The {ij}-merged game [...] considers that a pair of players merge or unite in a new player p which lies outside the original set of players. [...] In the {i ▹ j}-amalgamation game, player j delegates his/her role to player i, who belongs to the original set of players.
These lead to the corresponding properties:
A value, f , satisfies 2-merging efficiency if, for every (N,v)∈g and i,j∈N, fp(Np,vp)=fi(N,v)+fj(N,v).
A value, f, satisfies 2-amalgamation efficiency if, for every (N,v)∈g and i,j∈N, fi(Ni⊳j,vi⊳j)=fi(N,v)+fj(N,v).
So basically they’re just saying that players can’t artificially boost or reduce their assigned values by merging or amalgamating—the resulting reward is always just the sum of the individual rewards.
I don’t think it directly applies in the case of your professor and student case. The closest analogue would be if the professor and student were walking as part of a larger group. Then 2-efficiency would say that the student and professor collectively get X credit whether they submit their work under two names or one.
Sorry for the delayed reply. Does that help at all?
Thanks! Late replies are better than no replies ;)
I don’t think this type of efficiency deals with the practical problem of impact credit allocation though! Because there the problem appears to be that it’s difficult to find a common denominator for people’s contributions. You can’t just use man hours, and I don’t think the market value of man hours would do that much better (although it gets in the right direction).
This is the best explanation I could find: Notes on a comment on 2-efficiency and the Banzhaf value.
It describes two different kinds of 2-efficiency:
These lead to the corresponding properties:
So basically they’re just saying that players can’t artificially boost or reduce their assigned values by merging or amalgamating—the resulting reward is always just the sum of the individual rewards.
I don’t think it directly applies in the case of your professor and student case. The closest analogue would be if the professor and student were walking as part of a larger group. Then 2-efficiency would say that the student and professor collectively get X credit whether they submit their work under two names or one.
Sorry for the delayed reply. Does that help at all?
Thanks! Late replies are better than no replies ;)
I don’t think this type of efficiency deals with the practical problem of impact credit allocation though! Because there the problem appears to be that it’s difficult to find a common denominator for people’s contributions. You can’t just use man hours, and I don’t think the market value of man hours would do that much better (although it gets in the right direction).