Why does the probability in this maximization problem become theta hat?

Suppose we try to implement this social choice function. Assume that the buyers are expected utility maximizers. We now ask: If buyer 2 always announces his true value, will buyer 1 find it optimal to do the same? For each value of 0, buyer I's problem is to choose the valuation to announce, say , so as to solve or Max (0 0,) Prob (0, < ,) Max (0 )n, _ 81