<p>In games of imperfect information, one must reason about the belief on the state of nature, the beliefs of other player's beliefs of their beliefs, and so on. This is referred to as the belief hierarchy, which can stretch all the way to infinity. The <a href="http://www.ma.huji.ac.il/~zamir/documents/BayesianGames_ShmuelZamir.pdf" rel="nofollow noreferrer">universal belief space</a>, also referred to as the universal type space, is a compact way to model this.</p>

<p>Unfortunately, this is an unwieldy mathematical object. I realize that the value of it comes from the guarantee that in any game, one can represent the belief hierarchy by some set of types in a compact mathematical object. However, I am wondering if it is necessary to consider this level of detail in most games.</p>

<p>It seems that the higher one goes up the belief hierarchy, the less weight the information (beliefs of beliefs of...) carries. While this seems intuitive, I haven't seen any formalization of this in the literature. I realize this is a somewhat vague idea, but I am hopeful that there is some literature that formalizes this statement. Some papers I have seen only go up to second-order beliefs; it's unclear if this is because: it is sufficient for the game, it is a <em>good approximation</em> in some sense, or if the complexity of the reasoning precluded reasoning about higher levels.</p>

<p><strong>Question</strong>: Are there general results (or even results for specific classes of games) that state how many levels up the belief hierarchy that one must reason about?  </p>

CC BY-SA 4.0
2019-08-22T21:15:26.223Z
@afreelunch Sorry, that should read "belief on the state of nature", to describe some players' uncertainty over the true state of nature.
jonem
CC BY-SA 4.0
2019-08-23T17:27:26.033Z
I think in many, if not most, applied work involving games of incomplete information, people just assume common priors without worrying the epistemic foundations outlined in the article you linked. Your question is reminiscent of the [level-k and cognitive hierarchy theories](https://www.aeaweb.org/articles?id=10.1257/jel.51.1.5) in behavioral economics. Are you familiar with that literature? [This paper](https://www.aeaweb.org/articles?id=10.1257/aer.96.5.1737) for example suggests that empirically most people can be catergorized as level-1 and level-2 thinkers.
Herr K.
CC BY-SA 4.0
2019-08-23T17:38:07.710Z
You may also want to check out [Bergemann and Morris (2005)](http://www.princeton.edu/~smorris/pdfs/paper_43_Robust_Mechanism_Design.pdf).
Herr K.
<reference-request><game-theory><asymmetric-information><information>
How far must one reason up the belief hierarchy?
jonem
# How far must one reason up the belief hierarchy?

(Asked by: username_0 on 2019-08-22)

In games of imperfect information, one must reason about the belief on the state of nature, the beliefs of other player's beliefs of their beliefs, and so on. This is referred to as the belief hierarchy, which can stretch all the way to infinity. The [universal belief space](http://www.ma.huji.ac.il/~zamir/documents/BayesianGames_ShmuelZamir.pdf), also referred to as the universal type space, is a compact way to model this.

Unfortunately, this is an unwieldy mathematical object. I realize that the value of it comes from the guarantee that in any game, one can represent the belief hierarchy by some set of types in a compact mathematical object. However, I am wondering if it is necessary to consider this level of detail in most games.

It seems that the higher one goes up the belief hierarchy, the less weight the information (beliefs of beliefs of...) carries. While this seems intuitive, I haven't seen any formalization of this in the literature. I realize this is a somewhat vague idea, but I am hopeful that there is some literature that formalizes this statement. Some papers I have seen only go up to second-order beliefs; it's unclear if this is because: it is sufficient for the game, it is a *good approximation* in some sense, or if the complexity of the reasoning precluded reasoning about higher levels.

**Question**: Are there general results (or even results for specific classes of games) that state how many levels up the belief hierarchy that one must reason about?

**Tags:** reference-request, game-theory, asymmetric-information, information

> **Comments:**
> **username_0**: @afreelunch Sorry, that should read "belief on the state of nature", to describe some players' uncertainty over the true state of nature.  
> **username_1**: I think in many, if not most, applied work involving games of incomplete information, people just assume common priors without worrying the epistemic foundations outlined in the article you linked. Your question is reminiscent of the [level-k and cognitive hierarchy theories](https://www.aeaweb.org/articles?id=10.1257/jel.51.1.5) in behavioral economics. Are you familiar with that literature? [This paper](https://www.aeaweb.org/articles?id=10.1257/aer.96.5.1737) for example suggests that empirically most people can be catergorized as level-1 and level-2 thinkers.  
> **username_1**: You may also want to check out [Bergemann and Morris (2005)](http://www.princeton.edu/~smorris/pdfs/paper_43_Robust_Mechanism_Design.pdf).  

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## 0 answers

