The simplest idea in strategy
Some decisions look hard only because you are trying to guess what a rival will do. Often you do not need to guess. If one of your options is better than another no matter what the rival does, there is no reason to ever take the worse one. Spotting such options is the first and cheapest tool in game theory.
Two terms cover it:
- A strategy is dominated if some other strategy of yours gives a better payoff against every choice the rival might make.
- A strategy is a dominant strategy if it beats all your other strategies against every rival choice.
If you have a dominant strategy, you can simply play it. If you do not, you may still be able to cross off dominated options and make the problem smaller.
Strict and weak dominance
Strict dominance means the better option is strictly better in every case. Weak dominance means it is never worse and is better in at least one case. Strictly dominated options are safe to remove, since you would never prefer them under any belief about the rival. Weakly dominated options are trickier: they can still be best replies in some situations, and removing them can sometimes change which equilibria survive. A cautious approach is to remove strictly dominated options first and treat weak dominance as a tie-breaker.
Reading a payoff table
In the table below, you choose a row and your rival chooses a column. Each cell lists payoffs as (you, rival).
| X | Y | Z | |
|---|---|---|---|
| A | (3, 2) | (1, 0) | (0, 1) |
| B | (2, 1) | (2, 0) | (1, 0) |
| C | (1, 0) | (0, 2) | (0, 2) |
Step 1: look for dominated rows. Compare your payoffs in rows B and C, column by column. Against X, B gives 2 and C gives 1. Against Y, B gives 2 and C gives 0. Against Z, B gives 1 and C gives 0. B is better in every column, so C is strictly dominated and you would never play it.
Compare A and B too: against X, A gives 3 and B gives 2, but against Z A gives 0 and B gives 1. Neither dominates the other, so both stay.
Step 2: look for dominated columns, using the reduced table. Now suppose your rival also knows you will not play C. With only rows A and B left, compare the rival’s payoffs. Against A, the rival gets 2 from X, 0 from Y and 1 from Z. Against B, the rival gets 1 from X, 0 from Y and 0 from Z. X is better than both Y and Z against each of your remaining rows, so Y and Z are dominated and X remains.
Notice that this step was not available at the start. In the full table, row C gives the rival 0 from X but 2 from Y and 2 from Z, so X did not dominate. It became dominant only after C was removed.
Step 3: choose within what remains. The rival will play X. Against X, you get 3 from A and 2 from B, so you choose A. The outcome is (A, X) with payoffs (3, 2).
Check it: given the rival plays X, would you switch from A? No, since 3 beats 2. Given you play A, would the rival switch from X? No, since 2 beats 0 and 1. So neither wants to deviate, which makes this a Nash equilibrium as well.
Why the order of reasoning matters
The three steps above rely on a chain of assumptions. You remove C because it is bad for you. The rival can then reason that you will not play C. You can reason that the rival reasons that way, and so on. This is called iterated elimination of dominated strategies, and it assumes players are rational and believe each other to be rational. That is a strong assumption. If you suspect a rival may play a dominated option anyway, perhaps because they do not see the table the same way, the clean answer can fail.
The strictly dominated version has a pleasant property: the order in which you remove options does not change the final result. For weak dominance, order can matter, which is another reason to be careful.
A dominant choice that is still unhappy
A dominant strategy is not a promise of a good result. In the prisoner dilemma, defecting is dominant for both players, yet both defecting leaves them worse off than both cooperating. Dominance says what is best for you given the structure, not what is best for the group. So when you find a dominant strategy, you have a clear move, but you have not necessarily found a good outcome.
Dominance in actual games
Pure dominance is less common in games than in textbooks, but it appears in several forms:
- Game design. Designers try to avoid options that are strictly dominated, because they feel like traps. A card that costs more and does less than another card is a design flaw. Players who notice this can safely ignore it. When it is not strictly worse in every situation, the designer has created a real choice.
- Deck and build choices. Options may be dominated in a specific context, such as one that is worse than another at every price point. Judging this involves weighing probabilities, which is where deck consistency and metagame choices come in.
- Bidding and pricing. In some auctions a particular bid is dominant, which is why second-price formats are studied.
- Everyday tradeoffs. If a path takes longer and is also more dangerous than another, you never need to consider it.
A useful habit is to ask of any option, “Is there something that does at least as well in every case I can imagine?” If yes, drop it and put your attention on the real choices.
Dominance under uncertainty
When chance is involved, dominance can be stated in terms of outcomes. If option P gives a better result than option Q in every possible roll or draw, P dominates Q. A weaker, softer version compares expected value: P might have a higher average than Q without beating it in every case. That is a different kind of preference, and it is not dominance, since Q can still come out ahead in some outcomes. Keep the two ideas apart: dominance is about every case, expected value about the average.
Common mistakes
- Comparing across the wrong columns. Dominance requires comparing your options against the same rival choice, not against different ones.
- Ignoring that payoffs are not only points. If you value a particular outcome, such as denying a rival, that belongs in your payoff.
- Assuming the rival sees the same table. Their payoffs may differ from what you assume.
- Treating the process as a guarantee. It narrows choices, but if the table is not fully solved by elimination, you need other tools such as equilibrium analysis or mixed play.
Elimination is only a first pass, but it is a dependable one. Remove what is always worse, and what is left is often a much smaller decision.