? EXERCISE 280.4 (Adverse selection) Firm A (the “acquirer”) is considering taking
over firm T (the “target”). It does not know firm T’s value; it believes that this value, when firm T is controlled by its own management, is at least $0 and at most
3 $100, and assigns equal probability to each of the 101 dollar values in this range. Firm T 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 T, and firm T is worth x (under its own management). Then if T accepts A’s offer, A’s payoff is
2 x − y and T’s payoff is y; if T rejects A’s offer, A’s payoff is 0 and T’s payoff is
x. Model this situation as a Bayesian game in which firm A chooses how much to offer and firm T decides the lowest offer to accept. Find the Nash equilibrium (equilibria?) of this game. Explain why the logic behind the equilibrium is called adverse selection.