If all parents agree that school A is better than B, but parent 1 cares much more about A>B than parent 2 does, then the sum-of-utilities is different (so, not “zero sum”) depending on whether [ 1→A; 2→B ] or [ 1→B; 2→A ]. Every change in outcomes leads to someone losing (compared to the counterfactual), but the payoffs aren’t zero-sum.
That example is kind of useless, but if you have three parents and three schools (and even if parents agree on order), but each of the parents care about A>B and B>C in different ratios, then you can use that fact to engineer a lottery where all three parents are better off than if you assigned them to schools uniformly at random. (Sketch of construction: Start with equal probabilities, and let parents trade some percentage of “upgrade C to B” for some (different?) percentage of “upgrade B to A” with each other. If they have different ratios of their A>B and B>C preferences, positive-sum trades exist.)
Then, in theory, a set of parents cooperating could implement this lottery on their own and agree to apply just to their lottery-assigned school, and if they don’t defect in the prisoners’ dilemma then they all benefit. Not zero-sum.
Of course, it can also be the case that they value different schools different amounts and a bad mechanism can lead to an inefficient allocation (where pairs would just be better off switching), and I could construct such an example if this margin weren’t too narrow to contain it.
It is separately the case that if the administrators have meta-preferences over what parents’ preferences get satisfied, then they can make a choice of mechanisms (“play the metagame”, as you put it) that give better / worse / differently-distributed results with respect to their meta-preferences.
You shouldn’t take my claims on argument-from-authority alone, but it might help you have better priors about whether I’m right to know that I’ve published traditional-academic work in the specific field of matching theory.
With respect, I think that’s wrong.
If all parents agree that school A is better than B, but parent 1 cares much more about A>B than parent 2 does, then the sum-of-utilities is different (so, not “zero sum”) depending on whether [ 1→A; 2→B ] or [ 1→B; 2→A ]. Every change in outcomes leads to someone losing (compared to the counterfactual), but the payoffs aren’t zero-sum.
That example is kind of useless, but if you have three parents and three schools (and even if parents agree on order), but each of the parents care about A>B and B>C in different ratios, then you can use that fact to engineer a lottery where all three parents are better off than if you assigned them to schools uniformly at random. (Sketch of construction: Start with equal probabilities, and let parents trade some percentage of “upgrade C to B” for some (different?) percentage of “upgrade B to A” with each other. If they have different ratios of their A>B and B>C preferences, positive-sum trades exist.)
Then, in theory, a set of parents cooperating could implement this lottery on their own and agree to apply just to their lottery-assigned school, and if they don’t defect in the prisoners’ dilemma then they all benefit. Not zero-sum.
Of course, it can also be the case that they value different schools different amounts and a bad mechanism can lead to an inefficient allocation (where pairs would just be better off switching), and I could construct such an example if this margin weren’t too narrow to contain it.
It is separately the case that if the administrators have meta-preferences over what parents’ preferences get satisfied, then they can make a choice of mechanisms (“play the metagame”, as you put it) that give better / worse / differently-distributed results with respect to their meta-preferences.
You shouldn’t take my claims on argument-from-authority alone, but it might help you have better priors about whether I’m right to know that I’ve published traditional-academic work in the specific field of matching theory.
(Also in matching with monetary transfers: 1; 2)