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Strategic Persuasion Through Information Timeliness

Ahmet Bugra Gundogan, Melih Bastopcu

arXiv:2607.15939v1cs.ITcs.GTcs.NIcs.SYeess.SPeess.SYmath.IT

TL;DR

The paper studies how a sender can use the timing of truthful updates from binary Markov sources to influence a receiver while satisfying a participation constraint. It formulates a Stackelberg game and derives a single-source policy and algorithms for multiple sources and receivers.

  • Problem

    How can a sender use the timing of truthful updates from binary CTMC sources to influence a receiver while ensuring that following the messages does not worsen the receiver’s average utility relative to prior knowledge?

    Source: §I · p. 1

  • Method

    The paper formulates a Stackelberg persuasion framework, derives closed-form participation constraints, and uses convex optimization and branch-and-bound methods for multi-source and multi-receiver settings.

    Source: §I · p. 2, §IV · p. 6, §V · p. 8

  • Results

    When R > cmin, the single-source optimum assigns cmin to state-0 updates and the remaining budget to state-1 updates; the multi-source algorithm can search O(n) nodes under a total dominance order.

    Source: §I · p. 2

  • Takeaways & Limitations

    Controlling update timeliness alone provides a strategic way to influence how long the receiver estimates the state as favorable, subject to the participation constraint.

    Source: §I · p. 1

Abstract

from arXiv · show

We study a dynamic strategic communication problem in which a sender controls the timing of truthful updates from binary continuous-time Markov sources. The receiver chooses between a zero-order-hold estimator that follows the sender's updates and a prior-only default estimator, aiming to maximize a weighted correct-estimation utility. In contrast, the sender seeks to persuade the receiver to estimate the state as 1, regardless of the true state. This misalignment leads to a Stackelberg game in which the sender, as the leader, commits to state-dependent Poisson update rates, and the receiver, as the follower, decides whether to follow the sender's messages. The sender maximizes the long-term average time that the receiver's estimate equals 1, subject to a conditional intensity budget and a participation constraint (PC) ensuring that following the sender's messages does not degrade the receiver's average utility relative to its prior information. For a single source, we show that the sender's optimal policy allocates a minimum state-0 update intensity to the undesired state-0, just enough to satisfy the PC, and the remaining budget to the desired state-1. For multiple sources with heterogeneous minimum state-0 update intensities, we develop a branch-and-bound algorithm that typically avoids exhaustive search. Finally, we extend the solution to multiple receivers over dedicated channels. Our results show that controlling timeliness alone enables the sender to persuade the receiver and increase its utility.

I. INTRODUCTION

The paper asks whether a sender can persuade through the timing of truthful updates while respecting a receiver participation constraint. It models this as committed, state-dependent information release and develops solutions for single-source, multi-source, and multi-receiver settings.

I. INTRODUCTION

  • The motivating problem is to maximize the fraction of time the audience believes the fundamentals or campaign are doing well, subject to a rate budget and a participation constraint.The participation constraint requires that following messages not worsen the receiver’s long-run average utility relative to prior knowledge.Source: §I · p. 1
  • The sender commits to state-dependent Poisson update intensities, while the receiver chooses between following updates and a prior-only estimator.Unlike classical Bayesian persuasion, the sender controls when truthful information is released, not what information to reveal.Source: §I · p. 1
  • When R > cmin, the single-source optimum assigns cmin to state-0 updates and the remaining budget to state-1 updates.For R ≤ cmin, multiple Stackelberg equilibria exist, all yielding zero sender utility.Source: §I · p. 2
  • For multiple sources, each source may require a different minimum state-0 update rate, and state-1 rate allocation is convex for any fixed active-source set.A branch-and-bound algorithm finds the globally optimal allocation; dominance pruning can reduce the search to O(n) nodes under a total dominance order.Source: §I · p. 2
  • The framework extends to multiple receivers on dedicated private channels with heterogeneous utility biases, using the prior algorithmic framework to solve the resulting problem optimally..Source: §I · p. 2

II. SYSTEM MODEL AND PROBLEM FORMULATION

The section models independent binary CTMC sources and a Stackelberg game in which the sender controls truthful update timing, while the receiver seeks accurate estimates and the sender seeks an estimate of 1.

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • The sources are independent two-state CTMCs, with 0-to-1 and 1-to-0 transition rates λi and μi, respectively.The receiver knows the rates but not the current source state, so its prior knowledge is the CTMC steady-state distribution.Source: §II · p. 2
  • Receiver utility rewards correct 0 estimates by q and correct 1 estimates by 1−q, while sender utility is 1 whenever the receiver estimates 1.The prior-only default estimate is 0 under the stated assumption qπi0 ≥ 1−qπi1.Source: §II · p. 3
  • Under the stated ergodicity conditions, long-run average utilities do not depend on the initial source state or estimate.The receiver’s initial estimate is set equal to the initial source state for notational convenience.Source: §II · p. 3
  • The sender commits to state-dependent Poisson update rates, and the receiver follows messages with a zero-order-hold estimate or uses the prior-only default.The receiver follows only when doing so does not reduce its long-run average utility relative to the prior-only policy.Source: §II · p. 3
  • The interaction is a Stackelberg game: the sender chooses policies first, while receiver responses are restricted to following messages or defaulting to estimate 0.Ties favor following messages; the no-update policy instead uses the default response.Source: §II · pp. 3–4

A. The Receiver’s Average Utility Function

The receiver’s long-run utility is derived from the stationary behavior of the joint source-and-estimate process, then compared with its prior-only utility.

A. The Receiver’s Average Utility Function

  • Following updates produces a four-state CTMC for the source and receiver estimate: (0,0), (0,1), (1,0), and (1,1).The stationary distribution is obtained from the chain’s balance equations and normalization.Source: A. The Receiver’s Average Utility… · p. 4
  • The stationary probabilities yield the receiver’s long-term average utility under the message-following policy.The utility is calculated using the receiver’s state-dependent rewards.Source: A. The Receiver’s Average Utility… · p. 4
  • The participation comparison is between this message-following utility and the prior-only utility qμi/(μi+λi).If following messages does not meet that benchmark, the receiver uses the prior-only estimate of 0.Source: A. The Receiver’s Average Utility… · p. 4

B. Sender’s Persuasion Problem

The sender chooses state-conditioned update rates to maximize its own utility while ensuring that the receiver has no average-utility reason to reject the messages.

B. Sender’s Persuasion Problem

  • The sender’s objective is to influence the receiver’s estimate toward 1, even though the receiver evaluates message-following by its own average utility.The receiver follows only when its utility is at least the prior-only benchmark.Source: B. Sender’s Persuasion Problem · p. 4
  • For positive state-1 update intensity, the participation constraint requires a minimum state-0 update rate cmin.When the state-1 update rate is zero, the participation constraint is automatically satisfied for any state-0 rate.Source: B. Sender’s Persuasion Problem · p. 4
  • The sender maximizes its average utility subject to the aggregate conditional-intensity budget and the per-source participation and feasibility constraints.The budget bounds the sum of state-0 and state-1 update rates across sources.Source: B. Sender’s Persuasion Problem · p. 4

III. OPTIMAL INFORMATION REVELATION: SINGLE SOURCE AND SINGLE RECEIVER

For a single source, the optimal allocation uses the minimum state-0 update intensity needed for participation and assigns any remaining budget to state-1 updates; below that threshold, the sender’s optimal utility is zero.

III. OPTIMAL INFORMATION REVELATION: SINGLE SOURCE AND SINGLE RECEIVER

  • The threshold is cmin=qμ/(1−q)−λ, imposed by the participation constraint.The paper states that cmin is strictly positive.Source: §III · p. 4
  • With the participation constraint satisfied, sender utility increases strictly with s when c>0 and decreases strictly with c when s>0.This monotonicity explains why the optimal feasible allocation uses cmin and directs the remaining rate to s.Source: §III · p. 5
  • When R>cmin, equilibrium sets s=R−cmin and c=cmin, and the receiver follows; when R≤cmin, the sender’s optimal utility is zero.The R>cmin policy uses the full remaining budget for state-1 updates.Source: §III · p. 5
  • Under the optimal policy when R>cmin, the receiver’s equilibrium utility equals its prior-only utility, while the sender gains time with estimate 1.The receiver follows under the optimistic tie-breaking convention.Source: §III · p. 5

IV. OPTIMAL INFORMATION REVELATION: MULTI SOURCE AND SINGLE RECEIVER

In the multi-source setting, the sender chooses which sources to activate and how to allocate state-1 sampling under a total budget. Positive-utility policies use the minimum state-0 rate needed for each active source’s participation constraint, and fixing the active set yields a convex allocation problem.

IV. OPTIMAL INFORMATION REVELATION: MULTI SOURCE AND SINGLE RECEIVER

  • When R ≤ min_i c_i,min, the sender’s optimal utility is zero; any positive state-1 sampling makes that source’s participation constraint fail.A zero-utility equilibrium is attained by assigning no sampling rate to any source.Source: §IV · p. 5
  • When R > min_i c_i,min, every positive-utility optimal policy either leaves a source unsampled or assigns it s_i > 0 and c_i = c_i,min.Thus, any source receiving state-1 sampling uses exactly the minimum state-0 rate required by its participation constraint.Source: §IV · pp. 5–6
  • For any fixed active-source set, allocating state-1 sampling rates is a convex optimization problem.The fixed-set objective is concave in the sampling rates, and the budget and nonnegativity constraints define a convex feasible region.Source: §IV · p. 6
  • Bisection finds the budget-satisfying multiplier when activation costs fit the budget; sets whose minimum activation costs exceed R are infeasible.For a fixed active set, the total allocation varies continuously and strictly decreases with θ.Source: §IV · p. 7
  • For a fixed active set, the KKT solution assigns a positive rate only to sources with A_i/B_i > θ; sets returning a zero rate for an active source can be discarded.The multiplier θ is chosen to satisfy the sampling budget.Source: §IV · p. 7

V. AN EFFICIENT ALGORITHM TO FIND THE SENDER’S OPTIMAL INFORMATION REVELATION POLICY

The paper uses source dominance and branch-and-bound to find a globally optimal active set without always enumerating every subset. Dominance propagation can reduce the search to O(n) nodes for a total dominance order, while worst-case complexity matches exhaustive search.

V. AN EFFICIENT ALGORITHM TO FIND THE SENDER’S OPTIMAL INFORMATION REVELATION POLICY

  • An active source uses c_i,min in state-0 sampling and receives positive state-1 sampling; the sender prefers low activation costs and high returns.The sender’s utility rises with state-1 sampling and falls with state-0 sampling.Source: §V · p. 7
  • Larger λ_i and smaller μ_i improve marginal and asymptotic utility while lowering activation cost; smaller q has the same effect on all three quantities.The passage relates smaller q to a receiver placing less weight on correctly estimating state-0, which relaxes the participation constraint.Source: §V · p. 8
  • If source i dominates source j, no optimal solution activates j while leaving i inactive; incomparable sources must be evaluated through the objective.The dominance relation is a strict partial order.Source: §V · p. 8
  • Branch-and-bound prunes dominance-inconsistent decisions, over-budget active sets, and subtrees whose relaxed utility upper bound cannot beat the best solution.The upper bound relaxes the undecided sources’ activation costs when allocating their state-1 sampling rates.Source: §V · pp. 8–9
  • Algorithm 1 returns a globally optimal active set; its worst-case complexity matches exhaustive search, but a total dominance order reduces visits to O(n) nodes.In the worst case it visits O(2n) nodes, each requiring at most two KKT subproblems.Source: §V · p. 9, Figure 3 · p. 9

VI. THE OPTIMAL INFORMATION REVELATION POLICY FOR MULTIPLE RECEIVERS WITH DIFFERENT BIASES

With dedicated channels, the sender chooses separate state-dependent Poisson sampling rates for each source–receiver pair. Each active pair must meet that receiver’s participation constraint, and the resulting joint allocation treats the pairs as items in the branch-and-bound search.

VI. THE OPTIMAL INFORMATION REVELATION POLICY FOR MULTIPLE RECEIVERS WITH DIFFERENT BIASES

  • The sender maximizes total utility across pairs under a shared conditional-intensity budget, with separate state-0 and state-1 sampling rates for each pair.The budget sums the pairwise rates, and the pairwise state-0 rate must meet its minimum whenever state-1 sampling is positive.Source: §VI · p. 10
  • Each source–receiver pair is either inactive or active with its state-0 sampling rate set to the minimum required by that receiver’s participation constraint.The required minimum depends on the receiver bias and the source parameters; state-1 sampling is positive for an active pair.Source: §VI · pp. 9–10
  • For any fixed active set, the state-1 rates have a unique optimal allocation obtained through the inner KKT allocation.Each pair has a fixed participation-constraint cost and a concave utility function in its state-1 rate.Source: §VI · p. 10
  • The pairwise dominance test uses marginal utility, minimum state-0 sampling rate, and asymptotic sender utility, evaluated using each receiver’s bias and the source parameters.The test replaces the source index with the source–receiver pair index.Source: Figure 4 · p. 10
  • The multi-receiver problem applies the branch-and-bound procedure to N = nm source–receiver pairs using dominance propagation, feasibility pruning, and upper-bound pruning.Its worst-case complexity is O(2^N T_KKT(N)); with a perfect ordering of pairs, it reduces to O(N T_KKT(N)).Source: §VI · p. 10

VII. NUMERICAL RESULTS

The section presents numerical results to illustrate the paper’s theoretical results and allocation method.

VII. NUMERICAL RESULTS

  • The numerical results illustrate the theoretical results and the behavior of the proposed allocation method.The section passage does not specify particular numerical findings.Source: §VII · p. 10

A. Multi source and Single receiver Case

In the five-source example, branch-and-bound identifies the optimal active set while evaluating fewer sets than exhaustive search. The numerical results also show how the sender’s allocations and utility change with the budget and source heterogeneity.

A. Multi source and Single receiver Case

  • The marginal-utility and asymptotic-utility rankings order sources as 5, 1, 2, 4, 3, while the minimum-rate ranking orders them as 5, 1, 2, 3, 4.Sources 3 and 4 are incomparable under the stated dominance relation, and both are dominated by source 2.Source: A. Multi source and Single receiver Case · p. 10
  • Dominance reduces the five-source search to six dominance-closed active sets among 31 nonempty possibilities.The algorithm also prunes branches using optimistic upper bounds when dominance alone is insufficient.Source: A. Multi source and Single receiver Case · p. 11
  • The optimal active set is {1, 2, 4, 5}; branch-and-bound visits 11 nodes instead of evaluating all 31 nonempty sets.Source 4 is selected over source 3 despite its higher minimum state-0 rate: their asymptotic utilities are 0.605 and 0.359, respectively.Source: A. Multi source and Single receiver Case · p. 11
  • At R = 10, the optimal active set is {1, 2, 5}; at R = 20, it is {1, 2, 4, 5}.At R = 10, the active sources receive positive state-1 rates and state-0 rates equal to their minimum required values.Source: Figure 7 · p. 11, A. Multi source and Single receiver Case · p. 11
  • As the budget rises, the optimal active sets expand in order from {5} to {1, 5}, {1, 2, 5}, {1, 2, 4, 5}, and finally all five sources.Each transition occurs when activating the next source at its minimum state-0 rate and allocating it some state-1 sampling becomes more beneficial than increasing rates for already active sources; receiver utility remains constant.Source: A. Multi source and Single receiver Case · p. 11
  • With q = 0.5 and λ_i = 1, greater heterogeneity in µ_i increases sender utility but decreases receiver utility.Heterogeneity creates sources with µ_i close to 1, lowering their minimum state-0 sampling rates and making them easier to activate.Source: A. Multi source and Single receiver Case · p. 12

B. Multi source and Multi receiver Case

In the multi-source, multi-receiver example, the sender allocates utility across source–receiver pairs under different total budgets. Low budgets concentrate allocation on a few pairs; higher budgets activate additional pairs.

B. Multi source and Multi receiver Case

  • The example uses four sources and four receivers with heterogeneous source parameters and receiver utility weights, and computes optimal per-pair contributions under budget R.The allocation is obtained with Algorithm 1.Source: B. Multi source and Multi receiver Case · p. 12
  • At R = 4, the sender activates only a few source–receiver pairs, concentrating allocation on dominant pairs such as (4, 2) and (3, 2).Their utility contributions are relatively large compared with the number of active links.Source: B. Multi source and Multi receiver Case · p. 12
  • At R = 8, additional pairs become active while previously selected pairs with relatively high contributions remain active.At R = 14, some pairs continue to have higher utility contributions than others.Source: B. Multi source and Multi receiver Case · p. 12

C. Persuasion-Optimized versus Receiver-Optimized Sampling

The receiver-optimized benchmark achieves higher receiver utility in the plotted regime, while persuasion yields strictly higher sender utility by using only the minimum communication needed for receiver compliance.

C. Persuasion-Optimized versus Receiver-Optimized Sampling

  • For inactive sources, no updates are sent and the receiver uses the prior-only default estimator.Its utility is qµi/(µi + λi).Source: C. Persuasion-Optimized versus… · p. 12
  • Using ci = ci,min for active sources keeps the receiver at baseline utility while leaving more budget for state-1 sampling.The remaining budget is allocated through si to maximize the sender’s objective.Source: C. Persuasion-Optimized versus… · pp. 12–13
  • The sender’s persuasion utility increases with R, with visible breakpoints when the optimal active set changes.The receiver-optimal allocation’s sender utility has visible jumps when it activates a new source with nontrivial (si, ci).Source: C. Persuasion-Optimized versus… · pp. 12–13
  • Persuasion-model sender utility lies strictly above sender utility evaluated at the receiver-optimal allocation.The receiver-optimal benchmark instead has higher receiver utility in the plotted regime.Source: C. Persuasion-Optimized versus… · pp. 12–13

VIII. CONCLUSION

The paper derives participation constraints and an explicit single-source solution, then develops methods for multi-source and multi-receiver settings. Numerical results show that persuasion increases sender utility relative to receiver-optimal sampling.

VIII. CONCLUSION

  • The paper derives closed-form participation constraints and an explicit solution for the single-source case.Source: §VIII · p. 13
  • For multiple sources under a conditional-intensity budget, a bisection procedure computes the KKT multiplier for a fixed active set, alongside an algorithm to find optimal active sources.The active-source algorithm can have lower search complexity.Source: §VIII · p. 13
  • The model extends to multiple receivers with heterogeneous biases, using the same algorithmic framework.Source: §VIII · p. 13
  • Numerical results show that persuasion increases sender utility compared with receiver-optimal sampling.They also illustrate budget allocation across source–receiver pairs and active-set changes as the conditional-intensity budget varies.Source: §VIII · p. 13
  • The authors identify multi-sender environments with aligned and misaligned objectives as future work.Source: §VIII · p. 13
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