# Corporate takeovers and the dilution of shareholder property rights

In Chapter 7 we considered the problem of overlapping disposal rights in the context of corporate takeovers. One of the main conclusions there was that in contrast to the pessimistic result of Grossman and Hart (1980), there was a mixed strategy equilibrium in which a more efficient raider succeeded with some positive probability in taking over the corporation. Moreover, the equilibrium probability of a successful takeover turned out to be equal to the probability that each shareholder was pivotal. Finally, we showed that the raider's equilibrium expected profits depended positively on this probability, and so the raider's incentive was to maximise the probability that each shareholder would be pivotal.

More generally, suppose that there are *N* shareholders, each of whom holds one share. Suppose that the raider requires *K < N* shares to be tendered to gain control of the company. Then the probability that an individual shareholder is pivotal is:

Recall that for any *N*, the raider seeks to maximise this in any equilibrium. Hence the equilibrium probability that an individual shareholder will tender their shares is:

As an application of this result, recall that we provided an example in Chapter 7 with *N =* 3 shareholders and where the raider required that at least *K =* 2 two shareholders to tender in order to gain control. In that example, the equilibrium, profit-maximising probability of any individual

*к **2*

shareholder tendering was *p* = _ = _. Recall that this probability was

*N* 3

indirectly chosen by the raider by designing the offer price P_{R}.

More generally, suppose that *N* is even and that the raider requires at least 50 per cent of the shares to gain control of the company, so that

*K = ^{N}* . Then, according to the Bagnoli-Lipman result, the equilibrium

^{2} *N* 1

probability of each individual shareholder tending is = , and the

2N 2

probability that *at least* half of the shareholders tender their shares is:

Grossman and Hart (1980) argue that the free-rider problem can be overcome by *dilution* of shareholder rights. Dilution occurs when a raider reduces 'the value of post-raid company by a certain amount, which the raider is permitted to pay himself. For example, the raider can be allowed to pay himself a large salary or to issue a number of new shares to himself'.^{1}

Their simple argument runs as follows. Let w be the dilution factor, which the shareholders agree upon when the company's constitution is being drafted. Suppose that the raider makes a tender offer of *P _{R} > P*

_{S}, and suppose that a representative shareholder believes that the raid will succeed. Recall that in the Grossman and Hart framework, this means that each shareholder had no incentives to tender their shares, preferring to free-ride on the actions of other shareholders. As a consequence, in the absence of dilution, the takeover will fail.

Now consider the effect of dilution in this situation. Each individual shareholder can either tender or not. If each shareholder continues to believe that the takeover will succeed, then each shareholder's payoff is:

Thus, if the dilution factor is sufficiently large, we will have *P _{R} > V* - ф, and the shareholder will tender his shares.

Let us see how dilution works in our example from Chapter 7, with three symmetric shareholders: A, B and C. Let the dilution factor be such that *P _{R} > V* -ф. Is there a pure strategy equilibrium in which all shareholders tender their shares? Suppose that shareholders A and B have tendered their shares. Then shareholder C's payoff is exactly as it is in equation (10.26), namely:

Thus, C will tender his share. Since all shareholders are alike here, this shows that if the dilution factor is sufficiently large, there is a pure strategy equilibrium in which *all* shareholders tender their shares. In this example, the voluntary dilution of property rights increases the probability that resources move from low- to high-valued uses.