Timely Persuasion with Opposing Senders: When Does Competition Help the Receiver?

2026-10-03

AI Summary

This study addresses the competitive interaction between budget-constrained adversarial senders over a binary Markov chain, investigating how such competition affects the receiver’s state estimation utility. A Stackelberg game model is formulated, integrating Poisson rate modeling with a zero-order hold estimator for theoretical analysis. The work reveals that under finite-state conditions, a “zero-reporting” strategy exclusively frees budget resources for state intervention. Closed-form solutions for pure-strategy equilibria are derived, demonstrating that full-budget intervention outperforms both single-sender and silent strategies. Numerical evaluations further confirm that, beyond specific thresholds, sender competition significantly enhances the receiver’s long-term correct estimation utility.

Abstract

Status updates let a receiver track a changing process, but a sender that controls when they are sent can steer the receiver's estimate without misreporting. We study competition between two such senders with opposing preferences, who report on a binary continuous-time Markov chain using state-dependent Poisson rates under individual budgets. The receiver follows the reports through a zero-order-hold estimator only if this yields at least the long-run weighted correct-estimation utility of a prior-based default estimate. We model the interaction as a Stackelberg game in which the sender favored by this default leads, the opposing sender follows, and the receiver then selects its estimator. We characterize the pure-strategy equilibria in closed form. When the follower can obtain positive utility against a silent leader, limited state-$0$ reporting by the leader merely frees part of the follower's budget for state-$1$ reports, so every optimal leader policy is either silence or full-budget state-$0$ reporting, depending on explicit budget thresholds. A single sender with the same total budget leaves the receiver at its default utility, whereas the full-budget intervention outcome makes the receiver's participation constraint slack and strictly improves its utility. At the intervention threshold, silence and full-budget intervention yield identical sender payoffs but different receiver utilities. Numerical results illustrate these regimes and compare equilibrium performance with a receiver-optimal reporting benchmark.

Problem

• Timely Persuasion
• Information Design
• Competing Senders
• Status Updates
• Stackelberg Game

Innovation

• Timely Persuasion
• Stackelberg Game
• Competing Senders
• Continuous-Time Markov Chain
• Information Design