mergers

When two or more entities merge operations, the combined value often exceeds what each could produce alone. The excess is surplus - it comes from eliminating duplicate overhead, sharing infrastructure, or combining complementary capabilities.

The hard problem isn't creating surplus. It's dividing it.

Consider three common proposals for splitting shared costs or combined Profit:

• •Equal split: Everyone pays the same. Simple, but ignores who actually drives the cost.
• •Revenue-proportional: Bigger entity pays more. Sounds neutral, but it's a specific deal favoring one side.
• •Shapley value: Each party pays based on their average marginal contribution across every possible combination of participants.

Shapley value comes from Game Theory, and it's the only method that simultaneously satisfies three constraints:

1. 1)Efficiency - Allocations sum to the total. No surplus left on the table, no phantom surplus invented.
2. 2)Null player - If you contribute nothing, you pay nothing and receive nothing.
3. 3)Symmetry - If two parties make identical contributions, they get identical allocations.

These aren't aspirational principles. They're mathematical constraints. Any method that violates one of them produces outcomes that at least one party can prove is unfair.

After a merger or when shared services launch, someone decides how costs hit each division's P&L. That Allocation directly determines your EBITDA - and in PE-Backed companies, your EBITDA determines your Budget, your headcount, and your operating autonomy.

Three reasons this isn't just accounting:

1. Wrong Allocation creates perverse incentives. If your division gets overcharged for shared infrastructure, your rational move is to build your own - which destroys the surplus the merger was supposed to create. This is the most common failure mode in PE Portfolio Operations: the merger looks great on the Holding Company P&L but the divisions quietly sabotage it.

2. Allocation determines who "won" the merger. In a Multi-Brand Portfolio, each brand's GM is evaluated on their division's numbers. If the cost sharing formula shifts $200K/year from your P&L to theirs, you look like the growth story and they look like the problem child - regardless of actual performance.

3. It compounds. A bad Allocation formula doesn't just cost you money this year. It sets the precedent for every future shared service, every Capital Investment, every joint project. Get it right at the start.

The intuition behind Shapley value is simple: imagine all parties arriving one at a time, in random order. Each time someone arrives, measure the additional cost (or value) they bring. Average that marginal contribution over every possible arrival order.

Two-Party Case

For two parties, Shapley always splits surplus 50/50. The logic:

• •Party A arrives first: contributes their full standalone amount
• •Party A arrives second: contributes whatever the combination adds beyond B's standalone
• •Average across both orderings: A gets their standalone value plus half the surplus
• •Same logic applies to B

This means in any two-entity merger, the mathematically fair division gives each side their standalone value plus an equal share of the surplus they created together. If someone proposes Revenue-proportional, headcount-based, or any other formula, they're proposing a different deal - not a neutral default.

Three-or-More-Party Case

With three parties, there are 6 possible orderings (3! = 6). Asymmetries emerge: a party that creates large marginal contribution in many combinations gets a larger share. A party that mostly duplicates what others already bring gets less.

For each party: list their marginal cost (or value) in each of the 6 orderings, then average. The results sum to the total - guaranteed by the math.

Cost Sharing vs. Value Division

Shapley works in both directions:

• •Dividing value (merger surplus, joint project Revenue): higher Shapley share = larger piece of the pie
• •Allocating costs (shared warehouse, platform, finance team): higher Shapley share = larger piece of the bill

The math is identical. The only difference is whether a higher number is good or bad for you.

Use Shapley value when:

• •Merging portfolio companies - to allocate shared overhead and determine each division's post-merger EBITDA
• •Launching shared services - tech platforms, finance teams, warehouses that serve multiple business units
• •Joint projects - any deliverable where multiple teams contribute and the P&L impact needs to be attributed
• •consortium deals - multiple firms co-investing in infrastructure, where each partner needs a defensible cost share
• •Renegotiating existing splits - when an old equal-split or Revenue-proportional formula has become obviously unfair as the business has changed

Don't use Shapley when:

• •There are only two parties and you have strong Bargaining leverage - you can capture more than half the surplus through negotiation. Shapley is the fair answer, not the maximum answer.
• •The cost is trivial relative to the political cost of running the analysis
• •Contributions are genuinely equal and symmetric - Shapley will confirm equal split anyway

In practice: Use Shapley to establish the fair baseline, then negotiate from there. Knowing the fair answer gives you an Anchoring advantage even if the final deal deviates from it.

Mergers sits at the intersection of your two prerequisites. Valuation taught you that each party's worth depends on what an Asset produces for them specifically - in a merger, that standalone Valuation is each party's Outside Option, the floor below which they'd refuse to participate. Bargaining taught you how two parties divide surplus based on leverage, patience, and Anchoring - but when three or more parties combine, Bargaining alone can't produce a unique answer because every pair could negotiate a different split. Shapley value resolves this by computing each party's fair share from the principles of Game Theory. Looking forward, this framework underpins M&A due diligence (is the projected surplus real or invented to justify the deal?), PE Portfolio Operations (how do you run a Multi-Brand Portfolio without division-level P&L fights destroying the surplus?), cost sharing across any shared service, and the broader question of Efficient Allocation whenever multiple parties create value together.