I think the linearity property holds if the two value/payoff functions themselves can be added (because Shapley values are linear combinations of the value/payoff functions’ values with fixed coefficients for fixed sets of players), but usually not otherwise. Also, I think this would generally assume a common set of players, and that a player cooperates in one game iff they cooperate in the other, so that we can use (v+w)(S)=v(S)+w(S).
I think there’s the same problem that motivated the use of Shapley values in the first place. Just imagine multiple decisions one individual makes as part of 3 separate corresponding games:
Doing the basics to avoid dying, like eating, not walking into traffic (and then working, earning money and donating some of it)
Working and earning money (to donate, where and how much to work)
Donating (how much to donate, and optionally also where)
Let’s assume earning-to-give only with low impact directly from each job option.
1 and 2 get their value from eventually donating, which is the decision made in 3, but you’d already fully count the value of your donations in 3, so you shouldn’t also count it in 1 or 2. These can also be broken down into further separate games. It doesn’t matter for your donations if you avoid dying now if you die soon after before getting to donate. You won’t get to donate more if you do 1 more minute of work in your job before quitting instead of quitting immediately.
I think people wouldn’t generally make the mistake of treating these as separate games to sum value across, because the decisions are too fine-grained and because the dependence is obvious. Even if they were earning money to donate from impactful direct work, they still wouldn’t accidentally double count their earnings/donations, because they wouldn’t represent that with multiple games.
A similar example that I think could catch someone would be someone who is both a grant advisor and doing separate fundraising work that isn’t specific to their grants but raises more money for them to grant, anyway. For example, they’re both a grant advisor for an EA Fund, and do outreach for GWWC. If they treat these as separate coalition games they’re playing, there’s a risk that they’ll double count additional money that’s been raised through GWWC and was granted on their recommendation (or otherwise affected by their grantmaking counterfactually). Maybe assume that if they don’t make grant recommendations soon, there’s a greater risk the extra funds aren’t useful at all (or are much much less useful), e.g. the extra funding is granted prioritizing other things over potential impact, the funds are misappropriated, or we go extinct. So, they’re directly or indirectly counting extra funding in both games. This seems harder to catch, because the relationship between the two games isn’t as obvious, and they’re both big natural decisions to consider.
Another example: calculus was useful to a huge number of later developments. Leibniz “cooperated” in the calculus-inventing game, but we might say he also cooperated in many later games that depended on calculus, but any value we’d credit him with generated in those later games should already be fully counted in the credit he gets in the calculus-inventing game.
There are also more degenerate cases, like two identical instances of the same game, or artificial modifications, e.g. adding and excluding different players (but counting their contributions anyway, just not giving them credit in all games).
I think the linearity property holds if the two value/payoff functions themselves can be added (because Shapley values are linear combinations of the value/payoff functions’ values with fixed coefficients for fixed sets of players), but usually not otherwise. Also, I think this would generally assume a common set of players, and that a player cooperates in one game iff they cooperate in the other, so that we can use (v+w)(S)=v(S)+w(S).
I think there’s the same problem that motivated the use of Shapley values in the first place. Just imagine multiple decisions one individual makes as part of 3 separate corresponding games:
Doing the basics to avoid dying, like eating, not walking into traffic (and then working, earning money and donating some of it)
Working and earning money (to donate, where and how much to work)
Donating (how much to donate, and optionally also where)
Let’s assume earning-to-give only with low impact directly from each job option.
1 and 2 get their value from eventually donating, which is the decision made in 3, but you’d already fully count the value of your donations in 3, so you shouldn’t also count it in 1 or 2. These can also be broken down into further separate games. It doesn’t matter for your donations if you avoid dying now if you die soon after before getting to donate. You won’t get to donate more if you do 1 more minute of work in your job before quitting instead of quitting immediately.
I think people wouldn’t generally make the mistake of treating these as separate games to sum value across, because the decisions are too fine-grained and because the dependence is obvious. Even if they were earning money to donate from impactful direct work, they still wouldn’t accidentally double count their earnings/donations, because they wouldn’t represent that with multiple games.
A similar example that I think could catch someone would be someone who is both a grant advisor and doing separate fundraising work that isn’t specific to their grants but raises more money for them to grant, anyway. For example, they’re both a grant advisor for an EA Fund, and do outreach for GWWC. If they treat these as separate coalition games they’re playing, there’s a risk that they’ll double count additional money that’s been raised through GWWC and was granted on their recommendation (or otherwise affected by their grantmaking counterfactually). Maybe assume that if they don’t make grant recommendations soon, there’s a greater risk the extra funds aren’t useful at all (or are much much less useful), e.g. the extra funding is granted prioritizing other things over potential impact, the funds are misappropriated, or we go extinct. So, they’re directly or indirectly counting extra funding in both games. This seems harder to catch, because the relationship between the two games isn’t as obvious, and they’re both big natural decisions to consider.
Another example: calculus was useful to a huge number of later developments. Leibniz “cooperated” in the calculus-inventing game, but we might say he also cooperated in many later games that depended on calculus, but any value we’d credit him with generated in those later games should already be fully counted in the credit he gets in the calculus-inventing game.
There are also more degenerate cases, like two identical instances of the same game, or artificial modifications, e.g. adding and excluding different players (but counting their contributions anyway, just not giving them credit in all games).