Dominated Strategies: Identification and Elimination

Introduction

Welcome to our second lesson. In our previous session, we established the formal definition of a strategic game using players, strategy sets, and utility functions. We also acknowledged the core assumption that players are rational and that this rationality is common knowledge.

Today, we'll put that assumption to work. Our goal is to identify and eliminate strictly dominated strategies. This is our first "solution concept"—a method for simplifying a game by ruling out choices that a rational player would never make. Think of this as a powerful preprocessing step in an algorithm designed to find a game's solution. By pruning the search space of strategies, we can make the task of finding an equilibrium much more tractable.

What is a Strictly Dominated Strategy?

The most basic consequence of rationality is that a player will not choose an option if they have another option that is always better, no matter what their opponents do. This is the essence of a strictly dominated strategy.

Let's get the formal definition on the table. A strategy $s_i^{\prime }$ for player $i$ is strictly dominated by another strategy $s_i$ if playing $s_i$ yields a strictly higher payoff than playing $s_i^{\prime }$, for every possible combination of strategies chosen by the other players.

Mathematically, $s_i^{\prime }$ is strictly dominated by $s_i$ if:

Here, $s_{-i}$ represents a profile of strategies for all players except player $i$.

To see this definition explained and applied, let's watch a short video.

The key takeaway is in Claim 4.1 from the textbook Game Theory: An Introduction: "A rational player will never play a strictly dominated strategy." This is the logical foundation for our next step.

Iterated Elimination of Strictly Dominated Strategies (IESDS)

If we accept that no rational player will play a strictly dominated strategy, we can simply remove those strategies from the game matrix. But the logic doesn't stop there. This is where the assumption of common knowledge of rationality becomes critical.

1. I am rational, so I won't play my dominated strategies.
2. You are rational, so you won't play your dominated strategies.
3. I know you are rational, so I know you will eliminate your dominated strategies. This effectively creates a smaller game.
4. In this new, smaller game, some of my remaining strategies might now become dominated, even if they weren't in the original game. I can then eliminate them.
5. You know that I know that you are rational... and so on.

This recursive reasoning gives rise to the process of Iterated Elimination of Strictly Dominated Strategies (IESDS).

Let's see this process in action. The following video provides an excellent step-by-step walkthrough of IESDS on a 3x3 game.

As you saw, a complex 3x3 game was reduced to a single, unique prediction: (Down, Center). A game that can be solved to a unique outcome this way is called dominance solvable.

Here is another visual representation of the process, showing how the matrix shrinks at each step.

Two Crucial Properties of IESDS

For IESDS to be a reliable tool, we need to be sure of two things: that the order of elimination doesn't change the outcome, and that the process doesn't accidentally get rid of important solutions.

1. Order Independence: For strictly dominated strategies, the order in which you eliminate them (e.g., player 1 first or player 2 first) does not matter. You will always end up with the same set of remaining strategies. This is a powerful result that makes the process robust. (Note: this property does not hold for the related concept of weak dominance, which is one reason we focus on strict dominance).

2. Preservation of Nash Equilibria: This is the most critical property for your goal. The process of IESDS never eliminates a Nash Equilibrium. Any Nash equilibrium of the original game will survive every round of elimination and will be a Nash equilibrium of the final, reduced game. This is formally stated in Theorem 2 of the "Strategic games" lecture notes by Apt et al.

These properties mean you can confidently use IESDS as your first step when analyzing any game. It simplifies the game without losing any potential Nash equilibria, making your search for a solution much easier.

Practice Problem

Let's apply this. Consider the following game from the textbook by Tadelis. Your task is to find which strategy profiles survive the process of IESDS.

Player 2

Take a few minutes to work through this on your own. Look for a dominated strategy for either player, eliminate it, and then re-examine the smaller game. Repeat until you can't eliminate any more strategies.

Conclusion

Today we explored a powerful technique for simplifying games based on the assumption of common knowledge of rationality.

Key Takeaways:

• A strategy is strictly dominated if another single strategy provides a better payoff against all possible opponent moves.
• Rational players will never play strictly dominated strategies.
• Iterated Elimination of Strictly Dominated Strategies (IESDS) is the process of recursively removing these strategies from the game.
• IESDS is a valid simplification tool because the order of elimination doesn't matter for strict dominance, and it never removes a Nash Equilibrium.
• Some games are dominance solvable, meaning IESDS leads to a unique prediction. Many others are not, and the process terminates with multiple strategies remaining.

In the next lesson, we will shift our focus from eliminating "bad" strategies to identifying "good" ones. We will define and learn to compute the best response correspondence for each player. This will be our main tool for pinpointing pure strategy Nash equilibria by finding strategy profiles where every player is playing a best response to their opponents.