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A farmer has talent $\theta$ uniformly distributed on $[0,1]$. His payoff from farming his land is $\theta.$ Before setting up his farm, he chooses some $w\geq0$, which a plant owner can either accept or reject. If the owner rejects, then the farmer farms on his own, and the owner gets 0. If the owner accepts, then the farmer's payoff is $w$, while the owner's is $\frac{3}{2}\theta-w$.

I have found the following BNE: farmer offers some $w\geq\theta$ and the owner rejects all offers. Do others arrive at this as well? Thank you.

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  • $\begingroup$ Does the farmer offer $w$ before or after she knows what $\theta$ is? $\endgroup$
    – Walrasian Auctioneer
    Commented Jun 2, 2020 at 19:27
  • $\begingroup$ @WalrasianAuctioneer After she knows what $\theta$ is. $\endgroup$
    – Charles
    Commented Jun 2, 2020 at 20:30
  • $\begingroup$ What is the owner's payoff if they reject? $\endgroup$
    – Kitsune Cavalry ♦
    Commented Jun 3, 2020 at 0:01
  • $\begingroup$ @KitsuneCavalry 0. $\endgroup$
    – Charles
    Commented Jun 3, 2020 at 2:02
  • $\begingroup$ Another BNE (and infact a PBE) is one where a type $\theta$ farmer offers $w = \frac{3}{2}\theta$ and the owner always accepts. $\endgroup$
    – Walrasian Auctioneer
    Commented Jun 3, 2020 at 2:21

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