Three shareholders with overlapping disposal rights
In the preceding example there was a pure strategy Nash equilibrium in which the raider's takeover successfully went ahead because one
Table 7.3.1 The payoff matrix in the tender offer game with two shareholders
Shareholder B |
|||
Tender |
Don't Tender |
||
Shareholder A |
Tender |
(Pr, Pr) |
(Pr, Ps) |
Don't Tender |
(V, Pr) |
(Ps, Ps) |
shareholder was sufficiently large as to always be pivotal: B's action determined whether or not the takeover was successful. The situation with three or more shareholders is more complicated, however, because shareholders are usually only pivotal in some situations.
Consider, for example, the case of three shareholders (A, B and C), each of whom owns one share. The raider must obtain two shares to take control. We consider symmetric mixed strategy Nash equilibria. Suppose that each shareholder tenders their shares with probability p. For each shareholder to be willing to randomise in this way, the expected payoff from tendering to be equal to the expected payoff from not tendering. The expected payoff from tendering a share is P_{R}. The expected payoff from not tendering a share is equal to
Pr (Tender Offer is Successful) x V + Pr (Tender Offer is Unsuccessful) x P_{S}
= p^{2}V + [(1 -p)^{2} + 2p(1 -p)]P_{s}
Thus, if the raider has offered P_{R}, for the mixed strategy to be an equilibrium, the two pure strategies must be equally desirable, and we must have:
The right-hand side is a continuous function of p. If p = 0, then the right-hand side is equal to P_{S}, whereas if p = 1, the right-hand side is equal to V. Thus, by the intermediate value theorem, for any P_{R} there exists a value of p (which we denote by p*) such that equality holds in equation (7.2). Moreover, the derivative of the right-hand side with respect to p is:
which is positive if V > P_{S}. Thus, the number p* is unique. The second derivative of the right-hand side is also positive, so the equation in (7.2) is an increasing, convex function of p. Thus, we have the situation shown in Figure 7.3.6.
For any P_{R}, there is a symmetric mixed strategy Nash equilibrium in which each shareholder tenders his shares with probability p*, where this is the solution to:
Figure 7.3.6 The expected payoff to an individual shareholder
In such an equilibrium, for any offer P_{R}(p), the raider's expected profit is:
Plugging in the value of P_{R} from above gives us the raider's expected profit:
2
Thus, to maximise its expected profit, the raider should choose p = 3.
Notice also that the probability that a particular shareholder will be pivotal is p^{2}(1 -p). Thus, another interpretation of equation (7.3) is that the raider chooses a price that maximises the probability that each shareholder is pivotal. The expected profit as a function of p is shown in Figure 7.3.7.
4
In this equilibrium, the raider proposes a price of PR =— V + [(1 -p)^{2} +
- 8 ^{9}
- 2p(1 -p)]P_{S}, which gives an expected profit of —(V - P_{S}). Thus, the raider
expects to capture almost all of the economic value created by the takeover.
Note, however, that in this equilibrium there is some chance that the takeover will not succeed, even though it is efficient for the takeover to occur. The fact that it is not certain that the transaction will occur and
^{1}
Figure 7.3.7 The expected payoff to the raider
resources move from low- to high-valued uses occurs as a result of the existence of an externality between shareholders. When a shareholder is pivotal and decides to tender his shares, he exerts an uncompensated economic benefit on his fellow shareholders. Thus, there is an incentive for free-riding. As the number of shareholders rises, the incentive for free-riding becomes greater, and the likelihood of a successful takeover declines.