# Dictionary Definition

dominated adj

1 controlled or ruled by superior authority or
power

2 harassed by persistent nagging [syn: henpecked]

# User Contributed Dictionary

## English

### Verb

dominated- past of dominate

# Extensive Definition

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. 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 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 and A are intransitive: 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 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_\in S_\left[u_i(s^*,s_)\geq u_i(s^\prime,s_)\right] (With at least one strict inequality)

On the other hand, s^* strictly dominates
s^\prime if

- \forall s_\in S_\left[u_i(s^*,s_)> u_i(s^\prime,s_)\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 at 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)

The iterated elimination of dominated strategies, also known as the iterated deletion of dominated strategies, 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.)

## See also

## 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

dominated in French: Dominance stratégique

dominated in Polish: Strategia dominująca

dominated in Chinese: 支配性策略