Mathematicians Finally Know How Much Money Fixed Prices Quietly Lose
One set price throws away up to 26 cents of every dollar of a possible deal.
Mathematicians have pinned down exactly how much value one posted price leaves on the table. When a seller and a buyer with independent private values can trade only at a posted price, the best posted price always guarantees a 0.738024... fraction of first-best welfare — where that constant is the root of an explicit equation. The worst-case buyer is unique up to scaling, and no pair of distributions attains the worst case. The result closes the [0.7292, 0.73805] gap left by work running from SODA 2016 through two STOC 2023 papers and AAAI 2026. The proof is an explicit certificate of optimality: after one change of variables, the gap between the optimal value and that of any buyer distribution is a sum of nonnegative integrals. The same constant is also the exact guarantee of dominant-strategy mechanisms with individual rationality and strong budget balance in every realization — and for two units with increasing submodular valuations, an explicit finite instance falls below 0.7290804, so multi-unit trade is strictly harder than single-unit trade.
- Best single posted price captures 73.8% of the ideal deal value — about a quarter lost every time.
- The result closes a gap economists have argued over since a 2016 conference paper.
- Selling two items at once is provably harder, with a specific example scoring below 0.729.
Why It Matters
It puts a number on the hidden cost of one-price-fits-all shopping, ticketing and resale markets.