arXiv study: Toeholds don't deter bidders in multi-round takeover auctions
AI game solver shows toehold deterrence theory collapses after round two
Zain Naboulsi's new arXiv paper, "Does a Toehold Make a Bidder Bolder? Preemption and Multiplicity in Multi-Round Takeover Auctions," tackles a classic finance puzzle: why do bidders rarely acquire toeholds—small stakes in a target company—before launching a takeover? Classical theory says a toehold pays off twice: it makes the holder bid more aggressively and deters rivals from competing. Yet toeholds are rare, suggesting the deterrence effect may be weaker than assumed. To test this, Naboulsi models the contest as a multi-round auction game, solves it with computational game-theoretic solvers, and certifies the results to verifiable accuracy.
The findings are striking. First, the auction determines what a toehold-holder earns, but not how it bids: multiple equilibria exist, including an aggressive opening that scares off rivals and a cheap opening that invites competition, both yielding the same profit. Second, aggressive preemptive bidding appears even when the toehold is removed entirely—it comes from bidding in public and in turns, not from owning a stake. Third, the intuitive "bigger toehold, more deterrence" relationship only holds if the contest is cut short after one round; once a real second round is added, the relationship stops responding. So the profit motive for toeholds holds up, but the deterrence motive does not—helping explain why toeholds are rarer than theory predicts.
Naboulsi also issues a warning for anyone computing economics from game solvers: solve the auction once and you get a confident figure for preemptive bid value; solve it again from a different starting point and you get a different, equally converged answer. The paper reports which solvers handle contests of this shape, including versions too large to enumerate, and releases all code, games, and experiments. For M&A strategists and AI researchers, it's a reminder that game-theoretic results can be sensitive to model structure and solver initialization.
- Naboulsi's game model finds multiple equilibria: aggressive and cheap opening bids yield identical profits for toehold holders.
- Preemptive bidding emerges even without toeholds—driven by public, turn-based auction structure, not stake ownership.
- The 'bigger toehold, more deterrence' relationship breaks down once a second bidding round is added, explaining why toeholds are rare.
Why It Matters
For M&A strategists and AI game solvers: toehold deterrence is overrated, and solver outputs need rigorous validation.