Dominant Strategies and Nash Equilibrium

A dominant strategy gives a player at least as high a payoff as every alternative for every possible rival action, and a strictly higher payoff somewhere. If one strategy is worse in one case and better in another, it is not dominant.

Dominance is useful but not required for equilibrium. Many games have no dominant strategies.

Mutual best responses

A Nash equilibrium is a strategy profile in which each player's strategy is a best response to the others. Holding everyone else fixed, no player can improve by deviating alone.

In a matrix, mark each player's best responses. A cell with all players' marks is a Nash equilibrium. A game can have one, several, or no pure-strategy equilibria; mixed strategies may describe randomized choices when no pure equilibrium exists.

What equilibrium does not claim

Nash equilibrium is a consistency prediction, not a guarantee of fairness, social efficiency, or maximum joint payoff. Players may become stuck in a stable outcome that leaves all worse off than another feasible outcome.

An equilibrium also depends on the specified game. Change information, timing, payoffs, available actions, or enforcement, and best responses can change. Equilibrium selection becomes important when several equilibria exist: expectations, history, focal points, communication, and institutions can coordinate behavior.

Check equilibrium by unilateral deviations only. Moving both players at once may reveal a better joint outcome but does not disprove Nash stability.

Test every unilateral deviation

Consider a pricing game where High/High pays $(8,8)$, High/Low $(2,11)$, Low/High $(11,2)$, and Low/Low $(5,5)$. Low gives each player a higher payoff against either rival action, so Low is dominant for both. Low/Low is the unique Nash equilibrium.

High/High gives larger joint profit, 16 instead of 10, but it is not stable in the one-shot game. If one rival stays High, the other can switch alone to Low and raise its payoff from 8 to 11.

Multiple equilibria require coordination

Change Low/Low to $(1,1)$. Now each player prefers Low against High but High against Low. High/Low and Low/High are both Nash equilibria. Which one occurs may depend on history, communication, conventions, or a focal point.

Use the correct claim: Nash means no profitable unilateral deviation. It does not mean the equilibrium is unique, uses dominant strategies, maximizes joint payoff, or is morally desirable.