Adding such virtual players is an interesting idea. I guess it should be possible to properly add probability weighting, without virtual players, once one has fully understood the reason for Shapley’s permutation method of weighting. (Which I have not.)
Another thing I noticed: It seems one can generalize a “coalition” of “players” as a conjunction of propositions, where the propositions of the players not participating in the coalition are negated. So for players/propositions A, B, C, the coalition {A, C} would correspond to the proposition A∧¬B∧C. The Shapley value of a player would simply be the value of a proposition, like A. Since coalitions are just propositions as well, this has the (intuitive?) consequence that coalitions can also be players.
Furthermore, instead of talking about “values” one could talk about “utility”, as in expected utility theory. There is actually a formalization of utility theory, by Richard Jeffrey, which applies to a Boolean algebra of propositions (or “events”, as they are called in probability theory). So theoretically it should be possible to determine which restrictions placed on a Jeffrey utility function are equivalent to Shapley’s formula. Which could, presumably, aid understanding the assumptions behind it, and help with things like integrating probabilities into the Shapley formalism, since those are already integrated into Jeffrey’s theory.
Thanks Cubefox, very interesting ideas, I like the idea of generalising a coalition so that it can be treated as a player, that seems to make a lot of practical sense, I’ll look into Jeffrey to try and get my head around that.
If you want to look into Jeffrey, his utility theory is in his book “The Logic of Decision”, second/1983 edition, sections 4.4 and 5.5 to 5.9. It’s less than 10 pages overall, the rest isn’t really necessary.
Adding such virtual players is an interesting idea. I guess it should be possible to properly add probability weighting, without virtual players, once one has fully understood the reason for Shapley’s permutation method of weighting. (Which I have not.)
Another thing I noticed: It seems one can generalize a “coalition” of “players” as a conjunction of propositions, where the propositions of the players not participating in the coalition are negated. So for players/propositions A, B, C, the coalition {A, C} would correspond to the proposition A∧¬B∧C. The Shapley value of a player would simply be the value of a proposition, like A. Since coalitions are just propositions as well, this has the (intuitive?) consequence that coalitions can also be players.
Furthermore, instead of talking about “values” one could talk about “utility”, as in expected utility theory. There is actually a formalization of utility theory, by Richard Jeffrey, which applies to a Boolean algebra of propositions (or “events”, as they are called in probability theory). So theoretically it should be possible to determine which restrictions placed on a Jeffrey utility function are equivalent to Shapley’s formula. Which could, presumably, aid understanding the assumptions behind it, and help with things like integrating probabilities into the Shapley formalism, since those are already integrated into Jeffrey’s theory.
Of course this would probably be a lot of work...
Thanks Cubefox, very interesting ideas, I like the idea of generalising a coalition so that it can be treated as a player, that seems to make a lot of practical sense, I’ll look into Jeffrey to try and get my head around that.
If you want to look into Jeffrey, his utility theory is in his book “The Logic of Decision”, second/1983 edition, sections 4.4 and 5.5 to 5.9. It’s less than 10 pages overall, the rest isn’t really necessary.
Brilliant, thanks.