Can a strictly dominated strategy appear with positive probability in a Nash equilibrium?

In the standard finite expected-utility setting, no; it cannot be a best response.

Does every game contain a strictly dominant strategy?

No. Many games require other methods to analyze strategic choices.

Can a strategy become dominated only after other choices are removed?

Yes. Comparisons in the reduced game may reveal dominance that was absent before elimination.

AI Glossary / Graphs and symbolic AI / Intermediate

Strict dominance

Definition

Strict dominance is a comparison between two strategies available to the same player. One strategy strictly dominates another when it gives that player a strictly higher payoff for every possible combination of the other players' strategies. The dominated strategy is consistently worse under the stated game model, so choosing it cannot be a best response to the others' play.

What it means

The comparison concerns one player's alternatives, not whether one player earns more than another. In a payoff table, examine that player's payoffs while holding the opponents' choices fixed. The proposed dominating strategy must win every such comparison, with no ties.

A strategy that strictly dominates one alternative need not dominate all alternatives. Calling it a strictly dominant strategy makes the stronger claim that it beats every other available strategy against all opponents' choices. This distinction prevents a useful local comparison from being mistaken for a complete solution to the game.

How it works

Start with the player's strategy set and payoff function. For two strategies A and B, compare A's payoff with B's against each opponent strategy profile. If A's payoff is higher everywhere, A strictly dominates B. One equal or lower comparison is enough to defeat that particular strict-dominance claim.

For finite games with expected-utility evaluation, a pointwise strict advantage against every pure opponent profile also remains an advantage against any probability distribution over those profiles. Take the probability-weighted average of the payoff differences: all possible differences are positive.

Dominance analysis can also allow a mixed strategy to dominate a pure strategy. This broadens the search beyond comparisons between individual rows. When simplifying a game, strictly dominated strategies can be removed, and further dominance may become visible in the reduced game. Each conclusion must identify the alternatives still under consideration.

Three practical examples

Agent action review: identify a response option that produces less reward than another option in every modeled opponent scenario, helping simplify a simulation.
Negotiation training: distinguish an offer rule that is uniformly inferior in a toy game from one that is merely unattractive under a particular forecast.
Game-solving education: remove strictly dominated rows or columns before checking the remaining profiles for equilibrium, while retaining a record of the comparisons.

Think of it this way

Think of comparing two routes under every traffic condition listed in a planning table. If one is faster in every listed condition, the other is uniformly inferior on travel time. The analogy is bounded by the model: an unlisted closure, a different objective such as reliability, or a route-specific cost can reverse the real decision.

Worked example

Consider an original payoff table for one player with strategies A, B, and C. The opponent can choose Left or Right. Only the first player's payoffs are needed for these comparisons. A gives 5 against Left and 3 against Right. B gives 4 and 2. C gives 1 and 6.

A strictly dominates B because 5 > 4 and 3 > 2. If the opponent chooses Left with probability p, A's expected payoff is 5p + 3(1 − p) = 3 + 2p. B's is 4p + 2(1 − p) = 2 + 2p. A remains exactly one payoff unit better for every p between zero and one.

A does not dominate C: it wins against Left but loses against Right. Indeed, C's expected payoff is 1p + 6(1 − p) = 6 − 5p. A exceeds C only when p > 3/7, with equality at 3/7. That threshold comparison depends on beliefs and is not strict dominance.

For a separate mixed-strategy illustration, let D have payoffs (4, 0), E have (0, 4), and F have (1, 1). Neither D nor E strictly dominates F. However, a half-D, half-E mixture gives expected payoff (2, 2), strictly dominating F against either opponent choice. Randomization changes the available comparison without guaranteeing a higher realized payoff on every draw.

Illustrative case study

Fictional case: a team evaluates three scheduling agents in a small adversarial benchmark. An analyst calls the agent with the highest average score “dominant.” A reviewer asks for the scenario-by-scenario table and finds that the leading agent loses in one opponent configuration.

The team changes the description to best under the benchmark's stated weighting. A different agent is genuinely strictly dominated within the table and is removed from the next model comparison. This fictional example illustrates precise reporting. It does not establish that the removed agent is inferior under untested workloads, objectives, or implementation costs.

Common misconceptions and mistakes

Higher average payoff is not strict dominance. Better performance against the most likely opponent is not strict dominance either. Weak dominance allows ties under its usual definition, whereas strict dominance requires a strict advantage everywhere.

Do not compare different players' payoff numbers to establish dominance: their objectives and scales may differ. Do not treat a mixed strategy's expected superiority as a guarantee for each random outcome. Finally, failing to find a dominated strategy does not mean that every remaining profile is an equilibrium; dominance is only one analytical tool.

Business use

For business simulation, the useful question is whether an option can be rejected without needing to predict how other participants will choose. A valid strict-dominance comparison provides that conclusion within the model. Document the payoff definition, opponent possibilities, and whether mixtures are permitted.

Recheck the analysis when adding fees, constraints, new competitors, or uncertainty about the payoff table itself. A proof over a simplified table remains valuable, but its scope is the table, not every possible deployment environment.

Check your understanding

A gives payoffs (7, 3) and B gives (6, 3). Does A strictly dominate B?

Answer: No. The second comparison is a tie. A is never worse and is better in one case, fitting the usual weak-dominance definition instead.

Related terms

Dominant strategy
Mixed strategy
Expected utility
weak dominance
best response
payoff matrix
Nash equilibrium
iterated elimination

Learn next

Create a small payoff table and mark every strict row comparison for one player. Repeat using that player's payoff entries in the columns if appropriate. Then test whether a mixture beats an option that no single pure strategy dominates.

Pure strategy

Sources

Carnegie Mellon, 15-281 Game Theory course notes, dominance, best responses, and iterative removal: https://www.cs.cmu.edu/~15281/coursenotes/gametheory/ The payoff tables, threshold calculation, and fictional benchmark case are original teaching examples. Dominance statements assume a fixed finite game and expected-utility comparisons.

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