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In game theory, dominance (also called strategic 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. Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ...
A strategy in game theory is a sequence of activities and reactions, that fully determine an agents bahaviour in a game or a business situation. ...
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. - B dominates A: choosing B always gives at least a good an outcome as choosing A. There are 2 possibilities:
- B strictly dominates A: choosing B always gives a better outcome than choosing A, no matter what the other player(s) do.
- B weakly dominates A: There is at least one set of opponents' action for which B is superior, and all other sets of opponents' actions give A and B the same payoff.
- B is dominated by A: choosing B never gives a better outcome than choosing A, no matter what the other player(s) do. There are 2 possibilities:
- 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 and B the same payoff. (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 dominates all other strategies, but some are only weakly dominated.
- 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.
Mathematical definition In mathematical terms, For any player i, a strategy  weakly dominates another strategy  if ![forall s_{-i}in S_{-i}left[u_i(s^*,s_{-i})geq u_i(s^prime,s_{-i})right]](http://en.wikipedia.org/math/9/1/9/91946423568008a2eb3533449a356b2b.png) (Remember that S − i represents the product of all strategy sets other than i's)
On the other hand, s * strongly dominates  if
![forall s_{-i}in S_{-i}left[u_i(s^*,s_{-i})> u_i(s^prime,s_{-i})right]](http://en.wikipedia.org/math/8/8/3/883788c42b7e4dea56b51fee531bec21.png)
Dominance and Nash equilibria | C | D | | C | 1, 1 | 0, 0 | | D | 0, 0 | 0, 0 | 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 dominant strategy, the game has only one unique Nash equilibrium -- however, that Nash equilibrium is not necessarilly Pareto optimal, meaning that there may 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. In game theory, the Nash equilibrium (named after John Nash who proposed it) is a kind of optimal collective strategy in a game involving two or more players, where no player has anything to gain by changing only their own strategy. ...
Pareto efficiency, or Pareto optimality, is a central concept in economics with broad applications in game theory, engineering and the social sciences. ...
Will the two prisoners cooperate to minimise total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? The prisoners dilemma is a type of non-zero-sum game (game in the sense of Game Theory). ...
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 game pictured to the right.
Strategy C weakly dominates strategy D, but nonetheless, (D, D) is a Nash equilibrium.
Iterated elimination of dominated strategies (IEDS) One common technique for solving games involves iteratively removing dominated strategies. In the first step, all dominated strategies of the game are removed, since rational players will not play them. This results in a new, smaller game. Some strategies -- that were not dominated before -- may be dominated in the smaller game. These are removed, creating a new even smaller game, and so on. Iteration is the repetition of a process, typically within a computer program. ...
There are two versions of this process. One version involves only eliminating strictly dominated strategies. If, after completing this process, there is only one strategy for each player remaining, that strategy set is the unique Nash equilibrium.
Another version involves eliminating both strictly and weakly dominated strategies. If, at the end of the process, there is a single strategy for each player, this strategy set is also a Nash equilibrium. However, unlike the first process, elimination of weakly dominated strategies may eliminate some Nash equilibria. As a result, the Nash equilibrium found by eliminating weakly dominated strategies may not be the only Nash equilibrium. (In some games, if we remove weakly dominated strategies in a different order, we may end up with a different Nash equilibrium.)
External links and references - Fudenberg, Drew and Jean Tirole (1993) Game Theory MIT Press.
- Gibbons, Robert (1992) Game Theory for Applied Economists, Princeton University Press ISBN 0691003955
- Ginits, Herbert (2000) Game Theory Evolving Princeton University Press ISBN 0691009430
- Jim Ratliff's Game Theory Course: Strategic Dominance
This article incorporates material from Dominant strategy on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative, online mathematics encyclopedia. ...
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