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This is: Shapley values: Better than counterfactuals, published by NunoSempere on the Effective Altruism Forum.
[Epistemic status: Pretty confident. But also, enthusiasm on the verge of partisanship]
One intuitive function which assigns impact to agents is the counterfactual, which has the form:
CounterfactualImpact(Agent) = Value(World) - Value(World/Agent)
which reads "The impact of an agent is the difference between the value of the world with the agent and the value of the world without the agent".
It has been discussed in the effective altruism community that this function leads to pitfalls, paradoxes, or to unintuitive results when considering scenarios with multiple stakeholders. See:
Triple counting impact in EA
The counterfactual impact of agents acting in concert
In this post I'll present some new and old examples in which the counterfactual function seems to fail, and how, in each of them, I think that a less known function does better: the Shapley value, a concept from cooperative game theory which has also been brought up before in such discussions. In the first three examples, I'll just present what the Shapley value outputs, and halfway through this post, I'll use these examples to arrive at a definition.
I think that one of the main hindrances in the adoption of Shapley values is the difficulty in its calculation. To solve this, I have written a Shapley value calculator and made it available online: shapleyvalue.com. I encourage you to play around with it.
Example 1 & recap: Sometimes, the counterfactual impact exceeds the total value.
Suppose there are three possible outcomes: P has cost $2000 and gives 15 utility to the world Q has cost $1000 and gives 10 utility to the world R has cost $1000 and gives 10 utility to the world
Suppose Alice and Bob each have $1000 to donate. Consider two scenarios:
Scenario 1: Both Alice and Bob give $1000 to P. The world gets 15 more utility. Both Alice and Bob are counterfactually responsible for giving 15 utility to the world.
Scenario 2: Alice gives $1000 to Q and Bob gives $1000 to R. The world gets 20 more utility. Both Alice and Bob are counterfactually responsible for giving 10 utility to the world.
From the world's perspective, scenario 2 is better. However, from Alice and Bob's individual perspective (if they are maximizing their own counterfactual impact), scenario 1 is better. This seems wrong, we'd want to somehow coordinate so that we achieve scenario 2 instead of scenario 1.
Source
Attribution: rohinmshah
In Scenario 1:
Counterfactual impact of Alice: 15 utility.
Counterfactual impact of Bob: 15 utility.
Sum of the counterfactual impacts: 30 utility. Total impact: 15 utility.
The Shapley value of Alice would be: 7.5 utility.
The Shapley value of Bob would be: 7.5 utility.
The sum of the Shapley values always adds up to the total impact, which is 15 utility.
In Scenario 2:
Counterfactual impact of Alice: 10 utility.
Counterfactual impact of Bob: 10 utility.
Sum of the counterfactual impacts: 20 utility. Total impact: 20 utility.
The Shapley value of Alice would be: 10 utility.
The Shapley value of Bob would be: 10 utility.
The sum of the Shapley values always adds up to the total impact, which is 10+10 utility = 20 utility.
In this case, if Alice and Bob were each individually optimizing for counterfactual impact, they'd end up with a total impact of 15. If they were, each of them, individually, optimizing for the Shapley value, they'd end up with a total impact of 20, which is higher.
It would seem that we could use a function such as
CounterfactualImpactModified = CounterfactualImpact / NumberOfStakeholders
to solve this particular problem. However, as the next example shows, that sometimes doesn't work. The Shapley value, on the other hand, has the property that it always adds up to total value.
Property 1: T...
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