model is designed to illustrate.

Solving for mixed-strategy equilibria

When a game has no pure-strategy equilibrium (matching pennies, rock-paper-scissors) or you want the full equilibrium set of Battle of the Sexes, you need mixed strategies — probability distributions over pure strategies. The key solving technique is the indifference principle: in a mixed equilibrium, each player must be exactly indifferent between the pure strategies they mix over (otherwise they would deviate to the strictly better one).

For Battle of the Sexes, let Player 1 play Opera with probability p and Player 2 play Opera with probability q. Player 2 is indifferent between Opera and Football when:

Note the counter-intuitive result: Player 1 (who prefers Opera) mixes toward Opera less often relative to Player 2's threshold, and vice versa — each player's mixing probability is set by the other player's indifference condition, not their own preference. This is a general and often-missed feature of mixed equilibria that trips up newcomers.