Does a Toehold Make a Bidder Bolder? Preemption and Multiplicity in Multi-Round Takeover Auctions

This paper investigates the "toehold puzzle": why firms rarely acquire a pre-bid stake in a target company despite theoretical models suggesting that such a stake provides a dual advantage. Specifically, it tests whether a toehold truly deters rivals in a realistic, multi-round sequential auction, as opposed to the simplified single-round models often used in classical literature. The author models the takeover as a two-player extensive-form game of imperfect information, solving for Bayes-Nash equilibria using computational methods to ensure high precision (certified as epsilon-equilibria).

The study reveals that the deterrence effect—the idea that a toehold scares off rivals—is largely an artifact of restricting the auction to one or two rounds. Once the contest is extended to a genuine second or third round, the link between toehold size and rival deterrence breaks down. While the toehold continues to increase the holder's profit (the arithmetic effect), it fails to reliably influence the rival's behavior. Furthermore, the author identifies significant equilibrium multiplicity: the auction can support multiple stable outcomes where the toehold-holder earns the same profit but exhibits vastly different behaviors, ranging from aggressive jump-bidding to passive play. This suggests that preemption is often a feature of the sequential, public nature of bidding rather than a direct consequence of owning a stake.

These findings challenge the standard textbook justification for acquiring toeholds. If the deterrence benefit is not robust in multi-round contests, deal teams may be overestimating the strategic value of a toehold when weighing it against costs like regulatory disclosure and price impact. Additionally, the paper serves as a methodological warning for researchers using game solvers: because these auctions can have multiple equilibria, different solver runs may produce different, equally valid results, making it dangerous to rely on a single output as a definitive economic prediction.