We initiate the study of incentives in a general machine learning framework. We focus on a game-theoretic regression learning setting where private information is elicited from multiple agents with different, possibly conflicting, views on how to label the points of an input space. This conflict potentially gives rise to untruthfulness on the part of the agents. In the restricted but important case when every agent cares about a single point, and under mild assumptions, we show that agents are motivated to tell the truth. In a more general setting, we study the power and limitations of mechanisms without payments. We finally establish that, in the general setting, the VCG mechanism goes a long way in guaranteeing truthfulness and economic efficiency.

As kidney exchange programs are growing, manipulation by hospitals becomes more of an issue. Assuming that hospitals wish to maximize the number of their own patients who receive a kidney, they may have an incentive to withhold some of their incompatible donor-patient pairs and match them internally, thus harming social welfare. We study mechanisms for twoway exchanges that are strategyproof, i.e., make it a dominant strategy for hospitals to report all their incompatible pairs.

We study the problem of selecting a member of a set of agents based on impartial nominations by agents from that set. The problem was studied previously by Alon et al. and by Holzman and Moulin and has important applications in situations where representatives are selected from within a group or where publishing or funding decisions are made based on a process of peer review. Our main result concerns a randomized mechanism that in expectation selects an agent with at least half the maximum number of nominations. Subject to impartiality, this is best possible.

A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: Given a sequence of random variables X 1 , . . . , X n drawn independently from a distribution F, the goal is to choose a stopping time τ so as to maximize α such that for all distributions F we haveWhat makes this problem challenging is that the decision whether τ = t may only depend on the values of the random variables X 1 , . . . , X t and on the distribution F. For quite some time the best known bound for the problem was α ≥ 1 − 1/e ≈ 0.632 [Hill and Kertz, 1982]. Only recently this bound was improved by Abolhassani et al. [2017], and a tight bound of α ≈ 0.745 was obtained by Correa et al. [2017].The case where F is unknown, such that the decision whether τ = t may depend only on the values of the first t random variables but not on F, is equally well motivated (e.g., [Azar et al., 2014]) but has received much less attention. A straightforward guarantee for this case of α ≥ 1/e ≈ 0.368 can be derived from the solution to the secretary problem. Our main result is that this bound is tight. Motivated by this impossibility result we investigate the case where the stopping time may additionally depend on a limited number of samples from F. An extension of our main result shows that even with o(n) samples α ≤ 1/e, so that the interesting case is the one with Ω(n) samples. Here we show that n samples allow for a significant improvement over the secretary problem, while O(n 2 ) samples are equivalent to knowledge of the distribution: specifically, with n samples α ≥ 1 − 1/e ≈ 0.632 and α ≤ ln(2) ≈ 0.693, and with O(n 2 ) samples α ≥ 0.745 − ε for any ε > 0.

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