In game theory, strategic dominance (commonly called simply dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The opposite, intransitivity, occurs in games where one strategy may be better or worse than another strategy for one player, depending on how the player's opponents may play.
Terminology
When a player tries to choose the "best" strategy among a multitude of options, that player may compare two strategies A and B to see which one is better. The result of the comparison is one of: B is equivalent to A: choosing B always gives the same outcome as choosing A, no matter what the other players do. B strictly dominates A: choosing B always gives a better outcome than choosing A, no matter what the other players do. B weakly dominates A: choosing B always gives at least as good an outcome as choosing A, no matter what the other players do, and there is at least one set of opponents' action for which B gives a better outcome than A. (Notice that if B strictly dominates A, then B weakly dominates A. Therefore, we can say "B dominates A" as synonymous of "B weakly dominates A".) B and A are intransitive: B and A are not equivalent, and B neither dominates, nor is dominated by, A. Choosing A is better in some cases, while choosing B is better in other cases, depending on exactly how the opponent chooses to play. For example, B is "throw rock" while A is "throw scissors" in Rock, Paper, Scissors. B is weakly dominated by A: there is at least one set of opponents' actions for which B gives a worse outcome than A, while all other sets of opponents' actions give A the same payoff as B. (Strategy A weakly dominates B). B is strictly dominated by A: choosing B always gives a worse outcome than choosing A, no matter what the other player(s) do. (Strategy A strictly dominates B). This notion can be generalized beyond the comparison of two strategies. Strategy B is strictly dominant if strategy B strictly dominates every other possible strategy. Strategy B is weakly dominant if strategy B weakly dominates every other possible strategy. Strategy B is strictly dominated if some other strategy exists that strictly dominates B. Strategy B is weakly dominated if some other strategy exists that weakly dominates B.
Strategy: A complete contingent plan for a player in the game. A complete contingent plan is a full specification of a player's behavior, describing each action a player would take at every possible decision point. Because information sets represent points in a game where a player must make a decision, a player's strategy describes what that player will do at each information set.
Rationality: The assumption that each player acts in a way that is designed to bring about what he or she most prefers given probabilities of various outcomes; von Neumann and Morgenstern showed that if these preferences satisfy certain conditions, this is mathematically equivalent to maximizing a payoff. A straightforward example of maximizing payoff is that of monetary gain, but for the purpose of a game theory analysis, this payoff can take any desired outcome. E.g., cash reward, minimization of exertion or discomfort, promoting justice, or amassing overall “utility” - the assumption of rationality states that players will always act in the way that best satisfies their ordering from best to worst of various possible outcomes.
Common Knowledge: The assumption that each player has knowledge of the game, knows the rules and payoffs associated with each course of action, and realizes that every other player has this same level of understanding. This is the premise that allows a player to make a value judgment on the actions of another player, backed by the assumption of rationality, into consideration when selecting an action.
Dominance and Nash equilibria
If a strictly dominant strategy exists for one player in a game, that player will play that strategy in each of the game's Nash equilibria. If both players have a strictly dominant strategy, the game has only one unique Nash equilibrium, referred to as a "dominant strategy equilibrium". However, that Nash equilibrium is not necessarily "efficient", meaning that there may be non-equilibrium outcomes of the game that would be better for both players. The classic game used to illustrate this is the Prisoner's Dilemma.
Strictly dominated strategies cannot be a part of a Nash equilibrium, and as such, it is irrational for any player to play them. On the other hand, weakly dominated strategies may be part of Nash equilibria. For instance, consider the payoff matrix pictured at the right.