Iterated Dominance and Game Outcomes

1. The Logic: Common Knowledge of Rationality

The engine that drives IESDS is a core assumption in game theory: common knowledge of rationality. This isn't just about players being rational; it's a much deeper idea:

• Every player is rational (i.e., they maximize their own payoffs).
• Every player knows that every other player is rational.
• Every player knows that every player knows that every other player is rational.
• ...and so on, ad infinitum.

If Player 1 is rational, she will not play her strictly dominated strategy, $s_1^{\prime }$. If Player 2 knows Player 1 is rational, he can confidently rule out $s_1^{\prime }$ from consideration. He then analyzes the game as if that row/column does not exist. In this reduced game, one of his own strategies, $s_2^{\prime }$, might now be strictly dominated. Since he is also rational, he will eliminate $s_2^{\prime }$. Player 1, anticipating all of this, can then continue the cycle.

The following reading from an MIT course formalizes this link between common knowledge of rationality and the IESDS procedure.

2. The IESDS Algorithm

The logic of common knowledge translates into a straightforward, mechanical algorithm.

The Algorithm:

1. Scan: Examine the current game matrix. Identify any strategy for any player that is strictly dominated by another one of their pure strategies.
2. Eliminate: If you find a strictly dominated strategy, cross out that entire row (for the row player) or column (for the column player). The game is now reduced.
3. Iterate: Return to Step 1 and repeat the process on the new, smaller game.
4. Stop: The process ends when no player has any strictly dominated strategies in the remaining game.

The strategies (or single strategy profile) that survive this entire process are the solution of the game under IESDS.

Example Walkthrough

Let's see this in practice. The following image shows the first step in solving a 2x3 game.

After eliminating column R, we are left with a 2x2 game:

Now, we repeat the process. In this new game, let's look at the row player's strategies, U and D.

• Against L, U gives 4 and D gives 0 (4 > 0).
• Against M, U gives 1 and D gives 0 (1 > 0).
  Since U gives a strictly better payoff than D no matter what the column player does (between L and M), U strictly dominates D. We eliminate row D.

This leaves a 1x2 game:

Finally, in this trivial game, the column player simply has to choose between L (payoff 1) and M (payoff 2). Since 2 > 1, the player chooses M.

The single surviving outcome is (U, M).

For a more detailed walkthrough of a larger game, the following video is an excellent guide.

3. Application: The Median Voter Theorem

The power of IESDS extends far beyond simple matrix games. It can be used to analyze models of political competition, economics, and social behavior. A classic example from political science is the Median Voter Theorem, which predicts that in a two-candidate election, both candidates will position their platforms toward the political center.

Professor Ben Polak from Yale explains this model and solves it using IESDS. This example will show you how a seemingly complex behavioral phenomenon can be predicted with this rigorous, step-by-step logic.

This example vividly illustrates how a powerful prediction—candidates crowding the center—emerges directly from the assumption of common knowledge of rationality, applied iteratively.

4. Important Properties of IESDS

As you apply this method, keep two crucial properties in mind:

1. Order of Elimination Doesn't Matter (for Strict Dominance): When you are eliminating strictly dominated strategies, the order in which you remove them has no effect on the final set of surviving strategies. If you eliminate Player 1's dominated strategy first, or Player 2's, you will always arrive at the same solution. This makes IESDS a very robust solution concept.

2. The Solution Is Not Always a Single Outcome: In the examples we've seen, IESDS conveniently narrowed the game down to a single prediction. This is not always the case. The process might stop when there are still multiple strategies remaining for each player, none of which are strictly dominated. For instance, in the Battle of the Sexes game, neither strategy is strictly dominated, so IESDS eliminates nothing.

5. Practice

Now it's your turn to apply the algorithm. Consider the following game:

Apply the IESDS algorithm step-by-step. Which strategy profile, if any, is the unique survivor? Take a moment to work through it before reading the solution below.

Conclusion

Today, we've moved from simply identifying bad strategies to modeling a chain of logical deductions. This process, Iterated Elimination of Strictly Dominated Strategies (IESDS), is a fundamental tool for solving games.

Here are the key takeaways:

• IESDS is built on the assumption of common knowledge of rationality.
• It's a step-by-step algorithm: find a strictly dominated strategy, eliminate it, and repeat until no more can be found.
• For strict dominance, the order of elimination does not matter, making the solution robust.
• The process can lead to a unique outcome, as in the Median Voter Theorem, but it can also leave multiple strategies, telling us that rationality alone is not enough to yield a single prediction.

This raises a crucial question: What do we do when IESDS doesn't give us a single answer, or any answer at all? We need a more general concept of "stability." Our next lesson will introduce the most famous concept in all of game theory: Nash Equilibrium. We will define it as a profile of strategies where no player has a unilateral incentive to change their choice, providing a notion of equilibrium for any game, regardless of whether it's solvable by dominance.