What numerator and denominator? I am imagining that a single person could be a player in multiple cooperative games. The Shapley value for the person would be finite in each game, but if there are infinitely many games, the sum of all the Shapley values (adding across all games, not adding across all players in a single game) could be infinite.
Hmm I would guess that the number of realistic cooperative games in the world to grow ~linearly (or some approximation[1]) with the number of people in the world, hence the denominator.
[1] I suppose if you think the growth is highly superlinear and there are ~infinity people, than Shapley values can grow to be ~infinite? But this feels like a general problem with infinities and not specific to Shapleys.
I asked my question because the problem with infinities seems unique to Shapley values (e.g. I don’t have this same confusion about the concept of “marginal value added”). Even with a small population, the number of cooperative games seems infinite: for example, there are an infinite number of mathematical theorems that could be proven, an infinite number of Wikipedia articles that could be written, an infinite number of films that could be made, etc. If we just use “marginal value added”, the total value any single person adds is finite across all such cooperative games because in the actual world, they can only do finitely many things. But the Shapley value doesn’t look at just the “actual world”, it seems to look at all possible sequences of ways of adding people to the grand coalition and then averages the value, so people get non-zero Shapley value assigned to them even if they didn’t do anything in the “actual world”.
(There’s maybe some sort of “compactness” argument one could make that even if there are infinitely many games, in the real world only finitely many of them get played to completion and so this should restrict the total Shapley value any single person can get, but I’m just trying to go by the official definition for now.)
What numerator and denominator? I am imagining that a single person could be a player in multiple cooperative games. The Shapley value for the person would be finite in each game, but if there are infinitely many games, the sum of all the Shapley values (adding across all games, not adding across all players in a single game) could be infinite.
Hmm I would guess that the number of realistic cooperative games in the world to grow ~linearly (or some approximation[1]) with the number of people in the world, hence the denominator.
[1] I suppose if you think the growth is highly superlinear and there are ~infinity people, than Shapley values can grow to be ~infinite? But this feels like a general problem with infinities and not specific to Shapleys.
I asked my question because the problem with infinities seems unique to Shapley values (e.g. I don’t have this same confusion about the concept of “marginal value added”). Even with a small population, the number of cooperative games seems infinite: for example, there are an infinite number of mathematical theorems that could be proven, an infinite number of Wikipedia articles that could be written, an infinite number of films that could be made, etc. If we just use “marginal value added”, the total value any single person adds is finite across all such cooperative games because in the actual world, they can only do finitely many things. But the Shapley value doesn’t look at just the “actual world”, it seems to look at all possible sequences of ways of adding people to the grand coalition and then averages the value, so people get non-zero Shapley value assigned to them even if they didn’t do anything in the “actual world”.
(There’s maybe some sort of “compactness” argument one could make that even if there are infinitely many games, in the real world only finitely many of them get played to completion and so this should restrict the total Shapley value any single person can get, but I’m just trying to go by the official definition for now.)