The Mathematics of Dealmaking

In my previous column, I wrote about Arrow’s Impossibility Theorem and the structural reason no decision rule can satisfy every participant’s preferences at the same time. In a transaction, that shows up immediately. Sellers prioritize price, buyers prioritize return, lenders prioritize protection. Each group evaluates the same set of outcomes differently, and there is no mechanism that preserves all of those rankings simultaneously. Something has to give. That limitation is not the result of poor negotiation or incomplete information. It is embedded in the structure of the problem.

But transactions still happen. Companies are sold every day, often in competitive processes involving multiple bidders, financing sources, and advisors. If perfect alignment is impossible, the relevant question is what determines whether a deal exists, and if it does, where it lands. In practice, people talk about valuation gaps, market conditions, leverage availability, and execution risk. Those are real forces, but they are descriptions of a deeper structure. The process is not arbitrary. It follows a logic that can be written down with precision, and once written, it becomes clear that dealmaking is already behaving like a mathematical system whether anyone in the room represents it that way or not.

Every Deal Begins as a Set of Preferences

A transaction is often described as a negotiation between parties with different views of value. But “value” is not a single variable. In reality, every participant is balancing several objectives at once. A seller does not care only about headline valuation. The seller may also care about certainty of closing, how that value is paid (cash, stock, earnouts, or a combination), the extent of post-closing liability, the preservation of the company’s identity, and the speed of execution. Buyers face the same structural problem. Purchase price matters, but so do expected return, financing constraints, integration risk, and the quality of the asset under downside scenarios. Lenders add a third layer, where the focus is not price or upside, but coverage ratios, collateral, and the probability of default, all under downside conditions rather than base-case optimism.

The structure can be written formally. Let x represent a potential transaction structure. That includes price, cash at closing, deferred payments, earnouts, leverage, rollover equity, indemnities, and timing. Each participant evaluates x through its own value function:

V_S(x) for the seller, V_B(x) for the buyer, and V_L(x) for the lender. These are not abstractions added after the fact. They are a precise way of expressing how each participant already evaluates a deal. Each function captures how that participant values different combinations of outcomes. A seller prefers higher price and lower retained liability. A buyer prefers higher return, and lower risk. A lender prefers structures that maximize the likelihood of repayment under stress.

A participant does not need to love a structure for it to remain in play. It only needs to clear a minimum acceptable threshold. The seller will accept only structures where V_S(x) ≥ V_S^min, the buyer requires V_B(x) ≥ V_B^min, and the lender requires V_L(x) ≥ V_L^min. The set of transaction structures that satisfy all of these conditions simultaneously is:

F = {x : V_S(x) ≥ V_S^min, V_B(x) ≥ V_B^min, V_L(x) ≥ V_L^min}

This set is what practitioners refer to, without formal notation, when they say a deal is financeable, executable, and acceptable to both sides. The critical point is that a transaction exists if and only if this set is non-empty. If F = ∅, there is no deal. Once that condition is met, each participant solves their own problem inside that set. From the seller’s perspective:

max V_S(x) over x ∈ F

That is the core structure. There is no global optimum that simultaneously satisfies every participant’s ranking of outcomes. There is only a feasible set, and within that set each participant pushes toward the point most favorable to them. The transaction is the result of that interaction.

The Optimization of Dealmaking

Once a feasible set exists, the nature of the problem changes. The seller is no longer optimizing in isolation, because the seller’s ideal outcome may lie outside what the buyer and lender can support. The seller is not searching for an unconstrained maximum, but for the best outcome among those that can actually survive the system.

To see how this works in practice, it helps to make the seller’s objective more explicit. Suppose the seller evaluates outcomes based on four dimensions: price P(x), cash at closing C(x), execution certainty T(x), and seller’s retained risk or liability R(x). Then:

max V_S(x) over x ∈ F

where

V_S(x) = V_S(P(x), C(x), T(x), −R(x))

subject to

V_B(x) = V_B(IRR(x), L(x)) ≥ V_B^min

where IRR(x) = return on investment and L(x) = leverage ratio, and

V_L(x) = V_L(COV(x), D(x)) ≥ V_L^min

where COV(x) = coverage ratio and D(x) = default risk.

The seller prefers higher P, higher C, higher T, and lower R. But those dimensions do not move independently. A buyer may be willing to increase P(x), but only by replacing cash at closing with an earnout, which increases total potential value while reducing certainty of what will actually be received. A seller may reduce retained liability only by accepting a lower purchase price or a narrower buyer pool. A lender may cap leverage, which reduces the cash available at closing and forces the buyer to rebalance price, structure, or both. The transaction that emerges is therefore not the maximum of any one variable in isolation. It is the point in F where further improvement in one dimension would either violate a constraint or worsen another dimension materially. This is what practitioners are describing when they say a deal has been pushed to its limit.

Finding the Set and the Intersection

Consider again a business generating $8 million in EBITDA. A seller may believe the business supports a 10x (ten-times) multiple, implying an enterprise value of $80 million. A buyer evaluating the same company under a 20% return requirement and approximately 4x leverage would begin by determining how much debt the business can sustain. At 4x, an $8 million EBITDA business supports roughly $32 million of debt. The rest of the purchase price must be funded with equity, and that equity must generate the target return. Once those constraints are modeled, taking into account expected cash flow, growth, and exit assumptions, the maximum price the buyer can justify may be closer to $64 million. That difference is often described as a negotiation gap, but that description is incomplete. What it actually reflects is the distance between two sets of acceptable outcomes defined by different objective functions. In their initial form, those sets may not intersect.

If the seller’s acceptable set begins at $80 million under preferred terms, and the buyer’s acceptable set ends at $64 million under realistic financing constraints, the initial intersection may be empty. That does not end the process, it defines the problem. The work of dealmaking is to test whether changes in structure can move those acceptable regions into overlap. If leverage can expand, if contingent consideration can be introduced, if rollover equity is acceptable, or if risk can be redistributed in a way both sides can tolerate, then the set may become non-empty. If those adjustments still fail to produce overlap, then no transaction exists under the available conditions.

When the intersection does exist, the problem takes a different form. The objective is no longer to create a deal from nothing, but to move within the feasible set toward the most favorable available point. Structure becomes the mechanism through which that movement occurs. An earnout can increase total consideration while transferring part of the performance risk to the seller. A rollover can reduce the buyer’s upfront capital requirement while preserving exposure to future upside. Seller financing can bridge differences between available debt and desired proceeds. Each of these adjustments changes the mapping from x to V_S(x), V_B(x), and V_L(x), which is another way of saying that it changes how each participant evaluates the transaction. None of these mechanisms eliminates the underlying constraint, and none creates a global optimum. What they do is make it possible to search for a point inside the feasible set where the structure holds and the deal can close.

The final transaction is therefore not the maximum of any single variable, nor is it a compromise in the casual sense of the term. It is the point x ∈ F where further movement in one direction pushes the structure outside what the other participants can accept. That is the boundary practitioners reach when they say a deal has been fully negotiated. In formal terms, it is a boundary point of the feasible set. In practice, it is the moment when another turn of price, leverage, or protection breaks the deal.

What Better Dealmakers Actually See

Most people experience transactions through language, not structure. One side says the business is worth more, while the other says the risks are higher than expected. A lender says leverage cannot go that far. In ordinary conversation, that sounds like disagreement, posturing, or pressure. In mathematical terms, it is the system revealing its constraints. That distinction matters because it changes how the process is understood. If every difference in position is interpreted emotionally, the process becomes theatrical very quickly: a buyer might look like a lowballer, a seller might look unrealistic, and a lender might look obstructive. But once the problem is seen structurally, the issue is no longer who is right, but whether a feasible set exists under each party’s conditions, even if pushed by financial engineering. And if it does, the question becomes how the structure can be moved toward the most favorable point inside it.

That is where mathematics becomes useful to the practitioner. It does not turn dealmaking into a machine, and it does not eliminate judgment. It forces clarity about preferences, constraints, and tradeoffs. It makes it easier to separate structural limitations from noise and ego, and it can turn a bad process into a disciplined one. Arrow explains why no transaction can satisfy every participant’s ranking of outcomes at once. The optimization framework explains how to navigate this system and push it toward a viable closing. A deal exists only where acceptable regions overlap, and the final structure is the result of pushing as far as possible within that overlap. We may not speak in functions and sets when we conduct a sale process, but that is the logic we are following all the same.