Dominant Strategies and Iterated Elimination
A strategy is strictly dominant if it gives a player a higher payoff than every other strategy whatever the opponents do, and a strategy is strictly dominated if another strategy always does better; iterated elimination removes dominated strategies round after round, and the strategies that survive are the rational ones under common knowledge of rationality. In brief: in the prisoner’s dilemma, defecting strictly dominates cooperating for both players, so the outcome is (defect, defect). In the 3 × 3 game below, eliminating the dominated strategies one after another leaves the single profile (T, C).
Dominance is the simplest solution idea in game theory: a rational player never uses a dominated strategy, and a player who knows that the opponent is rational can reason that the opponent will not use theirs either. This guide defines strict and weak dominance, shows the procedure of iterated elimination of strictly dominated strategies (IESDS), explains why order matters for weak dominance, and gives applications such as the “guess two thirds of the average” game and the second-price auction, with worked examples and a table.
What is dominance?
Compare two strategies s and s′ of the same player. The strategy s strictly dominates s′ if s gives a strictly higher payoff than s′ against every strategy combination of the opponents. It weakly dominates s′ if it is never worse and sometimes strictly better. A strictly dominated strategy is never a best reply to anything, so it is never played by a rational player. To check dominance in a payoff matrix, compare two rows (or two columns) cell by cell, using only the payoff of the player who owns them.
What is a dominant-strategy equilibrium?
A strategy that strictly dominates all others is a strictly dominant strategy. If every player has one, the profile of these strategies is a dominant-strategy equilibrium, the most robust prediction: each player plays it regardless of beliefs about the others. The prisoner’s dilemma is the standard example, and it shows that such an equilibrium can be worse for everybody than another outcome.
Example 1: a dominant strategy in pricing
Two shops each choose a high or a low price, with the payoffs (row, column): (High, High) = (4, 4), (High, Low) = (1, 5), (Low, High) = (5, 1), (Low, Low) = (2, 2). For the row shop, Low gives 5 > 4 against High and 2 > 1 against Low, so Low is strictly dominant, and the same for the column shop. The equilibrium (Low, Low) yields (2, 2) instead of the (4, 4) of the mutually high prices.
How does iterated elimination work?
When no strategy is dominant, some may still be dominated. Remove all strictly dominated strategies, then examine the reduced game, where new dominances may appear because the opponent’s eliminated options no longer count, and repeat until nothing more can be removed. The order of elimination does not affect the final result for strict dominance. If one profile remains, the game is dominance solvable.
Example 2: a 3 × 3 game
| L | C | R | |
|---|---|---|---|
| T | (3, 1) | (4, 3) | (0, 1) |
| M | (1, 3) | (2, 2) | (5, 0) |
| B | (5, 2) | (3, 4) | (1, 1) |
- Round 1: no row is dominated (M beats T and B in column R). For the column player, R is strictly dominated by C: 3 > 1 in row T, 2 > 0 in row M, 4 > 1 in row B. Remove R.
- Round 2 (columns L, C): for the row player, M is strictly dominated by T (3 > 1 and 4 > 2) and by B (5 > 1 and 3 > 2). Remove M. (T does not dominate B: T is worse in L, 3 < 5.)
- Round 3 (rows T, B): for the column player, L gives 1 and 2, and C gives 3 and 4, so C strictly dominates L. Remove L.
- Round 4 (column C only): the row player compares T = 4 and B = 3, so B is dominated. Remove B.
Only (T, C) remains, with the payoffs (4, 3). This profile is also the unique Nash equilibrium; every dominance-solvable game has this property.
What goes wrong with weak dominance?
Iterated elimination of weakly dominated strategies can depend on the order and can eliminate Nash equilibria. Moreover, a strategy that is weakly dominated can be a perfectly reasonable choice if the opponent plays an unexpected action.
Example 3: weak dominance removes an equilibrium
Payoffs (row, column): (U, L) = (1, 1), (U, R) = (0, 0), (D, L) = (1, 1), (D, R) = (1, 1). For the row player, D weakly dominates U (equal in column L, better in R). If U is removed, the profiles (D, L) and (D, R) remain. But (U, L) is also a Nash equilibrium: row gets 1 from U against L, and 1 from D, so it has no reason to deviate. Weak dominance removes it, so it is a less reliable tool than strict dominance.
Which applications use dominance?
Example 4: guess two thirds of the average
Each player in a group names a number from 0 to 100; the winner is the one closest to 2/3 of the average. Any number above 2/3 · 100 ≈ 66.7 is dominated, because the target can never exceed 66.7. If all players know that, numbers above 2/3 · 66.7 ≈ 44.4 are dominated; next 29.6, then 19.8, and so on. The limit of the iterations (2/3)k · 100 → 0, so the iterated elimination leads to everyone naming 0. In experiments, however, the typical winning guess is around 20–35, indicating only a few rounds of reasoning.
Example 5: a second-price auction
In a sealed-bid second-price (Vickrey) auction the highest bidder wins and pays the second-highest bid. For a bidder whose value is v, bidding v weakly dominates every other bid: bidding less can only lose auctions that would have been profitable, and bidding more can only win auctions at a price above v. So truthful bidding is a weakly dominant strategy, which is why the design is used in practice (for example, in a variant for internet advertising).
Table of dominance concepts
| Concept | Condition | Safe to eliminate? | Example |
|---|---|---|---|
| strictly dominated | always strictly worse than another strategy | yes, in any order | R in Example 2 |
| weakly dominated | never better, sometimes worse | order may matter | U in Example 3 |
| strictly dominant | strictly best against everything | the other strategies are eliminated | Low in Example 1 |
| dominance solvable | one profile survives | it is the Nash equilibrium | (T, C) |
Where does the idea come from?
Dominance reasoning is implicit in the early work of von Neumann and Morgenstern (1944). The prisoner’s dilemma was devised by Merrill Flood and Melvin Dresher at RAND in 1950, and given its prison story by Albert Tucker. The second-price auction was analyzed by William Vickrey in 1961, work for which he shared the 1996 Nobel Prize in economics. The “two thirds of the average” game is a version of the beauty contest of Keynes, studied experimentally by Rosemarie Nagel in 1995.
Which mistakes are common?
Comparing the wrong payoffs
To test the row player’s strategies, compare only the first numbers of the cells in the same column. The column player’s payoffs are irrelevant for that comparison.
Eliminating a strategy that is only sometimes worse
Dominance requires being worse against every opponent choice. A strategy that is better in one column, like M in column R of Example 2, cannot be removed in that round.
Forgetting to re-examine after each round
New dominances appear after the opponent’s strategies are removed, so the procedure must be repeated until no more removal is possible.
Assuming the dominant-strategy outcome is efficient
It may be worse for all players, as in the prisoner’s dilemma.