I agree that this is unintuitive. Personally the part of that that I like less is that it feels like people could cheat that, by standing in line.
But they can’t cheat it! See this example: <http://shapleyvalue.com/?example=4>. You can’t even cheat by noticing that something is impactful, and then self-modifying so that in the worlds where you were needed you would do it, because in the worlds where you would be needed, you wouldn’t have done that modification (though there are some nuances here, like if you self-modify and there is some chance that you are needed in the future).
Not sure if that addresses part of what you were asking about.
I agree that SV’s don’t play nice with infinities, though I’m not sure whether there could be an extension which could (for instance, looking at the limit of the Shapley value).
I don’t think the example you give addresses my point. I am supposing that Leibniz could have also invented calculus, so v({2})=100. But Leibniz could have also invented lots of different things (infinitely many things!), and his claim to each invention would be valid (although in the real world he only invents finitely many things). If each invention is worth at least a unit of value, his Shapley value across all inventions would be infinite, even if Leibniz was “maximally unluckly” and in the actual world got scooped every single time and so did not invent anything at all.
I don’t understand the part about self-modifications—can you spell it out in more words/maybe give an example?
his Shapley value across all inventions would be infinite
Assuming an infinite number of players. If there are only a finite number of players, there are only finite terms in the Shapley value calculation, and if each invention has finite value, that’s finite.
The Shapley value is one way to distribute the total gains to the players, assuming that they all collaborate.
This means that there must be gains to distribute for anyone to get nonzero credit from that game, and that they in fact “collaborated” (although this could be in name only) to get any credit at all. Ignoring multiverses, infinitely many things have not been invented yet, but maybe infinitely many things will be invented in the future. In general, I don’t think that Leibniz cooperated in infinitely many games, or even that infinitely many games have been played so far, unless you define games with lots of overlap and double counting (or you invoke multiverses, or consider infinitely long futures, or some exotic possibilities, and then infinite credit doesn’t seem unreasonable).
Furthermore, in all but a small number of games, he might make no difference to each coalition even when he cooperates, so get no credit at all. Or the credit could decrease fast enough to have a finite sum, even if he got nonzero credit in infinitely many games, as it becomes vanishingly unlikely that he would have made any difference even in worlds where he cooperates.
I agree that this is unintuitive. Personally the part of that that I like less is that it feels like people could cheat that, by standing in line.
But they can’t cheat it! See this example: <http://shapleyvalue.com/?example=4>. You can’t even cheat by noticing that something is impactful, and then self-modifying so that in the worlds where you were needed you would do it, because in the worlds where you would be needed, you wouldn’t have done that modification (though there are some nuances here, like if you self-modify and there is some chance that you are needed in the future).
Not sure if that addresses part of what you were asking about.
I agree that SV’s don’t play nice with infinities, though I’m not sure whether there could be an extension which could (for instance, looking at the limit of the Shapley value).
I don’t think the example you give addresses my point. I am supposing that Leibniz could have also invented calculus, so v({2})=100. But Leibniz could have also invented lots of different things (infinitely many things!), and his claim to each invention would be valid (although in the real world he only invents finitely many things). If each invention is worth at least a unit of value, his Shapley value across all inventions would be infinite, even if Leibniz was “maximally unluckly” and in the actual world got scooped every single time and so did not invent anything at all.
I don’t understand the part about self-modifications—can you spell it out in more words/maybe give an example?
Assuming an infinite number of players. If there are only a finite number of players, there are only finite terms in the Shapley value calculation, and if each invention has finite value, that’s finite.
The Wikipedia page says:
This means that there must be gains to distribute for anyone to get nonzero credit from that game, and that they in fact “collaborated” (although this could be in name only) to get any credit at all. Ignoring multiverses, infinitely many things have not been invented yet, but maybe infinitely many things will be invented in the future. In general, I don’t think that Leibniz cooperated in infinitely many games, or even that infinitely many games have been played so far, unless you define games with lots of overlap and double counting (or you invoke multiverses, or consider infinitely long futures, or some exotic possibilities, and then infinite credit doesn’t seem unreasonable).
Furthermore, in all but a small number of games, he might make no difference to each coalition even when he cooperates, so get no credit at all. Or the credit could decrease fast enough to have a finite sum, even if he got nonzero credit in infinitely many games, as it becomes vanishingly unlikely that he would have made any difference even in worlds where he cooperates.