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Three towns want to share the cost of a runway. The core told us what is stable; it did not tell us what is fair. The Shapley value answers a different question: what does each party, on average, actually add to the group?

The Shapley value

The core can be a single harsh point, a whole region, or empty. When you want one canonical, fair division rather than a stability set, the Shapley value delivers it: give each player the average of the value they add as coalitions form in every possible order.

From earlier lessons

Two quick ones before we start.

The core of the seller-plus-two-buyers game is...

A player's marginal contribution to a coalition is...

Predict, then play

Same seller-plus-two-buyers game. Sample the orders in which the three players might join the coalition, record each one’s marginal contribution, and watch the averages converge.

Why did that happen?

Take a guess, then reveal the reason

The Shapley value credits each player with what they add, averaged fairly over every order of arrival. The seller is worth 100 whenever it joins after at least one buyer, but nothing if it arrives first with no buyer yet present. A buyer adds 100 only when it arrives to a lone seller, and nothing otherwise. Averaging over all six orders splits the credit so the seller gets the lion’s share but the buyers, who are sometimes pivotal, get something too. That is a fairness verdict, not a stability one.

The math

Definition, the Shapley value

The Shapley value gives player $i$ the average of its marginal contribution $v(S \cup \{i\}) - v(S)$ over all orders in which the players could join the grand coalition. It is the unique allocation satisfying four axioms: efficiency (the whole value is shared), symmetry (interchangeable players get equal shares), the null-player property (a player adding nothing gets nothing), and additivity across games.

For the seller (player $s$ ), consider the six orders. The seller’s marginal contribution is $100$ exactly when at least one buyer is already present, which happens in four of the six orders, and $0$ in the two where the seller is first. So

\varphi_s = \frac{0 + 0 + 100 + 100 + 100 + 100}{6} = \frac{400}{6} = \frac{200}{3} \approx 66.7,

and by symmetry each buyer gets $\varphi_b = 50/3 \approx 16.7$ . Note $\varphi_s + \varphi_{b_1} = 83.3 < 100$ , so the Shapley value lies outside the core: this game is not convex, and the fair division is not the stable one.

Work it out

Average the seller's contributions

List the six join orders and the seller’s marginal contribution in each. If the seller arrives first (orders $s,b_1,b_2$ and $s,b_2,b_1$ ), no coalition value exists yet, so it adds $0$ . In the other four orders a buyer is already present, so the seller completes a coalition worth $100$ and adds $100$ .

Average: $(0 + 0 + 100 + 100 + 100 + 100)/6 = 400/6 = 200/3 \approx 66.7$ . The two buyers split the rest by symmetry, each getting $50/3 \approx 16.7$ . Check that this sits outside the core: $66.7 + 16.7 = 83.3 < 100$ , so the coalition of the seller and one buyer could block it. Fairness (Shapley) and stability (core) genuinely diverge here.

Key idea

The Shapley value pays each player their average marginal contribution over all join orders, the unique fair division by four natural axioms. It need not lie in the core: fairness and stability can point to different splits.

Check yourself

Check yourselfapply

Why is the seller's Shapley value 200/3 rather than 100?

Check yourselfanalyse

The Shapley value here is (66.7, 16.7, 16.7). What does it mean that this is outside the core?

Check yourselfbrings back: the core

When would the Shapley value be guaranteed to lie inside the core?

Sources

Shapley, L. S. (1953). A value for n-person games. In Contributions to the Theory of Games II. Princeton University Press.