arXiv CSopen access

Strategic Persuasion Through Information Timeliness

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Strategic Persuasion Through Information Timeliness

arXiv:2607.15939v1 [cs.IT] 17 Jul 2026

Ahmet Bugra Gundogan , Melih Bastopcu , Member, IEEE

Abstract—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. Index Terms—Information design, persuasion through information timeliness, continuous-time Markov chains, Stackelberg games, strategic communication.

I. I NTRODUCTION N public markets, misrepresentation is illegal [2] while delaying truthful disclosure can be lawful under specified conditions [3]. This distinction creates a setting in which the timing of truthful information disclosure can itself become a strategic instrument. Motivated by this, we model a firm whose fundamentals switch between “favorable” (denoted as state-1) and “unfavorable” (denoted as state-0) regimes as a binary continuous-time Markov chain (CTMC). The sender (owners or management) cannot lie about outcomes, but can lawfully manage when to release verifiable updates by communicating more frequently when fundamentals are good (with rate s) and more slowly when they are bad (with rate c), subject to the owner’s total conditional-intensity budget R. As another example, we can consider a news provider that supports a campaign but reports only true updates. The campaign’s “momentum” can be similarly modeled using binary unfavorable (0) and favorable (1) states. The news provider cannot falsify content, but it can control when truthful reports are aired by releasing favorable news at a higher rate and unfavorable news at a lower rate. These state-dependent reporting rates are chosen subject

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The authors are with the Department of Electrical and Electronics Engineering, Bilkent University, Ankara, Türkiye (e-mails: [email protected]; [email protected]). This work was supported in part by the TUBITAK 2232-B program (Project No: 124C533). A part of this paper will be presented at the 23rd IFAC World Congress, Busan, Republic of Korea, August 23-28, 2026, [1].

to a conditional-intensity provisioning budget, which limits the total rate capacity allocated to the two reporting modes. Thus, the provider strategically controls the timing of truthful news rather than its content. The receivers (investors in the prior example, or the news followers in the latter example) can adopt the firm’s (or the news provider’s) messages only if doing so does not worsen their long-run average utility relative to their prior knowledge, which is a participation-type condition that we call the participation constraint (PC). However, the firm’s (or the news agency’s) goal is to use the timeliness of the information to maximize the fraction of time that the audience believes that the fundamentals (or the campaign, in the latter example) are doing well, subject to the budget R and the PC. Motivated by these examples, the key research question that we investigate in this work is: “By controlling only the timing of the information provided to the receiver, can the information provider (the sender) persuade the receiver to act in a way that the provider’s utility is maximized?” We study this question within a restricted real-time architecture in which the sender commits to state-dependent Poisson update intensities and the receiver chooses between a zeroorder-hold estimator based on the sender’s updates and a prioronly default estimator. The resulting equilibrium therefore characterizes persuasion within these stationary, memoryless policy classes. Our work is related to two distinct strands of strategic information transmission (SIT). Crawford and Sobel [4] study strategic communication through cheap talk, in which the sender does not commit in advance to a signaling rule. In contrast, Bayesian persuasion [5] considers a sender that commits to an information-revelation policy to influence the receiver’s action. Our model follows the latter commitment-based structure: the sender first commits to state-dependent information-revelation rates, and the receiver subsequently chooses its best response. However, unlike classical Bayesian persuasion, where the sender designs what information to reveal, the strategic instrument in our setting is timeliness: the sender controls when truthful information is released over a dynamically evolving CTMC. The receiver follows messages only when doing so does not worsen its long-run average utility relative to its prior information. In dynamic approaches, Che et al. [6] study frictions and timing costs, yielding Markov-perfect outcomes and links to static benchmarks. When the sender commits to a disclosure policy over a Markov-evolving state, greedy policies are optimal for a class of problems [7], while [8] studies sequential disclosure to a privately informed receiver. In more applied settings, dynamic persuasion has been studied with exogenous signals shaping timing incentives [9], receiver search and inspection generating persuasion-acquisition feedback [10], partial sender knowledge motivating “starting rough” [11],

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quadratic state-dependent costs in Gaussian models [12], and continuous-time filtering/control approaches that capture belief dynamics [13]. Related notions of “timeliness” arise in models of optimal waiting [14] and in interim disclosure between mandatory announcement times [15]. Closest to our setting, Ely [16] shows how a sender schedules truthful disclosures about an evolving state to shape the receiver’s actions over time. Ashkenazi-Golan et al. [17] study a two-state Markov environment with a myopic receiver and characterize intertemporal disclosure/silence rules. Farhadi and Teneketzis [18] study a principal who sequentially discloses information about a two-state Markov chain with an absorbing bad state, so as to delay a strategic detector from detecting the jump to the bad state. Lehrer and Shaiderman [19] analyze Markovian persuasion with stochastic revelations. This rate-based formulation can also model real-time systems in which the timing of information plays a critical role. Recently, age of information (AoI) has been introduced to measure the timeliness of information in communication systems [20], [21]. The timely remote estimation problem for a Wiener process under a sampling-rate constraint has been considered in the seminal work of [22]. The timely tracking of Poisson counting processes and of infection status with exponential time intervals has been studied in [23], [24]. Recently, information sources have been modeled as Markov chains, and remote estimation problems have been studied to minimize the age of incorrect information (AoII) and related semantic metrics in [25]–[29]. More specifically, [30] studies the minimization of the age of false and missed alarms in remote estimation of a binary Markov source. In [31], the authors study age-aware CSI acquisition over a finite-state Markovian channel, balancing data transmission against the acquisition of fresh CSI. Timely task processing under state-dependent worker performance has been analyzed in the context of task completion efficiency in [32]. Related work studies revenue-maximizing job submission to a queried Markov machine based on the age of its estimated state [33]. Unlike timely remote estimation problems in the AoI literature, in our work, the sender and the receiver have misaligned objectives; consequently, by strategically adjusting the timeliness of updates, the sender seeks to persuade the receiver to maintain its estimate in the desired state. We summarize our main contributions as follows: • We introduce a dynamic persuasion framework in which a sender influences a receiver’s real-time estimate of binary CTMC sources solely by controlling the timing of truthful updates. We formulate the interaction as a Stackelberg game and derive a closed-form PC, which reduces to a minimum state-0 sampling rate ci,min for each source (Section II). • In Section III, we consider the single source case, where we explicitly characterize the Stackelberg equilibrium: whenever R > cmin , the optimal sender policy allocates exactly the minimum rate cmin to state-0 updates to satisfy the PC and devotes the entire remaining budget to state-1 updates. When R ≤ cmin , we show that the game admits multiple Stackelberg equilibria, all yielding the sender zero utility.

0

µ1 λ1

1

source 1

x1 (t)

receiver’s estimates x̂1 (t) = 1

0

µ2 λ2

1

x2 (t)

source 2 xn (t)

µn

0 λn

sender

mij

receiver

x̂2 (t) = 0 x̂n (t) = 1

1

source n

Fig. 1. Communication system with n sources, a sender, and a receiver.

In Section IV, we extend our analysis to the multi source setting, where each source may require a different ci,min to satisfy its PC. We demonstrate that, for any fixed set of sources, the sender’s rate-allocation problem for state-1 updates is a convex optimization problem. • In Section V, we provide a branch-and-bound algorithm that finds the sender’s globally optimal update rate allocation; its worst-case complexity equals that of exhaustive search, while dominance-based pruning reduces the search to O(n) nodes when the sources admit a total dominance order, and eliminates provably suboptimal active sets in intermediate cases. • In Section VI, we extend the problem to a multi source and multi receiver setting, where each receiver communicates with the sender over a dedicated private channel and may have a heterogeneous utility bias. We then solve the resulting problem optimally using the algorithms developed in the previous section. • Finally, in Section VII, we provide illustrative numerical results demonstrating that the sender can achieve persuasion solely by controlling the timeliness of information.

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II. S YSTEM M ODEL AND P ROBLEM F ORMULATION In this work, we consider a system composed of n ≥ 1 information sources, a sender, and a receiver. Here, each source denoted by Ii for i = 1, . . . , n generates binary information streams (0’s or 1’s) with time-varying dynamics. More specifically, the binary information at source Ii follows a two-state CTMC where the transition from state-0 to state1 happens with rate λi > 0 and from state-1 to state-0 with rate µi > 0 as shown in Fig. 1. We assume that the source processes are mutually independent. Moreover, since λi > 0 and µi > 0, the two-state CTMC associated with each source i is irreducible and therefore admits a unique stationary distribution. We denote source i’s state at time t as xi (t) ∈ {0, 1}. Both the sender and the receiver know (λi , µi ) for all i, but only the sender is capable of continuously observing these sources and sharing them with the receiver. On the other hand, since the receiver does not observe xi (t), with its prior knowledge, the receiver will only know the steady-state distribution of the CTMC, which is given by λi µi , π1i = . (1) π0i = µi + λi µi + λi The receiver would like to estimate the sources’ states as accurately as possible. Based on the prior knowledge of the sources’ rates and the information obtained from the sender, the receiver forms a real-time estimate about source i’s state,

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TABLE I T HE RECEIVER ’ S UTILITY FUNCTION ui (t). xi (t)\x̂i (t) 0 1

0 q 0

1 0 1−q

TABLE II T HE SENDER ’ S UTILITY FUNCTION vi (t). xi (t)\x̂i (t) 0 1

0 0 0

1 1 1

denoted by x̂i (t), at time t. With the sources’ states and their corresponding estimates, the receiver will obtain the following utility ui (t) from source i: ui (t) = q 1(xi (t)= 0, x̂i (t)= 0)+(1−q)1(xi (t) = 1, x̂i (t) = 1), which is shown in Table I where where 1(·) denotes the indicator function, which is equal to 1 if the statement in its argument holds and to 0 otherwise. In other words, when xi (t) = 0 and x̂i (t) = 0, the receiver will obtain a weighted reward, that is, ui (t) = q and when xi (t) = 1 and x̂i (t) = 1, it will receive ui (t) = 1 − q from source i where 0 < q < 1. When there is no information provided, the receiver will only know the steady state distribution of xi (t), i.e., (π0i , π1i ) for all i in (1). We assume that (1−q)λi i i qπ0i = µiqµ +λi > µi +λi = (1 − q)π1 for all i. As a result, when there is no information provided, the receiver’s default estimation is x̂i (t) = 0 for all t. Similarly, the sender’s utility obtained from source i at time t is denoted as vi (t) and is given in Table II. As opposed to the receiver, the sender’s utility will be equal to 1 only when the receiver’s estimate is x̂i (t) = 1 irrespective of source i’s state. Different from most traditional communication literature where the sender and the receiver have aligned goals, here, we note from Tables I and II that the sender’s and the receiver’s utility functions are different and the sender’s utility also depends on the receiver’s estimate at time t. More specifically, while the receiver wants to know the states as accurately as possible and thus, maximize a weighted correct-estimation utility, the sender wants the receiver to always estimate the state as x̂i (t) = 1 for all t. To model such a system, we consider a setting where the sender shares the sources’ states with the receiver at random times. When source i’s state is equal to 0, we model the sender’s inter-transmission times as exponentially distributed with rate ci ≥ 0. Similarly, when the source i’s state is equal to 1, we model the sender’s inter-transmission times as exponentially distributed with rate si ≥ 0. Let us denote the time instant at which the sender sends the jth update (where j ≥ 1) about the ith source’s state as ti,j . By denoting the sender’s jth message about source i’s state as mij = xi (ti,j ), under the condition that the receiver follows the sender’s messages (which we will specify precisely), the receiver will form the following estimate x̂i (t) at time t based on the received messages: x̂i (t) = mij , ti,j ≤ t < ti,j+1 , (2) where we assume that the receiver knows the initial values of the sources, i.e., mi0 = xi (0) and ti,0 = 0 for all i. This initialization is adopted only for notational convenience. Under the ergodicity conditions considered below (in particular, for every source with si > 0 and ci ≥ ci,min > 0), the joint

process (xi (t), x̂i (t)) admits a unique stationary distribution. Consequently, the long-term average utilities of both the sender and the receiver are independent of the initial source state and the initial estimate. As the objectives of the sender and the receiver are different, we formulate the interaction between these agents as a Stackelberg game, in which the sender acts as the leader and the receiver as the follower. In this Stackelberg game, the sender commits to a strategy first by choosing β = {β1 , · · · , βn } where βi = (si , ci ) for all i.1 Then, the receiver observes the sender’s information-revelation policy and selects its best response. At this point, the receiver has two options: (i) if the sender’s messages about source i lead to an estimate no worse than the initial knowledge in terms of maximizing R T the receiver’s long-term average utility, i.e., limT →∞T1 t=0 ui (t)dt, the receiver will follow the sender’s messages as in (2). We denote this policy as σsender . (ii) If following the sender’s messages leads to a lower average utility compared to the default policy which uses only the prior information, then the receiver will ignore the sender’s messages for source i and use the estimate x̂i (t) = 0 for all t which will give the utility of i qπ0i = µiqµ +λi from source i. We denote this policy as σdefault . Remark 1: We restrict the receiver’s strategy space to the two low-complexity estimators σsender and σdefault : under σsender , the receiver holds the most recently received message as its estimate as in (2), which is the standard zero-orderhold estimator widely adopted in the remote estimation and AoII literature [24], [25], [30], [34]. Accordingly, the PC in (10) compares the long-term average utilities of the two admissible estimators. Thus, the PC is a participation-type constraint rather than a per-history obedience requirement as in classical Bayesian persuasion [5]. We note that, since the sampling rates are state-dependent, silence itself carries information: a fully Bayesian receiver could track its posterior belief between updates and revert its estimate once the belief drops below its decision threshold, thereby achieving a utility no smaller than that of σsender and weakening the sender’s persuasion power. Together with the restriction of the sender to Poisson sampling policies (see footnote 1), this defines a Stackelberg game between two stationary, memoryless policy classes. Characterizing the equilibrium under a belief-tracking Bayesian receiver is an interesting direction for future work. Finally, we represent the sender’s utility function R T obtained from source i as JS,i (βi , BRi (βi )) = limT →∞ T1 t=0 vi (t)dt and the sender’s total average utility as JS (β, BR(β)) = P n i=1 JS,i (βi , BRi (βi )) which depends on the sender’s committed policy β and the receiver’s best response to β given by BRi (βi ) ∈ {σsender , σdefault }. Similarly, based on the sender’s policy, if the receiver follows the sender’s messages as in (2), the receiverR will obtain the average utility of T JR,i (βi ) = limT →∞ T1 t=0 ui (t)dt from source i where the receiver’s estimate x̂i (t) is determined in (2). Thus, we define the Stackelberg equilibrium as JS (β ∗ , BR(β ∗ )) ≥JS (β, BR(β)) for all β

(3)

1 In this work, we restrict the sender’s policy space to Poisson sampling and

characterize the corresponding Stackelberg equilibrium. Focusing on Poisson sampling policies allows for analytical tractability and has been considered in the literature, such as [23], [34].

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ci

(0,0)

(0,1)

µi

λi

µi

λi

(1,1)

(1,0) si

Fig. 2. Continuous-time Markov chain for (xi (t), x̂i (t)).

where

qµi , µi + λi BRi (βi ) = (4) qµ i σdefault , JR,i (βi ) < . µi + λi In other words, the Stackelberg equilibrium within the stated sender and receiver policy classes in (3) and (4) is achieved when the sender commits to a policy that will maximize its own utility function by also considering how the receiver will respond to the sender’s committed policy. In (4), we adopt optimistic leader-favorable tie-breaking: if the two admissible receiver policies yield the same long-term average receiver utility, the receiver selects σsender . For the degenerate noupdate policy βi = (0, 0), we adopt the convention that the receiver uses σdefault , since the sender provides no updates. A. The Receiver’s Average Utility Function As a result of following the sender’s messages, source i’s state and its estimate at the receiver (xi (t), x̂i (t)) form a CTMC with four states given as {(0, 0), (0, 1), (1, 0), (1, 1)} as shown in Fig. 2. By dropping source i’s index to derive a general expression and assuming that (s, c) 6= (0, 0), similar to the steps in [24], we find the unique stationary distribution of the CTMC given by π = {π00 , π01 , π10 , π11 }. For that, we first write the stationary balance equations as: π00 λ = π10 µ + π01 c, (5) π10 µ + π10 s = π00 λ, (6) π01 c + π01 λ = π11 µ, (7) π11 µ = π10 s + π01 λ. (8) P1 P1 π = 1, we Using the above equations and n=0 mn m=0 find the steady-state distribution of the CTMC as: µλs µλc λs(λ+c) µc(µ+s) , π01 = , π10 = , π11 = , π00 = κ κ κ κ where κ = (µ + λ)(µc + λs + cs). Then, from Table I, we can find the receiver’s long-term average utility obtained from source i as i i JR,i (βi ) = q π00 + (1 − q) π11 q µi ci (µi + si ) + (1 − q) λi si (λi + ci )
. (9) = (µi + λi ) µi ci + λi si + ci si Note that if the messages mij help the receiver to form an estimate no worse than the initial knowledge (which will be the compliance/participation condition for the two admissible receiver policies in our Stackelberg game), the receiver will follow the messages mij as in (2) and as a result obtain JR,i (βi ) in (9). Otherwise, the receiver will use x̂i (t) = 0 for i all t and obtain the utility of µiqµ +λi from source i.  σsender ,

JR,i (βi ) ≥

B. Sender’s Persuasion Problem As noted earlier, the sender’s utility function vi (t) provided in Table II depends on the receiver’s estimation. As a result, the sender wants to influence the receiver to follow its messages and affect x̂i (t) in a way to maximize its own utility. To do

that, the sender should commit to a policy βi = (si , ci ) such that the receiver’s utility JR,i (βi ) is greater than or equal to qµi µi +λi which is the PC. When we substitute JR,i (βi ) in (9) into i JR,i (βi ) ≥ µiqµ +λi and perform some algebraic manipulations, under the assumption that si > 0, we obtain qµi − λi . (10) ci ≥ ci,min = 1−q Due to our assumption that the receiver’s initial estimation without the sender’s information is equal to 0, that is qπ0i = (1−q)λi qµi i µi +λi > µi +λi = (1 − q)π1 , ci,min will always be positive, i.e., ci,min > 0. On the other hand, when si = 0, this constraint is automatically satisfied for all ci . We impose an additive budget on the state-conditioned update-intensity parameters, Pn 2 i=1 (ci + si ) ≤ R. Then, the sender’s persuasion problem becomes: n n X X λi si (ci + λi + µi ) i i (π01 + π11 )= max (µi + λi )(µi ci + λi si + ci si ) {si ,ci } i=1 i=1 n X (ci + si ) ≤ R s.t. i=1

ci ≥ 1(si > 0)ci,min , si ≥ 0, i ∈ {1, . . . , n}. (11) We note from (11) that Pnthe sender has the conditional intensity budget R such that i=1 ci + si ≤ R. The second constraint (ci ≥ 1(si > 0)ci,min ) in (11) is the PC for each source, and the third constraint is the feasibility constraint. 3 In the next section, we provide the Stackelberg equilibrium of the game formulated in (3) and (4) for a single source.

III. O PTIMAL I NFORMATION R EVELATION : S INGLE S OURCE AND S INGLE R ECEIVER In this section, we provide an explicit solution to the sender’s information-revelation problem in (11) and thereby characterize a Stackelberg equilibrium of the game formulated in (3) and (4) for a single source. For convenience, by dropping the source index i, we rewrite the sender’s optimization problem for a single source, i.e., when n = 1, as λs(c + λ + µ) max JS (s, c) = π01 + π11 = (µ + λ)(µc + λs + cs) {s,c} s.t. c + s ≤ R (12) c ≥ 1(s > 0)cmin , s ≥ 0. As seen in (12), there is a minimum sampling rate for c, qµ − λ, arising from the PC, which denoted as cmin = 1−q is strictly positive as mentioned before. In order to find the sender’s optimal solution, in the next lemma, we characterize how the sender’s utility function behaves with respect to the sampling rates s and c. 2 Thus, R prices the two conditional intensity controls rather than the realized stationary number of transmissions. Under policy (si , ci ), the realized average update intensity of source i is π0i ci + π1i si . 3 In the Stackelberg game formulated in (3) and (4), the sender may also choose si > 0 with ci < ci,min for some source i. In that case, the receiver’s best response is σdefault , and the sender obtains zero utility from source i while consuming the positive rate si + ci . Such a policy is weakly dominated by allocating (si , ci ) = (0, 0) to source i, which yields the same zero utility while freeing the rate si + ci ; hence, the constraint ci ≥ 1(si > 0) ci,min in (11) entails no loss of optimality. Similarly, the sources from which the sender seeks no utility are represented by (si , ci ) = (0, 0), in which case the corresponding summand in (11) is defined to be zero, consistent with the zero utility obtained under σdefault . Consequently, the optimal values of (11) and the Stackelberg game coincide.

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Lemma 1: Under the assumption that the PC holds, the sender’s utility JS (s, c) in (12) is strictly increasing in s when c > 0, and strictly decreasing in c when s > 0. Proof: We begin the proof by showing that JS (s, c) is an increasing function of s when c > 0. For that, the partial derivative of JS (s, c) with respect to s is given by ∂JS (s, c) λµc(λ + µ + c) . = ∂s (µ + λ)(µc + λs + cs)2 (s,c) > 0 when c > 0, which is indeed the Thus, we have ∂JS∂s case by the PC, as it implies that c ≥ cmin > 0. Similarly, when s > 0, JS (s, c) is decreasing in c since ∂JS (s, c) λµs(λ + µ + s) . =− ∂c (µ + λ)(µc + λs + cs)2 (s,c) < 0 when s > 0, which As a result, we have ∂JS∂c completes the proof.  Thus, whenever the PC can be met, to maximize its own utility the sender should allocate most of its sampling rate to s and just enough of its sampling rate to c to meet the PC. In the next theorem, we characterize the Stackelberg equilibrium of the game for the single source case. Theorem 1: The single source game admits the following Stackelberg equilibrium characterization: (i) If R ≤ cmin , the sender’s optimal utility is zero. One such Stackelberg equilibrium attaining this utility is (β ∗ , BR(β ∗ )) = ((0, 0), σdefault ). (ii) If R > cmin , the sender’s optimal policy is β ∗ = (s̄, cmin ) with s̄ = R − cmin > 0, to which the receiver’s best response is BR(β ∗ ) = σsender under the tie-breaking convention in (4). At this equilibrium, the sender obtains the utility JS (β ∗ , BR(β ∗ )) =

λs̄ (cmin + λ + µ)
. (µ + λ) µcmin + λs̄ + cmin s̄

In both cases, the receiver’s equilibrium utility equals its priorqµ . only utility, JR (β ∗ , BR(β ∗ )) = µ+λ Proof: We begin our proof by considering the case when R ≤ cmin and show that every feasible policy yields the sender zero utility. Consider any policy β = (s, c) with c + s ≤ R. If s > 0, then c ≤ R − s < R ≤ cmin , so the sender cannot meet the PC in (10). Consequently, the receiver would not follow the sender’s messages, i.e., BR(β) = σdefault , and would keep its estimate x̂(t) = 0 for all t, in which case the sender would obtain zero utility. If instead s = 0, the sender never reports state-1, and thus obtains zero utility under any best response of the receiver, since x̂(t) = 1 can hold at most during an initial transient; in this case, the receiver obtains its default utility under either response. In particular, when R = cmin , the only policies satisfying the PC are those with s = 0 and c = cmin , which fall into the second case. Since all feasible policies yield the sender the same (zero) utility when R ≤ cmin , we choose (β ∗ , BR(β ∗ )) = ((0, 0), σdefault ) as one such Stackelberg equilibrium, at which the sender obtains JS (β ∗ , BR(β ∗ )) = 0 qµ , which and the receiver obtains JR ((0, 0), σdefault ) = µ+λ establishes part (i). Next, we consider the case when R > cmin . In this case, the sender has a sufficient conditional intensity budget to

persuade the receiver to follow its messages while allocating s̄ = R−cmin > 0 to state-1 updates. By Lemma 1, the sender’s utility is an increasing function of s and a decreasing function of c when s > 0 and c > 0. As a result, the sender should set c = cmin to satisfy the PC and then allocate the remaining sampling rate to s to maximize its own utility. Hence, when R > cmin , the Stackelberg equilibrium is achieved at (β ∗ , BR(β ∗ )) = ((R − cmin , cmin ), σsender ), where BR(β ∗ ) = σsender follows from the tie-breaking convention embedded in (4). The corresponding utilities of the sender and the receiver λs̄(cmin +λ+µ) with are given by JS (β ∗ , BR(β ∗ )) = (µ+λ)(µc min +λs̄+cmin s̄) qµ s̄ = R − cmin and JR (β ∗ , BR(β ∗ )) = µ+λ , respectively, which establishes part (ii).  When R > cmin , as a result of applying the sender’s optimal policy found in Theorem 1, the receiver follows the sender’s messages, i.e., σsender , and the receiver’s utility is the same as in the case with only prior information due to the optimistic leader-favorable tie-breaking assumption. However, the sender benefits from applying this policy, as the receiver’s estimate equals 1 for some portion of the time. Building on these insights, we next generalize the results from the single source case to the multi source setting. IV. O PTIMAL I NFORMATION R EVELATION : M ULTI S OURCE AND S INGLE R ECEIVER In this section, we extend our analysis to a setting where the sender reveals information about multiple sources to the receiver. Our goal is to solve the general persuasion problem with n ≥ 1 sources in (11). The sender’s informationrevelation policy is more involved since the sender should decide which information source it should sample and at which rates. In order to characterize the sender’s optimal informationrevelation policy, we start with the setting in which the sender’s total budget is limited by R ≤ mini∈{1,...,n} ci,min . Lemma 2: When R ≤ mini∈{1,...,n} ci,min , the sender’s optimal utility is zero. One such Stackelberg equilibrium ∗ achieving this utility is βi∗ = (0, 0) and i ) = σdefault PnBRi (β qµi for all i, at which the receiver obtains i=1 µi +λi .

Proof: For any feasible policy, every source i with si > 0 satisfies ci ≤ R − si < ci,min , so its PC fails. As a result, the receiver chooses BRi (βi ) = σdefault and the sender obtains JS,i (βi , BRi (βi )) = 0. On the other hand, every source with si = 0 yields JS,i (βi , BRi (βi )) = 0 under any best response of the receiver, by the argument in the proof of Theorem 1(i). Hence the sender’s utility at equilibrium is zero, and it can be attained by βi∗ = (0, 0) with BRi (βi∗ ) = σdefault for all i.  In the remaining part of this section, we focus our attention on the setting where R > mini∈{1,...,n} ci,min . Thus, at least by allocating all of its sampling rate, the sender is capable of persuading the receiver to follow its messages for some sources. To characterize the sender’s optimal informationrevelation policy, next, we state that the sender allocates a positive sampling rate si > 0 to source i’s state-1 if and only if the sampling rate for source i’s state-0 satisfies ci = ci,min . Lemma 3: When R > mini∈{1,...,n} ci,min , any optimal policy with positive sender utility satisfies the following: for every source i, exactly one of a) (s∗i , c∗i ) = (0, 0); or b) s∗i > 0 and c∗i = ci,min holds.

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Proof: Since R > mini ci,min , the policy that assigns cm = cm,min and sm = R − cm,min > 0 to a source m ∈ arg mini ci,min and (0, 0) to all other sources is feasible and, by Lemma 1, yields a strictly positive utility. Hence, the sender’s optimal policy satisfies JS (β ∗ , BR(β ∗ )) > 0, and there exists a source j with JS,j (βj∗ , BRj (βj∗ )) > 0, which requires s∗j > 0 and c∗j ≥ cj,min . Moreover, by (10), the PC threshold cj,min does not depend on sj ; thus, increasing sj alone never violates source j’s PC. First, suppose that s∗i > 0 and c∗i < ci,min for some i. Then, the PC fails for source i, so JS,i (βi∗ , BRi (βi∗ )) = 0 while source i consumes the rate δ = s∗i + c∗i > 0. Note that i 6= j since JS,j (βj∗ , BRj (βj∗ )) > 0. By reallocating this rate, i.e., setting (si , ci ) = (0, 0) and sj = s∗j + δ, the total rate and all other sources are unchanged, source j’s PC still holds, and JS,j strictly increases by Lemma 1 since c∗j ≥ cj,min > 0, contradicting optimality. Second, suppose that s∗i > 0 and c∗i > ci,min for some i. By setting ci = ci,min and si = s∗i +(c∗i −ci,min ), the total rate is unchanged and the PC for source i still holds with equality. Since JS,i is strictly decreasing in ci and strictly increasing in si by Lemma 1, this modification strictly increases JS,i , again contradicting optimality. Therefore, if s∗i > 0, then c∗i = ci,min . It remains to consider the case s∗i = 0 and c∗i > 0 for some source i. Since s∗i = 0, source i yields zero sender utility while consuming the positive budget c∗i > 0. Since the optimal sender utility is strictly positive, there exists at least one source j with s∗j > 0 and, from the preceding arguments, c∗j = cj,min > 0. We can therefore set (si , ci ) = (0, 0) and increase sj to s∗j + c∗i , while keeping all other variables unchanged. This modification preserves the total budget and source j’s PC, since c∗j = cj,min remains unchanged. By Lemma 1, increasing sj strictly increases the sender’s utility, contradicting the optimality of the original policy. Hence, s∗i = 0 implies c∗i = 0. Combining the above cases, every source i at an optimal policy satisfies either (s∗i , c∗i ) = (0, 0), or s∗i > 0 and c∗i = ci,min , which completes the proof.  Thus, Lemma 3 shows that, at an optimal policy with positive sender utility, each source is either not sampled, or sampled with si > 0 and ci = ci,min . Accordingly, we introduce a binary activation variable zi ∈ {0, 1} for each source: zi = 1 indicates that source i is active, i.e., sampled with si > 0 and ci = ci,min , whereas zi = 0 indicates that it is inactive, i.e., not sampled, with (si , ci ) = (0, 0). The sender’s problem in (11) can then be reformulated as n X si (ci,min + µi + λi ) λi ˆ zi max JS (z, s) = λi + µi ci,min µi + si λi + ci,min si {zi ,si } i=1 n X (zi ci,min + si ) ≤ R s.t. i=1

zi ∈ {0, 1}, si ≥ 0, i ∈ {1, . . . , n}. (13) Thus, zi captures whether source i is activated, while the corresponding state-0 update rate of an active source is fixed at the minimum value required by the PC. Therefore, multiplying the objective function by zi in (13) appropriately captures this behavior. Similarly, conditional intensity budget constraint Pthe n can be written as i=1(zi ci,min +si ) ≤ R. To solve the optimization in (13), we fix the values of zi and then solve the resulting optimization problem over the

variables si . With this goal, for a given set of zi ’s, we first analyze the convexity of the sender’s persuasion problem in (13) with respect to si . Lemma 4: For a given set of zi ’s, the sender’s informationrevelation problem in (13) is a convex optimization problem. Proof: The first and second derivatives of the sender’s utility function are given by λi ci,min µi (ci,min + µi + λi ) ∂ JˆS (z, s) = zi . ∂si λi + µi (ci,min µi + si λi + si ci,min )2 λi ci,min µi (λi +ci,min )(ci,min + µi + λi ) ∂ 2 JˆS (z, s) = −2zi . ∂s2i λi + µi (ci,min µi + si λi + si ci,min )3 Since the first derivative is non-negative and the second deriva2 ˆ ˆ S (z,s) ≤ 0, ≥ 0 and ∂ J∂s tive is non-positive, that is, ∂ JS∂s(z,s) 2 i i respectively, we can conclude that the sender’s utility is a concave non-decreasing function P of si . Since the conditional intensity budget constraint, ni=1 (zi ci,min + si ) ≤ R,4 and the feasibility constraint, si ≥ 0, define a convex feasible region, the optimization problem in (13) is convex for any fixed set of zi values.  For a given set of zi ’s, let us denote S as the set of active source indices such that zi = 1. Then, the complement S c is the set of inactive source indices with zi = 0. For sources in S c , we have si = 0 due to Lemma 3. For the remaining sources in S, we introduce the Lagrangian function [35] for (13) to find their optimum update rates si : X λi si (ci,min + µi + λi ) L=− λi + µi ci,min µi + si λi + ci,min si i∈S ! X X +θ (ci,min + si ) − R − νi s i , i∈S

i∈S

where θ ≥ 0 and νi ≥ 0 for all i. Next, the KKT conditions are given by λi ci,min µi (ci,min + µi + λi ) ∂L =− +θ−νi=0, (14) ∂si λi + µi (ci,min µi +si λi +si ci,min )2 for all i ∈ S. Then, the complementary slackness (C.S.) conditions can be stated as follows: ! X θ (ci,min + si ) − R =0, (15) i∈S

νi si =0, for all i ∈ S. By solving (14) for si , we obtain s ! Ai −1 , si = Ci Bi (θ − νi )

(16)

(17)

where Ai = λi (ci,min + µi + λi ), Bi = ci,min µi (λi + µi ), c µi and Ci = λii,min +ci,min . By the C.S. condition (16), either si > 0 (which implies νi = 0) or si = 0 (with νi ≥ 0). Thus, the optimal values of si , denoted by s∗i , are equal to !+ r Ai ∗ si = Ci −1 , (18) Bi θ

where (x)+ = max{x, 0}. Although the closed-form solution for each s∗i is given in (18), it depends on the Lagrange multiplier θ, which 4 For some given sets of z ’s, we may have R − i

which case the problem is infeasible.

Pn

i=1 zi ci,min < 0, in

7

must be chosen such that the conditional intensity budget P constraint in (13) is satisfied with equality, P i.e., i∈S (ci,min + s∗i ) = R. From (18), the total allocation i∈S (ci,min + s∗i ) is a continuous and strictly decreasing function of θ over + i 0 < θ ≤ maxi∈S A Bi . As θ → 0 , the total allocation Ai approaches infinity, whereas atPθ = maxi∈S B , it is equal i P to i∈S ci,min . Therefore, if i∈S ci,min < R, there exists a unique θ satisfying the conditional intensity budget, which can be obtained via bisection. At each iteration, we evaluate the total allocation at the midpoint of the current interval and update the bounds according to whether the resulting allocation is greater than or less than R. On the other hand, P if i∈S ci,min > R, the active set S is infeasible and can be discarded directly. From the expression of s∗i in (18), the optimal allocation for a given active set S exhibits a threshold structure: only Ai the sources with B > θ receive a positive update rate. By i Lemma 3, a globally optimal policy cannot allocate ci = ci,min and si = 0 to any source; hence, whenever the solution in (18) returns s∗i = 0 for some i ∈ S, the candidate set S can be discarded without loss of optimality, since by Lemma 3 no optimal solution of (13) activates a source with a zero state-1 sampling rate. The sender’s globally optimal policy can therefore be found by enumerating the candidate active sets, solving (18) for each, and eliminating the discarded sets. V. A N E FFICIENT A LGORITHM TO F IND THE S ENDER ’ S O PTIMAL I NFORMATION R EVELATION P OLICY In this section, we develop a branch-and-bound algorithm that finds the sender’s globally optimal information-revelation policy with typically lower computational complexity than the exhaustive search described in Section IV. Recall from Lemma 1 that, when si > 0 and ci > 0, the sender’s utility is strictly increasing in si and strictly decreasing in ci , whereas the PC requires the sender to allocate ci = ci,min to every source whose messages the receiver follows. Hence, the sender should ideally activate the sources that require a small ci,min while providing a large utility in return. To identify such sources, we first express the sender’s utility obtained from an active source i (i.e., a source with ci = ci,min and si > 0) in a more informative form. Recall from (13) that the sender’s utility obtained from source i when zi = 1 (i.e., ci = ci,min ) is given by JS,i (si , ci,min ) =

si (ci,min + µi + λi ) λi . (19) λi + µi ci,min µi + si (λi + ci,min )

Since ci is fixed to ci,min for all active sources, with a slight abuse of notation, in the remainder of this section we write JS,i (si ) instead of JS,i (si , ci,min ). By using the constants Ai , Bi , and Ci defined in (17), we can rewrite (19) as JS,i (si ) =

si Ai Ci . Bi Ci + si

(20)

We observe from (20) that the utility contribution of an active source i is fully characterized by three quantities: the i marginal utility A Bi , the minimum state-0 sampling rate ci,min , and the asymptotic ceiling of the sender’s utility from source

Ai i given by ABi Ci i . The first quantity is the marginal utility B , i which determines the initial rate of return at si = 0, i.e.,

∂JS,i (si ) Ai λi = = . (21) ∂si Bi (λi + µi ) (qµi − (1 − q)λi ) si =0 Ai is also the Since JS,i (si ) is concave in si by Lemma 4, B i largest marginal return that source i can offer. Moreover, we recall from the KKT solution in (18) that, for a fixed active Ai set, only the sources with B > θ receive a nonzero allocation i ∗ si > 0. The second quantity is the minimum state-0 sampling qµi − λi , which is the fixed sampling rate that rate ci,min = 1−q the sender must allocate to source i’s state-0 information in order to satisfy the PC. The third quantity is the asymptotic sender utility ABi Ci i , which is the maximum utility that source i can provide as si → ∞, i.e., Ai Ci λi lim JS,i (si ) = = . (22) si →∞ Bi q(λi + µi ) i Thus, while A Bi and ci,min determine whether source i receives a positive allocation, ABi Ci i limits how much utility source i can ultimately contribute, even under an unlimited sampling rate. In the next lemma, we characterize how these three quantities vary with respect to λi , µi , and q. i Lemma 5: The marginal utility A Bi and the asymptotic utility Ai C i Bi are both increasing in λi and decreasing in µi and q, whereas the activation cost ci,min is decreasing in λi and increasing in µi and q. i Proof: We first show the monotonicity of A Bi . The partial Ai derivatives of Bi with respect to λi and µi are given by   Ai qµ2i + (1 − q)λ2i ∂ = > 0, ∂λi  Bi  (λi + µi )2 (qµi − (1 − q)λi )2 λi ((2q − 1)λi + 2qµi ) Ai ∂ =− 2 < 0, ∂µi Bi (λi + µi )2 (qµi − (1 − q)λi ) where the latter is strictly negative since (2q − 1)λi + 2qµi = 2 (qµi − (1 − q)λi ) + λi > 0 due to our initial assumption that qµi − (1 − q)λi > 0 (i.e., the receiver’s default estimate is x̂i (t) = 0). In addition, we have   ∂ Ai λi =− 2 < 0, ∂q Bi (qµi − (1 − q)λi )

Ai since λi > 0. Thus, B is increasing in λi and decreasing in i i µi and q. Next, for the asymptotic utility ABi Ci i = q(λiλ+µ , i) the partial derivatives are given by   µi Ai Ci ∂ = > 0, ∂λi Bi q(λi + µi )2   λi Ai Ci ∂ =− < 0, ∂µi Bi q(λi + µi )2   λi ∂ Ai Ci < 0. =− 2 ∂q Bi q (λi + µi ) Thus, the asymptotic utility ABi Ci i exhibits the same monotonic i behavior as the marginal utility A Bi . ∂ci,min ∂ci,min q Finally, since ∂λi = −1 < 0, ∂µ > 0, and = 1−q i ∂ci,min µi = (1−q)2 > 0, the minimum state-0 sampling rate ∂q ci,min is decreasing in λi and increasing in µi and q, which completes the proof. 

8

Lemma 5 shows that a larger λi and a smaller µi (i.e., a source that spends a larger fraction of time in state-1) Ai simultaneously improve the initial slope B and the asymptotic i Ai C i sender utility Bi while reducing ci,min . Similarly, a smaller q makes persuasion easier in all three aspects: the marginal and asymptotic utilities increase while ci,min decreases. This is consistent with the interpretation that a smaller q means the receiver places less weight on correctly estimating state-0, which relaxes the PC and makes persuasion less costly for the sender. Based on these three quantities, we can now formally characterize when one source dominates another. Lemma 6 (Source Dominance): If source i satisfies the following conditions relative to source j, with at least one of them being strict Ai Aj ≥ , (23) Bi Bj ci,min ≤ cj,min , (24) Aj Cj Ai Ci ≥ , (25) Bi Bj then no optimal solution to (13) can activate source j while leaving source i inactive. In this case, we say that source i dominates source j and express the dominance as i ≻ j. Proof: We prove the result in two steps. First, we show that conditions (23) and (25) imply the point-wise utility dominance JS,i (s) ≥ JS,j (s) for all s ≥ 0. Then, by using a swap argument together with condition (24), we prove the dominance claim. Step 1 (Point-wise utility dominance): At s = 0, both utilities are equal to zero, and thus the inequality trivially holds. For any s > 0, from (20), the condition JS,i (s) ≥ JS,j (s) is equivalent to s s Aj Cj Ai Ci ≥ . Bi Ci + s Bj Cj + s Since s > 0, Ci + s > 0, and Cj + s > 0, canceling s and cross-multiplying yields the equivalent condition Ai Ci Aj Cj (Cj + s) ≥ (Ci + s). Bi Bj . Ak k Ck By noting that Ck = AB Bk for k ∈ {i, j} and rearranging k the terms, theconditionJS,i (s) ≥ J S,j (s) becomes Ai Aj Ai Ci Aj Cj Ai Ci Aj Cj Ai Aj − s − + ≥ 0. (26) Bi Bj Bi Bj Bi Bj Bi Bj A C The first term in (26) is non-negative since ABi Ci i , Bj j j > 0 A

j i and A Bi ≥ Bj by (23). Similarly, the second term is non-

A

A C

Ai C i j i negative since A ≥ Bj j j by (25). Bi , Bj , s > 0 and Bi Hence, we have JS,i (s) ≥ JS,j (s) for all s ≥ 0. Furthermore, when at least one of the inequalities in (23) or (25) is strict, the corresponding term in (26) is strictly positive, and thus JS,i (s) > JS,j (s) for all s > 0. Step 2 (Swap argument): Assume by contradiction that, in an optimal solution, source j is active with cj = cj,min and sj > 0 while source i is inactive. Consider the following alternative policy: we deactivate source j, which frees a total sampling rate of cj,min +sj , and activate source i by allocating ci = ci,min and si = sj + (cj,min − ci,min ), while keeping the allocations of all the other sources unchanged. By condition (24), we have cj,min ≥ ci,min , and thus si ≥ sj > 0. Moreover,

the total sampling rate consumed by the new allocation is ci,min + si = cj,min + sj , which is exactly equal to the freed rate; hence, the total sampling constraint remains satisfied. We now distinguish two cases depending on which of the conditions (23)–(25) holds with strict inequality. If at least one of (23) or (25) is strict, then Step 1 gives JS,i (sj ) > JS,j (sj ); since JS,i (·) is strictly increasing by Lemma 1 and si ≥ sj , we obtain JS,i (si ) ≥ JS,i (sj ) > JS,j (sj ). If instead (23) and (25) hold with equality while (24) is strict, then (26) holds with equality, i.e., JS,i (s) = JS,j (s) for all s ≥ 0, but we have si = sj + (cj,min − ci,min ) > sj ; hence, JS,i (si ) > JS,i (sj ) = JS,j (sj ) again by the strict monotonicity of JS,i (·). In both cases, the alternative policy yields a strictly higher utility for the sender, contradicting the optimality of the original solution. Therefore, no optimal solution can activate source j while leaving source i inactive, which completes the proof.  The dominance relation ≻ defined through (23)–(25) forms a strict partial order among the sources, since it is irreflexive by the requirement of at least one strict inequality and transitive as the weak inequalities in (23)–(25) are preserved along a chain of comparisons. When the three conditions do not consistently favor one source over the other, the two sources are incomparable under ≻; in this case, their relative contributions cannot be determined a priori and must be evaluated directly through the objective function in (13). Based on the dominance relation ≻ established in Lemma 6, we now develop a branch-and-bound algorithm that finds the optimal active set S ∗ without exhaustively enumerating all 2n possible subsets. Throughout the search, the algorithm maintains three disjoint sets: Sin , the set of sources decided to be active; Sout , the set of sources decided to be inactive; and Sund , the set of sources yet to be decided. At each step, the algorithm selects one source from Sund and branches into two subproblems: one in which the selected source is moved to Sin and one in which it is moved to Sout . This branching induces a binary tree whose leaves correspond to fully determined active sets. To avoid exploring the entire tree, the algorithm employs three pruning mechanisms, which we describe next. The first mechanism is dominance propagation, which exploits the partial order ≻ obtained in Lemma 6. Whenever a source j is moved to Sin , every source i ∈ Sund with i ≻ j must also be moved to Sin , since by Lemma 6 no optimal solution can activate source j while leaving source i inactive. Conversely, whenever a source i is moved to Sout , every source j ∈ Sund with i ≻ j must also be moved to Sout , since if the dominating source i is not worth activating, then neither is the dominated source j. These forced decisions can cascade: a single inclusion or exclusion may trigger a chain of further forced decisions until no additional propagation is possible. After the propagation, if there exist i ∈ Sout and j ∈ Sin with i ≻ j, then the current partial assignment contradicts Lemma 6, and the corresponding node is pruned. The second mechanism is feasibility pruning. If the total activation cost of the sources in Sin exceeds the conditional P c intensity budget, i.e., i∈Sin i,min > R, then no feasible completion of the current partial assignment exists, and the node is pruned. The third mechanism is upper bound pruning. At each node,

9

Algorithm 1 Branch-and-Bound for the Optimal Active Set Require: {(λi , µi )}ni=1 , q, R Ensure: Optimal active set S ∗ and utility J ∗ A Ai C i 1: Compute Bi , ci,min , B for all i; build the dominance i i relation ≻ via Lemma 6 2: J ∗ ← 0, S ∗ ← ∅; S EARCH (∅, ∅, {1, . . . , n}) 3: procedure S EARCH (Sin , Sout , Sund ) 4: Propagate ≻ until stable: if j∈Sin , i∈Sund , i≻j, move i→Sin ; if i∈Sout , j∈Sund , i≻j, move j→Sout 5: if ∃ i ∈ Sout , j ∈ Sin with i ≻ j then return 6: end Pif 7: if i∈Sin ci,min > R then return 8: end if P 9: if Sin 6= ∅ and i∈Sin ci,min < R then solve (18) over Sin to obtain J(Sin ); 10: if s∗i > 0 for every i ∈ Sin and J(Sin ) > J ∗ then ∗ J ←J(Sin ), S ∗ ←Sin ; 11: end if 12: end if 13: if Sund = ∅ then return 14: end if P 15: Rrem ← R − i∈Sin ci,min ; if Rrem = 0 then return 16: Solve (18) over Sin ∪ Sund with rate Rrem for the si ’s to obtain UB; if UB ≤ J ∗ then return i 17: k ← arg maxi∈Sund A Bi 18: S EARCH(Sin ∪ {k}, Sout , Sund \ {k}) 19: S EARCH(Sin , Sout ∪ {k}, Sund \ {k}) 20: end procedure we compute an upper bound, denoted by UB, on the utility achievable by any completion of the current partial assignment. If UB does not exceed the utility of the best solution found so far, then no completion can yield a higher utility, and the entire subtree rooted at the current node is pruned. To compute UB, we solve the KKT allocation in (18) over Sin ∪ Sund , where the P total rate available for the sampling rates si is Rrem = R− i∈Sin ci,min , that is, without charging the activation costs ci,min of the undecided sources. This relaxation yields a valid upper bound as follows. Any feasible completion of the current node that activates aP subset T ⊆ SP und can allocate a total rate of at most R − i∈Sin ci,min − i∈T ci,min ≤ Rrem to its sampling rates si , and only to the sources in Sin ∪ T ⊆ Sin ∪Sund . Hence, the si allocation of every feasible completion is also feasible for the relaxed problem, and consequently, UB is greater than or equal to the utility of the best achievable completion. While the branching order does not affect the correctness of the algorithm, it influences its practical efficiency. As a heuristic, we branch on the sources in decreasing order of their i marginal utility A Bi . This choice tends to discover high-quality solutions early in the search, which in turn strengthens the upper bound pruning in the subsequent branches. The complete procedure is given in Algorithm 1. Since the pruning mechanisms only eliminate subtrees that provably cannot contain an optimal active set, Algorithm 1 is guaranteed to return the globally optimal solution of (13). Regarding the computational complexity, each node requires

receivers’ estimates x̂11 (t) = 1

0

µ1 λ1

receiver 1

1

source 1

x1 (t)

x̂n1 (t) = 1 m1ij

0

µ2 λ2

1

x2 (t)

sender

m2ij

x̂12 (t) = 1

receiver 2

source 2 xn (t)

x̂21 (t) = 0

x̂22 (t) = 0 x̂n2 (t) = 1

mm ij

x̂1m (t) = 1 µn

0 λn

1

source n

receiver m

x̂2m (t) = 0 x̂nm (t) = 1

Fig. 3. Communication system with n sources, a sender, and m receivers.

solving at most two KKT subproblems via the bisection search described in Section IV, each with a cost of TKKT (n), where TKKT (n) denotes the complexity of a single bisectionbased KKT solution over at most n sources. In the worst case, the algorithm visits O(2n ) nodes, resulting in a total runtime of O(2n TKKT (n)). In this case, Algorithm 1 has the same complexity as the exhaustive search described in Section IV. In contrast, when the dominance relation ≻ forms a total order (i.e., a chain), the dominance propagation resolves all branching decisions, and the algorithm visits only O(n) nodes. For the intermediate cases, the number of visited nodes depends on the structure of the partial order, where the pruning mechanisms eliminate the subtrees corresponding to dominance-violating and suboptimal active sets. VI. T HE O PTIMAL I NFORMATION R EVELATION P OLICY FOR M ULTIPLE R ECEIVERS WITH D IFFERENT B IASES In this section, we extend the single receiver model by considering m ≥ 2 receivers indexed by r ∈ {1, . . . , m} and n independent sources indexed by i ∈ {1, . . . , n} as shown in Fig. 3. Receiver r’s utility is characterized by substituting the parameter qr ∈ (0, 1) in Table I, which plays the same role as q in the single receiver formulation. The sender observes the source(s) and commits to Poisson sampling policies. If the sender satisfies the PC, each receiver r forms an estimate about source i’s state denoted by x̂ir (t) using the most recent message it receives, following the same procedure as in the single receiver model provided in (2). If the sender fails to satisfy the PC for receiver r regarding the state of source i, then receiver r follows the default policy σdefault , under which it sets x̂ir (t) = 0 for all t. We focus on the dedicated communication channel setting in which the sender can communicate with each receiver over a private channel, and thus can select receiver-specific sampling rates. Throughout this section, we use the same notation and utility definitions as in the single receiver model, with the only modification that the receiver parameter is now indexed by r. In particular, whenever the sender intends receiver r to follow the information provided by the sender (i.e., when a positive sampling rate for state-1 is used), receiver r’s PC requirement induces a minimum state-0 sampling rate that depends on qr and source i’s parameters λi and µi . We assume that qr µi > (1 − qr )λi for all (i, r), so that each receiver’s default estimate for every source is x̂ir (t) = 0, consistent with the single receiver assumption. Under dedicated communication channels and the Poisson sampling policy introduced earlier, the sender can select a distinct Poisson sampling rate for each source–receiver

10

pair (i, r), such that we have (sir , cir ) for i = 1, . . . , n and r = 1, . . . , m. Here, cir is the sampling rate when xi (t) = 0 and sir is the sampling rate when xi (t) = 1. The sender to the conditional intensity budget Pnis subject Pm given by i=1 r=1 (sir + cir ) ≤ R with the feasibility constraints sir ≥ 0 and cir ≥ 0. For each pair (i, r), the same participation constraint argument as in the single receiver model applies, with q replaced by qr . In particular, whenever we have sir > 0, it is necessary that cir ≥ cir,min where qr µi cir,min , 1−q − λi . Since the sender utility for a fixed source r is increasing in s and decreasing in c for s > 0, any feasible solution can be improved by setting cir = cir,min whenever sir > 0. Thus, the sender either keeps pair (i, r) inactive by selecting (sir , cir ) = (0, 0) or activates it with cir = cir,min and sir > 0. Under dedicated communication channels with multiple receivers, the sender maximizes the total utility across all pairs: n X m X λi sir (cir + λi + µi )
JS,ir (sir ,cir)= {sir ,cir} (λ +µi) µi cir+λi sir+cir sir i=1 r=1 i=1r=1 i

max s.t.

n X m X

n X m X

(sir + cir ) ≤ R

i=1 r=1

cir ≥ 1(sir > 0)cir,min ,

sir ≥ 0 ∀i, r.

(27)

Similar to Section IV, by introducing the binary activation variables zir ∈ {0, 1}, we define the active set of pairs given by S , {(i, r) : zir = 1}. Then, if a source-receiver pair is selected to be active, we set zir = 1, in which case the sender allocates cir = cir,min and sir > 0. Otherwise, zir = 0, and no resources are allocated, i.e., cir = sir = 0. Following the approach in Section IV, we reformulate the optimization problem in (27) as n X m X sir (cir,min +µi +λi ) λi max JˆS (z, s) = zir λ +µ c µi +sir λi +cir,min sir {zir ,sir} i i ir,min i=1 r=1

s.t.

n X m X

(zir cir,min + sir ) ≤ R

i=1 r=1

zir ∈ {0, 1}, sir ≥ 0,

∀i, r.

(28)

The optimization problem in (28) has the same structure as the multi source single receiver problem, with the difference that each decision variable is now over source-receiver pairs (i, r). In particular, each pair (i, r) is characterized by a fixed PC cost cir,min and a concave utility function JS,ir (sir , cir,min ) in terms of sir . As in the previous analysis, for a fixed active set S, the optimal allocation of {sir }(i,r)∈S admits a unique optimal solution. In particular, for each pair (i, r), we define the constants which are obtained by substituting cir = cir,min into the single receiver definitions and Air = λi (cir,min +µi + c µi λi ), Bir = cir,min µi (λi + µi ), and Cir = λiir,min +cir,min . Then, for a fixed S, there exists θ ≥ 0 such that !+ r Air ∗ −1 , (i, r) ∈ S, (29) sir = Cir Bir θ P P with θ selected so that (i,r)∈S s∗ir = R − (i,r)∈S cir,min . Consequently, each pair (i, r) can be expressed by the same three quantities that govern the branch-and-bound procedure in

source rankings Ai /Bi (desc.): 5, 1, 2, 4, 3 ci,min (asc.): 5, 1, 2, 3, 4 Ai Ci /Bi (desc.): 5, 1, 2, 4, 3 dominance: 5 ≻ 1 ≻ 2 ≻ {3, 4}

Sund = {1, . . . , 5} 5 in

5 out

Sin = {5} Sund = {1, 2, 3,∗ 4} evaluate Sin , update J 1 in Sin = {1, 5} Sund = {2, 3, 4} evaluate Sin , update J ∗

5 out 5 ≻ {1, 2, 3, 4}: all out

1 out

1 out 1 ≻ {2, 3, 4}: all out 2 out 2 in Sin = {1, 2, 5} Sund = {3, 4} 2 out evaluate Sin , update J ∗ 2 ≻ {3, 4}: both out 4 out 4 in Sin = {1, 2, 4, 5} Sund = {3} 4 out UB ≤ J ∗ : pruned evaluate Sin , update J ∗ 3 in 3 out Sin = {1, 2, 3, 4, 5} Sund = ∅ evaluate Sin , update J ∗

Sin = {1, 2, 4, 5} Already evaluated Sin .

Fig. 4. Branch-and-bound search for the five-source example.

Algorithm 1. The first quantity is the marginal utility Air /Bir , the second is the minimum state-0 sampling rate cir,min , and the third is the asymptotic sender utility Air Cir /Bir from source i and receiver r, all of which depend on both the receiver bias qr and the source parameters (λi , µi ). The dominance relation in Lemma 6 applies directly after replacing the source index i by the pair index (i, r): pair (i1 , r1 ) dominates pair (i2 , r2 ) if the three conditions (23)–(25) hold with at least one strict inequality when evaluated at the pairwise quantities. The optimization problem in (28) can be viewed as a multi source-receiver pair selection problem with N = nm items, indexed by (i, r). For each source-receiver pair, the utility function in (28) is of the same form as in the multi source single receiver formulation. Algorithm 1 is then applied by treating each pair (i, r) as an individual item, with the same dominance propagation, feasibility pruning, and upper bound pruning operating over the N pairs. Thus, Algorithm 1 has the following search complexity: i) in the worst case when there is no dominance relation between the sourcereceiver pairs, its search complexity is O(2N TKKT (N )); and ii) in the best case when the (i, r) pairs admit a perfect ordering, the search complexity reduces to O(N TKKT (N )). The corresponding objective values are computed by solving the inner KKT allocation in (29) for each candidate active set under consideration. VII. N UMERICAL R ESULTS In this section, we provide numerical results to illustrate the theoretical results and the behavior of the proposed allocation method. A. Multi source and Single receiver Case In our first numerical example, we consider n = 5 sources with transition rates λ = [1.3, 1.8, 0.7, 2.3, 1.5], µ = [2.3, 3.8, 3.2, 5.3, 2.0], and receiver bias q = 0.5. For these parameters, we compute the three source parameters introduced in Section V, namely the marginal utility Ai /Bi , the minimum state-0 sampling rate ci,min , and the asymptotic sender utility Ai Ci /Bi obtained from source i. The corresponding rankings are shown in Fig. 4. In particular, the marginal-utility ranking and the asymptotic-utility ranking both yield the order 5, 1, 2, 4, 3, whereas the minimum state-0 sampling ci,min rate ranking yields 5, 1, 2, 3, 4. Hence, sources 5, 1, and 2 admit a consistent dominance ordering, while sources 3 and 4 are incomparable under Lemma 6. The resulting dominance structure is therefore 5 ≻ 1 ≻ 2, with both sources 3 and 4 dominated by source 2 but incomparable with each other.

4

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3

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allocation

allocation

11

2

1

2

1

0

0 1

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5

1

2

3

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source index

Fig. 6. Utility functions with respect to the conditional intensity budget R. 3

Fig. 5. The optimal resource allocation of the sender for R = {10, 20}.

sender utility receiver utility

2.5 2

utility

This dominance structure substantially reduces the effective search space. Indeed, among the 25 − 1 = 31 nonempty active sets, only the following six sets are dominance-closed: {5}, {1, 5}, {1, 2, 5}, {1, 2, 3, 5}, {1, 2, 4, 5}, and {1, 2, 3, 4, 5}. The branch-and-bound search tree generated by Algorithm 1 is shown in Fig. 4. The algorithm branches according to decreasing marginal utility Ai /Bi , starting from source 5. The leftmost explored (blue) branch corresponds to the successive inclusion of sources 5, 1, 2, 4, and 3. At each visited blue node, the current active set Sin is evaluated and the best known value J ∗ is updated whenever an improvement is obtained. The main computational savings arise from dominance propagation. Whenever a source is excluded, all sources dominated by it are also excluded, and the corresponding subtree (purple boxes) is collapsed without further evaluation. For example, excluding source 5 at the root forces the exclusion of all remaining sources, since 5 ≻ 1 ≻ 2 and 2 ≻ {3, 4}. Similarly, excluding source 1 forces the exclusion of sources 2, 3, and 4, while excluding source 2 forces the exclusion of sources 3 and 4. These cascaded eliminations account for most of the pruning observed in Fig. 4. Upper-bound pruning (red boxes) provides an additional reduction when dominance alone is insufficient. In particular, after reaching the node with Sin = {1, 2, 5} and exploring the branch in which source 4 is excluded, the algorithm computes an optimistic upper bound by solving the relaxed KKT allocation over {1, 2, 3, 5}, i.e., without considering c3,min of the undecided source 3. Since this upper bound does not exceed the best known value already attained at Sin = {1, 2, 4, 5}, that sub-tree is pruned. The node corresponding to the active set {1, 2, 4, 5} in Fig. 4 is the optimal solution for this example which highlights the importance of the asymptotic utility Ai Ci /Bi . Although source 4 has a higher minimum state-0 sampling rate than source 3, namely c4,min = 3.0 versus c3,min = 2.5, it is preferred in the optimal active set because its asymptotic utility is substantially larger, i.e., A4 C4 /B4 = 0.605 versus A3 C3 /B3 = 0.359. Overall, the branch-and-bound algorithm visits only 11 search-tree nodes for this example, whereas exhaustive search would require evaluating all 31 nonempty active sets. With the same (λi , µi ) sets given previously and q = 0.5, we plot the optimal (si , ci ) allocations in Fig. 5 when the conditional intensity budget is R = {10, 20}. For R = 10, the optimal active set is {1, 2, 5}, i.e., the three most dominant sources shown in Fig. 4, and the corresponding bars show positive si only on these indices with ci = ci,min . When we increase the budget to R = 20, the sender starts to

1.5 1 0.5 0 0.2

0.4

0.6

0.8

1

k

Fig. 7. Agents’ utilities with respect to heterogeneity parameter of k.

send updates about source 4’s state and the optimal set becomes {1, 2, 4, 5}. This example also illustrates the necessity of our branch-and-bound algorithm. If sources were ranked solely based on their ci,min values, source 4, which has a relatively large ci,min , would appear to be the least efficient source. We would therefore consider {1, 2, 3, 5} followed by {1, 2, 3, 4, 5}, thereby missing the optimal set for R = 20, namely {1, 2, 4, 5}. Next, we plot the sender’s optimal utility JS∗ as the budget R increases, together with the receiver’s benchmark utility, which remains constant, as shown in Fig. 6. We consider the same set of λi and µi as before and choose q = 0.5. The colored bands highlight the different regions which are the intervals of R where the optimal active source set remains the same. For example, when the conditional intensity budget is very limited, i.e., 0.5 < R ≤ 2.55, the sender can only send updates about the most dominant source (source 5). As R increases, the sender begins to transmit updates about a larger set of sources, specifically in the following order: {5}, {1, 5}, {1, 2, 5}, {1, 2, 4, 5}, and finally {1, 2, 3, 4, 5}. Each transition occurs exactly when allocating ci,min to the next most dominant source, along with some si on that source, becomes more beneficial than further increasing the si of currently active sources. The receiver’s utility curve remains constant by the construction of the problem due to the optimistic leader-favorable tie-breaking in (4). As we increase the sender’s conditional intensity budget R, the sender can obtain a higher utility whereas the receiver’s default utility remains constant across all R. In our next simulation result, we choose n = 5, q = 0.5, R = 15, and λi = 1 for all i. To ensure the PC, we impose µi ≥ 1. We generate heterogeneous µi -profiles by ki µi = 1 + (C − n) Pn , j j=1 k P so that i µi = C = 20 and each µi ≥ 1. We vary the parameter k ∈ [0.2, 1] to control heterogeneity of the µi distribution. For example, k = 1 yields the uniform case where all µi ’s are equal, while smaller k values produce a

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R = 4.00 4

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(c)

Fig. 8. Multi source multi receiver: per-pair utility heatmaps under different budgets.

more uneven distribution, with some µi values close to 1 and others significantly larger. The blue solid curve in Fig. 7 shows the optimal sender utility JS∗ as a function of k at the equilibrium. The minimum sender utility occurs at k = 1 (where µi = 4 for all i), and the sender’s utility increases as k decreases (which leads to a more heterogeneous distribution of µi ). Intuitively, with q = 0.5 and λi = 1, the PC for source i is ci,min = µi − 1. A more heterogeneous distribution of µi creates some sources with µi close to 1, hence leads to very small ci,min values for some sources. These sources are easier to activate as they do not require high ci,min and also yield high marginal utility for persuasion. Thus, the sender benefits from the heterogeneous distribution of µi ’s. The dashed curve in Fig. 7 shows the receiver’s utility P red q µi f (µi ) = µi /(µi + 1) is concave in µi , the i µi +λi . Because P receiver’s utility (q i f (µi )) is maximized with the uniform µi distribution and decreases with heterogeneity in µi . Thus, the heterogeneity in µi helps the sender (by lowering some ci,min ) but hurts the receiver, yielding the opposing trends shown in Fig. 7. B. Multi source and Multi receiver Case We consider the multi source multi receiver setting and illustrate the optimal pairwise utility contributions under different total budgets. We use 4 sources with parameters λ = [0.25, 0.4, 0.8, 1.1] and µ = [1.1, 1.4, 1.5, 1.82], and 4 receivers with utility weights qr = [0.47, 0.38, 0.62, 0.55]. Here, we solve the joint source–receiver allocation problem using Algorithm 1 and compute the optimal per-pair utility values under the conditional intensity budget R. Fig. 8 shows the resulting heatmaps of per-pair utility contributions for R = {4, 8, 14}. Each nonzero cell in the heatmap corresponds to an active source-receiver pair, and the number written in the cell is the utility contribution of that pair to the sender’s total utility. In the low-budget regime (when R = 4), as shown in Fig. 8(a), the optimal policy is highly selective and activates only a small number of source-receiver pairs. The allocation is concentrated on a few dominant pairs such as (i, r) = (4, 2) and (3, 2). The corresponding utility values are relatively large compared to the number of active links. This behavior is consistent with the insight from the single receiver case: when the total budget is limited, the sender prioritizes only the most dominant source-receiver pairs and leaves the remaining pairs inactive. When the budget increases to R = 8 (shown in Fig. 8(b)), additional source-receiver pairs become active. In this example, the sender first retains the previously selected pairs with relatively high utility contributions and then expands the active set by including additional pairs with smaller but still positive contributions. For the larger budget R = 14 (as shown in Fig. 8(c)), some source-receiver pairs continue to have higher utility contributions than others while

sender utility under persuasion-optimized sampling sender utility under receiver-optimal sampling receiver utility under persuasion-optimized sampling receiver utility under receiver-optimal sampling

0.5 0 0

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Fig. 9. Utility comparison under sender-optimal persuasion and receiveroptimal sampling.

increasing their total contributed utility compared to the lower budget cases. The persistence of several high-valued cells across all three panels shows that certain source-receiver pairs, such as (4, 2) and (3, 2), receive larger utility allocations for the chosen (λi , µi , qr ) values. C. Persuasion-Optimized versus Receiver-Optimized Sampling In Fig. 9, we compare the optimal utilities under our persuasion model with a receiver-optimized sampling benchmark in which the sender directly maximizes the receiver’s weighted steady-state correct estimation probability by solving problem (30) via exhaustive search for the same n = 5 sources and (λi , µi ) parameters as in Subsection VII-A, with receiver bias q = 0.5. For the receiver-optimized benchmark, we define the extended per-source receiver utility as  c)   qµi ci (µi + si ) + (1 − q)λi si (λi +
i ,    (µi + λi ) µi ci + λi si + ci si J¯R,i (si , ci ) , if (si , ci ) 6= (0, 0),   qµ  i  , if (si , ci ) = (0, 0).  µi + λi

For an inactive source with (si , ci ) = (0, 0), no updates are transmitted, and the receiver therefore employs the prior-only default estimator σdefault . Accordingly, its utility is defined as the prior utility qµi /(µi + λi ). Then, we write the receiveroptimized sampling rate allocation problem as: n X J¯R,i (si , ci ) max {si ,ci}

s.t.

i=1

n X

(ci + si ) ≤ R

i=1

ci ≥ 0, si ≥ 0, i ∈ {1, . . . , n}. (30) Fig. 9 shows that the receiver utility under the receiver-optimal sampling benchmark (purple dash-dotted curve) is higher (in the plotted regime) than the receiver utility in the persuasion setup (yellow dashed curve), since without persuasion the budget can be used entirely to optimize the correct state estimation accuracy. However, in the persuasion setup the sender chooses ci = ci,min for active sources and, under the optimistic assumption we have adopted in this problem, the sender keeps the receiver exactly at its baseline utility while using the remaining budget to maximize its own objective through the si allocations. Turning to the sender utilities, the persuasion model (blue solid curve) exhibits the active-set structure identified earlier: the utility increases with R, with visible breakpoints at each change of the optimal active set. The red solid curve shows the sender utility in (11) evaluated at the receiver-optimal allocation, that is, the (si , ci ) obtained by

13

solving problem (30), when the receiver employs σsender . This curve lies strictly below the persuasion-model utility, since the receiver-optimal allocation is not designed to favor the sender; moreover, its visible jumps occur when the receiver-optimal exhaustive solution activates a new source with nontrivial (si , ci ), which creates a discrete improvement in sender utility, while the receiver-optimal curve itself remains comparatively smooth since it is the quantity being directly optimized. Overall, Fig. 9 shows the main strategic benefit of the timely persuasion: by paying only the minimum communication cost needed to secure receiver compliance, the sender can use more of the budget for state-1 sampling and thereby achieve substantially higher sender utility, even though this keeps the receiver at its minimum acceptable base utility level. VIII. C ONCLUSION In this paper, we studied a dynamic persuasion problem in which a sender influences a receiver by controlling the timing of information updates from binary CTMC sources. We derived closed-form participation constraints and obtained an explicit solution for the single source case. For the multi source setting under a conditional intensity budget, we developed a bisection procedure for the unique KKT multiplier to compute the optimal state-dependent update rates for a fixed active set, and designed an efficient algorithm to determine the optimal active sources with potentially lower search complexity. We further extended the model to a multi receiver setting with heterogeneous receiver biases and solved the problem using the same algorithmic framework. Numerical results illustrated how the sender allocates the budget across source–receiver pairs, how active sets evolve with the conditional intensity budget, and how persuasion increases the sender’s utility compared to the receiver-optimal sampling benchmark. As a future direction, we plan to extend the model to multi sender environments and study equilibrium policies under both aligned and misaligned sender objectives. R EFERENCES

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