Determine the balanced sequence that maximizes adaptive profit

Determine, among all 1-balanced sequences of sellers and buyers containing m buyers and m sellers (each agent trades a single identical item; seller values are i.i.d. from a log-concave distribution F_S and buyer values are i.i.d. from a monotone hazard rate distribution F_B), the ordering of sellers and buyers that maximizes the expected profit of the optimal adaptive posted-price mechanism.

Background

In the paper’s study of α-balanced sequences, the authors analyze profit for online posted-price mechanisms and compare adaptive and non-adaptive strategies. They show that using prices derived from an optimal fractional relaxation yields an online mechanism that is asymptotically competitive.

Within this context, they observe that, among 1-balanced sequences, moving some buyers earlier can slightly improve adaptive profit compared to the canonical sequence S^mB^m. Although their results show this difference is asymptotically insignificant, they explicitly note that precisely identifying which balanced sequence achieves the maximum profit remains unresolved.