11 research outputs found
Convergence and representation theorems for set valued random processes
AbstractIn this paper we study set valued random processes in discrete time and with values in a separable Banach space. We start with set valued martingales and prove various convergence and regularity results. Then we turn our attention to larger classes of set valued processes. So we introduce and study set valued amarts and set valued martingales in the limit. Finally we prove a useful property of the set valued conditional expectation
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Essays on Learning and Induction
What is the correct way to respond to newly acquired information? What methods for updating beliefs and other attitudes are rational? And what makes them rational? This dissertation is a collection of independent essays, each of which addresses these questions. Among other things, I investigate the extent to which Bayesian learning can be considered objective, the circumstances in which rational learning reduces uncertainty and produces consensus, whether rational learning is compatible with disagreement and polarization, and the relationship between long-run and short-run norms for learning
Efficient Approximation Schemes for Stochastic Probing and Prophet Problems
Our main contribution is a general framework to design efficient polynomial
time approximation schemes (EPTAS) for fundamental classes of stochastic
combinatorial optimization problems. Given an error parameter ,
such algorithmic schemes attain a -approximation in only
time, where is some function that depends
only on . Technically speaking, our approach relies on presenting
tailor-made reductions to a newly-introduced multi-dimensional extension of the
Santa Claus problem [Bansal-Sviridenko, STOC'06]. Even though the
single-dimensional problem is already known to be APX-Hard, we prove that an
EPTAS can be designed under certain structural assumptions, which hold for our
applications.
To demonstrate the versatility of our framework, we obtain an EPTAS for the
adaptive ProbeMax problem as well as for its non-adaptive counterpart; in both
cases, state-of-the-art approximability results have been inefficient
polynomial time approximation schemes (PTAS) [Chen et al., NIPS'16; Fu et al.,
ICALP'18]. Turning our attention to selection-stopping settings, we further
derive an EPTAS for the Free-Order Prophets problem [Agrawal et al., EC'20] and
for its cost-driven generalization, Pandora's Box with Commitment [Fu et al.,
ICALP'18]. These results improve on known PTASes for their adaptive variants,
and constitute the first non-trivial approximations in the non-adaptive
setting.Comment: 33 page
Prophet Inequalities: Separating Random Order from Order Selection
Prophet inequalities are a central object of study in optimal stopping
theory. A gambler is sent values online, sampled from an instance of
independent distributions, in an adversarial, random or selected order,
depending on the model. When observing each value, the gambler either accepts
it as a reward or irrevocably rejects it and proceeds to observe the next
value. The goal of the gambler, who cannot see the future, is maximising the
expected value of the reward while competing against the expectation of a
prophet (the offline maximum). In other words, one seeks to maximise the
gambler-to-prophet ratio of the expectations.
The model, in which the gambler selects the arrival order first, and then
observes the values, is known as Order Selection. Recently it has been shown
that in this model a ratio of can be attained for any instance. If the
gambler chooses the arrival order (uniformly) at random, we obtain the Random
Order model. The worst case ratio over all possible instances has been
extensively studied for at least years. Still, it is not known if
carefully choosing the order, or simply taking it at random, benefits the
gambler. We prove that, in the Random Order model, no algorithm can achieve a
ratio larger than , thus showing for the first time that there is a
real benefit in choosing the order.Comment: 36 pages, 2 figure
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Simple vs. Optimal Mechanism Design
Mechanism design has found various applications in today\u27s economy, such as ad auctions and online markets. The goal of mechanism design is to design a mechanism or system such that a group of strategic agents are incentivized to choose actions that also help achieve the designer’s objective. However, in many of the mechanism design problems, the theoretically optimal mechanisms are complex and randomized, while mechanisms used in practice are usually simple and deterministic. The focus of this thesis is to resolve the discrepancy between theory and practice by studying the following questions: Are the mechanisms used in practice close to optimal? Can we design simple mechanisms to approximate the optimal one? In this thesis we focus on two important mechanism design settings: multi-item auctions and two-sided markets. We show that in both of the settings, there are indeed simple and approximately-optimal mechanisms. Following Myerson\u27s seminal result, which provides a simple and revenue-optimal auction when a seller is selling a singleitem to multiple buyers, there has been extensive research effort on maximizing revenue in multi-item auctions. However, the revenue-optimal mechanism is proved to be complex and randomized. We provide a unified framework to approximate the optimal revenue in a fairly general setting of multi-item auctions with multiple buyers. Our result substantially improves the results in the literature and applies to broader cases. Another line of works in this thesis focuses on two-sided markets, where sellers also participate in the mechanism and have their own costs. The impossibility result by Myerson and Satterthwaite shows that even in the simplist bilateral trade setting (1 buyer, 1 seller, 1 item), the full welfare is not achievable by a truthful mechanism that does not run a deficit. In this thesis we focus on a more challenging objective gains from trade --- the increment of the welfare, and provide simple mechanisms that approximate the optimal gains from trade, in bilateral trade and many other two-sided market settings