Two owners with overlapping disposal rights
The conclusions of Grossman and Hart seem to be intuitively attractive but they have been challenged in an important paper by Bagnoli and Lipman (1988). To understand their criticism, we analyse the case of a small number of non-atomistic shareholders. To see how numbers matter, note that the problem associated with overlapping disposal rights does not arise if there is only a single shareholder, and is also not a problem with two shareholders if one of them is sufficiently large. To see the latter result, suppose that there are two shareholders, labelled A and B. Suppose that the fraction of shares owned by B is sB > 1/2. Then the payoffs from tendering and not tendering can be illustrated in the payoff matrix in Table 7.3.1.
Since PR > PS shareholder B has a dominant strategy here: to tender his shares. Since A knows this, A can free ride, not tender, and obtain V once the takeover is completed. The Nash equilibrium is therefore for B to tender and A not to tender. But since sB > 1/2, this means that the takeover succeeds. The increase in economic value is realised and there is no efficiency loss. B is always pivotal in this example - his action always determines the outcome of the tender offer.