We consider a revenue-maximizing seller with n items facing a single buyer. We introduce the notion of symmetric menu complexity of a mechanism, which counts the number of distinct options the buyer may purchase, up to permutations of the items. Our main result is that a mechanism of quasi-polynomial symmetric menu complexity suffices to guarantee a (1−ε)-approximation when the buyer is unit-demand over independent items, even when the value distribution is unbounded, and that this mechanism can be found in quasi-polynomial time.
Our key technical result is a polynomial time, (symmetric) menu-complexity-preserving black-box reduction from achieving a (1−ε)-approximation for unbounded valuations that are subadditive over independent items to achieving a (1−O(ε))-approximation when the values are bounded (and still subadditive over independent items). We further apply this reduction to deduce approximation schemes for a suite of valuation classes beyond our main result.
Finally, we show that selling separately (which has exponential menu complexity) can be approximated up to a (1−ε) factor with a menu of efficient-linear (f(ε)⋅n) symmetric menu complexity.