5 research outputs found
Dynamic Pricing with Finitely Many Unknown Valuations
Motivated by posted price auctions where buyers are grouped in an unknown
number of latent types characterized by their private values for the good on
sale, we investigate revenue maximization in stochastic dynamic pricing when
the distribution of buyers' private values is supported on an unknown set of
points in [0,1] of unknown cardinality . This setting can be viewed as an
instance of a stochastic -armed bandit problem where the location of the
arms (the unknown valuations) must be learned as well. In the
distribution-free case, we prove that our setting is just as hard as -armed
stochastic bandits: no algorithm can achieve a regret significantly better than
, (where T is the time horizon); we present an efficient algorithm
matching this lower bound up to logarithmic factors. In the
distribution-dependent case, we show that for all our setting is strictly
harder than -armed stochastic bandits by proving that it is impossible to
obtain regret bounds that grow logarithmically in time or slower. On the other
hand, when a lower bound on the smallest drop in the demand curve is
known, we prove an upper bound on the regret of order . This is a significant improvement on previously known
regret bounds for discontinuous demand curves, that are at best of order
. When in the distribution-dependent case, the
hardness of our setting reduces to that of a stochastic -armed bandit: we
prove that an upper bound of order (up to factors)
on the regret can be achieved with no information on the demand curve. Finally,
we show a upper bound on the regret for the setting in which the
buyers' decisions are nonstochastic, and the regret is measured with respect to
the best between two fixed valuations one of which is known to the seller
Dynamic Pricing with Finitely Many Unknown Valuations
Motivated by posted price auctions where buyers are grouped in an unknown number of latent types characterized by their private values for the good on sale, we investigate regret minimization in stochastic dynamic pricing when the distribution of buyers\u2019 private values is supported on an unknown set of points in [0, 1] of unknown cardinality K
Optimal No-regret Learning in Repeated First-price Auctions
We study online learning in repeated first-price auctions with censored
feedback, where a bidder, only observing the winning bid at the end of each
auction, learns to adaptively bid in order to maximize her cumulative payoff.
To achieve this goal, the bidder faces a challenging dilemma: if she wins the
bid--the only way to achieve positive payoffs--then she is not able to observe
the highest bid of the other bidders, which we assume is iid drawn from an
unknown distribution. This dilemma, despite being reminiscent of the
exploration-exploitation trade-off in contextual bandits, cannot directly be
addressed by the existing UCB or Thompson sampling algorithms in that
literature, mainly because contrary to the standard bandits setting, when a
positive reward is obtained here, nothing about the environment can be learned.
In this paper, by exploiting the structural properties of first-price
auctions, we develop the first learning algorithm that achieves
regret bound when the bidder's private values are
stochastically generated. We do so by providing an algorithm on a general class
of problems, which we call monotone group contextual bandits, where the same
regret bound is established under stochastically generated contexts. Further,
by a novel lower bound argument, we characterize an lower
bound for the case where the contexts are adversarially generated, thus
highlighting the impact of the contexts generation mechanism on the fundamental
learning limit. Despite this, we further exploit the structure of first-price
auctions and develop a learning algorithm that operates sample-efficiently (and
computationally efficiently) in the presence of adversarially generated private
values. We establish an regret bound for this algorithm,
hence providing a complete characterization of optimal learning guarantees for
this problem