Dominant & Dominated Strategies

Strict dominance and iterated elimination of dominated strategies.

The idea

Dominance compares two strategies of the same player across every possible choice of the opponents, and identifies strategies a rational player never uses.

Fix a player and two of that player's strategies, $s$ and $t$. Strategy $s$ strictly dominates $t$ when $s$ pays strictly more than $t$ against every combination of choices the opponents could make; $t$ is then strictly dominated. A player who maximizes payoff never plays a strictly dominated strategy — whatever the opponents do, $s$ pays more — so we may delete $t$ from the game.

Deletion can expose further dominance. Knowing $t$ will not be played, the opponents choose within a smaller game, and a strategy that survived only by its payoff against $t$ may now be strictly dominated itself. Deleting repeatedly, until no strictly dominated strategy remains, is iterated elimination of strictly dominated strategies.

Strategy $s$ weakly dominates $t$ when $s$ never pays less than $t$ and pays strictly more against at least one choice of the opponents. Weak dominance does not justify deletion: against some opponent choices $t$ pays exactly as much as $s$, so a rational player may still play it.

Ways to work on it

• Walkthrough. Strict dominance and one round of iterated elimination.
• Practice. Spot the strictly dominated strategy in a 2x2 game.
• Hardest. Run full iterated elimination on a 3x3 game and find the survivor.

Not sure where to start? Take the ten-question placement test.