In one view of the classical game of prediction with expert advice with binary outcomes, in each round, each expert maintains an adversarially chosen belief and honestly reports this belief. We consider a recently introduced, strategic variant of this problem with selfish (reputation-seeking) experts, where each expert strategically reports in order to maximize their expected future reputation based on their belief. In this work, our goal is to design an algorithm for the selfish experts problem that is incentive-compatible (IC, or \emph{truthful}), meaning each expert's best strategy is to report truthfully, while also ensuring the algorithm enjoys sublinear regret with respect to the expert with the best belief. Freeman et al. (2020) recently studied this problem in the full information and bandit settings and obtained truthful, no-regret algorithms by leveraging prior work on wagering mechanisms. While their results under full information match the minimax rate for the classical ("honest experts") problem, the best-known regret for their bandit algorithm WSU-UX is $O(T^{2/3})$, which does not match the minimax rate for the classical ("honest bandits") setting. It was unclear whether the higher regret was an artifact of their analysis or a limitation of WSU-UX. We show, via explicit construction of loss sequences, that the algorithm suffers a worst-case $\Omega(T^{2/3})$ lower bound. Left open is the possibility that a different IC algorithm obtains $O(\sqrt{T})$ regret. Yet, WSU-UX was a natural choice for such an algorithm owing to the limited design room for IC algorithms in this setting.

We study online learning settings in which experts act strategically to maximize their influence on the learning algorithm's predictions by potentially misreporting their beliefs about a sequence of binary events. Our goal is twofold. First, we want the learning algorithm to be no-regret with respect to the best fixed expert in hindsight. Second, we want incentive compatibility, a guarantee that each expert's best strategy is to report his true beliefs about the realization of each event. To achieve this goal, we build on the literature on wagering mechanisms, a type of multi-agent scoring rule. We provide algorithms that achieve no regret and incentive compatibility for myopic experts for both the full and partial information settings. In experiments on datasets from FiveThirtyEight, our algorithms have regret comparable to classic no-regret algorithms, which are not incentive-compatible. Finally, we identify an incentive-compatible algorithm for forward-looking strategic agents that exhibits diminishing regret in practice.

Wagering mechanisms are one-shot betting mechanisms that elicit agents' predictions of an event. For deterministic wagering mechanisms, an existing impossibility result has shown incompatibility of some desirable theoretical properties. In particular, Pareto optimality (no profitable side bet before allocation) can not be achieved together with weak incentive compatibility, weak budget balance and individual rationality. In this paper, we expand the design space of wagering mechanisms to allow randomization and ask whether there are randomized wagering mechanisms that can achieve all previously considered desirable properties, including Pareto optimality. We answer this question positively with two classes of randomized wagering mechanisms: i) one simple randomized lottery-type implementation of existing deterministic wagering mechanisms, and ii) another family of simple and randomized wagering mechanisms which we call surrogate wagering mechanisms, which are robust to noisy ground truth. This family of mechanisms builds on the idea of learning with noisy labels (Natarajan et al. 2013) as well as a recent extension of this idea to the information elicitation without verification setting (Liu and Chen 2018). We show that a broad family of randomized wagering mechanisms satisfy all desirable theoretical properties.

We consider the design of forecasting competitions in which multiple forecasters make predictions about one or more independent events and compete for a single prize. We have two objectives: (1) to award the prize to the most accurate forecaster, and (2) to incentivize forecasters to report truthfully, so that forecasts are informative and forecasters need not spend any cognitive effort strategizing about reports. Proper scoring rules incentivize truthful reporting if all forecasters are paid according to their scores. However, incentives become distorted if only the best-scoring forecaster wins a prize, since forecasters can often increase their probability of having the highest score by reporting extreme beliefs. Even if forecasters do report truthfully, awarding the prize to the forecaster with highest score does not guarantee that high-accuracy forecasters are likely to win; in extreme cases, it can result in a perfect forecaster having zero probability of winning. In this paper, we introduce a truthful forecaster selection mechanism. We lower-bound the probability that our mechanism selects the most accurate forecaster, and give rates for how quickly this bound approaches 1 as the number of events grows. Our techniques can be generalized to the related problems of outputting a ranking over forecasters and hiring a forecaster with high accuracy on future events.