Only in the short run it isn't. On the long run, school trading creates incentives which make the entire system worse.
So you're not just optimizing for a single round of students, you're optimizing for the future rounds of students. And pareto-optimal makes no sense when you're looking at future rounds with different students.
But you could create a pareto-optimal analogy: allowing two students now to trade place in a seemingly pareto-optimal way would actually harm other students when you look at the future rounds, so the trade actually isn't pareto-optimal.
It isn't Pareto-optimal in the long run either. Pareto-optimal doesn't mean "optimal" or perfect. It just means there is a way to increase the satisfaction of someone without decreasing the satisfaction of someone else.
It's not even designed to deal with repeat situation, incentives and strategizing. It just looks at the outcome and says: is this optimal?
> It's not even designed to deal with repeat situation, incentives and strategizing. It just looks at the outcome and says: is this optimal?
That's exactly what I'm saying:
> And pareto-optimal makes no sense when you're looking at future rounds with different students.
And that's why I talked about a time-series analogy of pareto-optimality:
> But you could create a pareto-optimal analogy: allowing two students now to trade place in a seemingly pareto-optimal way would actually harm other students when you look at the future rounds, so the trade actually isn't pareto-optimal.
In that sense, you don't just look at the students selecting now, but at the future students as well. In that case, allowing trades now leads to dissatisfaction in the future, so while allowing trades now might seem to lead to a pareto-optimal scenario, it doesn't lead to a time-series-pareto-optimal scenario.
I think he's saying that the error is in the original assignment algorithm: it produces a solution that is not Parto-optimal. If it was, there would be no mutually advantageous trades. One way it could achieve that is by randomly doing mutually improving trades until none are left.
But you can't consider just the current round of students when designing such a system, you need to consider all subsequent rounds.
In that case, allowing trades now leads to dissatisfaction in the future. So, while allowing trades now might seem to lead to a pareto-optimal scenario, it doesn't lead to a time-series-pareto-optimal scenario.
The alternate system I described doesn't involve "allowing trades" and therefore can't create perverse incentives or affect subsequent rounds.
The idea is to make an algorithm that performs a Pareto-optimal assignment algorithmically, so that no possible mutually advantageous trades exist when it is done. It's totally possible for an algorithm to do that, and the current Dutch one clearly doesn't.
One possible way is to run the current algorithm, and then as long as the solution is not Pareto-optimal, perform a swap at random from the set of all pairs of students where swapping would improve both of their preferences. This can be proven to converge with sufficient steps. Doing this kind of swap algorithmically has nothing to do with "allowing trades". I'm calling them "swaps" instead of "trades" here to make clear there is no manual action involved on the part of the students.
I hope this clarifies what I meant, and why objecting to "allowing trades" does not make sense as a response.
Only in the short run it isn't. On the long run, school trading creates incentives which make the entire system worse.
So you're not just optimizing for a single round of students, you're optimizing for the future rounds of students. And pareto-optimal makes no sense when you're looking at future rounds with different students.
But you could create a pareto-optimal analogy: allowing two students now to trade place in a seemingly pareto-optimal way would actually harm other students when you look at the future rounds, so the trade actually isn't pareto-optimal.
So it isn't an error, it's working as intended.