Optimal Efficiency-Envy Trade-Off via Optimal Transport

We consider the problem of allocating a distribution of items to $n$ recipients where each recipient has to be allocated a fixed, prespecified fraction of all items, while ensuring that each recipient does not experience too much envy. We show that this problem can be formulated as a variant of the semi-discrete optimal transport (OT) problem, whose solution structure in this case has a concise representation and a simple geometric interpretation. Unlike existing literature that treats envy-freeness as a hard constraint, our formulation allows us to \emph{optimally} trade off efficiency and envy continuously. Additionally, we study the statistical properties of the space of our OT based allocation policies by showing a polynomial bound on the number of samples needed to approximate the optimal solution from samples. Our approach is suitable for large-scale fair allocation problems such as the blood donation matching problem, and we show numerically that it performs well on a prior realistic data simulator

Essays on Online Learning and Resource Allocation

This thesis studies four independent resource allocation problems with different assumptions on information available to the central planner, and strategic considerations of the agents present in the system. We start off with an online, non-strategic agents setting in Chapter 1, where we study the dynamic pricing and learning problem under the Bass demand model. The main objective in the field of dynamic pricing and learning is to study how a seller can maximize revenue by adjusting price over time based on sequentially realized demand. Unlike most existing literature on dynamic pricing and learning, where the price only affects the demand in the current period, under the Bass model, price also influences the future evolution of demand. Finding arevenue-maximizing dynamic pricing policy in this model is non-trivial even in the full information case, where model parameters are known. We consider the more challenging incomplete information problem where dynamic pricing is applied in conjunction with learning the unknown model parameters, with the objective of optimizing the cumulative revenues over a given selling horizon of length . Our main contribution is an algorithm that satisfies a high probability regret guarantee of order ²/³; where the market size is known a priori. Moreover, we show that no algorithm can incur smaller order of loss by deriving a matching lower bound. We then switch our attention to a single round, strategic agents setting in Chapter 2, where we study a multi-resource allocation problem with heterogeneous demands and Leontief utilities. Leontief utility function captures the idea that for certain resource allocation settings, the utility of marginal increase in one resource depends on the availabilities of other resources. We generalize the existing literature on this model formulation to incorporate more constraints faced in real applications, which in turn requires new algorithm design and analysis techniques. The main contribution of this chapter is an allocation algorithm that satisfies Pareto optimality, envy-freenss, strategy-proofness, and a notion of sharing incentive. In Chapter 3, we study a single round, non-strategic agent setting, where the central planner tries to allocate a pool of items to a set of agents who each has to receive a prespecified fraction of all items. Additionally, we want to ensure fairness by controlling the amount of envy that agents have with the final allocations. We make the observation that this resource allocation setting can be formulated as an Optimal Transport problem, and that the solution structure displays a surprisingly simple structure. Using this insight, we are able to design an allocation algorithm that achieves the optimal trade-off between efficiency and envy. Finally, in Chapter 4 we study an online, strategic agent setting, where similar to the previous chapter, the central planner needs to allocate a pool of items to a set of agents who each has to receive a prespecified fraction of all items. Unlike in the previous chapter, the central planner has no a priori information on the distribution of items. Instead, the central planner needs to implicitly learn these distributions from the observed values in order to pick a good allocation policy. Additionally, an added challenge here is that the agents are strategic with incentives to misreport their valuations in order to receive better allocations. This sets our work apart both from the online auction mechanism design settings which typically assume known valuation distributions and/or involve payments, and from the online learning settings that do not consider strategic agents. To that end, our main contribution is an online learning based allocation mechanism that is approximately Bayesian incentive compatible, and when all agents are truthful, guarantees a sublinear regret for individual agents' utility compared to that under the optimal offline allocation policy

Practical Algorithms for Resource Allocation and Decision Making

Algorithms are widely used today to help make important decisions in a variety of domains, including health care, criminal justice, employment, and education. Designing \emph{practical} algorithms involves balancing a wide variety of criteria. Deployed algorithms should be robust to uncertainty, they should abide by relevant laws and ethical norms, they should be easy to use correctly, they should not adversely impact user behavior, and so on. Finding an appropriate balance of these criteria involves technical analysis, understanding of the broader context, and empirical studies ``in the wild''. Most importantly practical algorithm design involves close collaboration between stakeholders and algorithm developers. The first part of this thesis addresses technical issues of uncertainty and fairness in \emph{kidney exchange}---a real-world matching market facilitated by optimization algorithms. We develop novel algorithms for kidney exchange that are robust to uncertainty in both the quality and the feasibility of potential transplants, and we demonstrate the effect of these algorithms using computational simulations with real kidney exchange data. We also study \emph{fairness} for hard-to-match patients in kidney exchange. We close a previously-open theoretical gap, by bounding the price of fairness in kidney exchange with chains. We also provide matching algorithms that bound the price of fairness in a principled way, while guaranteeing Pareto efficiency. The second part describes two real deployed algorithms---one for kidney exchange, and one for recruiting blood donors. For each application cases we characterize an underlying mathematical problem, and theoretically analyze its difficulty. We then develop practical algorithms for each setting, and we test them in computational simulations. For the blood donor recruitment application we present initial empirical results from a fielded study, in which a simple notification algorithm increases the expected donation rate by $5\%$. The third part of this thesis turns to human aspects of algorithm design. We conduct several survey studies that address several questions of practical algorithm design: How do algorithms impact decision making? What additional information helps people use complex algorithms to make decisions? Do people understand standard algorithmic notions of fairness? We conclude with suggestions for facilitating deeper stakeholder involvement for practical algorithm design, and we outline several areas for future research