Binary-Report Peer Prediction for Real-Valued Signal Spaces

TL;DR

This paper extends peer prediction theory by modeling real-valued signals with binary reports and analyzing threshold-based equilibria across OA, DG, DMI, and RBTS. It derives necessary and sufficient conditions for finite equilibria, highlights stability properties, and shows that real-valued signals fundamentally alter incentive alignment compared to the classical discrete-signal setting. Through Gaussian and multimodal/skewed distributions, the work demonstrates that uninformative equilibria can be stable in OA while multi-task mechanisms (DG, DMI, RBTS) can yield stable informative equilibria under certain conditions, but with varying degrees of robustness and design rigidity. The results reveal significant practical limitations for deploying standard peer-prediction mechanisms in realistic settings and motivate new approaches to mechanism design and equilibrium refinement. Overall, the work provides a framework to understand how richer information structures reshape incentive compatibility and how dynamics drive systems toward different equilibria.

Abstract

Theoretical guarantees about peer prediction mechanisms typically rely on the discreteness of the signal and report space. However, we posit that a discrete signal model is not realistic: in practice, agents observe richer information and map their signals to a discrete report. In this paper, we formalize a model with real-valued signals and binary reports. We study a natural class of symmetric strategies where agents map their information to a binary value according to a single real-valued threshold. We characterize equilibria for several well-known peer prediction mechanisms which are known to be truthful under the binary report model. In general, even when every threshold would correspond to a truthful equilibrium in the binary signal model, only certain thresholds remain equilibria in our model. Furthermore, by studying the dynamics of this threshold, we find that some of these equilibria are unstable. These results suggest important limitations for the deployment of existing peer prediction mechanisms in practice.

Figures (7)

• Figure 1: Plots of $F$, $G$, and $Q$ in the Gaussian model (§ \ref{['subsec:gaussian-intro']}) for the parameters $a = b = 1$.
• Figure 2: Plot of $G-F$ in the Gaussian model (§ \ref{['subsec:gaussian-intro']}), for parameters $a = b = 1$. $G(x)-F(x)$ is decreasing at $x=0$, meaning the corresponding equilibrium in DG is stable under the dynamics.
• Figure 3: Equilibria in OA and DG when signals are noisy versions of a skewed Normal, across different values of the skewness parameter $\alpha$.
• Figure 4: $F$, $G$, and $Q$ over signals $x$ in the two-component Gaussian mixture setting with means $-2, 2$ and precision 2.
• Figure 5: Bifurcation diagrams for two- and three-component Gaussian mixtures. (Top) Two-component case: OA (left) exhibits three equilibria at high precision, while DG (right) maintains a single stable equilibrium at $\tau=0$. (Bottom) Three-component case: OA (left) has five equilibria, with two stable at $\tau=\pm1$; DG (right) has three equilibria, with stability reversed from the two-component OA case.

Theorems & Definitions (59)

• Definition 1
• Proposition 1
• proof
• Theorem 1
• proof
• Proposition 2
• proof
• Theorem 2
• proof
• Corollary 1