# Strategic predominance

In Game theory, the strategic predominance appears when a Stratégie is better for a player than another strategy, whatever the strategy of the adversary.

## Definitions

That is to say two strategies has and B possible by a player in a given play with perfect information. B dominates A' if the payment associated with B is equal to or higher than that associated with for any strategy of the adversary has. If there exists moreover a strategy of the adversary such as the payment associated with B is strictly higher than that associated with has for this strategy, then B dominates strictly has .

These definitions spread naturally with any whole of strategies:

• B is strictly dominant if it dominates all the other possible strategies strictly.
• B is slightly dominant if it dominates all the other strategies but that there is at least one which is not strictly dominated by B .

## Mathematical formulation

That is to say a player $i$, $S_i$ the whole of its strategies and $S_ \left\{- I\right\}$ the whole of the strategies of its adversaries. A strategy $s^* \ in S_i$ slightly dominates another statégie $s^ \ premium \ in S_i$ if

$\ forall s_ \left\{- I\right\} \ in S_ \left\{- I\right\} \ left u_i \left(s^ \ premium, s_ \left\{- I\right\}\right) \ right$
with at least a strict inequality.

In the same way, $s^*$ dominates $s^ \ prime$ strictly if

$\ forall s_ \left\{- I\right\} \ in S_ \left\{- I\right\} \ left u_i \left(s^ \ premium, s_ \left\{- I\right\}\right) \ right$

## Strategic predominance and balance of Nash

If there exists a dominant strategy for a player in a play, then this strategy will be played by this player with the balance of Nash of the play. If the two players have a dominant strategy, then the balance of nash is single and is with the meeting of these two strategies. This balance can however not be Pareto - optimal, like in the case of the Dilemme of the prisoner.

Reciprocally, a strictly dominated strategy cannot be played balance of Nash of the play, but a slightly dominated strategy can the being, like in the case of the play opposite: C dominates slightly D , but (D, D) is a balance of Nash.

## Repeated elimination of the dominated strategies

A technique of resolution of the plays is the repeated elimination of the dominated strategies. At the first stage, all the dominated strategies are eliminated from the play, since the player will not choose them with balance. That led to a new handset, more reduced. Strategies which were not dominated before can become it, because of elimination of nonrelevant situations. One eliminates them in their turn, until there does not exist any more strategy dominated in the play.

If one eliminated thus only the strictly dominated strategies and that it does not remain any more that one strategy for each player, one determined the balance of Nash of the play.

If one also eliminated the strategies slightly dominated with each stage and that there remains only one strategy per player, one also obtains a balance of Nash. However, it can then exist in the set of other balances of Nash, and balances it obtained at the end it proceeded can be different according to the order from elimination of the strategies retained.

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