Dominant Strategies and Nash Equilibrium

Game theory looks at situations where one firm's best move depends on what a rival does. A dominant strategy is the strongest kind of answer: one choice that beats the alternative no matter what the rival picks. If a firm earns more from cutting price whether its rival cuts price or not, cutting price is dominant for that firm. When both firms in a duopoly have a dominant strategy, you can predict the outcome without any guessing games, just by checking each firm's own payoffs against each possible rival move.

A Nash equilibrium is a looser, more common finding. You test it by freezing one player's choice and asking if the other player regrets theirs, then switching which player you freeze. If neither side wants to move given what the other actually chose, you have found a Nash equilibrium. This does not require either strategy to be dominant. A player might only prefer their choice because of what the specific rival move is, not because it beats everything else in every situation. That is why a Nash equilibrium can exist even when nobody has a dominant strategy at all, which happens often once payoffs get uneven or one player's best move depends heavily on guessing the other's intent.

This matters for oligopoly because two firms can each play their dominant strategy, land in a Nash equilibrium, and both end up worse off than if they had cooperated. That mismatch between individual logic and joint outcome is exactly why cartels are tempting and why they are hard to sustain: each firm still wants to defect even after agreeing not to.