We present a general, efficient technique for providing contextual predictions that are "multivalid" in various senses, against an online sequence of adversarially chosen examples (x, y). This means that the resulting estimates correctly predict various statistics of the labels y not just marginally -as averaged over the sequence of examples -but also conditionally on x ∈ G for any G belonging to an arbitrary intersecting collection of groups G.We provide three instantiations of this framework. The first is mean prediction, which corresponds to an online algorithm satisfying the notion of multicalibration from Hébert-Johnson et al. [2018]. The second is variance and higher moment prediction, which corresponds to an online algorithm satisfying the notion of mean-conditioned moment multicalibration from Jung et al. [2020]. Finally, we define a new notion of prediction interval multivalidity, and give an algorithm for finding prediction intervals which satisfy it. Because our algorithms handle adversarially chosen examples, they can equally well be used to predict statistics of the residuals of arbitrary point prediction methods, giving rise to very general techniques for quantifying the uncertainty of predictions of black box algorithms, even in an online adversarial setting. When instantiated for prediction intervals, this solves a similar problem as conformal prediction, but in an adversarial environment and with multivalidity guarantees stronger than simple marginal coverage guarantees.

All proper scoring rules incentivize an expert to predict accurately (report their true estimate), but not all proper scoring rules equally incentivize precision. Rather than treating the expert's belief as exogenously given, we consider a model where a rational expert can endogenously refine their belief by repeatedly paying a fixed cost, and is incentivized to do so by a proper scoring rule.Specifically, our expert aims to predict the probability that a biased coin flipped tomorrow will land heads, and can flip the coin any number of times today at a cost of per flip. Our first main result defines an incentivization index for proper scoring rules, and proves that this index measures the expected error of the expert's estimate (where the number of flips today is chosen adaptively to maximize the predictor's expected payoff). Our second main result finds the unique scoring rule which optimizes the incentivization index over all proper scoring rules.We also consider extensions to minimizing the ℓ ℎ moment of error, and again provide an incentivization index and optimal proper scoring rule. In some cases, the resulting scoring rule is differentiable, but not infinitely differentiable. In these cases, we further prove that the optimum can be uniformly approximated by polynomial scoring rules.Finally, we compare common scoring rules via our measure, and include simulations confirming the relevance of our measure even in domains outside where it provably applies. CCS Concepts: • Theory of computation Algorithmic mechanism design.

We present a deterministic polynomial-time algorithm for computing [Formula: see text]-approximate (pure) Nash equilibria in (proportional sharing) weighted congestion games with polynomial cost functions of degree at most [Formula: see text]. This is an exponential improvement of the approximation factor with respect to the previously best deterministic algorithm. An appealing additional feature of the algorithm is that it only uses best-improvement steps in the actual game, as opposed to the previously best algorithms, that first had to transform the game itself. Our algorithm is an adaptation of the seminal algorithm by Caragiannis at al. [Caragiannis I, Fanelli A, Gravin N, Skopalik A (2011) Efficient computation of approximate pure Nash equilibria in congestion games. Ostrovsky R, ed. Proc. 52nd Annual Symp. Foundations Comput. Sci. (FOCS) (IEEE Computer Society, Los Alamitos, CA), 532–541; Caragiannis I, Fanelli A, Gravin N, Skopalik A (2015) Approximate pure Nash equilibria in weighted congestion games: Existence, efficient computation, and structure. ACM Trans. Econom. Comput. 3(1):2:1–2:32.], but we utilize an approximate potential function directly on the original game instead of an exact one on a modified game. A critical component of our analysis, which is of independent interest, is the derivation of a novel bound of [Formula: see text] for the price of anarchy (PoA) of [Formula: see text]-approximate equilibria in weighted congestion games, where [Formula: see text] is the Lambert-W function. More specifically, we show that this PoA is exactly equal to [Formula: see text], where [Formula: see text] is the unique positive solution of the equation [Formula: see text]. Our upper bound is derived via a smoothness-like argument, and thus holds even for mixed Nash and correlated equilibria, whereas our lower bound is simple enough to apply even to singleton congestion games.

We give a simple, generic conformal prediction method for sequential prediction that achieves target empirical coverage guarantees against adversarially chosen data. It is computationally lightweight -comparable to split conformal prediction -but does not require having a heldout validation set, and so all data can be used for training models from which to derive a conformal score. It gives stronger than marginal coverage guarantees in two ways. First, it gives threshold calibrated prediction sets that have correct empirical coverage even conditional on the threshold used to form the prediction set from the conformal score. Second, the user can specify an arbitrary collection of subsets of the feature space -possibly intersecting -and the coverage guarantees also hold conditional on membership in each of these subsets. We call our algorithm MVP, short for MultiValid Prediction. We give both theory and an extensive set of empirical evaluations.