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

Ahmet Bugra Gundogan, Melih Bastopcu

arXiv:2610.04468v1cs.ITcs.GTcs.NIcs.SYeess.SPeess.SYmath.IT

TL;DR

The paper asks when competition between opposing senders who control truthful-report timing can improve a receiver’s estimate. It characterizes their Stackelberg equilibria and finds that full-budget intervention can improve receiver utility beyond the same-budget single-sender default. At the intervention threshold, the two optimal leader policies give identical sender payoffs but different receiver utilities.

  • Problem

    The paper asks when an opposing sender intervenes and whether that intervention improves receiver estimation when senders control truthful-report timing.

    Source: §I · p. 1

  • Method

    The paper models two opposing senders’ state-dependent Poisson reporting policies under individual budgets as a Stackelberg game followed by the receiver’s estimator choice.

    Source: §I · p. 2, A. Average Utilities and the… · p. 3

  • Results

    At R2 = 2, both Os and Oi are equilibria, giving receiver utilities 0.3 and 0.3286, respectively; above 2, Oi utility rises toward 0.38.

    Source: §V · p. 6

  • Takeaways & Limitations

    A full-budget intervention can make the participation constraint slack and strictly improve receiver utility, whereas a same-total-budget single sender leaves it at the default.

    Source: §VI · p. 7, §V · p. 7

Abstract

from arXiv · show

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.

I. INTRODUCTION

The paper asks when a sender with opposing preferences intervenes and whether that intervention improves receiver estimation. It analyzes competition in which senders control truthful-report timing, then characterizes equilibrium policies and receiver benefits.

I. INTRODUCTION

  • The motivating question is whether an opposing sender’s reports can both induce intervention and improve estimation when the receiver may ignore updates.A sender can report favorable states promptly and unfavorable ones less often, but must report unfavorable states often enough to keep the receiver following them [3].Source: §I · p. 1
  • Unlike remote-estimation work that designs sampling to improve estimates,,, this paper treats estimation performance as an outcome of opposing senders’ equilibrium choices.The senders commit to timing truthful reports rather than altering information content; the single-sender timely-persuasion model leaves the receiver no gain when persuasion is feasible [3].Source: §I · pp. 1–2
  • The leader chooses between silence and full-budget state-0 reporting, with budget thresholds determining when intervention is optimal.When intervention is optimal, it makes the participation constraint slack and strictly improves receiver utility; a same-budget single sender leaves utility at the default.Source: §I · p. 2
  • At the intervention threshold, silence and full-budget intervention yield identical sender payoffs but different receiver utilities.Numerical results illustrate the regimes and show that an appropriate budget division can attain the receiver-optimal reporting benchmark for the considered parameters.Source: §I · p. 2

II. SYSTEM MODEL AND PROBLEM FORMULATION

The model has two senders that observe a binary continuous-time Markov source and send truthful reports under individual budgets. The receiver chooses between following reports and a default estimate, after sequential sender commitments.

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • Each sender uses state-dependent Poisson reporting rates, with reports revealing the current state and budgets constraining the sum of its rates.Conditional on the source trajectory, the senders’ reporting processes are independent.Source: §II · p. 2
  • The receiver chooses between a hold-last estimate updated by either sender’s reports and a constant default estimate of 0.The default is optimal under the assumed stationary-prior weighting, qπ0 > (1 − q)π1.Source: §II · p. 2
  • The receiver’s participation constraint compares its utility from following reports with the utility of its admissible constant default.The receiver selects one of the two policies before reporting begins and uses it thereafter; the held estimate need not be optimal after every observation history.Source: §II · p. 2
  • Sender 2 commits first, sender 1 observes and responds, and the receiver then selects its estimator.The default estimate favors sender 2, so it acts as leader against sender 1, who prefers estimate 1.Source: §II · p. 3
  • The senders’ payoffs sum to one for every strategy profile, making their game constant-sum.The source is a binary CTMC with transition rates λ from 0 to 1 and µ from 1 to 0.Source: §II · pp. 2–3

A. Average Utilities and the Participation Constraint

Under the report-following policy, the receiver responds to the aggregate reporting rates, and its participation constraint reduces to a minimum state-0 reporting requirement whenever state-1 reports occur.

A. Average Utilities and the Participation Constraint

  • The two senders’ reports combine into a conditionally Poisson process with aggregate rates s = s1 + s2 and c = c1 + c2.Because each report reveals the current state, the estimate and payoffs under report-following depend on the aggregate rates, not on which sender generated each report.Source: A. Average Utilities and the… · p. 3
  • With no reports, the receiver’s estimate remains at 0 and its utility equals the default utility; ignoring reports yields that same utility for every sender policy.The receiver follows reports only when their utility is at least the default utility.Source: A. Average Utilities and the… · p. 3
  • For positive aggregate state-1 reporting, participation requires aggregate state-0 intensity of at least cmin.The requirement is met jointly by both senders’ state-0 reports, unlike the single-sender model where one sender must meet it alone.Source: A. Average Utilities and the… · p. 3
  • When the participation constraint binds, the receiver is indifferent between following reports and the default, and the tie-breaking rule selects report-following except when both aggregate rates are zero.This tie-break favors sender 1 and ensures its maximum is attained when the constraint binds.Source: A. Average Utilities and the… · p. 3

B. Stackelberg Game Formulation

The game specifies sequential sender commitments followed by the receiver’s estimator choice. The leader minimizes the follower’s best-response payoff, with a stated selection when the follower’s optimal payoff is zero.

B. Stackelberg Game Formulation

  • The follower’s strategy maps the leader’s policy to a response, while the receiver’s strategy maps both sender policies to an estimator.The estimator set consists of report-following and default policies.Source: B. Stackelberg Game Formulation · p. 3
  • Because the senders’ payoffs sum to one, the leader’s equilibrium choice minimizes the follower’s optimal payoff.All maximizers give the leader the same payoff, so no tie-breaking rule is needed for the follower.Source: B. Stackelberg Game Formulation · p. 3

III. EQUILIBRIUM CHARACTERIZATION

The equilibrium is characterized by backward induction: first derive sender 1’s best response to a fixed leader policy, then determine sender 2’s optimal commitment.

III. EQUILIBRIUM CHARACTERIZATION

  • Backward induction derives sender 1’s best response to a fixed leader policy before determining sender 2’s optimal commitment.The receiver’s response is taken as given in both steps.Source: §III · p. 3

A. Sender 1’s Best Response

For a fixed leader policy, the follower must supply enough state-0 reporting to meet the receiver’s participation constraint. When a positive payoff is attainable, the unique best response meets that minimum and spends the remaining budget on state-1 reports.

A. Sender 1’s Best Response

  • If R1 < δ, every follower policy yields zero payoff because the participation constraint fails whenever reports arrive.With no reports, the receiver retains the default estimate and the follower also receives zero payoff.Source: A. Sender 1’s Best Response · p. 4
  • When the follower’s optimal payoff is positive, its unique best response uses state-0 intensity δ and state-1 intensity R1 − δ.Here δ = (cmin − c2)+ is the minimum state-0 intensity required for the receiver to follow the reports.Source: A. Sender 1’s Best Response · p. 4
  • Leader state-0 reports lower δ until c2 reaches cmin, freeing follower budget for state-1 reports; when δ = R1 and s2 > 0, the selected response has zero state-1 intensity but positive payoff.If the follower’s optimal payoff is zero, every feasible follower policy is a maximizer.Source: A. Sender 1’s Best Response · p. 4

B. Sender 2’s Optimal Policy

The leader’s optimal policy reduces to silence or full-budget state-0 reporting. Budget thresholds determine which policy is optimal, with both outcomes possible at the intervention threshold.

B. Sender 2’s Optimal Policy

  • The leader’s minimum follower-payoff choice is either silence or full-budget state-0 reporting, since the follower payoff increases on [0, cmin] and decreases thereafter.For c2 < cmin, additional state-0 reports free follower budget for state-1 reports; for c2 ≥ cmin, the follower already spends its budget on state-1 reports.Source: B. Sender 2’s Optimal Policy · p. 4
  • Intervention is strictly preferred only when R1 > Rcrit_1 and R2 > Rcrit_2; at R2 = Rcrit_2, silence and intervention are both optimal.Rcrit_1 is the follower-budget threshold above which intervention can be strictly preferred; Rcrit_2 is the leader-budget threshold for strict preference.Source: B. Sender 2’s Optimal Policy · pp. 4–5
  • When R1 ≤ cmin, the equilibrium gives sender 1 payoff 0 and sender 2 payoff 1; silence is one such equilibrium outcome.In this regime, the stated equilibrium policies depend on whether R1 < δ or R1 = δ with s2 = 0.Source: B. Sender 2’s Optimal Policy · p. 5
  • For R1 > cmin, silence yields follower policy (R1 − cmin, cmin), while intervention yields (R1, 0); the receiver follows reports in both outcomes.The leader’s respective policies are (0, 0) and (0, R2).Source: B. Sender 2’s Optimal Policy · p. 5
  • At R2 = Rcrit_2, the two outcomes give identical sender payoffs but different receiver utilities; intervention gives the receiver a strictly positive gain over silence.The schematic payoff comparison shows that, when R1 > Rcrit_1, the leader’s payoff exceeds silence exactly above the threshold.Source: B. Sender 2’s Optimal Policy · p. 5, Figure 2 · p. 5

IV. RECEIVER BENEFIT FROM COMPETITION

Competition can raise the receiver’s utility above the single-sender default when the intervention equilibrium makes the participation constraint slack. The improvement requires positive state-1 reporting and state-0 reporting above the participation threshold.

IV. RECEIVER BENEFIT FROM COMPETITION

  • A single sender with sender 2’s preferences optimally remains silent because the default estimate already matches its preference, leaving receiver utility at qπ0.Source: §IV · p. 5
  • A single sender with sender 1’s preferences leaves the receiver at qπ0 even when it controls the full budget, choosing state-0 intensity cmin when R > cmin.That choice makes the participation constraint bind.Source: §IV · p. 6
  • ZOH improves strictly over the default exactly when aggregate rates satisfy s > 0 and c > cmin; competition helps only through an equilibrium with both conditions.Source: §IV · p. 6
  • At R2 = Rcrit, silence and intervention are both equilibria with identical sender payoffs, but the receiver gets qπ0 under silence and strictly more under intervention.Below the threshold, the unique equilibrium yields qπ0; above it, the unique intervention equilibrium yields the higher utility.Source: §IV · p. 6
  • Under Oi, sender 2’s state-0 reports exceed cmin and cannot be offset by sender 1, making the participation constraint slack and strictly raising receiver utility above qπ0.Oi is an equilibrium when R1 > cmin and R2 ≥ Rcrit.Source: §IV · p. 6

V. NUMERICAL RESULTS

The numerical examples illustrate how the budget split determines whether competition improves receiver utility and show that an appropriate split can attain the receiver-optimal benchmark. With R1 fixed at 2, receiver utility jumps at R2 = 2 while sender payoffs remain continuous.

V. NUMERICAL RESULTS

  • With λ = 2, µ = 3, and q = 0.5, the example has cmin = 1, default receiver utility 0.3, and Rcrit = 2 when R1 = 2.Source: §V · p. 6
  • 0.4 and 0.6 are the sender payoffs for R2 ≤ 2; above 2, the follower’s payoff falls toward 0.16 as the leader’s rises toward 0.84.The sender payoffs are continuous at R2 = 2, unlike receiver utility.Source: §V · p. 6
  • 0.3 below R2 = 2 rises to 0.3286 at R2 = 2 under Oi, then increases toward 0.38 as R2 grows, with R1 fixed at 2.At R2 = 2, silence and intervention are both equilibria, yielding receiver utilities 0.3 and 0.3286, respectively.Source: §V · p. 6, Figure 5 · p. 7
  • At budget division (R1, R2) = (3.5, 4.5), Oi implements aggregate rates (3.5, 4.5), the receiver-optimal benchmark maximizer for total budget 8.Moving away from this division reduces receiver utility; if either sender receives too little budget, utility returns to the default 0.3.Source: §V · p. 7

VI. CONCLUSION

The closed-form Stackelberg equilibria make the leader’s optimal policy either silence or full-budget state-0 reporting when the follower can obtain positive utility against a silent leader. Receiver utility stays at its default in the low-budget regime, but improves under the optimal full-budget intervention.

VI. CONCLUSION

  • When the follower can obtain positive utility against a silent leader, every optimal leader policy is silence or full-budget state-0 reporting, with explicit budget thresholds determining which.The leader’s state-0 reports initially help the follower by reducing the reporting effort needed for participation.Source: §VI · p. 7
  • In the low-budget regime, equilibrium leader policies can be nonunique, but all leave the receiver at its default utility.Source: §VI · p. 7
  • When the full-budget intervention outcome is optimal, it makes the participation constraint slack and strictly improves receiver utility relative to a single sender with the same total budget.Source: §VI · p. 7
  • In the numerical example, an appropriate budget division attains a receiver-optimal reporting benchmark; future work considers heterogeneous sources, alternative commitment orders, and receivers using report absence.Source: §VI · p. 7
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