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Dominance and Iterated Elimination

Some conclusions in a game need a forecast of the other player. A few do not, and those are worth finding first, because they survive being wrong about everyone else.

Strict dominance

An action is strictly dominated if some other action pays you more against every single thing the opponent might do.

Strict dominance

\pi_i(a_i, a_{-i}) > \pi_i(a_i', a_{-i}) \quad \text{for every } a_{-i}

Better in every column, not on average across columns. The distinction matters: an action can lose to another on average and still be worth playing against a particular opponent.

Here $a_i'$ is dominated and should never be played. Notice what is absent: any probability that the opponent does one thing rather than another. You do not need beliefs, so you cannot have wrong ones.

In the quoting game from what is a strategic game, tightening paid more whether the competitor quoted wide or tight. Quoting wide was strictly dominated, and both firms tightened without either needing a view.

The quoting game, cell by cell

You, them	They Wide	They Tight
You Wide	120, 120	0, 180
You Tight	180, 0	90, 90equilibrium

Bold marks the best response: yours in flame, theirs in ink. A cell where both are bold is an equilibrium, because neither side wants to move. Tightening is better in both columns, so it wins without any view on what they will do. You both tighten and take 90 each, when staying wide would have paid 120. Nobody made a mistake.

Bold marks the best response: yours in flame, theirs in ink. A cell where both are bold is an equilibrium, because neither side wants to move. Tightening no longer wins in both columns: at 1.50 ticks against 2, undercutting buys flow too cheaply to be worth it. Dominance is gone and there are now two equilibria, so what you should do depends entirely on what you think they will do.

Daily flow120 lots

Wide quote: half-spread2.00 ticks

Tight quote: half-spread1.50 ticks

Both quote wide

120

Both quote tight

Cost of competing

−30

Bold is the best response. Drag the tight half-spread below half the wide one and the dominance argument disappears.

Drag the tight half-spread below half the wide one and the dominance vanishes. That is worth internalising early: dominance is a property of the numbers in the matrix, not of the game's story.

Weak dominance

An action weakly dominates another if it is at least as good in every column and strictly better in at least one. Weaker grounds, and worth treating with more care, because the order in which you eliminate weakly dominated actions can change what you are left with. Strict elimination has no such problem.

Iterating

Removing a dominated action changes the game, and a strategy that was not dominated before can become dominated once the opponent's worst options are gone. So you look again, and repeat.

Worked example: two-thirds of the average

Everyone picks a number from 0 to 100. Whoever is closest to two thirds of the group average wins.

The average cannot exceed 100, so the target cannot exceed $\tfrac{2}{3} \times 100 \approx 66.7$ . Every guess above 66.7 is therefore dominated by 66.7 itself, and no rational player picks one.

Now assume everyone has worked that out. The average cannot exceed 66.7, so the target cannot exceed 44.4, and guesses above that go too. Repeat the argument and the surviving set shrinks each round, converging on 0. Zero is the unique equilibrium.

Play this with real people and the winning guess is usually between 20 and 35.

The gap is not irrationality. It is that the argument requires everyone to iterate, and to believe everyone else iterates, without limit. Most people go one or two rounds. Someone reasoning one step deeper than the field guesses around 33, someone two steps deeper guesses around 22, and the winner is whoever judged the field's depth correctly rather than whoever solved the game.

Key takeaway

The equilibrium answer is 0, and 0 loses. Both facts are true, and knowing which one the question is asking for is the actual skill. Against equilibrium players, play the equilibrium. Against a real field, model the field.

Tip

Asked this in an interview, give both: the iterated argument down to 0, then the observation that real fields land near 20 to 35 and why. Stopping at 0 reads as recall. Stopping at 25 reads as a guess.

Test your knowledge

What makes an argument from strict dominance stronger than an argument from expected value across the opponent's likely actions?

It holds against every action the opponent might take, so it needs no view of them

It produces a higher payoff than any expected value calculation could

It applies to games with more than two players, whereas expected value arguments do not

It guarantees the outcome will be good for both players rather than only for you

In the two-thirds of the average game played against a normal group of people, why does the iterated elimination answer of 0 usually lose?

Because the iterated argument contains an arithmetic error that only shows up after several rounds

Because the target is two thirds of the average, and an average of real numbers can never reach 0

Because it needs unlimited rounds of mutual reasoning, and real players stop after one or two

Because players deliberately choose high numbers to punish anyone who has studied the game

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