# What is the difference between the cognitive hierarchies and level k model?

Both these models are hierarchical models of strategic behaviour, where the k'th level plays a best response to the earlier levels. What exactly is the difference, mathematically? A precise definition of each and how they differ would clarify this.

## 2 Answers

There are several variants of level-$$k$$ models and cognitive hierarchy models present in the literature. What the level-$$k$$ models have in common is that level-$$k$$ players best respond to the assumption that all other players are level-$$(k-1)$$, while cognitive hierachy models assume that a level-$$k$$ player best responds to the assumption that all other players are of some lower level.

@VARulle's answer covers the gist of the two models. I'll fill in some of the details.

The level-$$k$$ model assumes that players can be divided into types labeled $$L0, L1, L2,\dots$$ etc., where

• the $$L0$$ players are non-strategic in the sense that they do not play a best response to actions of any other type of players (an $$L0$$ player is usually, though not necessarily, assumed to randomly select from his set of available actions);
• an $$Lk$$ player (for any $$k\ge 1$$) plays a best response to the belief that all other players are of type $$L(k-1)$$.

The cognitive hierarchy (CH) model generalizes the level-$$k$$ model by allowing an $$Lk$$ player to hold the belief that there is a distribution of lower type players $$L(k-1), L(k-2),\dots,L0$$, where each type best responds to a distribution of their respective lower types. This generalization addresses an apparent issue in level-$$k$$; that is, as $$k$$ gets larger, players in the level-$$k$$ model would seem increasingly irrational as they ignore the presence of an increasingly large number of types.

One way to model the CH beliefs mathematically is to truncate the type distribution. Let $$f(\cdot)$$ denote the true distribution of types. Under CH, an $$Lk$$ player can be assumed to perceive a proportion $$$$g_k^{\text{CH}}(l)=\frac{f(l)}{\sum_{i=0}^{k-1}f(i)}, \qquad \text{for }l=0,1,\dots,k-1$$$$ of players of a lower type $$Ll$$ ($$l). In contrast, an $$Lk$$ player in the level-$$k$$ model would have a belief over types given by $$$$g_k^\text{Lk}(l)=\begin{cases}1&\text{if }l=k-1\\0 & \text{otherwise}\end{cases}$$$$ Observe that $$g_k^\text{CH}$$ converges to $$f$$ as $$k$$ gets larger, so that more sophisticated types have more accurate beliefs about the others.

• Another assumption typically made in level-$k$ models (but not in CH models) is that $L0$ types do not really exist (frequency $=0$) but are just there to anchor the beliefs of $L1$. Commented Oct 18, 2022 at 7:55