Weakly Dominated Strategy in Game Theory Explained

In game theory, strategic decision-making relies on understanding how different choices compare to one another. This article defines the concept of a weakly dominated strategy, explains how it differs from a strictly dominated strategy, provides a clear payoff matrix example, and discusses its significance in solving games and predicting rational outcomes.

Understanding Weakly Dominated Strategies

A strategy is considered weakly dominated if there is another strategy available to the player that yields a payoff at least as high in all possible scenarios, and a strictly higher payoff in at least one scenario.

In simpler terms, if Strategy A is weakly dominated by Strategy B: * Strategy B is never worse than Strategy A. * Strategy B is sometimes better than Strategy A, depending on what the other players choose to do.

Because Strategy B is always an equal or superior choice, a rational player has no strong incentive to choose the weakly dominated Strategy A.

Weak vs. Strict Dominance

To fully grasp weak dominance, it is helpful to contrast it with strict dominance:

• Strictly Dominated Strategy: Strategy A is strictly dominated by Strategy B if Strategy B always yields a strictly higher payoff than Strategy A, regardless of the opponent’s actions. Rational players will never play a strictly dominated strategy.
• Weakly Dominated Strategy: Strategy A is weakly dominated by Strategy B if Strategy B yields a payoff that is greater than or equal to Strategy A’s payoff for all opponent moves, and strictly greater for at least one opponent move.

A Practical Example

Consider a two-player game where Player 1 must choose between Strategy Up and Strategy Down, while Player 2 chooses between Left and Right.

Let’s look at Player 1’s payoffs (the first number in each pair):

To determine if Player 1 has a weakly dominated strategy, compare the payoffs of Up and Down:

1. If Player 2 plays Left: Player 1 gets 5 by playing Up, and 5 by playing Down. (Payoffs are equal).
2. If Player 2 plays Right: Player 1 gets 3 by playing Up, and 1 by playing Down. (Up is strictly better).

Since playing Up is equal to Down in one scenario and strictly better than Down in the other, Down is a weakly dominated strategy for Player 1. Conversely, Up weakly dominates Down.

Solving Games Using Weak Dominance

In game theory, analysts often simplify complex games using a process called Iterated Elimination of Dominated Strategies (IEDS). By systematically removing dominated strategies, you can narrow down the potential rational outcomes of a game.

However, eliminating weakly dominated strategies comes with a major caveat that does not apply to strictly dominated ones: the order of elimination matters.

When you eliminate strictly dominated strategies, you will always arrive at the same final set of strategies regardless of the order in which you remove them. With weakly dominated strategies, changing the order of elimination can actually change the final equilibrium of the game. For this reason, game theorists handle the elimination of weakly dominated strategies with extra caution.