A Geometric Perspective on Minimal Peer Prediction

Abstract

Minimal peer prediction mechanisms truthfully elicit private information (e.g., opinions or experiences) from rational agents without the requirement that ground truth is eventually revealed. In this article, we use a geometric perspective to prove that minimal peer prediction mechanisms are equivalent to power diagrams, a type of weighted Voronoi diagram. Using this characterization and results from computational geometry, we show that many of the mechanisms in the literature are unique up to affine transformations. We also show that classical peer prediction is “complete” in that every minimal mechanism can be written as a classical peer prediction mechanism for some scoring rule. Finally, we use our geometric characterization to develop a general method for constructing new truthful mechanisms, and we show how to optimize for the mechanisms’ effort incentives and robustness.

When is the mode functional the Bayes classifier?

“…Lambert and Shoham () state it as Proposition 7.2 in the setting of truthful elicitation of answers to multiple‐choice questions. A related, more general finding is Theorem 6 of Frongillo and Witkowski ().…”

Power Diagram Detection with Applications to Information Elicitation

“…To make headway, one typically assumes some structure about the underlying Bayesian (or pseudo-Bayesian) process through which agents form their beliefs. In particular, one typically assumes that agents receive some signal S ∈ O about the true outcome (like the quality of a hotel, or the correct label for an image classification task), and from this signal form a posterior belief [18]. Their work shows that there exists a mechanism with a truthful equilibrium if and only if the sets {P o } form a power diagram [18,Corollary 3.5].…”

“…In particular, one typically assumes that agents receive some signal S ∈ O about the true outcome (like the quality of a hotel, or the correct label for an image classification task), and from this signal form a posterior belief [18]. Their work shows that there exists a mechanism with a truthful equilibrium if and only if the sets {P o } form a power diagram [18,Corollary 3.5]. (More precisely, there must exist a power diagram with sites s o such that P o ⊆ cell(s o ) for all o ∈ O, but we restrict attention to maximal constraints, where every distribution is allowed to be the posterior following some signal.)…”

Incentive-Compatible Surveys via Posterior Probabilities

“…The players are indexed by π ∈ R, where R is infinite and countable. 10 The state of nature is an r.v. Ω taking values in {1, .…”

“…13 9 Many more examples can be found in [24] and [25]. 10 We need the assumption that there are infinitely many players for several reasons: first, we do not want to impose assumptions on the form of the payoffs outside of equilibrium; for this, we use the fact that, with an infinite number of players, the form of equilibrium payoff does not change when a player of one type mimics the equilibrium strategy of another type; second, achieving truth-telling of types is much harder with finitely many players, as is the implementation of equilibrium payoffs using practical inputs. We postpone for future research the analysis of the setup with finitely many players; finally, we need an infinite number of players because we invoke de Finetti's theorem in our model setup.…”