In a matching market, envy is the awareness that someone else got a better assignment than you. The question is how envy distributes across participants.
Ortega, Ziegler, Arribillaga, and Zhao (arXiv:2603.23385) study the Deferred Acceptance algorithm in random markets. The key quantity: how many proposing agents are unenvied — assigned something that nobody else wanted more? The answer connects to the coupon collector problem, yielding an exact finite-market expression.
The surprise is in the comparison. Random Serial Dictatorship (RSD) lets agents pick in random order — the first picks their favorite, the second picks from what's left, and so on. A constant fraction of agents get their top choice. DA, by contrast, produces stable matchings through iterated proposals and rejections. Despite their different structures and different properties, both algorithms leave exactly H_n (the n-th harmonic number) proposing agents unenvied in expectation.
H_n grows logarithmically. As markets scale, the unenvied agents become a vanishing fraction. In a market of a million participants, only about 14 are unenvied — each assigned something so mediocre that no one else covets it. The rest envy someone. The algorithm distributes envy almost universally.
The implication for mechanism design: the most advantaged participants are not the ones everyone else notices. They comprise an increasingly negligible share of the market. DA's famed stability property — no pair prefers each other to their assigned partners — constrains outcomes enough that almost no one ends up in an envy-free position. Stability and envy-freedom pull in the same direction: both reduce extreme outcomes. But they don't eliminate envy. They distribute it.
The coupon collector connection is elegant. The unenvied agents are like the last few coupons — the ones no one has collected. In a random matching market, the “coupon” is an assignment that happens to be no one's preferred improvement. The logarithmic count of such assignments is the same log that governs the coupon collector's waiting time. The structure of envy in matching markets inherits the structure of coverage in collecting.