IESDS and Nash Equilibrium - same solution

Question

Applying the Iterated Elimination of Strictly Dominated Strategies (IESDS) to a game resulted with the same solution of the Nash Equilibrium.

What does this imply? Actually that specific "quadrant" of the matrix is the:

• Pareto optimal
• Nash Equilibrium
• Dominant strategies (through IESDS).

EDIT: This is a Matrix that shows what I'm talking about:

Game I'm talking about here.

Quadrant (1,1) is a Nash Equilibrium and the solution of IESDS as well as the Pareto optimum scenario.

What I'm trying to ask is: are my results wrong or this can actually happen?

Thanks!

Answer 1

Not only are your results correct, but you should have expected that Nash equilibrium and IESDS survival would coincide. Here are two facts which might illuminate the connection:

1. If IESDS eliminates all but one strategy profile, that strategy profile is the unique Nash equilibrium.

2. Nash equilibria are often Pareto optimal. They are not always Pareto optimal (see the Prisoner's Dilemma) for such an example, but it is not universally true that Nash equilibria are suboptimal.

So, indeed, your results are perfectly reasonable (and correct).

A further point about IESDS (which sometimes goes by other acronyms, FYI) is that it's a useful procedure to do even if it doesn't result in just one surviving strategy profile. Strategies that survive IESDS are rationalizable, and strategies that aren't rationalizable are never played with positive probability in a (mixed) Nash equilibrium. Therefore, IESDS is often used as a first step to get rid of strategies that won't be in NE, and then slightly more difficult analysis can be used to find the Nash equilibrium in the remaining set of strategy profiles.