This is a basic question, but I cannot find the answer to it. In short, equilibria are often conditioned on a specific strategy. E.g., given A's strategy, B updates her beliefs to xyz. My problem is that often an action could be a part of the equilibrium path for many different strategies, so how does B know which strategy A is following?
Below is an example that might clarify the question.
Consider a classic signaling game (2 types, 2 actions) à la Spence. Suppose furthermore that there are two equilibria: one pooling, one separating. In the pooling eq., say, both types of senders send "Low". In the separating, the "Strong" type sends High, the Weak type sends Low.
The part that I struggle with is: how does the receiver know what equilibrium path they are in? For example, say the receiver observes "Low". If we are in "pooling world", then the receiver's posteriors are just going to be her priors. If, however, we are in "separating world", then she can update to p(Weak|Low)=1. But observing "Low" alone does not tell the receiver what strategy the sender is following, so how can she possibly update her beliefs? It seems to me that she would need to have beliefs not only about types, but also about strategies being followed.
Sorry if this is idiotic, but this has puzzled me for a while.