As I said I’m skating on thin ice, but the theorem says you can convert any positive or negative sum game into a zero sum game. (its due to von Neumann or nash, but i think i saw it in books on evolutionary game theory . i think there are analogs in physics , and even ecology, etc. ).
Again, i think that may be related to the counterfactual/shapley conversion i ‘see’ or think exists, but can’t prove it----i’d have to look at the definitions again.
To possibly fall through more holes in the ice , i think the prisoner’s dillema might be the simplest example.
(I’m just not fluent in the definitions since i didn’t learn them when i was studying some game theory; but i looked at many game theory texts where they did occur—mostly for more complex situations than i was dealing with.
Also the term ‘counterfactual’ i only learned from a history book by Niall Ferguson (not a big hero of mine but had what seemed like worthwhile ideas--- he wrote ‘counterfactual history’—eg ‘what would be state of the world if Germany had won WW2?’ )
as noted , i also find examples which use ‘vignettes’ or ‘scenarios’, fractions, whole numbers like ‘7 EA candidates’, ’60 million$ ′ , along with the names of countries (India) and organizations, make it difficult (or time consuming for me) to process. but this is just a stylisitic or personal issue.
I wonder if you think an excercize trying to compare the shapley vs counterfactual value of the 2 cases for WW2 is meaningful—ie would money spent by UK/USA/etc fighting the war have been better spent another way?
i may even put this question to myself to see if its meaningful in your framework. i spend a bit of time on questionable math/logic problems (some of which have solutions, but i try to find different proofs because i dont understand the existing ones, and occasionaly do. Many theorems have many correct proofs which look very different and use different methods, and often have been discovered by many people on different continents at the same time (eg the renormalization group in physics was discovered by Feynman and Nambu (japan) about the same time) . I wish i had a study group who shared my interests in various problems like this one; the few aquaintances i have who work on math/logic basically work on problems that interest them, and don’t find mine interesting or relevant. )
P.S. I just re-skimmed your article and see you dealt in Scenario 6 with ‘tragedy of the commons’ which i view as an n-person variant of the 2 -person prisoner’s dillema.
also your example 2 (Newton and Leibniz ) is an example which is sort of what i was thinking. The theorem i was thinking of would add to the picture and have something like a ‘god’ who would create either Newton, Leibniz, or both of them. Shapley value would be the same in all cases. (unless 2 calculus discoveries are better than 1----in sciences sometimes this is seen as true (‘replication’), or having ‘multiple witnesses’ in law as opposed to just an account by one (who is the victim and may not be believed )).
(its also the case for example that the 3 or 4 or even 5 early versions of quantum mechanics—schrodinger, heisenberg, dirac, feynman, bohm—though some say debroglie anticipated bohm , and feynman acknolwedged that he found his idea in a footnote in a book by Dirac—although redundant in many ways, each have unique perspectives . the golden rule also has many formulations i’ve heard)
(In my scenario, with ‘god’ , i think the counterfactual value of either newton or leibniz would be 1---because without either or both there would be no calculus with shapley value 1. god could have just created nothing---0 rather than 1).
In a way what you seem to be describing is how to avoid the ‘neglectedness’ problem of EA theory. This overlaps with questions in politics—some people vote for people in a major party who may win anyway, rather than vote for a ‘minor party’ they may actually agree with more. This might be called the ‘glow effect’—similarily some people will support some rock or sports star partly just to be in the ‘in crowd’. So they get ‘counterfactual value’ even if the world is no better off-voting for someone who will win any way is no better than voting for one who will lose—or rather they actually get additional Shapley value because they are ‘happier’ being in the ‘in crowd’ rather than being a less favored minority—but this involves a different calculation for the Shapley value, including ‘happiness’ and not just ‘who won’. But, some people are happier being in ‘minorities’, so thats another complication in the calculations.
(eg the song by Beck ‘i’m a loser’ comes to mind. pays to be a loser some times or support an unpopular cause because its actually a neglected one—people just didn’t know its actual or Shapley value. )
would surprise me; can you think of a source?
As I said I’m skating on thin ice, but the theorem says you can convert any positive or negative sum game into a zero sum game. (its due to von Neumann or nash, but i think i saw it in books on evolutionary game theory . i think there are analogs in physics , and even ecology, etc. ).
Again, i think that may be related to the counterfactual/shapley conversion i ‘see’ or think exists, but can’t prove it----i’d have to look at the definitions again.
To possibly fall through more holes in the ice , i think the prisoner’s dillema might be the simplest example.
(I’m just not fluent in the definitions since i didn’t learn them when i was studying some game theory; but i looked at many game theory texts where they did occur—mostly for more complex situations than i was dealing with.
Also the term ‘counterfactual’ i only learned from a history book by Niall Ferguson (not a big hero of mine but had what seemed like worthwhile ideas--- he wrote ‘counterfactual history’—eg ‘what would be state of the world if Germany had won WW2?’ )
as noted , i also find examples which use ‘vignettes’ or ‘scenarios’, fractions, whole numbers like ‘7 EA candidates’, ’60 million$ ′ , along with the names of countries (India) and organizations, make it difficult (or time consuming for me) to process. but this is just a stylisitic or personal issue.
I wonder if you think an excercize trying to compare the shapley vs counterfactual value of the 2 cases for WW2 is meaningful—ie would money spent by UK/USA/etc fighting the war have been better spent another way?
i may even put this question to myself to see if its meaningful in your framework. i spend a bit of time on questionable math/logic problems (some of which have solutions, but i try to find different proofs because i dont understand the existing ones, and occasionaly do. Many theorems have many correct proofs which look very different and use different methods, and often have been discovered by many people on different continents at the same time (eg the renormalization group in physics was discovered by Feynman and Nambu (japan) about the same time) . I wish i had a study group who shared my interests in various problems like this one; the few aquaintances i have who work on math/logic basically work on problems that interest them, and don’t find mine interesting or relevant. )
P.S. I just re-skimmed your article and see you dealt in Scenario 6 with ‘tragedy of the commons’ which i view as an n-person variant of the 2 -person prisoner’s dillema.
also your example 2 (Newton and Leibniz ) is an example which is sort of what i was thinking. The theorem i was thinking of would add to the picture and have something like a ‘god’ who would create either Newton, Leibniz, or both of them. Shapley value would be the same in all cases. (unless 2 calculus discoveries are better than 1----in sciences sometimes this is seen as true (‘replication’), or having ‘multiple witnesses’ in law as opposed to just an account by one (who is the victim and may not be believed )).
(its also the case for example that the 3 or 4 or even 5 early versions of quantum mechanics—schrodinger, heisenberg, dirac, feynman, bohm—though some say debroglie anticipated bohm , and feynman acknolwedged that he found his idea in a footnote in a book by Dirac—although redundant in many ways, each have unique perspectives . the golden rule also has many formulations i’ve heard)
(In my scenario, with ‘god’ , i think the counterfactual value of either newton or leibniz would be 1---because without either or both there would be no calculus with shapley value 1. god could have just created nothing---0 rather than 1).
In a way what you seem to be describing is how to avoid the ‘neglectedness’ problem of EA theory. This overlaps with questions in politics—some people vote for people in a major party who may win anyway, rather than vote for a ‘minor party’ they may actually agree with more. This might be called the ‘glow effect’—similarily some people will support some rock or sports star partly just to be in the ‘in crowd’. So they get ‘counterfactual value’ even if the world is no better off-voting for someone who will win any way is no better than voting for one who will lose—or rather they actually get additional Shapley value because they are ‘happier’ being in the ‘in crowd’ rather than being a less favored minority—but this involves a different calculation for the Shapley value, including ‘happiness’ and not just ‘who won’. But, some people are happier being in ‘minorities’, so thats another complication in the calculations.
(eg the song by Beck ‘i’m a loser’ comes to mind. pays to be a loser some times or support an unpopular cause because its actually a neglected one—people just didn’t know its actual or Shapley value. )