# Why system of beliefs, $\mu(x)$, needed in dynamic incomplete information games

We need to make statements about the sequential rationality of players who are in information sets that are not singletons. In these non-trivial information sets, we need to be able to say something like ...

"In this information set, player 2 is not playing a best response, and therefore his behavior is not sequentially rational"

But how does this motivate us to develop the system of beliefs? I don't exactly see why making every player to have a well-defined belief over where she is in each of her information sets do in dynamic games of incomplete information.

In some ways, beliefs are a more natural way of interpreting solution concepts like Nash equilibrium and its refinements.

For example, consider a simple simultaneous-move coordination game as follows: $$\begin{array}{c|cc} &L&R\\\hline T&1,1&0,0\\ D&0,0&2,2 \end{array}$$ We say that $(D,R)$ is a Nash equilibrium because $D$ is a best response to $R$ and vice versa. But this is a simultaneous-move game; how does player 1 know that player 2 will choose $R$? Well, she doesn't need to know, she just need to believe that player 2 will choose $R$ (with sufficiently high probability), and such a belief alone can justify the play of $D$ as a best response.

Thus, we can think of a Nash equilibrium as one in which every player is best responding to their beliefs about what other players will do, and that their beliefs are correct, in the sense that other players choices confirm their beliefs.

To the extent that a simultaneous-move game can be modeled using a game tree with imperfect information, we can talk about beliefs at non-trivial information sets where a player is uncertain about which strategy the previous player has chosen. Furthermore, this framework can be generalized to include incomplete information games if we let this previous player be Nature/Chance.

By dividing Nash equilibrium into these two elements --- (i) best response to one's belief and (ii) beliefs are correct --- we obtain a more nuanced understanding of the concept, which in turn shed light on the ways we can refine this solution concept by placing restrictions on either or both elements.

Solution concepts like the (weak) perfect Bayesian equilibrium and sequential equilibrium require that beliefs off-equilibrium path cannot be arbitrary (they must follow Bayes' rule whenever possible). Others such as quantal response equilibrium maintain the requirement that players have correct beliefs but relaxes the supposition that people always best respond to their (correct) beliefs.

• Herr, thanks for the answer. I have a question. How do you construct a game tree in TeX? In your other response, it was beautifully drawn. Can you share the method? :) – Frank Swanton Feb 10 '18 at 21:14
• @FrankSwanton: Glad I could help :) For drawing game trees, please my answer here. Also, this question on TeX.SX also has answers that suggest several alternative methods. – Herr K. Feb 10 '18 at 21:33