Definition
The Shapley value is the rule, developed by Lloyd Shapley, that divides the gain from cooperation by giving each participant a share equal to their average contribution across all possible orderings in which the coalition could have been built up.
Why it matters
When a pool produces more income than its members could produce alone, the additional gain has to be divided according to some rule. The Shapley value is the rule that satisfies a particular set of fairness principles — equal treatment of equivalent contributions, full distribution of the gain, additivity across separable components, and nothing for participants who contribute nothing. It is the canonical fairness benchmark in cooperative game theory and the principal reference point against which any actual allocation rule can be evaluated.
How it works
The Shapley value computes each participant's share by asking, for every possible ordering in which the coalition could have been formed, how much that participant added when they joined. The participant's Shapley value is the average of these marginal contributions across all orderings.
A concrete illustration. Three retirees pool to produce $60,000 of total lifetime income, against $14,000 each on their own and $36,000 for any pair pooling together. With three participants, there are six possible orderings in which the coalition could form. In each ordering, each participant's marginal contribution is calculated — what the coalition could produce after they joined minus what it could produce before they joined. If all three participants are identical, the calculation produces $20,000 each, because by symmetry every participant contributes equally on average. If the three participants differ — say, one has a longer expected lifespan or a different contribution structure — the Shapley value calculation produces unequal shares that reflect each participant's distinct marginal contribution profile.
The Shapley value has four characterizing properties. Efficiency: the shares sum to the total gain available. Symmetry: participants whose marginal contributions are identical across all orderings receive identical shares. Additivity: when a cooperative situation can be decomposed into separable components, each participant's Shapley value in the full situation equals the sum of their Shapley values in the components. Null player: a participant who adds nothing to any coalition they join receives nothing. These four properties together uniquely identify the Shapley value rule — no other allocation rule satisfies all four.
In practice
For an individual considering a pool, the Shapley value is the formal answer to "what would my fair share of the pool's gains be?" Most pool designs do not implement Shapley value allocations directly — the calculation is computationally demanding and not transparent to participants — but the principles the Shapley value embodies (equal treatment of equivalent contributions, full distribution of gain, no reward for null contribution) are relevant criteria for any pool's actual allocation rules. Professionals evaluating a pool's redistribution rule against fairness principles are asking, implicitly, how the actual rule compares to the Shapley benchmark.
In the Longevity Standard Framework
The Shapley value is supporting vocabulary in the Longevity Standard framework, providing the canonical fairness benchmark against which a pool's actual redistribution rules can be evaluated. The four claim properties characterize a pool's structure; they do not specify how pool gains are divided among members. The Shapley value is one principled answer to that distributional question. In LS pool design consulting, the Shapley value is one of several allocation benchmarks against which proposed redistribution rules can be tested — particularly relevant when a pool brings together members of meaningfully different risk classes, where simpler proportional rules can produce allocations that members of one class find unacceptable.
Related terms
- Cooperative game theory
- Core
- Nash equilibrium
- Solidarity principle
- Actuarial fairness
- Pool governance
- Mutualization
- Risk pooling