Pure Strategy Nash Equilibrium Quiz

A pure strategy Nash equilibrium is a strategy profile where every player chooses one definite strategy with certainty rather than randomizing. At this profile, no individual player can increase their payoff by switching to a different strategy unilaterally, given the strategies chosen by all other players. The pure strategy requirement distinguishes it from mixed strategy equilibria where players randomize, and the Nash condition ensures no deviation incentive exists.

To identify a pure strategy Nash equilibrium, analysts check each cell of the payoff matrix to determine whether any player could earn a higher payoff by switching to a different strategy while all other players remain at their current strategies. If no player can profitably deviate at a particular cell, that cell is a Nash equilibrium. This best-response checking method is the standard technique applied across all types of strategic games.

The defining feature of a pure strategy is certainty: each player commits fully to one strategy rather than randomly selecting between options. A pure strategy Nash equilibrium is a profile of such certain choices where the mutual best-response condition holds. This contrasts with mixed strategy Nash equilibria where players assign probabilities to multiple strategies. Both types satisfy the Nash equilibrium condition, but pure strategy equilibria are more transparent and easier to identify directly from the payoff matrix.

Row 1 strictly dominates Row 2 for Player 1 (6 versus 4 under any rival choice), and Column 1 strictly dominates Column 2 for Player 2 (5 versus 3 under any rival choice). Rational players always choose dominant strategies. The pure strategy Nash equilibrium is (Row 1, Column 1) where both players play their dominant strategies. At this cell, neither player benefits from deviating, confirming it satisfies the Nash equilibrium no-deviation condition.

Not every finite game has a pure strategy Nash equilibrium. Some games, such as Matching Pennies, have no cell where both players are simultaneously playing best responses in pure strategies. Nash's existence theorem guarantees that a Nash equilibrium always exists if mixed strategies are allowed, but pure strategy equilibria may not exist in every game. When no pure strategy equilibrium exists, players must mix across strategies with specific probabilities to reach equilibrium.

When a game has a unique pure strategy Nash equilibrium, game theory produces a clear and unambiguous prediction. Rational players who understand the payoff structure will independently identify and select the unique equilibrium through best-response reasoning without needing to coordinate. Uniqueness eliminates the equilibrium selection problem that arises with multiple equilibria, making the unique Nash equilibrium a particularly powerful and actionable prediction in applied strategic analysis.

Pure strategy Nash equilibria involve certain strategy choices rather than randomization, may fail to exist in some games requiring mixed strategy analysis, and are identified through the no-unilateral-deviation check. The claim that they always produce maximum payoffs for all players is false: the Prisoners Dilemma has a pure strategy Nash equilibrium at mutual defection that leaves both players worse off than the cooperative outcome, directly contradicting this claim.

In Matching Pennies, at any pure strategy profile, one player always has an incentive to switch. If both show Heads, Player 2 wants to switch to Tails. If Player 1 shows Heads and Player 2 shows Tails, Player 1 wants to switch to Tails. This cycle of deviation continues through every cell, meaning no pure strategy profile satisfies the Nash equilibrium condition simultaneously. The only equilibrium is a mixed strategy where each player randomizes with equal probability.

A pure strategy Nash equilibrium driven by dominant strategies is the strongest possible prediction in game theory. Dominant strategies are best responses under every rival strategy, not just at one specific equilibrium profile. When both players have dominant strategies pointing to the same cell, the resulting Nash equilibrium is uniquely compelling: each player should choose their dominant strategy regardless of what they believe rivals will do, making the prediction robust to any assumption about rival rationality or expectations.

When more than one cell in a payoff matrix satisfies the condition that no player can profitably deviate, each qualifying cell is independently a pure strategy Nash equilibrium. Games with multiple pure strategy Nash equilibria are common, particularly in coordination games. Each equilibrium is self-enforcing on its own. The existence of multiple equilibria creates an equilibrium selection challenge that the basic Nash concept alone cannot resolve.

A pure strategy Nash equilibrium requires that no player can improve their payoff by switching strategies unilaterally. At (Top, Left), both Player 1 and Player 2 can earn more by switching. Since at least one player has a deviation incentive, the no-deviation condition is violated and (Top, Left) cannot be a Nash equilibrium. This is the direct application of the Nash equilibrium verification test: any profitable unilateral deviation disqualifies a strategy profile as an equilibrium.

Dominant strategies are by definition best responses under all rival strategy combinations, so any strategy profile where all players play dominant strategies satisfies the Nash equilibrium condition. However, pure strategy Nash equilibria also arise in games where best responses hold only at specific rival strategies, not universally. Many coordination games have pure strategy Nash equilibria without any player having a dominant strategy, confirming that Nash equilibrium is the broader concept encompassing but not limited to dominant strategy equilibria.

The Cournot Nash equilibrium is the intersection of both firms' best-response functions. At this quantity pair, Firm 1 is maximizing profit given Firm 2's equilibrium output, and Firm 2 is doing the same. Neither firm can increase profit by changing output unilaterally. This intersection represents the pure strategy Nash equilibrium of the Cournot model, where both firms' strategies are specific quantities chosen with certainty and each is simultaneously a best response to the other.

The Prisoners Dilemma has a unique pure strategy Nash equilibrium at mutual defection since defection dominates cooperation. The Battle of the Sexes has two pure strategy Nash equilibria at the two coordination outcomes. The Cournot duopoly pure strategy Nash equilibrium is found at the best-response function intersection. Matching Pennies has no pure strategy Nash equilibrium at all: the claim that (Heads, Heads) is one is false since Player 2 would always prefer to switch to Tails.

Pure strategy Nash equilibria correspond to firms making specific, observable choices rather than randomly mixing between options. When a firm sets a particular price, chooses a definite output level, or decides whether to advertise, it is making a pure strategy choice. These concrete commitments are observable and directly testable against real market data. Mixed strategy equilibria, while theoretically important, describe randomized behavior that is harder to observe and verify in practice, making pure strategy equilibria more practically applicable.