An acquirer considers making a public offer to acquire shares of a target. The target has N shares outstanding, where N is an even natural number. Conditional on observing the acquirer's offer, the target's shareholders decide whether to accept it or not. The value of the target is V(succes) if the acquirer obtains at least 50% of the shares and V(failure) < V(succes) if otherwise.
a. Suppose that the target's shareholders are nonpivotal, that is, each of them takes the aggregate decisions of all shareholders to accept or reject the acquirer's offer as given. Show that there is no Nash equilibrium in which the target acquires at least 50% of the shares for a per-share price of p < V(succes) / N.
b. For letter b) and c) below, assume that each of the target's shareholders holds one indivisible share.
On the acquirer's behalf, design an offer such that, under a Nash equilibrium, the acquirer obtains 50% of the synergies V(succes) - V(failure). (Hint: it could help to draw the two-shareholder normal form of the game)
c. On the acquirer's behalf design an offer such that, under a Nash equilibrium, the acquirer obtains 100% of the synergies V(succes) - V(failure). (Hint: it could help to condition the payoffs of the two-shareholder normal form of the game on the offer designed)
d. Given your answers in a), b), and c), what allows the acquirer to appropriate 100% of synergies in c) that is not in either a) or b)?