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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, 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?

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  • $\begingroup$ Interesting question but why must one 'reason about the belief of the state of nature'? Does the state of nature have beliefs? $\endgroup$ – afreelunch Aug 22 at 12:10
  • $\begingroup$ @afreelunch Sorry, that should read "belief on the state of nature", to describe some players' uncertainty over the true state of nature. $\endgroup$ – jonem Aug 22 at 21:15
  • $\begingroup$ 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 in behavioral economics. Are you familiar with that literature? This paper for example suggests that empirically most people can be catergorized as level-1 and level-2 thinkers. $\endgroup$ – Herr K. Aug 23 at 17:27
  • $\begingroup$ You may also want to check out Bergemann and Morris (2005). $\endgroup$ – Herr K. Aug 23 at 17:38

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