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Iterated dominance

Iterated dominance repeatedly prunes a strategic game. A three-by-two example shows strict elimination; a tied game reveals weak dominance and order dependence.

Iterated dominance is a game-theoretic procedure that repeatedly removes dominated strategies and then checks the smaller game for further deletions. In a model of competing trading decisions, it can rule out choices before solving for equilibrium. The crucial word is repeatedly: an action that survives the original payoff table may become inferior once some opponent actions are excluded.
The deletion rule must be specified. Iterated elimination of strictly dominated strategies removes choices that pay strictly less than some alternative against every remaining opponent strategy. Replacing “strictly” with “weakly” changes what the procedure can conclude.

Compare one player's payoffs at a time

Write ui(si,s−i)u_i(s_i,s_{-i}) for player ii's payoff from its strategy sis_i and the other players' strategy profile s−is_{-i}. An available alternative tit_i strictly dominates sis_i in the current reduced game when

ui(ti,s−i)>ui(si,s−i)for every remaining s−i.u_i(t_i,s_{-i})>u_i(s_i,s_{-i}) \qquad\text{for every remaining }s_{-i}.
The same alternative must win every comparison. Choosing a different alternative after seeing each opponent action does not establish dominance. Nor do we compare one player's payoff with another player's: each player maximises its own objective. Yildiz's notes, §2.1, give the strict-dominance definition, including domination by a mixed strategy.

Why the second round matters

Consider this fictional one-round, simultaneous game. The rules and payoff table are known to both players; each prefers a higher own utility score. Each cell lists (Row utility, Column utility). The numbers are stipulated utility units, not estimated trading returns.
Row's strategyColumn: LColumn: R
A(3,2)(3,2)(2,1)(2,1)
B(2,1)(2,1)(1,2)(1,2)
C(1,0)(1,0)(0,3)(0,3)

For Row, A beats B whether Column chooses L or R: 3>23>2 and 2>12>1. A also beats C: 3>13>1 and 2>02>0. Delete B and C.

Column cannot initially delete R. Its L payoffs across A, B and C are 2,1,02,1,0, while its R payoffs are 1,2,31,2,3. L is better against A, but R is better against B and C. Neither column strictly dominates the other in the original game.
After excluding B and C, only Row's A remains. Column now compares its second entries: L gives 22, R gives 11. In this reduced game, L strictly dominates R. The sole surviving profile is (A, L), with payoffs (3, 2).
A fictional game shrinks from Row strategies A, B and C against Column L and R to A against L and R, then to A against L. A gives Row 3 versus L and 2 versus R, strictly beating B and C. With only A left, Column prefers L's 2 to R's 1. The final payoffs are 3 for Row and 2 for Column.
A fictional game shrinks from Row strategies A, B and C against Column L and R to A against L and R, then to A against L. A gives Row 3 versus L and 2 versus R, strictly beating B and C. With only A left, Column prefers L's 2 to R's 1. The final payoffs are 3 for Row and 2 for Column.
Original calculation for the stated utility game. First remove Row's B and C; only then does Column's L dominate R. Deletion rounds describe reasoning, not successive moves by the players.
Row needs no prediction of Column's choice to reject B and C. Column's second-round inference additionally relies on Row avoiding those inferior choices. More rounds require further layers of reasoning about the other players' rationality and knowledge of the game. Polak's Yale lecture 2 develops this distinction. If a trading counterparty may use an excluded rule, the reduced table has discarded a behaviour that can still matter.

Surviving does not mean equilibrium

In a finite game, strict elimination terminates, and exhaustive deletion gives the same surviving pure-strategy sets regardless of deletion order, under the standard rule allowing available mixed strategies to dominate. It preserves every Nash equilibrium: an equilibrium cannot put positive probability on a strictly dominated strategy. These properties are set out in Manea's MIT slides 12–18 and 33.
A single surviving profile, as above, therefore solves the finite game. With several survivors, elimination alone generally leaves work to do. Survival does not make every surviving combination a mutual best response. In matching pennies, for example, neither pure action is dominated, yet no pure profile is a Nash equilibrium.
Checking only pure alternatives can also stop too early. Suppose Row's utility vectors against L and R are X: (4,0)(4,0), Y: (0,4)(0,4) and Z: (1,1)(1,1). Neither X nor Y strictly dominates Z. A half-X, half-Y mix gives expected utility (2,2)(2,2), strictly above Z in both columns. This comparison uses expected-utility payoffs and the player's own randomisation; it does not average the opponent's actions.

A tie changes the logic

An alternative weakly dominates a strategy when it pays at least as much against every remaining opponent profile and strictly more against at least one. Ties mean a weakly dominated strategy can still be a best response. Iterating weak deletion can remove Nash equilibria and can depend on the deletion order; Marx and Swinkels' institutional abstract explicitly identifies the order problem.

Here is a separate fictional example, again with Row's payoff first:

Row's strategyColumn: LColumn: R
U(2,2)(2,2)(2,1)(2,1)
D(2,1)(2,1)(1,1)(1,1)

U weakly dominates D, and L weakly dominates R. Two legal sequences lead to different surviving sets:

  • Delete D first. With only U, L strictly dominates R. Only (U, L) remains.
  • Delete R first. With only L, U and D both pay Row 22. Neither now weakly dominates the other, because there is no strict improvement anywhere. Both (U, L) and (D, L) remain.

The original profile (D, L) is a Nash equilibrium: Row is indifferent between U and D, and Column is indifferent between L and R when Row plays D. The first sequence deletes that equilibrium; the second retains it. This is why a solver must distinguish strict comparisons from ties and document its weak-deletion convention. A numerical tolerance that silently turns a tiny strict advantage into equality can change the reduction.

For a trading application, the demanding step comes before deletion: specify feasible strategies, information and each player's utility, including relevant costs and constraints. A table built from noisy backtest averages does not establish dominance across every opponent response. Iterated dominance gives conditional logical restrictions for the specified game; it does not show that actual traders follow the reasoning.

Sources

  • Muhamet Yildiz, Graduate Game Theory, §§2.1–2.2, especially Definitions 2.2 and 2.4: strict dominance, mixed dominators and iterative elimination. Notes dated Spring 2022, hosted with MIT's Spring 2024 course.
  • Mihai Manea, Strategy & Information, MIT 14.16 (10 February 2016), slides 12–18 and 33: finite-game elimination, mixed dominance, order independence and preservation of Nash strategies.
  • Ben Polak, Putting Yourselves into Other People's Shoes, Yale ECON 159 (2007), iterative-deletion discussion: successive reasoning about other players and common knowledge.
  • Leslie M. Marx and Jeroen M. Swinkels, Order Independence for Iterated Weak Dominance, Games and Economic Behavior 18(2), 219–245 (1997), institutional abstract: weak deletion is generally order dependent; additional conditions can restore independence. The abstract, rather than the full article, was used here.
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