New Math Trick Makes Online Shopping Prices Fairer and Faster

Researchers have introduced a unified potential method for contextual search and bilateral trade with one-bit feedback, obtaining order-tight algorithms for both problems. Their framework starts from a Bayesian model in which the unknown target, or the unknown pair of valuations, is sampled from a known prior; the decision maker bases predictions on the posterior distribution conditioned on previous feedback.

For Bayesian contextual search, they give an algorithm with total loss bounded by O(Tr sqrt Σ). For Bayesian contextual bilateral trade, regret is bounded by O(Tr sqrt Σ_S + Tr sqrt Σ_B).

They then adapt a dual trace-square-root potential to the classical worst-case setting, where the target and valuations are chosen adversarially from the unit Euclidean ball, obtaining tight O(sqrt d)-regret algorithms for both. These results match existing lower bounds up to a constant factor and improve previous guarantees of O(d log d) for contextual search and O(d6^d) for bilateral trade.