In the context of EA, Shapley values are a method for assigning credit for the impact of an intervention to each of a set of actors collaborating to make it happen. The concept of a Shapley value comes from cooperative game theory, where it is a general solution to the problem of distributing gains from cooperation. In impact assessment, it is often compared to the calculation of the raw counterfactual impact of an intervention.

Loosely, each Shapley value (each associated with a specific actor) corresponds to the average of the counterfactual impact produced by the actor, considering each possible way in which the agents could have assembled themselves to cooperate. Formally, we define a coalition, that is a potential set of actors cooperating with each other, and define each actor's Shapley value as^[1]:

φ_{i} (v) = \frac{1}{number of actors} \sum coalitions including i \frac{counterfactual impact of i under this coalition}{number of coalitions excluding i of this size}

This solution to the problem is known to fulfill some desirable properties, including that the sum of the values adds up to the total value provided, that agents that contribute the same amount get the same amount of credit, and that it doesn't matter in which order actors choose to collaborate.

Furthermore, when used to assess the impact of projects, Shapley values can be helpful as adjustment factors to prevent coordination issues arising from multiple actors acting based on the estimated counterfactual impact of their contributions. Not using these adjustments can lead to, for example, the double counting of impact, because actors fail to consider what would happen if other actors didn't cooperate.

Further reading

Sempere, Nuño (2019) Shapley values: Better than counterfactuals, Effective Altruism Forum, October 10.

Pinsent, Stan (2023) Shapley values: an introductory example, Effective Altruism Forum, November 2023.

Sempere, Nuño (2020) Shapley Values II: Philantropic Coordination Theory & other miscellanea, Effective Altruism Forum, March 10.

Related entries

^{^}
Or using the notation of so-called coalitional games and permutations:
$φ_{i} (v) = \frac{1}{n} \sum S \subseteq N ∖ {i} \frac{(v (S \cup {i}) - v (S))}{(\frac{n - 1}{n - | S | - 1})}$
Where $v (S)$ denotes the worth of coalition $S$ , that is, the total expected value

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