No, I think it’s a classic scarce resource allocation, and among the parents it’s a zero-sum game in the stronger technical sense too. It’s the metagame the administrators play of choosing the game the parents play where it’s possible to do better, where the “zero” the parents compete around can be increased.
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.
No, I think it’s a classic scarce resource allocation, and among the parents it’s a zero-sum game in the stronger technical sense too. It’s the metagame the administrators play of choosing the game the parents play where it’s possible to do better, where the “zero” the parents compete around can be increased.
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)