🤖 machine learning

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

By modeling multi-round takeover auctions as a computable game, this paper demonstrates that while toeholds provide a clear arithmetic profit advantage, they fail to deliver the expected strategic deterrence against rivals, thereby explaining the rarity of toeholds in practice and highlighting the critical importance of initial conditions when using game solvers to analyze economic equilibria.

Original authors: Zain Naboulsi

Published 2026-08-11

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Zain Naboulsi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a high-stakes game of poker, but instead of chips, the players are bidding for a mysterious treasure chest. The twist? No one knows exactly how much gold is inside; they only have a hunch based on clues they found in the dark. This is the world of game theory, a branch of science that studies how people make decisions when their outcomes depend on what others do. In this specific corner of the field, researchers look at auctions, where people bid for something valuable. A classic idea in auction theory is the "toehold." Imagine a bidder quietly buying a small piece of the treasure chest before the auction even starts. The old-school theory says this little piece gives them a superpower: it makes them willing to bid more aggressively because they already own a slice of the prize, and it scares their rival into quitting because the rival thinks, "Wow, that guy is so confident, I'd better not fight him." This paper asks a simple but tricky question: Does that scary "superpower" actually work if the auction isn't just one quick offer, but a long, back-and-forth battle with many rounds?

The paper, titled "Does a Toehold Make a Bidder Bolder?", dives into this question by building a computer model of a multi-round takeover auction. Think of it as a video game simulation where two AI bidders fight over a company. One bidder starts with a small "toehold" (a pre-bought stake), and they take turns raising the price or walking away. The researcher ran this simulation thousands of times to see what happens when the fight drags on for several rounds, rather than ending in a single snap.

Here is the big surprise: The toehold loses its magic when the game gets longer.

In the old, simple models where the auction is just one quick move, having a toehold does indeed make the bidder bolder and the rival more likely to quit. It's like having a secret weapon that makes your opponent tremble. But the author found that once they added a second and third round to the game, that connection broke. The toehold still helps the bidder make more money if they win (the "profit" reason), but it stops working as a reliable scare tactic (the "deterrence" reason).

In fact, the simulation showed something even weirder: The rival might quit just as often even if the bidder has zero toehold. The researcher found that the game has multiple "solutions" or ways it can play out. In one version of the game, the bidder jumps in with a huge, scary bid, and the rival runs away. In another version, the bidder starts with a tiny, cheap bid, and the rival stays and fights. Both versions are stable and make sense mathematically, but they lead to completely different behaviors. The only thing that changes between these two versions is the "coordination" between the players, not the size of the toehold.

The paper explicitly rules out the idea that a toehold is a guaranteed ticket to scaring off a rival in a long auction. It argues that the "bigger toehold, more fear" rule is just an artifact of looking at too-short games. In the real world, where auctions can drag on, owning a small piece of the company doesn't automatically make your opponent fold. The "fear" factor is actually a result of the players coordinating on a specific type of aggressive behavior, which can happen even without a toehold.

The author is very careful to say they didn't prove this happens in every single real-world takeover. They simulated a specific game with specific rules (like a price grid with 9 steps and 3 rounds). However, within their simulation, the results were clear and repeatable. They found that the "deterrence" benefit of a toehold is not a fixed number you can count on; it's a fickle thing that disappears as soon as the auction gets a genuine second round.

So, if you are a deal team thinking about buying a toehold to scare your competition, this paper suggests you might be betting on a ghost. The math says the toehold will still help you if you win, but it won't reliably make your rival quit. The "scary" part of the strategy is actually just a game of chicken that can happen with or without the extra stake. The paper concludes that the reason toeholds are so rare in real life might not just be because they are expensive or risky to buy, but because the strategic advantage everyone thought they had might not be as solid as we once believed.

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

Problem Statement
The paper investigates the strategic value of a "toehold"—a pre-acquired stake in a target company—within the context of multi-round takeover auctions. Classical auction theory (e.g., Bulow et al., 1999; Fishman, 1988) posits two primary benefits to holding a toehold:

Arithmetic Profit: The holder profits regardless of the outcome (paying less if they win, selling shares at a higher price if they lose).
Deterrence (Aggressiveness): The toehold allows the holder to bid more aggressively, signaling strength and frightening rivals into conceding immediately.

While the arithmetic benefit is well-established, the deterrence benefit is the subject of an empirical "puzzle": toeholds are observed far less frequently in practice than theory predicts, despite their supposed strategic advantage. Existing models often assume single-round or two-move exchanges. This paper asks whether the deterrence channel survives when the contest is modeled as a genuine multi-round, sequential, ascending auction with imperfect information.

Methodology
The author models the takeover as a two-player, extensive-form game of imperfect information with the following primitives:

Common Value: The target has a value $W$ unknown to both bidders, drawn from a discrete grid.
Private Signals: Each bidder receives noisy private signals regarding $W$ .
Toehold: Bidder 0 owns a fraction $\theta$ of the target.
Protocol: An ascending auction with up to $R$ rounds. Bidders alternate turns, able to "jump bid" (raise to any higher price on a discrete grid) or pass.
Payoffs: General-sum (non-zero-sum). The winner pays the bid price; the toehold holder retains their share value.

Computational Approach
Given the general-sum nature of the game, exact equilibrium computation is PPAD-hard. The author employs the following computational strategy:

Solver: They use fictitious play (specifically, own-profit fictitious play) to compute strategies. Unlike zero-sum solvers that maximize the difference in payoffs, this method maximizes each player's individual profit.
Certification: Solutions are not claimed as exact Nash equilibria but as $\epsilon$ -equilibria. The author certifies that no player can improve their profit by more than $\epsilon$ (ranging from $5 \times 10^{-7}$ to $8 \times 10^{-5}$ ) by deviating.
Multiplicity Detection: To avoid the solver arbitrarily selecting one equilibrium from a set, the author runs multiple independent restarts (seeds). They also employ a forced-opening best-response mechanism: forcing the toehold holder to open at a specific price and calculating the exact optimal response of the rival. This allows them to price the value of a preemptive bid without relying on the solver's convergence to a specific profile.
Benchmarking: The study benchmarks specialized solvers (Counterfactual Regret Minimization - CFR) against general reinforcement learning methods (PPO, PPG, NFSP) on both tractable and intractable (large state space) instances.

Key Results
The study yields three primary findings that challenge classical intuition:

Equilibrium Multiplicity and Non-Identification of Deterrence:
In multi-round auctions (specifically $R=3$ ), the game admits multiple equilibria that yield identical profits for the toehold holder but exhibit radically different behaviors.
- Example: At a toehold of $\theta=0.05$ , one equilibrium profile shows the holder jump-bidding and the rival conceding with probability 0.333. A second, equally stable profile shows the holder opening cheaply with the rival conceding with probability 0.002.
- The deterrence rate is not pinned down by the toehold size. The model cannot predict whether a toehold will deter a rival; it depends on which equilibrium the players coordinate on.
Preemption Persists Without a Toehold:
Preemptive jump bidding (signaling strength to induce a rival to fold) occurs even when the toehold is zero ( $\theta=0$ ). This confirms that preemption is a feature of the sequential structure and the common-value inference (winner's curse), not a specific benefit purchased by the toehold. The toehold does not "buy" the aggressiveness channel; the channel exists independently.
The Two-Move Artifact:
The classical "larger toehold $\rightarrow$ higher deterrence" relationship holds only in single-round or two-move models ( $R=1$ ).
- At $R=1$ , deterrence increases with toehold size.
- At $R=2$ , deterrence saturates and stops responding to toehold size.
- At $R=3$ , deterrence becomes unidentified (multiplicity emerges).
  The "deterrence edge" is an artifact of truncating the game too early.

Solver Performance

Specialized vs. General: On small, enumerable instances, exact CFR finds equilibria faster than deep RL methods. However, PPG converges to near-exact equilibria ( $\epsilon \approx 0.0007$ ) and outperforms NFSP, which fails to converge. PPO is competitive but seed-sensitive.
Scalability: In intractable regimes (large state spaces with multiple private signals), exact solvers cannot run. PPO and PPG successfully drive learned-best-response exploitability estimates down to near-zero, demonstrating their utility for analyzing complex auction mechanisms where exact enumeration is impossible.

Significance and Claims
The paper claims to offer a "model-side reason" for the rarity of toeholds in practice, complementing existing explanations based on disclosure costs and market impact.

Theoretical Contribution: It demonstrates that the strategic "bidding edge" attributed to toeholds in classical literature is not robust in sequential, multi-round contests. The deterrence channel is not a reliable function of toehold size once the game extends beyond two moves.
Methodological Contribution: It highlights the danger of relying on single solver runs in general-sum games, where different initializations can lead to different behavioral predictions (multiplicity) despite identical profit outcomes. The author advocates for using forced-opening best-response calculations to isolate economic variables from solver artifacts.
Limitations: The author explicitly states that their findings are specific to the parameterization used (discrete grids, specific signal noise). They do not claim to prove that deterrence is unidentifiable in all market data, but rather that the theoretical prediction of a smooth deterrence curve is an artifact of short-horizon models.

In conclusion, the paper argues that while a toehold remains profitable, it does not reliably confer a strategic deterrence advantage in multi-round auctions. The "edge" is not a property of the toehold itself, but a coordination outcome of the sequential game that can exist without a toehold and vanish with a longer contest horizon.

More like this