I second Michael Dennis’ comment below, that the infinite regress of priors is avoided in standard game theory by specifying a common prior. Indeed the specification of this prior leads to a prior selection problem.
Just to make sure that I was understood, I was also pointing out that “you can have a well-specified Bayesian belief over your partner” even without agreeing on a common prior, as long as you agree on a common set of possibilities or something effectively similar. This means that talking about “Bayesian agents without a common prior” is well-defined.
When there is not a common prior, this lead to an arbitrarily deep nesting of beliefs, but they are all well-defined. I can refer to “what A believes that B believes about A” without running into Russell’s Paradox. When the priors mis-match then the entire hierarchy of these beliefs might be useful to reason about, but when there is a common prior, it allows much of the hierarchy to collapse.
Just to make sure that I was understood, I was also pointing out that “you can have a well-specified Bayesian belief over your partner” even without agreeing on a common prior, as long as you agree on a common set of possibilities or something effectively similar. This means that talking about “Bayesian agents without a common prior” is well-defined.
When there is not a common prior, this lead to an arbitrarily deep nesting of beliefs, but they are all well-defined. I can refer to “what A believes that B believes about A” without running into Russell’s Paradox. When the priors mis-match then the entire hierarchy of these beliefs might be useful to reason about, but when there is a common prior, it allows much of the hierarchy to collapse.