Power of Second Opportunity: Dynamic Pricing with Second Chance

In this paper, we consider the following dynamic pricing problem. Suppose the market price $v_{t}$ of an item arriving at time $t$ is determined by $v_{t}=\pmb{\theta}^{\mathrm{T}}\pmb{x}_{t}$, where $\pmb{x}_{t}$ is the feature vector of that item and $\pmb{\theta}$ is an unknown vector parameter. The seller has to post prices without knowing $\pmb{\theta}$ such that the total regret in time span $T$ is minimized. Considering real-world scenarios in which people may negotiate prices, we propose a model called Second Chance Pricing, in which a seller has a second opportunity to post a price after the first offer is declined. Theoretical analysis shows that a second chance of pricing results in a total regret between $o(\frac{\ln T}{n\ln n}+\frac{1}{n})$ and $O(n^{2}\ln T)$, where $n$ is the dimension of the feature space. Experiments on both synthetic data and real data demonstrate significant benefits brought about by the second chance where the regret is only 13% of that of one chance.