Game Theory Question

I’m working on a Economics question and need guidance to help me study.

Hello – can you please help me solve the below question?

A firm’s has quality x ∈ {0,…,100}, each of which is equally likely. Regardless of x, with probability 1/10 , the firm can only send message m = ∅ while with probability 9/10 it can send one of two messages m = ∅ or m = x. Consumer observes the firm’s message and forms a belief about the firm’s expected quality b = E [x|m]. The firm wants to maximize b, i.e., it wants to maximize the consumer’s expectation of its quality.

Note the distinction between 0, which is a potential value that x can take and ∅, which is not a potential value for x but rather a message that is “silent” about x.

Is there an equilibrium with a cutoff type x∗ such that: (i) every firm with x < x∗ sends a message m=∅, (ii) every firm with x≥x∗ sends a message m=x if it can? If so, what is x∗?

Thanks!

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