Dominance (game theory)

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.

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 as 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 B at least the same payoff as A.
 * 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 at least 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 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 $$s^*\in S_i$$ weakly dominates another strategy $$s^\prime\in S_i$$ if
 * $$\forall s_{-i}\in S_{-i}\left[u_i(s^*,s_{-i})\geq u_i(s^\prime,s_{-i})\right]$$ (With at least one strict inequality)

(Remember that $$S_{-i}$$ represents the product of all strategy sets other than $$i$$'s)

On the other hand, $$s^*$$ strictly dominates $$s^\prime$$ if
 * $$\forall s_{-i}\in S_{-i}\left[u_i(s^*,s_{-i})> u_i(s^\prime,s_{-i})\right]$$

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 -- however, that Nash equilibrium is not necessarily Pareto optimal, 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 to the right.

Strategy C weakly dominates strategy D. Consider playing C: if one's opponent plays C one gets 1; if one's opponent plays D one gets 0. Compare this to D where one gets 0 regardless. Since in one case one does better by playing C instead of D and never does worse, C weakly dominates D. Despite this, (D, D) is a Nash equilibrium. Suppose both players choose D. Neither player will do any better by unilaterally deviating -- if a player switches to playing C, they will still get 0. This satisfies the requirements of a Nash equilibrium.

Iterated elimination of dominated strategies (IEDS)
Also known as the iterated deletion of dominated strategies, it is one common technique for solving games that 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.

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 0-691-00395-5
 * Ginits, Herbert (2000) Game Theory Evolving Princeton University Press  ISBN 0-691-00943-0
 * Rapoport, A. (1966) Two-Person Game Theory: The Essential Ideas University of Michigan Press.
 * Jim Ratliff's Game Theory Course: Strategic Dominance

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