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Computing the Domination-Based Concepts

• “Dominance”

• “Iterated Elimination of Dominated Strategies”

Game TheoryLecture 10

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Identifying dominated strategies• Recall that one strategy dominates another when the first

strategy is always at least as good as the second, regardlessof the other players’ actions.

• In this lecture, we discuss some computational tools foridentifying dominated strategies, and consider thecomputational complexity of this process.

• Recall: Iterated removal of strictly dominatedstrategies eliminates the same set of strategies regardless of the

elimination order, and

all Nash equilibria of the original game will be containedin the remaining set.

Thus, this method can be used to narrow down the setof strategies to consider before attempting to identify asample Nash equilibrium.

In the worst case, this procedure will have no effect—many games have no dominated strategies.

In practice, however, it can make a big difference toiteratively remove dominated strategies beforeattempting to compute an equilibrium.

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Identifying dominated strategies (Cont’d)• Recall: Iterated removal of weakly dominated

strategiesElimination order does make a difference: the set of

strategies that survive iterated removal can differdepending on the order in which dominatedstrategies are removed.

Removing weakly dominated strategies caneliminate some equilibria of the original game.

There is still a computational benefit to this technique:

Since no new equilibria are ever created by thiselimination (and since every game has at least oneequilibrium), at least one of the original equilibriaalways survives.

This is enough if all we want to do is to identify asample Nash equilibrium.

Furthermore, iterative removal of weakly dominatedstrategies can eliminate a larger set of strategiesthan iterative removal of strictly dominatedstrategies and so will often produce a smaller game.

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Domination by a pure strategy• Checking whether a (not necessarily pure) strategy si

for player i is (strictly; weakly) dominated by any purestrategy for i.

• Let us consider the case of strict dominance.

To solve the problem we must check every pure

strategy ai for player i and every pure-strategyprofile for the other players to determinewhether there exists some ai for which it is neverweakly better for i to play si instead of ai. If so, si isstrictly dominated.

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Domination by a pure strategy

• The case of weak dominance can be tested usingessentially the same algorithm, except that we musttest the condition ui(si,a−i) > ui(ai,a−i). Also, we needto do a bit more book-keeping:

• We must also set dom ← false if there is not atleast one a−i for which ui(si, a−i) < ui(ai, a−i).

• For both definitions of domination, the complexityof the procedure is O(|A|), linear in the size of thenormal-form game.

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Domination by a mixed strategy• Recall that sometimes a strategy is not

dominated by any pure strategy, but isdominated by some mixed strategy.

• We cannot use a simple algorithm like before totest whether a given strategy si is dominated bya mixed strategy because these strategiescannot be enumerated.

• However, it turns out that we can still answerthe question in polynomial time by solving alinear program.

To this end, we will assume that player i’sutilities are strictly positive.

This assumption is without loss ofgenerality since if any player i’s utilitieswere negative, we could add a constant toall payoffs without changing the game.

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Domination by a mixed strategy• Each flavor of domination requires a somewhat

different linear program.

First, let us consider strict domination by amixed strategy. This would seem to have thefollowing straightforward LP formulation(indeed, a mere feasibility program).

• While the constraints do indeed describestrict domination by a mixed strategy, they donot constitute a linear program.

The problem is that the constraints in linearprograms must be weak inequalities.

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Strict Domination by a mixed strategy

• This LP simulates the strict inequality of constraintthrough the objective function.

Because no constraints restrict the pj’s fromabove, this LP will always be feasible.

However, in the optimal solution the pj’s may notsum to 1; indeed, their sum can be greater than 1or less than 1.

In the optimal solution, the pj’s will be set so thattheir sum cannot be reduced any further withoutviolating constraint (*).

• Instead, we must use the LP that follows:

(*)

Thus for at least some a−i ∈ A−i we will have:

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In this case, we can add a positiveamount to each pj in order to causeconstraint (*) to hold in its strict versioneverywhere while achieving thecondition

Strict Domination by a mixed strategy

(*)

• A strictly dominating mixed strategy thereforeexists if and only if the optimal solution to theLP has objective function value strictly lessthan 1.

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Weak Domination by a mixed strategy• Again our inability to write a strict inequality will make

things more complicated. However, we can derive anLP by adding an objective function to the feasibilityprogram.

• Because of constraint (*), any feasible solution willhave a nonnegative objective value.

• If the optimal solution has a strictly positive objective,the mixed strategy given by the pj’s achieves strictlypositive expected utility for at least one a−i∈A−i,meaning that si is weakly dominated by this mixedstrategy.

(*)

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• Finally, we consider the iterated removal ofdominated strategies.

• We only consider pure strategies as candidatesfor removal;

indeed, as it turns out, it never helps toremove dominated mixed strategies whenperforming iterated removal.

It is important, however, that we considerthe possibility that pure strategies may bedominated by mixed strategies.

Iterated dominance

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• Each step removes one pure strategy for oneplayer, so there can be at most

steps.

• A single step of this process consists ofchecking whether every pure strategy of everyplayer is dominated by any other mixedstrategy, which requires us to solve at worst

linear programs.

• For both flavors of domination, it requires onlypolynomial time to iteratively removedominated strategies until the game has beenmaximally reduced (i.e., no strategy isdominated for any player).

Iterated dominance (Cont’d)