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A seller offers a price $P \in \mathbb{R}^{+}$. Buyers can have type $\theta$ distributed uniformly on $ [0,1]$. If a buyer of type $\theta$ accepts, then he gets $\theta-p$, and the seller gets $P$. If he rejects, both the buyer and the seller get 0. Find all PBEs. Here is my solution:

Seller's expected utility from offering $p\in[0,1]$ is $(1-p)p.$ This is maximised at $p*=1/2.$ So in the PBE a seller will offer $p=1/2$ and all buyers of type $\theta>1/2$ will accept. Is this correct?

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  • $\begingroup$ What is $w$? A typo? $\endgroup$
    – VARulle
    Jun 15 '20 at 17:40
  • $\begingroup$ @VARulle Sorry, edited. $\endgroup$
    – George
    Jun 15 '20 at 17:42
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You are correct if there is only one buyer or one good for each buyer. If there are $n$ buyers, but only one good, the seller maximizes $$(1-p^n)p,$$ which has FOC $p^* = (\frac{1}{n+1})^{1/n}$.

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