About the Project
Applicants should have a first-class degree in a quantitative discipline (Mathematics, Physics, Engineering, Theoretical Computer Science) and an aptitude for mathematics (with a view to developing new theory and proofs). A strong candidate will have substantial background and interest in probability and mathematical statistics; the ability to pursue independent study; and an ambition to pursue an academic career.
The exact nature of the investigation will depend on the student’s strengths and interest. The problem is likely to feature an exploration-exploitation tradeoff. Relevant literature includes:
1. Pricing models in which an unknown demand function is estimated (non-parametrically or parametrically) from independent, identically distributed demand observations [Besbes and Zeevi 2009, den Boer and Zwart 2015]. From the methodology viewpoint, large-deviation results [Dembo and Zeitouni 1998, Borovkov 1998] appear to be essential to such investigations.
2. Stochastic multi-armed bandits [Lai and Robbins 1985, Burnetas and Katehakis 1996].
The primary project supervisor is Athanassios (Thanos) N. Avramidis. Recent work has been on the development of algorithms and the study of convergence of the expected loss (the regret) for a finite-state, finite-action Markov Decision Process model similar to that in [den Boer and Zwart 2015], but where finitely many unknown purchase probabilities are estimated non-parametrically. While stochastic-bandit methods apply in principle to this setting [Burnetas and Katehakis 1996], the exponential growth of the number of policies in the problem size (inventory level and number of selling periods) appears to pose an insurmountable obstacle. An initial investigation could focus in a setting similar to those discussed above.
Borovkov, A. A., Mathematical Statistics, Gordon and Breach: Amsterdam, 1998.
Burnetas, A. N. and M. N. Katehakis, ``Optimal Adaptive Policies for Sequential Allocation Problems'', Advances in Applied Mathematics, Vol. 17, 1996, pp. 122--142.
Dembo, A. and O. Zeitouni, ``Large Deviation Techniques and Applications'', Springer, 1998.
den Boer, A. V. and B. Zwart, ``Dynamic pricing and learning with finite inventories'', Operations Research, Vol. 63, No. 4, 2015, pp. 965--978.
Lai, T. L. and H. Robbins, ``Asymptotically efficient adaptive allocation rules'', Advances in Applied Mathematics Vol. 6, 1985, pp. 4--22.
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