<p><a href="https://economics.stackexchange.com/a/53151/42">@VARulle's answer</a> covers the gist of the two models. I'll fill in some of the details.</p>
<p>The level-<span class="math-container">$k$</span> model assumes that players can be divided into types labeled <span class="math-container">$L0, L1, L2,\dots$</span> etc., where</p>
<ul>
<li>the <span class="math-container">$L0$</span> players are <em>non-strategic</em> in the sense that they do not play a best response to actions of any other type of players (an <span class="math-container">$L0$</span> player is usually, though not necessarily, assumed to randomly select from his set of available actions);</li>
<li>an <span class="math-container">$Lk$</span> player (for any <span class="math-container">$k\ge 1$</span>) plays a best response to the belief that <em>all other players are of type <span class="math-container">$L(k-1)$</span></em>.</li>
</ul>
<p>The cognitive hierarchy (CH) model generalizes the level-<span class="math-container">$k$</span> model by allowing an <span class="math-container">$Lk$</span> player to hold the belief that there is <em>a distribution of lower type players <span class="math-container">$L(k-1), L(k-2),\dots,L0$</span></em>, where each type best responds to a distribution of their respective lower types. This generalization addresses an apparent issue in level-<span class="math-container">$k$</span>; that is, as <span class="math-container">$k$</span> gets larger, players in the level-<span class="math-container">$k$</span> model would seem increasingly irrational as they ignore the presence of an increasingly large number of types.</p>
<p>One way to model the CH beliefs mathematically is to truncate the type distribution. Let <span class="math-container">$f(\cdot)$</span> denote the true distribution of types. Under CH, an <span class="math-container">$Lk$</span> player can be assumed to perceive a proportion
<span class="math-container">\begin{equation}
g_k^{\text{CH}}(l)=\frac{f(l)}{\sum_{i=0}^{k-1}f(i)}, \qquad \text{for }l=0,1,\dots,k-1
\end{equation}</span>
of players of a lower type <span class="math-container">$Ll$</span> (<span class="math-container">$l&lt;k$</span>). In contrast, an <span class="math-container">$Lk$</span> player in the level-<span class="math-container">$k$</span> model would have a belief over types given by
<span class="math-container">\begin{equation}
g_k^\text{Lk}(l)=\begin{cases}1&amp;\text{if }l=k-1\\0 &amp; \text{otherwise}\end{cases}
\end{equation}</span>
Observe that <span class="math-container">$g_k^\text{CH}$</span> converges to <span class="math-container">$f$</span> as <span class="math-container">$k$</span> gets larger, so that more sophisticated types have more accurate beliefs about the others.</p>

CC BY-SA 4.0
2022-10-18T07:55:45.637Z
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 $L1$.
VARulle
CC BY-SA 4.0
2022-10-17T22:39:06.967Z
2022-10-17T22:39:06.967Z
Herr K.
<p>There are several variants of level-<span class="math-container">$k$</span> models and cognitive hierarchy models present in the literature. What the level-<span class="math-container">$k$</span> models have in common is that level-<span class="math-container">$k$</span> players best respond to the assumption that all other players are level-<span class="math-container">$(k-1)$</span>, while cognitive hierachy models assume that a level-<span class="math-container">$k$</span> player best responds to the assumption that all other players are of <em>some</em> lower level.</p>

CC BY-SA 4.0
2022-10-17T13:00:23.173Z
2022-10-17T13:00:23.173Z
VARulle
<p>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.</p>

<game-theory><behavioral-economics>
What is the difference between the cognitive hierarchies and level k model?
user56834
# What is the difference between the cognitive hierarchies and level k model?

(Asked by: username_0 on 2022-10-17)

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.

**Tags:** game-theory, behavioral-economics

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## 2 answers

### Answer by username_2 on 2022-10-17. Score: 3 | Accepted answer

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.


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### Answer by username_1 on 2022-10-17. Score: 2

[@VARulle's answer](https://economics.stackexchange.com/a/53151/42) 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
\begin{equation}
g\_k^{\text{CH}}(l)=\frac{f(l)}{\sum\_{i=0}^{k-1}f(i)}, \qquad \text{for }l=0,1,\dots,k-1
\end{equation}
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
\begin{equation}
g\_k^\text{Lk}(l)=\begin{cases}1&\text{if }l=k-1\\0 & \text{otherwise}\end{cases}
\end{equation}
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.

> **Comments:**
> **username_2**: 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 $L1$.  

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