# Bayesian Nash Equilibrium

Firm A is considering taking over firm B. It does not know firm B’s value; it believes that this value, when firm B is controlled by its own management, is at least €0 and at most €150, and assigns equal probability to each of the 151 euro values in this range. Firm B knows its own value. Firm B will be worth 50% more under firm A’s management than it is under its own management. Suppose that firm A bids 𝑦 to take over firm B, and firm B is worth 𝑥 (under its own management). Then if B accepts A’s offer, A’s payoff is (1.5𝑥 − 𝑦) and B’s payoff is 𝑦; if B rejects A’s offer, A’s payoff is 0 and B’s payoff is 𝑥. Model this situation as a Bayesian game with simultaneous moves in which firm A chooses how much to offer and firm B decides the lowest offer to accept. Find the Bayesian Nash equilibrium (equilibria) of this game.

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