The bidder can win when the gain on its toehold plus the amount it can expropriate from all other shareholders is more than what the bidder must pay in the higher front-end bid price necessary to overcome the pivotal shareholder's private tendering costs.
Profits to the bidder are equal to the value of the firm under the bidder's management less the cost of purchasing a controlling interest at the first-tier price, B; the cost of purchasing the remaining shares not in the toehold at the second-tier price, [V.sub.b]; and the opportunity cost of holding a toehold that could otherwise be sold into the market.
The last term represents the opportunity cost of the toehold shares.
In this case, for tipping to be profitable for the bidder, the increase in the back-end price induced by tipping the arbitrageurs must be less than the private tendering costs of the marginal shareholder, weighted by a function of the bidder's toehold. (Of course, it is also necessary that profits, given by Equations (9) or (11), depending on whether tipping occurs, be positive for the bid to occur.)
That is, the incentive to tip decreases with increases in the toehold. In addition, [Delta][Delta (difference)][[Pi].sub.1]/[Delta]([[V.sub.b].sup.*] - [V.sub.b]) = - (1 - [Alpha])N, which is negative.
In this case, for tipping to be profitable for the bidder, the increase in the back-end price induced by tipping arbitrageurs must be less than the difference between the second-tier price when the bidder does not tip and the value of the firm under incumbent management, weighted by a factor that depends on the bidder's toehold.