Strategic dominance
Strategic dominance
In game-theory, a strategy A dominates another strategy B if A will always produce a better result than B, regardless of how any other player plays. Some very simple games (called straightforward games) can be solved using dominance. ## Terminology A player can compare two strategies, A and B, to determine which one is better. The result of the comparison is one of: 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' actions 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" to mean "B weakly dominates A".) 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 B the same payoff as A. (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). Neither A nor B dominates the other: 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|Rock, Paper, Scissors]]. 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—cash reward, minimization of exertion or discomfort, or promoting justice can all be modeled as amassing an overall “utility” for the player. 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.
Moreover, although any strategy profile that survives IEWDS must satisfy a certain cautious-rationality condition (players never play weakly dominated strategies given full-support beliefs about opponents) , the elimination process can exclude some Nash equilibria of the original game.
Use‐cases of the method include simplification of games in which weak dominance relations exist, and the analysis of games under “cautious belief” assumptions. For instance, recent work shows that in certain well-founded extensive-form games the maximal application of IEWDS can reduce the game to a trivial subgame containing a unique subgame-perfect equilibrium.