# Difference between equilibrium and k-rationalizability

I'm reading a Structural Models of Nonequilibrium Strategic Thinking: Theory, Evidence, and Applications by Crawford, Costa-Gomes and Iriberri. They write the following:

In two-person games, a player can find his set of k-rationalizable strategies via k rounds of iterated strict dominance, without the need for fixed-point reasoning. Thus, k-rationalizability is cognitively less taxing than equilibrium, especially for small k.

K rationalizable strategies are those that survive k-rounds of iterated strict dominace (for n=2 player games). For n >2, k-rationalizable strategies are those that survive k-rounds of iterative never best response elimination.

What is meant by fixed point reasoning here? Is there a difference between fixed point reasoning and iterative never-best-response elimination?

Relevant section is Section 2.2 found on page 12.

• Specifying the paper (or linking to it) may help. – Giskard Nov 20 '19 at 22:13
• @Giskard Right, I linked the paper. Thanks – Amin Sammara Nov 20 '19 at 22:33

What is meant by fixed point reasoning here?

Specifically, Nash equilibria. One way to define a Nash equilibrium in words is "a strategy profile from which no player can be made better-off by unilaterally deviating to a different action". In other words, a Nash equilibrium is a "fixed point" because if the game ends up there, play stops because there are no more optimal moves.

Is there a difference between fixed point reasoning and iterative never-best-response elimination?

Yes, they are different solution concepts. The key is the last paragraph on page 12 of the article - basically, think of k-rationalizability as half-assing the game while still keeping the assumptions of full info, rationality, belief in mutual rationality, and so on. Instead of stopping play when there are no optimal moves, play stops when some kind of decision-making resource is exhausted. It is interesting precisely to the extent that it may or may not model how real people, who face real information costs and processing time limitations, might approach games.

• Thank you for your answer. I suspected fixed point reasoning to be related to NE but the wording suggests some sort of process relevant to finding a NE. Thus it would be comparable to iterative elimination, which is a method of finding a (possibly non equilibrium) solution. – Amin Sammara Nov 20 '19 at 23:26
• @heh A Nash equilibrium is a fixed poind because it is literally a fixed point of the best response mapping $BR(s) = \left(BR_1(s_{-1}),BR_2(s_{-2}),BR_3(s_{-3}),...,BR_n(s_{-n})\right)$. – Giskard Nov 21 '19 at 5:49
• @Giskard that tautology does not answer the question that is asked. It does not explain why a Nash equilibrium is a fixed point; it just explains in symbols what it means for a Nash equilibrium to be a fixed point. The crucial distinction between Nash and truncated iteration is why play stops. The math is important, but (in my view) in game theory the intuition should come first. – heh Nov 21 '19 at 16:06
• @AminSammara yes, as the paper explains, your suspicion is correct for the 2-player case. More generally, think of it as a truncated process of iterative elimination. A player sets "k" according to what is "good enough". As the paper notes, it's not a great solution concept, but it is probably closer to how real-world agents behave given finite decision-making resources and lack of perfect information. – heh Nov 21 '19 at 16:12
• @heh thank you everything makes sense. I'm also very grateful that you responded given how badly written my comments were. And I agree that random was not the best word to use there but your answer confirms my intuition. Thanks again. – Amin Sammara Nov 22 '19 at 1:27