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.