Game Theory Applications
Lecture 6
Takeover Game
Hsiang-Lin Chih
02-8674-6874(O)
hlchih@mail.ntpu.edu.tw
當公司股權非常分散時，股東將無誘因去監督公司，從而經營階層與股東間的代理
問題可能非常嚴重，亦即經營階層可能不會去追求股東利益最大，反而會濫用公司資
源，追求自己的利益。
然而，如果經營階層真的只追求自己的利益，公司獲利與股價必然因此降低，從而
該公司反而容易被其它公司所購併，該經營階層將會被撤換。因此，在此種被購併的
威脅下，該經營階層將會努力追求股東利益最大，從而代理問題會消失。
 請問上述的故事正確嗎？
 Grossman and Hart (1980) 說上述故事是錯的，因為公司股東具有坐享其成的現象。
 實際上，Grossman and Hart (1980) 的說法並不完全符合實際世界的現象。為什麼？
1. Grossman and Hart (1980; Bell J. E.)
1.1 It is commonly thought that a widely held corporation that is not being run
in the interest of its shareholders will be vulnerable to a takeover bid by a
raider, but Grossman and Hart (1980) argued that is false. Under assumptions
that shareholders are atomistic1 and the only successful bids are those, which
are expected to be successful with certainty 2 , Grossman and Hart (1980)
advanced the proposition, which subsequent work accepted, that successful
takeover bids must be made at or above the expected value of minority shares.
This proposition led to Grossman and Hart’s insightful observation that a
free-rider problem exists.
1.2 Consider the problem of a firm with disperse ownership that has received a
takeover offer from a raider who can improve the operating efficiency of the
firm. Suppose that the raider, with no initial shares, makes an offer
1That
is, the probability that any shareholder’s tender decision will be decisive in determining
the success or failure of the bid is negligible. Thus, each shareholder will ignore his impact on the
outcome of the bid in making his tender decision.
2Shareholders
and the raider are assumed to have rational expectations about the outcomes.
And in Grossman and Hart’s paper, they do not consider bids with stochastic outcomes, i.e., bids
that succeed some fraction of the time and fail the remaining fraction of the time.
1
(unconditional or conditional)3 at a price, p, that is between the current value
of the firm under the incumbent’s management, q ($100 per share) and its
value under the raider’s management, v ($120 per share). Furthermore, we
assume the shareholders are atomistic and have rational expectations, and the
incumbent cannot make counter tender offers.
First, we consider the case that the raider makes a tender offer at p = $105
per share, as shown in table 1.4(1)(conditional offer) and 1.5(1)(unconditional
offer). If we assume that the only successful bids are those, which are expected
to be successful with certainty, i.e., the atomistic shareholder conjectures that
others are going to tender and the offer will win with certainty, then he can do
better by holding on (he gets v = $120) than by tendering (he gets p = $105).
Since no shareholder tender his shares, a bid with tender price p<v will, of
course, fail. Therefore a successful bid such that p<v is not an equilibrium
since that violates the assumption that shareholders have rational expectations.
Then, we can conclude that if the raider’s bid is going to win, no
shareholder will sell unless he is offered at least the posttakeover value of his
stock. Consequently, the raider cannot purchase a share unless he pays at
least $120 per share (as shown in table 1.4(2)). If he does so, the atomistic
shareholders will free ride on all the improvement generated by the raider. Then
even ignoring any cost of the tender offer, the raider cannot earn any profit by
taking over the firm. Therefore, there are no takeover bids since the raider
won’t provide public goods (ie, to improve the firm value), and bad
management is not penalized. This is an outcome, which is highly undesirable
for all shareholders. Grossman and Hart (1980) named this situation as a
“free-rider problem”.
3The
offer is unconditional (any-and-all) in that tendered shares are always purchased by the
raider, regardless of the outcome of the offer.
By contrast, in a conditional offer, shares are only purchased when the offer succeeds. With such
a bid, the raider chooses three objects. First, he chooses a number p that he will pay for shares
offered to him. Second, he chooses an integer number of shares that he wishes to acquire; that is,
he announces that he will pay p for K shares. If fewer than this number is offered to the raider, he
buys no shares at all. If this number is offered, he buys all of them and pays p for each. The third
object he chooses is a rationing device that determines how many he buys from whom in the event
that more than K shares are offered.
Noe (1995) said that unconditional tender offers are currently the most common form of tender
offer.
2
Table 1.4 Conditional Tender Offer
(1) The Raider bids for $105 per share
Win
Atomistic
Tender
Shareholder
Not Tender
Lose
(2) The Raider bids for $120+ε per share
Win
Atomistic
Tender
Shareholder
Not Tender
Lose
Table 1.4 Unconditional Tender Offer
(1) The Raider bids for $105 per share
Win
Atomistic
Tender
Shareholder
Not Tender
Lose
(2) The Raider bids for $120+ε per share
Win
Atomistic
Tender
Shareholder
Not Tender
Lose
1.3 In practice the free-rider problem is not so severe as in Grossman and Hart
(1980)’s model since raids do take place. What changes are necessary to allow
for this possibility? The most obvious modification is to let the raider obtain
some private benefits (z per share), which the minority shareholders can be
excluded from sharing with the raider.4
2. Bebchuk (1989; JFQA)
2.1 Bebchuk (1989) shows that Grossman and Hart (1980)’s proposition of a
free-rider problem does not always hold once he drops their assumption that
only successful bids are those for which success could have been conjectured
with certainty.
4Hirshleifer
and Titman (1990) and Shleifer and Vishny (1986) have demonstrated that, when
the raider has a toehold stake in the target firm at the time the takeover offer is made, the raider’s
capital gains on his own shares will allow him to earn profits even when the offer price equals the
postacquisition value of the firm.
3
2.2 Consider the case that the raider makes an unconditional offer at p = $105
per share, as shown in table 2.3(1). If the atomistic shareholder conjectures
that other shareholders are not going to tender, implying that the offer will
fail, then he is better off tendering his shares. On the other hand, if he
conjectures that others are going to tender, implying that the offer will win,
then he is better off not tendering and, thus, receiving the value under the
raider’s management. In this scenario, it’s easy to see that neither tendering
nor not tendering (henceforth referred to as free riding) is a dominant strategy,
and thus the shareholder will play mixed strategies.
2.3 From 2.2, we can see that any unconditional bid that is below v but above q
will succeed with a positive probability, that the raider’s expected payoff from
such a bid (not counting the transaction costs of making the bid) is always
positive, and that the raider might elect to make such a bid in the absence of
expropriation (or exclusion), even if he has no initial stake in the target firm..
Table 2.3 Unconditional Tender Offer
(1) The Raider bids for $105 per share
Win
Atomistic
Tender
Shareholder
Not Tender
Lose
(2) The Raider bids for $120 per share
Win
Atomistic
Tender
Shareholder
Not Tender
Lose
3. Bagnoli and Lipman (1988; RFS)
Let’s recall Grossman and Hart (1980). In the case of an unconditional bid
below v but above q, they argue that certain success of the bid is not a rational
equilibrium outcome. But Bagnoli and Lipman (1988) pointed out that this
argument is perfectly symmetric. That is, it is also the case that, if shareholders
conjecture that the offer will fail with certainty, then all shareholders will
strictly prefer tendering, and thus certain failure of the bid is also not a rational
equilibrium. Therefore, if there’s no equilibrium at all, the fact that there is no
equilibrium with a successful bid means nothing. Therefore Bagnoli and Lipman
(1988) think that the analysis of Grossman and Hart (1980) is “informal” since
4
Grossman and Hart (1980)’s analysis ruled out mixed-strategy equilibria and
didn’t identify any equilibrium.
4. 創造囚犯的困境 – The two-tier tender offer
(1) 假設： q = 100 (pretakeover)，v = 120 (posttakeover後)
(2) 併購公司 (raider) 所提的two-tier tender offer
* 以每股105元的價格收購至50％之股權
* 若收購成功，則以每股90元的價格收購其餘股權
⇒ 令tender之比例為x%，且x > 50，則每股價值為
(50/x)*105 + ((x - 50)/x)*90 = 90 + (50/x)*15
(3) 小股東會願意賣？
Tender offer 成功
Tender offer 失敗
小股東tender
90 + (50/x)*15
105
小股東不tender
90
100
(4) Tender offer後小股東權益受損：囚犯困境!
Tender offer前：每股價值100元
Tender offer後：每股價值90 + (50/100)*15 = 97.5元
5
Why x%=100%?