Shapley value

You already know that the marginal contribution of player i to coalition S is v(S ∪ {i}) - v(S). The problem: that number changes depending on what's already in S. Paid ads add $200K when they're the first channel live, but $400K when content and sales are already running (because the traffic has somewhere to convert).

The Shapley value resolves this by asking: what if we averaged over every possible order the contributors could have arrived? For each ordering (permutation), you record what i adds when it's their turn to join. Then you take the Expected Value across all orderings.

Formally, for n contributors, there are n! orderings. The Shapley value of contributor i is:

φ_i = (1/n!) × Σ [marginal contribution of i in each ordering]

There's also an equivalent exact formula that groups by coalition size instead of enumerating orderings:

φ_i = Σ over S⊆N\{i} of [ |S|! × (n-|S|-1)! / n! ] × [ v(S ∪ {i}) - v(S) ]

Both give the same number. The ordering version is easier to reason about. The coalition formula is what you'd precompute in a spreadsheet or script. For 3 contributors, the ordering approach walks through 6 permutations. The coalition formula evaluates, for each contributor, the 4 subsets of the other two players - one of size 0, two of size 1, and one of size 2. Both are tractable by hand. For 10+ contributors, you'd sample thousands of random orderings rather than enumerate all 3.6 million, and average the samples.

Three P&L problems that Shapley solves cleanly:

1. Attribution across channels. Giving 100% credit to the last channel that touched a deal ignores every channel that built awareness and interest upstream. Giving 100% credit to the first channel ignores everything that converted and closed. Both are wrong when channels have synergies. Shapley gives you a defensible split that sums to the actual total - whether you're attributing Pipeline, Revenue, or any other shared outcome. No double-counting, no undercounting.

2. Cost sharing across business units. If three product lines share a platform team, how do you allocate that Cost Center? Splitting evenly ignores that one product line drives 80% of the platform complexity. Splitting by usage ignores that the platform wouldn't exist without the first product line's initial investment. Shapley accounts for both by considering every possible ordering of who showed up first.

3. Headcount and Budget decisions. When you're running a P&L and need to cut one channel or team, the naive move is to cut whoever has the lowest standalone value. But standalone value ignores synergies. Shapley tells you who's genuinely contributing the least in the context of everything else you're running. That's the difference between cutting the channel that looks cheap and cutting the channel that actually adds the least marginal value.

Step 1: Define the value function. For every possible subset (coalition) of your contributors, determine the total value that subset produces. This is v(S). With 3 contributors (A, B, C) you need 8 values: the empty set, three singletons, three pairs, and the full set.

Step 2: Enumerate orderings (or use the formula). Write out every permutation. For each, walk through the ordering left to right, recording what each contributor adds when they join.

Step 3: Average. Each contributor's Shapley value is the mean of their marginal contributions across all orderings.

Step 4: Sanity check. The Shapley values must sum to v(N) - the total value of the full coalition. This is called the efficiency property. If they don't sum correctly, you have a calculation error.

Key properties that make Shapley the right choice:

• •Efficiency: Allocations sum to total value. No value left on the table, no double-counting.
• •Symmetry: If two contributors add the same marginal value in every coalition, they get the same Shapley value.
• •Null player: A contributor who adds zero marginal value to every coalition gets zero. No participation trophies.
• •Additivity: If you run two independent value functions and add them, each contributor's Shapley value is the sum of their Shapley values from each function. This means you can decompose complex Allocation problems into simpler ones.

Use Shapley when:

• •Contributors have synergies or redundancies - meaning marginal contribution depends on context
• •You need the Allocation to sum exactly to the total (Budget, Pipeline, cost sharing)
• •Fairness matters for incentives - if people feel the split is arbitrary, they'll game whatever metric you pick instead (a direct instance of Goodhart's Law)
• •You have a small enough number of contributors (under ~15) to compute exactly, or you're willing to approximate with sampling

Don't use Shapley when:

• •Contributors are independent (no synergies). If v(S) is just the sum of individual values, marginal contribution is constant across orderings and Shapley reduces to each contributor's standalone value. You don't need the machinery.
• •You can't measure coalition values. Shapley requires v(S) for subsets. If you can't measure what happens when you remove a channel - what Pipeline looks like without channel X - you can't populate the value function. This often requires Sensitivity Analysis: toggling contributors on and off and measuring the result. Consider a simpler Scoring Model or Conjoint Analysis instead.
• •Speed matters more than fairness. For real-time Pricing or auction decisions, Shapley's computation cost may be too high. Use a simpler decision rule and reserve Shapley for quarterly P&L reviews.
• •The number of contributors is huge (100+ Marketing Spend line items). The exact computation scales as 2^n. Even sampling thousands of random orderings takes time to converge. Group contributors into logical clusters first.

Shapley value directly extends the marginal contribution concept - instead of measuring contribution relative to one specific coalition, it averages across all possible coalitions weighted by ordering. This averaging is itself an Expected Value computation: if you pick a random ordering uniformly, the Shapley value is the expected marginal contribution of player i. It inherits from Game Theory the core insight that your value depends on what others are doing - Shapley gives you a principled way to collapse that dependency into a single number.

Downstream, Shapley connects to Efficient Allocation and cost sharing problems. When you're doing Capital Allocation across business units, Shapley-based cost sharing ensures every unit pays less than their standalone cost (everyone benefits from joining the coalition) and the total is fully covered (efficiency). It also informs weighted voting in consortium decisions and M&A due diligence scenarios where you need to determine how much value each acquired entity contributes to the combined Portfolio. In practice, the hardest step is building the value function - which often requires Sensitivity Analysis (toggling contributors on and off) or backtesting against historical periods where different subsets were active.