Identifying Dominant Strategies

1. Strictly Dominated and Strictly Dominant Strategies

A rational player's goal is to maximize their payoff. Consider a scenario where you have two strategies, A and B. If strategy A gives you a better payoff than strategy B no matter what your opponent does, it would be irrational to ever choose B. In this case, we say strategy A strictly dominates strategy B.

Formal Definition

Let's make this precise. You'll recall our notation from the previous module, which is standard in academic texts. A strategy $s_i^{\prime }$ for player $i$ is strictly dominated by another strategy $s_i$ if, for any combination of strategies the other players might choose, $s_{-i}$, player $i$'s payoff from playing $s_i$ is strictly greater than the payoff from playing $s_i^{\prime }$.

Mathematically:

Conversely, a strategy $s_i^{\ast }$ is strictly dominant if it strictly dominates every other strategy available to player $i$.

A core assumption in game theory is that rational players never play strictly dominated strategies.

To delve into the formal definition and its immediate implications, please read the following section from a Princeton University text on game theory.

How to Identify a Strictly Dominant Strategy

The definition points us to a clear procedure: to check if a strategy is dominant for a player, you must compare it against all of their other strategies, one by one. For each comparison, you check the payoffs against every possible action by the opponent.

Let's see this in action. Suppose you are Player 1 (the row player). To see if your strategy 'Top' is dominant, you would:

1. Compare 'Top' vs. 'Bottom' (assuming these are your only other options).
2. For this comparison, you check: Is $u_1(\text{Top},\text{Left})>u_1(\text{Bottom},\text{Left})$?
3. Then you check: Is $u_1(\text{Top},\text{Right})>u_1(\text{Bottom},\text{Right})$?
4. If the answer to all these checks (across all of your opponent's columns) is 'yes', then 'Top' strictly dominates 'Bottom'. If 'Top' dominates all other rows in this way, it is a strictly dominant strategy.

The following video provides a very clear, practical walkthrough of this pairwise comparison process.

As the video shows, if a player does not have one strategy that is best in all situations, they do not have a dominant strategy.

2. Weakly Dominated and Weakly Dominant Strategies

The condition for strict dominance is very demanding. A more common situation is weak dominance. Here, a strategy is at least as good as another in all situations, and strictly better in at least one.

Formal Definition

A strategy $s_i^{\prime }$ for player $i$ is weakly dominated by another strategy $s_i$ if:

1. The payoff from $s_i$ is always greater than or equal to the payoff from $s_i^{\prime }$ for all of opponent strategies $s_{-i}$.
2. There is at least one opponent strategy profile $s_{-i}^{\ast }$ for which the payoff is strictly greater.

Mathematically:

1. $u_i(s_i,s_{-i})\ge u_i(s_i^{\prime },s_{-i})$ for all $s_{-i}\in S_{-i}$
2. There exists an $s_{-i}^{\ast }\in S_{-i}$ such that $u_i(s_i,s_{-i}^{\ast })>u_i(s_i^{\prime },s_{-i}^{\ast })$

A weakly dominant strategy is one that weakly dominates all other strategies. The argument for playing a weakly dominant strategy is that it's safer; it never does worse than other strategies and sometimes does better. A rational and cautious player would avoid a weakly dominated strategy.

The following video introduces this concept using a compelling historical example before giving the formal definition.

3. A Visual Summary and Unified Procedure

Let's consolidate our understanding. A strategy dominates another if it's always better. The "always" is what differentiates strict and weak dominance.

• Strict Dominance: Payoff is strictly greater (>) for ALL opponent actions.
• Weak Dominance: Payoff is greater than or equal to (≥) for ALL opponent actions, AND strictly greater (>) for AT LEAST ONE opponent action.

This image provides a concise visual summary with a clear example for each type of dominance.

The Algorithm for Finding Dominated Strategies

Whether checking for strict or weak dominance, the process is the same. For a given player (e.g., the row player):

1. Select a pair of strategies (rows) to compare. Let's call them Row A and Row B.
2. Go column by column, comparing the player's payoff from Row A to the payoff from Row B.
3. Check for dominance:
   • If Payoff(A) > Payoff(B) in every column, then Row A strictly dominates Row B.
   • If Payoff(A) ≥ Payoff(B) in every column AND there is at least one column where Payoff(A) > Payoff(B), then Row A weakly dominates Row B.
4. Conclude: The dominated strategy (Row B in these cases) is one a rational player would seek to avoid.
5. Repeat this process for all possible pairs of rows for the row player, and all pairs of columns for the column player.

Conclusion

In this lesson, you've learned how to identify strategies that are fundamentally better or worse than others, regardless of an opponent's specific action. This is a crucial first step in simplifying complex strategic situations.

The key takeaways are:

• A strategy is strictly dominated if another strategy provides a strictly higher payoff against all possible actions by opponents. Rational players will never play a strictly dominated strategy.
• A strategy is weakly dominated if another strategy provides at least an equal payoff against all opponent actions and a strictly higher payoff against at least one. Cautious, rational players will avoid these as well.
• The process of identifying dominance involves a systematic, pairwise comparison of a player's strategies against every possible action by their opponents.

Identifying and eliminating dominated strategies is a powerful way to reduce the complexity of a game. But what happens after we eliminate one? Sometimes, in the newly simplified game, another strategy might suddenly become dominated. This suggests an iterative process.

In our next lesson, we will explore this exact idea: the iterated elimination of dominated strategies (IEDS). We will see how this step-by-step reduction can often shrink a complex game down to a single, predictable outcome.