Industrial Internet-of-Things (IIoT) deployments increasingly rely on learning: predictive-maintenance models, anomaly detectors, and digital-twin controllers are updated as operating conditions drift, sensors degrade, and production lines are reconfigured. In this environment, federated learning (FL) is not only a technical architecture but also a procurement problem. A server (the plant operator, OEM, or platform) must model updates from a population of edge nodes whose data quality is heterogeneous and whose ability to produce timely updates varies with local compute, connectivity, and operational constraints. When updates arrive too slowly or reflect stale local states, model performance degrades in precisely the regimes where IIoT systems are most valuable: rare events, transient faults, and fast-changing processes. We therefore view IIoT-FL as a market for , in which the central platform designs incentives to elicit both higher update effort and truthful participation under severe observability limits.
Two frictions make this market distinctive. First, the platform does not observe the node’s action (how aggressively it refreshes, retrains, or uploads) nor its private “type” (how valuable its data are and how costly effort is). The platform instead sees coarse telemetry: noisy freshness and responsiveness signals (e.g., AoI-like timestamps, service latency), and minimal metadata about the update payload. Second—and increasingly binding in 2026—privacy regulation and engineering practice restrict what telemetry can be collected and how it can be used in contracts. Data minimization mandates, sectoral rules for critical infrastructure, and the growing standardization of privacy-preserving FL (secure aggregation, differential privacy, trusted execution) collectively imply that the platform cannot condition payments on raw local traces or rich per-example statistics. Even when richer signals exist technically, they are often legally or reputationally unusable, or they are deliberately by design (e.g., quantized timestamps, delayed reporting, bounded counters) to mitigate leakage.
These frictions turn “pay for performance” into “pay for what you are allowed to observe.” Our starting point is a simple operational interpretation: each node chooses an update intensity (shorter cycle, more frequent retraining, more aggressive communication), which tends to improve freshness but may worsen responsiveness because it loads local resources and networks. The platform cares about both. In many IIoT pipelines, user-facing or mission-facing utility is well summarized by satisfaction-style objectives that penalize delay while rewarding accurate, up-to-date models. This makes freshness metrics such as Age-of-Information (AoI) and service latency natural primitives: they are interpretable, widely logged, and already used in SLAs. The difficulty is that the platform observes only noisy, privacy-filtered versions of these metrics and must design transfers measurable with respect to them.
Our central claim is that the economic logic of IIoT-FL contracting can be organized around three questions: (i) (ii) and (iii) To address these, we build a repeated contracting model in which a risk-neutral platform posts a stationary payment rule based on observables, nodes best-respond by choosing update intensity, and privacy enters as an explicit information constraint on what can be conditioned upon. While the environment is motivated by FL, the mechanism-design problem is classical: incentivize hidden action under limited observability, with the additional twist that “limited observability” is itself policy-relevant and technologically controllable.
The first contribution is a monotonicity result for performance pay in a setting with multiple observables. Under a multidimensional monotone likelihood ratio / supermodularity condition—economically, that “better outcomes” (lower observed AoI, lower observed latency, higher observed metadata) become more likely when a node exerts higher update effort—we show that the platform can restrict attention to contracts that are monotone in each coordinate. Intuitively, if higher effort shifts the distribution toward fresher and faster telemetry, then paying for fresher/faster realizations is incentive compatible and avoids wasting transfers on outcomes that are unlikely under effort. This yields a clear design principle: even when the platform cannot reduce observables to a single verified score, optimal schemes respect the partial order of outcomes and do not perversely reward staleness or slowness. This principle connects directly to practice: many deployed reward heuristics already assign points for “freshness” and “responsiveness,” but our result clarifies when such heuristics are not merely intuitive but without loss of optimality.
The second contribution formalizes the using Blackwell’s informativeness order. Privacy strengthening is modeled as a garbling of the platform’s observable signal: the platform sees a coarsened, noisier, or otherwise less informative version of performance. We show that optimal attainable welfare is monotone in this sense: any privacy rule that makes the observable less informative weakly reduces the best achievable surplus, and under mild conditions strictly reduces it. This is not an argument against privacy; it is a precise statement of the tradeoff. It implies that privacy choices have an economic shadow price: tighter privacy reduces the feasible set of incentive-compatible contracts and thereby changes equilibrium update effort, freshness, and latency. Framed this way, policy discussions about permissible telemetry can be connected to quantifiable operational outcomes (staler models, slower anomaly detection) and to budgetary implications (higher payments needed to induce a given effort).
The third contribution provides tractable, closed-form predictions in
a quadratic-Gaussian specialization that is well suited for estimation
and calibration. By reducing multidimensional telemetry to a
one-dimensional sufficient statistic (a linear index of noisy
AoI/latency/metadata), and restricting attention to affine payment
rules,
p(S) = α0 + α1S,
we obtain sharp comparative statics: the induced update intensity scales
with the incentive slope, and the optimal slope shrinks as measurement
noise increases. This captures a practical phenomenon: when telemetry
becomes noisier—whether due to sensor limitations, network jitter, or
deliberate privacy noise—the platform rationally “turns down”
pay-for-performance because it would otherwise pay for randomness. The
model also clarifies when stronger latency emphasis can reduce freshness
effort (and conversely), thereby linking contract parameters to
engineering tradeoffs in scheduling, compression, and local compute
allocation.
Finally, we connect welfare losses from privacy to information-theoretic quantities. In the Gaussian case, the welfare gap relative to a full-information benchmark can be bounded explicitly by a decreasing function of mutual information (or SNR). This yields a quantitative “price of privacy”: as privacy mechanisms reduce informativeness, the equilibrium moves away from the first-best action, and the associated loss can be summarized in a single, estimable statistic. For practitioners, this suggests a measurement agenda: rather than debating privacy in abstract terms, one can ask how a proposed telemetry constraint reduces the mutual information between effort (or type) and observable signals, and then translate that reduction into predicted degradation in freshness/latency or increased procurement cost.
Empirically, the framework implies several testable patterns in IIoT-FL deployments. We should observe weaker performance pay (flatter reward curves) and lower update intensity in environments with higher telemetry noise or stronger privacy filters; sharper responses of update frequency to budgets when limited liability or payment caps bind; and systematic substitutions between freshness and latency targets as platform weights change. It also suggests that “privacy-preserving by design” systems should be evaluated not only for leakage risk but also for how their observables support incentive provision—an evaluation criterion that is often missing from technical privacy audits.
We close with a limitation that also motivates the next section. We intentionally abstract from many FL-specific constraints—nonconvex optimization, client drift, aggregation bias, and strategic manipulation of local training—so that the contracting logic is transparent. These issues matter, and they enter naturally as refinements to the signal structure and the server’s value function. The goal of this paper is to illuminate the core tradeoff: . The remainder of the paper situates this approach within related work on AoI/latency satisfaction metrics, Stackelberg incentives in IIoT-FL, and classic results on informativeness and contracting.
Our modeling choice to treat update procurement through observables such as Age-of-Information (AoI) and service latency is motivated by a large measurement and control literature in networking and cyber-physical systems. AoI was introduced to quantify how stale a remote estimate is, and has become a standard performance criterion when the of information matters as much as (or more than) raw throughput . In IIoT, staleness enters directly into fault detection and predictive maintenance: a slightly more accurate model trained on outdated conditions can be operationally inferior to a slightly noisier but up-to-date model when processes drift or faults emerge abruptly. In parallel, service latency captures the responsiveness of the learning pipeline (e.g., end-to-end delay from local compute and uplink through aggregation and deployment), which matters for closed-loop decision making and for “time-to-mitigation” in safety-critical environments.
A key practical point is that AoI and latency are and in many
deployments. They are therefore natural candidates for contractible
performance proxies when richer signals (e.g., raw gradients,
per-example losses, or device traces) are restricted by privacy or
security. This complements a growing set of “satisfaction” objectives in
systems papers that combine freshness/quality rewards with latency
penalties—often in a scalarized form such as
utility ≈ τ ⋅ (quality
proxy) − λ ⋅ (delay),
which aligns with how platform operators reason about service-level
objectives. Our contribution is not to introduce these metrics, but to
embed them into a principal–agent framework where they are of hidden
effort and are potentially . This lens clarifies when paying more for
“better telemetry” is without loss of optimality, and when telemetry
degradation necessarily reduces the achievable surplus.
There is now an extensive literature on incentivizing participation and effort in federated learning (FL). Much of it adopts a Stackelberg structure: the server announces rewards (prices, bonuses, or auctions), and clients respond by choosing participation, local compute, or reporting actions . These models differ in their strategic primitives—some emphasize and budget allocation; others focus on (e.g., number of local epochs, CPU cycles, transmit power) and energy costs; still others consider multi-dimensional private information about data quality and resource constraints. Mechanism proposals include posted-price schemes, VCG-style auctions for client selection, reputation systems, and learning-based pricing rules when client responses are unknown.
Two gaps motivate our approach. First, many incentive schemes implicitly assume the platform can verify a meaningful contribution score (accuracy improvement, gradient quality, or loss reduction) at the client level. In IIoT-FL, such verification may be technically unavailable (secure aggregation hides individual updates) or institutionally constrained (data minimization and critical-infrastructure rules restrict telemetry). Second, even when verification is possible, it is often noisy and delayed, and can be gamed by overfitting or adversarial behavior. By centering the contract on observables like AoI/latency/metadata—quantities that remain meaningful under privacy-preserving pipelines—we focus on a constraint that is increasingly binding in practice. The Stackelberg logic remains: the server commits to a payment rule p(⋅), clients best-respond with an update intensity x, and the outcome is mediated by a signal channel P(O ∣ x, t). What changes is that O is explicitly restricted, and privacy enters as a reduction in the informativeness of that channel.
On the FL side, the optimization literature highlights constraints that naturally map into our reduced-form technology. Device heterogeneity (compute, bandwidth), partial participation, stragglers, and non-i.i.d. data all generate variable completion times and variable effectiveness of an update . Asynchrony and stale updates have been studied as fundamental challenges: when clients train on outdated global models, the server’s aggregation can degrade, and when communication is slow, global progress slows. From our perspective, these features justify two modeling choices in Section~3. First, we allow the action x (update intensity) to jointly affect freshness and latency—capturing the fact that more aggressive update cycles can congest local resources or networks, even as they reduce staleness. Second, we treat observed metrics as noisy proxies: Ã and Ẽ are affected by jitter, batching, quantization, and privacy noise, while D̃ captures coarse payload descriptors that survive telemetry minimization (e.g., sample counts binned into ranges, compression levels, or declared update sizes).
We emphasize that we are not proposing AoI/latency as sufficient statistics for the full FL learning dynamic. Rather, they are contractible proxies that correlate with the underlying contribution in environments where direct measurement is unavailable or undesirable. In Section~3 we formalize this by letting the server’s gross value v(t, x) be increasing in “better” freshness and responsiveness, while allowing types t = (q, σ) to shift both the value of induced effort (through q) and the marginal cost of effort (through σ).
Our treatment of privacy as a constraint on contractibility connects to classic results on informativeness and decision making. Blackwell’s ordering formalizes when one signal is more informative than another: O is more informative than O′ if O′ can be generated from O by garbling (a stochastic kernel). In statistical decision theory, more informative signals weakly improve the value of optimal decisions; in mechanism design and contracting, more informative performance measures expand the feasible set of incentive-compatible contracts . This is the conceptual backbone of our Proposition~2: strengthening privacy by reducing telemetry is modeled as a Blackwell garbling, and thus weakly reduces the best achievable welfare.
A related but distinct strand concerns contracting with performance measures. In one-dimensional moral hazard, monotone likelihood ratio (MLR) conditions deliver monotone optimal payments and tractable incentive schemes . Our setting is multidimensional (Ã, Ẽ, D̃), which is precisely where “naively reward better outcomes” can fail without structure. We therefore lean on multidimensional generalizations—supermodularity and log-supermodularity/MLR with respect to a partial order—to justify coordinate-wise monotonicity (Proposition~1). This aligns with the monotone comparative statics toolkit (Topkis, Milgrom–Shannon) and with the broader idea that when higher effort shifts the distribution toward better outcomes in an ordered sense, incentive pay can be chosen to respect that order without sacrificing optimality.
Relative to the FL incentives literature, we intentionally abstract from strategic gradient manipulation, poisoning, and collusion, as well as from dynamic learning externalities across rounds. These are important in adversarial or competitive settings, but our objective is to isolate a procurement problem that persists even with honest clients: . Relative to the AoI/latency literature, we abstract from detailed queueing and scheduling microfoundations, and instead use reduced-form mappings from update intensity to freshness/latency that can be calibrated or learned. The payoff from this abstraction is that we can cleanly connect (i) telemetry constraints, (ii) incentive power, and (iii) welfare loss through a common language—Blackwell informativeness and, in the Gaussian specialization, mutual information.
Section~3 now formalizes the primitives: types, actions, the signal/garbling structure induced by privacy, and the resulting contracting problem, together with a first-best benchmark that clarifies what is lost when action and type are unobservable.
We model the procurement of privacy-preserving federated-learning (FL) updates as a repeated principal–agent problem between a server (principal) and an IIoT node (agent). The key friction is that the server cannot contract on the node’s private information nor on its effort choice, and is further restricted—by privacy regulation and system design—to conditioning payments only on a limited set of auditable telemetry.
In each round, the node has private type
t = (q, σ) ∈ 𝒯,
where q captures the
usefulness of the node’s data to the global model (e.g.,
representativeness, label quality, or task relevance), and σ captures the node’s marginal
inefficiency/cost of producing timely updates (e.g., energy cost,
opportunity cost, or a composite of compute and uplink constraints). We
allow q and σ to be statistically related under
a common prior F, but they are
not observed by the server.
After observing t and the server’s posted contract, the node chooses an (effort) x ∈ 𝒳 ⊂ ℝ+. Higher x corresponds to more aggressive freshness effort: shorter local update cycles, more frequent participation, or higher priority scheduling for training and communication. In some deployments it is convenient to parametrize by a cycle length $\theta\in[\underline\theta,\bar\theta]$ with x = 1/θ; we treat x as the primitive.
The node’s real cost of effort is c(t, x). Our analysis will assume c is increasing and convex in x, and that the marginal cost shifts with σ in a single-crossing way: higher σ makes high intensity disproportionately more expensive. This captures the empirical regularity that heterogeneity is often about the slope of the effort–resource tradeoff rather than a fixed participation fee.
The node’s action x affects two operational outcomes: freshness of the update stream and responsiveness of the service pipeline. We denote by A the (true) Age-of-Information (AoI) associated with the node’s contribution in that round, and by E the (true) end-to-end service latency (or completion delay). Lower A and lower E are both operationally better, but they need not move in the same direction when x increases: more aggressive update cycles can improve freshness while exacerbating congestion, queueing, or straggler effects that increase latency.
To keep the contracting problem tractable while preserving this
tension, we use a reduced-form mapping from x to outcomes,
A = Ā − αx, E = Ē + βx,
with α > 0 and β ≥ 0. The parameter β is allowed to be small (nearly
independent latency) or large (strong congestion/straggler effect). We
interpret this mapping as a local linearization of a more complex
queueing and scheduling environment, potentially learned from
operational logs. Nothing in the contract-theoretic structure requires
linearity; what matters is that outcomes shift systematically with x.
In addition to timing, we allow a coarse component D that is correlated with the magnitude or potential usefulness of an update. Examples include binned sample counts, declared payload sizes after compression, or a coarse ``work report’’ that survives privacy minimization. We write D = D(x, q), allowing the metadata to increase with x (more frequent or larger updates) and with q (more informative data may generate larger effective updates even after compression). Crucially, D is not assumed to be a sufficient statistic for contribution; it is simply an additional contractible proxy.
The server cannot directly observe (A, E, D).
Instead, it observes a verifiable but noisy/coarsened vector
O = (Ã, Ẽ, D̃) ∈ 𝒪,
generated by a known signal channel P(O ∣ x, t).
A canonical measurement model is additive noise,
à = A + εA, Ẽ = E + εE, D̃ = D(x, q) + εD,
where the noise terms capture timestamp jitter, batching, quantization,
secure-aggregation delays, and privacy noise (e.g., differential privacy
mechanisms that coarsen or randomize reported metadata). We allow P to be multidimensional and
correlated; we use the reduced-form notation P(O ∣ x, t)
to keep the analysis agnostic about the exact measurement stack.
We interpret and as restrictions on what is contractible. Formally,
the server must choose a payment rule measurable with respect to O:
p : 𝒪 → ℝ+, and
optionally a selection rule s : 𝒪 → {0, 1}.
Stronger privacy corresponds to observing a signal O′ that is less
informative than O in the
Blackwell sense. That is, there exists a stochastic kernel K such that
P(O′ ∣ x, t) = ∑O ∈ 𝒪K(O′ ∣ O) P(O ∣ x, t).
This captures, in a unified way, coarser binning (e.g., reporting AoI in
buckets), randomized response, suppressing certain fields, or routing
telemetry through a trusted privacy layer before it is logged.
Importantly, we treat the garbling as exogenous from the contracting
perspective: it is pinned down by policy and system architecture, and
the server optimizes subject to it.
The agent is risk-neutral with per-round expected utility
U(t; x, p) = 𝔼[p(O) ∣ x, t] − c(t, x),
and participates if this utility exceeds an outside option ū(t) (which may include
energy costs, privacy disutility, or a fallback workload). The server is
risk-neutral with per-round payoff
Π(t; x, p) = 𝔼[v(t, x) ∣ t] − 𝔼[p(O) ∣ x, t],
where v(t, x) is the
server’s gross value from the node’s induced contribution, before
transfers. The total welfare from a realized action is
W(t; x) = v(t, x) − c(t, x),
so transfers net out; informational frictions matter because they
distort x, not because paying
the agent is intrinsically costly.
The timing in each round is standard: (i) t ∼ F is realized privately by the node; (ii) the server posts p(⋅) (and potentially a participation/selection rule); (iii) the node chooses x (and whether to participate); (iv) O ∼ P(⋅ ∣ x, t) is realized and becomes verifiable; (v) the server pays p(O). We focus on stationary contracts in the repeated interaction, which is appropriate when rounds are i.i.d. or when a Markov environment admits a stationary optimum under standard regularity.
Given p(⋅), the node’s best
response is
x*(t; p) ∈ arg maxx ∈ 𝒳 𝔼[p(O) ∣ x, t] − c(t, x),
with the interior first-order condition
$$
\frac{\partial}{\partial x}\mathbb E\big[p(O)\mid
x,t\big]=\frac{\partial}{\partial x}c(t,x),
$$
whenever differentiability and interiority hold. The server chooses
p anticipating this response,
potentially subject to a per-round budget constraint 𝔼[p(O)] ≤ B.
To isolate the role of privacy-limited observability, we define a
full-information benchmark in which the server can contract directly on
(t, x) (or
equivalently observes x and
can tailor the transfer to t).
In that case, transfers can implement the welfare-maximizing action for
each type:
xFB(t) ∈ arg maxx ∈ 𝒳 v(t, x) − c(t, x).
If v is concave in x and c is convex, the first-best
satisfies the pointwise condition
$$
\frac{\partial}{\partial x}v(t,x^{FB}(t))=\frac{\partial}{\partial
x}c(t,x^{FB}(t)).
$$
Let W* denote the
expected welfare achieved under this benchmark. Our object of interest
is the best attainable welfare when contracts must be measurable with
respect to privacy-preserving observables, denoted W*(O), and how
W*(O)
changes as privacy strengthens via Blackwell garbling. Section~4 studies
the second-best contracting problem under these constraints and
characterizes the structure of optimal payments when the signal family
satisfies an ordered likelihood condition.
We now characterize the server’s optimal contract when transfers must be measurable with respect to privacy-preserving telemetry O = (Ã, Ẽ, D̃). The core difficulty is that the server values a latent contribution that depends on the node’s private type t = (q, σ) and unobserved action x, yet it can reward only those features of performance that survive auditing and privacy minimization. This section makes two points. First, under mild regularity the server’s optimization over measurable payment rules admits an optimal solution and can be studied using the agent’s incentive (best-response) problem. Second, under an ordered-likelihood (MLR/supermodularity) condition, we can w.l.o.g. restrict attention to contracts that pay more for better realized freshness/latency/metadata, even though the signal is multidimensional.
Fix an admissible payment rule p : 𝒪 → ℝ+ (and, if used,
a selection rule s : 𝒪 → {0, 1}). For a given type
t, the node chooses
x*(t; p) ∈ arg maxx ∈ 𝒳 𝔼 [p(O) ∣ x, t] − c(t, x),
subject to participation 𝔼[p(O) ∣ x*(t; p), t] − c(t, x*(t; p)) ≥ ū(t).
The server anticipates this mapping t ↦ x*(t; p)
and solves
maxp ∈ 𝒫(O) 𝔼t[v(t, x*(t; p)) − 𝔼[p(O) ∣ x*(t; p), t]],
optionally subject to 𝔼t𝔼[p(O) ∣ x*(t; p), t] ≤ B.
Because transfers are not intrinsically costly in welfare terms,
distortions arise only through how imperfect observability limits the
set of incentive-compatible x*(⋅). In particular,
even if the first-best xFB(t)
is interior and smoothly varying in t, it may not be implementable when
p can condition only on O.
A useful way to interpret incentive compatibility here is marginal:
for differentiable environments, an interior best response
satisfies
$$
\frac{\partial}{\partial x}\mathbb E\!\left[p(O)\mid
x,t\right]=\frac{\partial}{\partial x}c(t,x).
$$
Thus, the entire incentive content of a contract is summarized by how
the expected payment changes with x. When privacy or coarsening makes
𝔼[p(O) ∣ x, t]
relatively flat in x, the
node’s chosen intensity falls; conversely, more informative telemetry
allows the server to create steeper marginal incentives without
excessive expected transfers.
We do not attempt maximal generality, but the economically relevant cases satisfy standard conditions ensuring existence of an optimal contract in the measurable class. Suppose 𝒳 is compact (or effectively compact after bounding feasible update rates operationally), c(t, x) is continuous and convex in x, v(t, x) is continuous and concave in x, and the channel P(⋅ ∣ x, t) varies continuously with x (weak continuity is sufficient). If we additionally restrict p to lie in an integrable envelope (e.g., 𝔼[p(O)] ≤ B or p(O) ∈ [0, p̄]), then the principal’s objective is well-defined and upper-semicontinuous over a compact set of contracts, yielding existence by a standard Weierstrass argument. Practically, the same restrictions are natural: real systems impose payment caps, budgets, or limited liability, and telemetry itself is bounded/quantized.
The main structural question is whether an optimal contract can be taken to be monotone in the observables. We formalize “better performance” by the partial order O ≼ O′ if à ≥ Ã′, Ẽ ≥ Ẽ′, and D̃ ≤ D̃′. Intuitively, O′ is if it exhibits fresher updates, lower latency, and richer (coarse) metadata.
Assumption (iii) in the enclosing scope posits a multidimensional MLR / log-supermodularity property: higher effort x shifts the likelihood of outcomes upward in this order. Operationally, this says that more aggressive update intensity makes “good telemetry” relatively more likely. Under this ordered-likelihood condition, the expected payment inherits in (x, O) whenever p is monotone in the order. Increasing differences is precisely what we need for robust incentives: it implies that improving the realized telemetry raises the marginal return to effort rather than merely the level of compensation.
This yields a monotonicity principle: if a proposed payment rule is non-monotone (e.g., it sometimes pays more for worse à holding the other coordinates fixed), then we can replace it by an order-preserving rearrangement (an isotonic regression with respect to ≼) that (weakly) strengthens incentives while not increasing expected transfers for any fixed induced distribution over O. The economic content is straightforward. Paying “against” the signal order wastes money on outcomes that are comparatively more likely under low effort, and it blunts the slope of 𝔼[p(O) ∣ x, t] with respect to x. By contrast, a payment that is weakly decreasing in à and Ẽ and weakly increasing in D̃ aligns the agent’s private choice with the direction in which effort moves the distribution of telemetry.
Accordingly, we can w.l.o.g. restrict attention to contracts
satisfying
Ã′ ≤ Ã, Ẽ′ ≤ Ẽ, D̃′ ≥ D̃ ⇒ p(Ã′, Ẽ′, D̃′) ≥ p(Ã, Ẽ, D̃).
This does claim that the optimal contract is additively separable across
Ã, Ẽ, D̃, nor
that it is strictly monotone; rather, it identifies a large class of
“reasonable” contracts that are without loss and are consistent with
common engineering practice (bonuses for freshness and throughput,
penalties for delay).
While monotonicity is a coordinate-wise statement, implementation
often benefits from compressing O into a one-dimensional score. When
the measurement model is approximately linear-Gaussian (or when the
principal restricts attention to linear indices for simplicity and
auditability), we can take
S = wD D̃ − wA Ã − wE Ẽ,
with weights (wA, wE, wD)
reflecting both the technology (how x moves each coordinate in
expectation) and the noise levels (how informative each coordinate is).
A monotone contract in the partial order is then equivalently a
nondecreasing function of S
for appropriate positive weights. This bridges the multidimensional MLR
condition with a familiar “scorecard” implementation: the server
publishes a scalar performance score and pays a bonus increasing in that
score. The Gaussian specialization in Section~5 will make this reduction
exact within an affine class, but the qualitative message already
applies here.
Two constraints are especially important in IIoT procurement. First, limited liability (or nonnegativity of payments) requires p(O) ≥ 0. This restriction interacts with monotonicity in a natural way: the contract typically decomposes into a baseline transfer plus a performance bonus, with the baseline pinned down by participation and the bonus component providing incentives. When performance is poor (high Ã, high Ẽ, low D̃), the payment is driven toward the lower bound, which is often zero in practice.
Second, a per-round budget 𝔼[p(O)] ≤ B trades off incentive power against expected transfers. In Lagrangian form, the budget adds a multiplier on expected payments, effectively raising the marginal “cost” of incentive provision. Under ordered-likelihood, tightening the budget weakly flattens the optimal payment schedule (reduces the spread between good and bad outcomes), which in turn lowers the induced x* and worsens freshness or latency according to the underlying technology. This is precisely the operational tradeoff faced by platform designers: if payments cannot be made sufficiently sensitive to auditable outcomes, the system relies on voluntary effort and will generally underprovide timely updates.
Our monotonicity conclusion rests on an ordered signal condition that is empirically plausible but not automatic: if telemetry is heavily filtered, if timestamps are batched adversarially, or if measurement noise dominates the action effect, then the likelihood order may fail and monotone contracts may cease to be without loss. Even then, the analysis still clarifies : it should preserve a performance order that effort shifts in a consistent direction. This observation will be central in the next section, where we formalize how further privacy strengthening—modeled as Blackwell garbling—reduces attainable welfare by shrinking the set of incentive-feasible payment rules.
We now formalize a basic but powerful comparative statics result: . The logic is not tied to a particular functional form for costs or values; rather, it is an information-ordering statement about what contracts are feasible when transfers must be measurable with respect to privacy-preserving observables.
Let O ∈ 𝒪 denote the
richest auditable telemetry the platform could (in principle) make
available for contracting, and let O″ ∈ 𝒪″ denote
a privacy-strengthened version obtained by additional minimization,
aggregation, or noise injection. We say that O″ is a of O if there exists a stochastic
kernel K(⋅ ∣ ⋅) such that, for
every (x, t),
P(O″ ∣ x, t) = ∑O ∈ 𝒪K(O″ ∣ O) P(O ∣ x, t) (discrete
case)
(with the analogous integral representation in continuous spaces).
Economically, the kernel K
captures everything privacy regulation might do measurement: rounding
timestamps, batching reports, reporting only bins of AoI/latency, adding
differential-privacy noise, suppressing rare metadata fields, or
publishing only an aggregate score. The key restriction is that the
additional transformation does not depend on (x, t) except through O; it is a purely informational
degradation.
This modeling choice aligns with how privacy rules are implemented in practice. Compliance teams typically do not rewrite the underlying physics P(O ∣ x, t); instead they constrain what is logged, retained, and surfaced for downstream uses (including payments), which is naturally represented as post-processing of an internal signal.
Let W*(O) denote the maximal attainable expected welfare (equivalently, expected server value net of real effort costs) when contracts p must be measurable with respect to O, subject to the agent’s incentive and participation constraints and any payment feasibility constraints (limited liability, budget B, etc.). Likewise define W*(O″) when contracts must be measurable with respect to the coarser O″. Transfers cancel in welfare, but they matter indirectly because they are the only instrument through which the server can influence the unobserved action x.
Importantly, we can interpret W*(⋅) as the : it is the welfare produced by the best incentive scheme consistent with the privacy regime.
The central implication of garbling is that it can only shrink the
feasible set of contracts. Formally, any payment rule measurable with
respect to O″ can
be implemented using O by with
the garbling kernel. Concretely, for any p″ : 𝒪″ → ℝ+,
define a derived payment rule p : 𝒪 → ℝ+ by
p(O) = 𝔼 [p″(O″) ∣ O] = ∑O″ ∈ 𝒪″p″(O″) K(O″ ∣ O),
again with the integral analogue in continuous spaces. Then for every
(x, t),
𝔼 [p(O) ∣ x, t] = ∑Op(O) P(O ∣ x, t) = ∑O″p″(O″) P(O″ ∣ x, t) = 𝔼 [p″(O″) ∣ x, t].
Thus, , the expected payment schedule as a function of x and t is identical under (O, p) and under (O″, p″).
It follows that the agent’s best response x*(t) and
participation decision are preserved. Because the induced effort and the
physical welfare component v(t, x) − c(t, x)
are unchanged, any outcome achievable under O″ is also achievable
under O. Therefore the
principal can do (weakly) better with the more informative signal:
W*(O) ≥ W*(O″).
This is Proposition~2 in the global context: .
Weak inequality is essentially unconditional; strictness requires that the extra information in O is . If the garbling is nontrivial but the first-best action xFB(t) is constant across types (or the optimal second-best happens to be constant because incentives are irrelevant), then additional observability may not change anything. In the economically interesting cases, however, the optimal action varies with type or state—e.g., high-q nodes are worth pushing harder, low-σ nodes can be induced cheaply, or congestion states call for different update intensities. When O allows the server to differentially reward outcomes that are informative about the underlying x (or about t), while O″ collapses those distinctions, the feasible set of incentive-compatible allocations strictly contracts, and so W*(O) > W*(O″) under mild regularity.
This strictness criterion is operational: privacy harms welfare precisely to the extent it destroys of behaviors or states that the platform would otherwise like to induce.
The Blackwell result reframes telemetry design as a governance
problem with a clear economic object: information is valuable not
because it is intrinsically monetizable, but because it expands the
space of implementable incentive schemes. From this viewpoint, common
privacy interventions have interpretable effects:
(i) (coarse bins for Ã, Ẽ) reduces the
sensitivity of payments to marginal improvements;
(ii) (reporting only batch averages) blurs attribution, weakening
incentives at the node level;
(iii) (including differential privacy) reduces effective signal-to-noise
ratio, forcing flatter contracts;
(iv) (dropping metadata fields) can eliminate precisely those correlates
of effort that are cheapest to measure and easiest to audit.
At the same time, the theorem does say that ``more data is always better’’ in an unconstrained sense; it says that, holding fixed the legal and ethical constraints that bind payments to observables, . This provides a transparent normative benchmark: if a proposed privacy rule is a pure garbling of already-compliant telemetry, any welfare gains must come from elsewhere (e.g., reduced breach risk, higher participation, or dynamic trust effects), not from improved incentives.
Blackwell monotonicity is a partial order and therefore qualitative: it ranks privacy regimes but does not measure how costly a given garbling is. Two different garblings are often incomparable, and even when comparable, the welfare gap can be small or large depending on curvature, noise levels, and how strongly v(t, x) responds to effort. This motivates our next step. In Section~6 we impose a quadratic-Gaussian specialization in which telemetry compresses into a sufficient statistic S and the optimal contract is affine/threshold in S. In that setting, we can convert ``informativeness’’ into a scalar (mutual information or SNR) and obtain explicit welfare-loss bounds—a quantitative, interpretable that complements the Blackwell ordering developed here.
To move beyond the qualitative Blackwell ranking, we now impose a tractable quadratic–Gaussian specialization in which (i) the agent’s effort choice admits a closed-form best response under simple contracts, and (ii) the informational content of privacy-preserving telemetry can be summarized by a single scalar signal-to-noise ratio (SNR), or equivalently by mutual information. This delivers an explicit, interpretable : how much welfare we lose as telemetry is noisified, quantized, or otherwise garbled.
We adopt the standard linear-quadratic primitives
(cf. Proposition~3):
$$
c(t,x)=\tfrac{1}{2}\sigma x^2,\qquad
v(t,x)=a(q)+b(q)x-\tfrac{1}{2}\kappa x^2,
$$
where b(q) captures
the marginal value of update intensity for higher-quality data, and
κ ≥ 0 captures diminishing
returns (or congestion externalities) on the server side. Absent
informational frictions, the (type-contingent) first-best would be
$$
x^{FB}(t)=\frac{b(q)}{\kappa+\sigma}\quad \text{(clipped to $\mathcal X$
if needed).}
$$
Thus, heterogeneity in either q or σ makes the target action vary
across nodes/states, precisely the case where privacy-induced
information loss is economically salient.
We compress the privacy-preserving observable vector O = (Ã, Ẽ, D̃)
into a scalar statistic
S = γx + η, η ∼ 𝒩(0, ση2),
which can be interpreted as an optimally weighted linear index of
``better outcomes’’ such as (−Ã, −Ẽ, D̃) under
a local linearization and joint normality (or as the best linear
predictor when the platform reports only a score). Importantly, many
privacy interventions map naturally into this reduced form: rounding
timestamps, batching, and differential-privacy noise all increase ση2
and therefore reduce effective informativeness.
Within this specialization, and consistent with Proposition~1’s monotonicity logic, we restrict attention to contracts that are monotone in S. Two practically relevant forms are:
p(S) = α0 + α1S.
p(S) = (α0 + α1S)+, which is equivalent to a linear bonus schedule above a threshold S̄ = −α0/α1 and 0 otherwise.
The second form captures the institutional reality that many IIoT nodes cannot be fined (no negative payments), and that budgets/approval processes often create implicit floors at 0. This limited-liability feature is also what makes measurement noise even with risk-neutral agents: when pay is truncated, noisier telemetry forces the principal to ``pay for luck’’ more frequently to deliver a given marginal incentive.
Under the pure affine rule and ignoring clipping to 𝒳, the agent’s problem is
$$
\max_x\;\; \mathbb E[\alpha_0+\alpha_1(\gamma
x+\eta)]-\tfrac{1}{2}\sigma x^2
\;=\;\alpha_0+\alpha_1\gamma x-\tfrac{1}{2}\sigma x^2,
$$
so the best response is
$$
x^*(q,\sigma)=\frac{\alpha_1\gamma}{\sigma}.
$$
Thus incentive is governed by α1γ, and higher
marginal cost σ attenuates
induced intensity mechanically.
Under the limited-liability affine contract p(S) = (α0 + α1S)+,
expected pay is the truncated-normal moment
$$
\mathbb E[p(S)\mid x]
=
s\,\Phi\!\Big(\frac{s}{\alpha_1\sigma_\eta}\Big)+\alpha_1\sigma_\eta\,\phi\!\Big(\frac{s}{\alpha_1\sigma_\eta}\Big),
\qquad s:=\alpha_0+\alpha_1\gamma x,
$$
and the marginal incentive becomes
$$
\frac{\partial}{\partial x}\mathbb E[p(S)\mid x]
=\alpha_1\gamma\;\Phi\!\Big(\frac{\alpha_0+\alpha_1\gamma
x}{\alpha_1\sigma_\eta}\Big).
$$
The first-order condition ∂x𝔼[p] = σx
therefore features an Φ(⋅)
that decreases as ση2
rises (holding the contract fixed), reflecting that a larger fraction of
realizations are pushed into the zero-pay region. Consequently, to
sustain a given x under
noisier telemetry, the principal must raise α1 and/or lower the
threshold—both of which increase expected transfers. Optimally, the
principal trades off higher induced effort against higher expected
payments, and the optimal incentive slope α1* is
(generically) in ση2
(Proposition~3(ii)).
The quadratic structure implies that welfare is locally a quadratic
loss in deviations from the first-best action. In particular, completing
the square yields (suppressing type arguments)
$$
W(t;x) \;=\; a(q)+\frac{b(q)^2}{2(\kappa+\sigma)} \;-\;
\frac{\kappa+\sigma}{2}\,\big(x-x^{FB}(t)\big)^2,
$$
up to constants independent of x. Hence the welfare loss from any
second-best policy is controlled by the mean-squared distortion in
induced intensity:
$$
W^*-W^*(O)\;\le\;\frac{\kappa+\bar\sigma}{2}\,\mathbb
E\big[\big(x-x^{FB}(t)\big)^2\big],
$$
for an appropriate upper bound σ̄ on marginal costs over
participating nodes (or more generally via curvature bounds, as in
Proposition~4).
The informational step is to relate this distortion to what can be
inferred from privacy-preserving telemetry. When S is Gaussian and x is (approximately) Gaussian in the
induced equilibrium cross-section—as occurs when (q, σ) are Gaussian and
x*(t) is
linear in types—the Bayes-optimal estimator is linear and the estimation
error is explicit:
$$
\mathrm{mmse}(x\mid S)=\frac{\mathrm{Var}(x)}{1+\mathrm{SNR}},
\qquad \mathrm{SNR}:=\frac{\gamma^2\mathrm{Var}(x)}{\sigma_\eta^2}.
$$
Moreover, mutual information is
$$
I(x;S)=\tfrac{1}{2}\log(1+\mathrm{SNR}),
$$
so the same object can be written as an information index. Combining
these relationships with curvature bounds yields the quantitative
privacy loss bound in Proposition~4:
$$
0\le W^*-W^*(O)\;\le\; C\cdot \mathrm{mmse}(x\mid O)
\;\le\; C''\cdot \frac{1}{e^{2I(x;O)}-1}.
$$
This expression formalizes a simple operational message: privacy
interventions that reduce effective SNR (more noise, coarser bins,
heavier aggregation) raise mmse and
therefore impose a welfare cost that scales like an inverse-exponential
function of mutual information.
The bound is not merely qualitative. In the Gaussian channel, mmse(x ∣ S) is pinned down exactly by SNR, and a linear (or threshold-linear, under limited liability) contract can implement an induced x that is proportional to the posterior mean of the target action. By choosing a Gaussian environment in which (i) xFB(t) is linear in a Gaussian type index and (ii) privacy acts as additive Gaussian noise on that index, one can make the welfare gap equal (up to constant curvature factors) to (κ + σ)mmse(x ∣ S)/2. In this sense, the information-theoretic upper bound is tight up to constants: it is achieved by the natural linear score S and a monotone pay-for-score scheme.
This specialization clarifies privacy bites: not through transfers per se (they net out of welfare), but through the informational constraint that forces the principal to infer effort/type from noisy, privacy-compliant telemetry. At the same time, the results inherit the limitations of the LQG benchmark: linearization, Gaussianity, and the restriction to simple monotone contracts. We view these as useful approximations for engineering practice (where contracts are rarely fully nonlinear), while the Blackwell result in the previous section provides the more general, form-free direction of the privacy–welfare tradeoff.
We now outline three extensions that preserve the core economic logic——while bringing the model closer to IIoT practice. We keep these extensions modular: each can be added without disturbing the main comparative statics, but each highlights a distinct way in which privacy, budgets, and strategic behavior interact.
A realistic server contracts with many nodes i ∈ {1, …, I} per round,
often under a hard accounting constraint on total rewards. A
parsimonious formulation is separable values and costs with a shared
budget:
$$
\max_{\{p_i,s_i\}}\;\; \mathbb E\Big[\sum_{i=1}^I
s_i(O_i)\,v(t_i,x_i)\;-\;\sum_{i=1}^I s_i(O_i)\,p_i(O_i)\Big]
\quad \text{s.t.}\quad
\mathbb E\Big[\sum_{i=1}^I s_i(O_i)\,p_i(O_i)\Big]\le B,
$$
where si(Oi) ∈ {0, 1}
is an acceptance/selection rule (e.g., whether an update is admitted or
whether the node is paid that round), and each node chooses xi to maximize
𝔼[si(Oi)pi(Oi) ∣ xi, ti] − c(ti, xi)
plus an outside option ū(ti).
Two forces emerge. First, the budget turns procurement into mechanism
design even when v and c are separable: the shadow value of
budget couples agents through a Lagrange multiplier μ ≥ 0, pushing the platform toward a
architecture. In the quadratic–Gaussian specialization, for example, it
is natural to implement a one-dimensional score Si and select
the top performers subject to expected spend; a simple approximation
is
si(Oi) = 1{Si ≥ S̄}, pi(Oi) = (α0 + α1Si)+,
where S̄ (or equivalently α0) is adjusted to
satisfy the budget in expectation. This mirrors the operational reality
of federated learning where only a subset of clients are sampled or
admitted each round due to bandwidth and energy limits.
Second, selection changes incentives because payment becomes rather than a pure transfer. Under MLR/supermodularity (as in Proposition~1), we can still choose si(⋅) and pi(⋅) to be monotone in the same partial order (lower Ã, lower Ẽ, higher D̃). Intuitively, if higher effort shifts the distribution of Oi toward better outcomes, then both and should load positively on those outcomes. Endogenous participation then takes a familiar form: the platform can either (i) raise expected rents to attract high-q/low-σ nodes, or (ii) keep rents low and instead rely on selection to concentrate budget on those whose telemetry indicates high marginal product. Privacy tightens this tradeoff because selection must be justified by noisy signals: stronger garbling forces either weaker screening or higher payments to obtain the same effective pool.
Many IIoT deployments are not i.i.d. across rounds. Network
congestion, battery state, local data drift, and even policy constraints
evolve, so the node’s effective type tr = (qr, σr)
may follow a Markov process. A minimal dynamic extension is:
tr + 1 ∼ M(⋅ ∣ tr), Or ∼ P(⋅ ∣ xr, tr), r = 0, 1, 2, …
with per-period payoffs as before and common discount factor δ ∈ (0, 1).
If we restrict attention to contracts p(O) (and possibly stationary selection rules s(O)), the node’s best response remains myopic in x whenever (i) the action affects only current-period observables through P(O ∣ x, t) and (ii) the transition M does not depend on x (or depends weakly enough that we treat it as negligible). This is a useful benchmark: in many engineering settings, a device’s update intensity changes immediate telemetry but does not materially alter its longer-run cost parameter σ (at least over the contracting horizon). Under this separation, the static incentive constraints and the monotonicity conclusions carry over round-by-round, and the welfare comparison under Blackwell garblings continues to hold , hence also in discounted sums.
When actions do affect future states (e.g., aggressive updating depletes battery and raises future σ), then x creates an intertemporal externality on the agent’s own continuation value. Stationary contracting remains attractive operationally, but it is no longer generally optimal. A tractable approach is to augment the observable with a privacy-compliant state proxy (e.g., battery bucket, connectivity class) and interpret the environment as a controlled Markov process with a low-dimensional score Sr; then the platform can implement monotone contracts p(Sr, ẑr) where ẑr is a coarse state report. Privacy enters twice: it garbles both performance and state, reducing the achievable dynamic insurance/incentives tradeoff. We view this as a natural place where the information-theoretic bounds (via I(t; O) or I(x; O)) can be reinterpreted as bounds on the platform’s ability to track latent operating regimes over time.
A recurring concern in pay-for-telemetry schemes is that nodes may rather than exert real effort. In our baseline, O is generated by a known channel P(O ∣ x, t) and is verifiable under the privacy constraint. In practice, however, nodes may have limited ability to manipulate coarse metadata (e.g., inflate D̃), to shape timing artifacts (e.g., batch transmissions to affect measured AoI/latency), or to exploit model-specific loopholes.
We can capture this by adding a manipulation choice m ∈ ℳ with cost k(m) and modified
signals
O ∼ P(⋅ ∣ x, m, t), U = 𝔼[p(O) ∣ x, m, t] − c(t, x) − k(m).
Even when k(m) is
convex and m is “mild,” the
key design objective becomes between the server’s value driver v(t, x) and the
contractible score extracted from O. Three robustification tools are
particularly compatible with privacy constraints:
(i) If manipulation can improve one coordinate (say D̃) while degrading another (say Ẽ), then monotone contracts that load on coordinates (or on a sufficient statistic S combining them) reduce the profitability of gaming. Formally, one can choose S = ωDD̃ − ωAÃ − ωEẼ with weights reflecting both marginal value and susceptibility to manipulation.
(ii) Since manipulation typically operates through extreme reported values, bounding p(O) (or using (α0 + α1S)+ with an upper cap) can make the gain from gaming second-order while keeping the first-order effort incentive. This is also attractive for compliance: privacy policies often favor predictable, non-explosive payment rules.
(iii) If the platform can occasionally run a privacy-compliant audit (e.g., secure enclave verification, randomized spot-check of timing logs, or cross-validation with network-side telemetry), then an audit probability π with penalty F for detected manipulation effectively adds an expected cost πF to m. Even small π can deter manipulation when F is contractual and credible; importantly, audits need not reveal raw data, only violations of reported constraints. Conceptually, this restores a stronger effective channel P(O ∣ x, t) by discouraging adversarial deviations.
These extensions underscore a common theme: while transfers net out of welfare, is economically central. Budgeted selection, dynamics, and manipulation each magnify the role of information constraints, and each suggests operational design responses (score-based admission, stationary state-contingent rules, and robust scoring with audits) that remain consistent with the monotonicity logic developed earlier.
The propositions above are qualitative—more informative telemetry supports stronger incentives and higher welfare—but the practical question is quantitative: efficiency is lost when AoI/latency are noisy or coarsened for privacy, and (if any) are worth the engineering and compliance cost. In this section we outline a numerical illustration that maps our reduced-form parameters to IIoT-relevant magnitudes and provides a calibration workflow a platform could execute using privacy-compliant logs.
A convenient bridge from engineering measurement error to our welfare
bounds is the Gaussian specialization in which the contract loads on a
one-dimensional score
S = ωDD̃ − ωAÃ − ωEẼ,
interpreted as the platform’s . Under the reduced-form observation model
à = (Ā − αx) + εA
and Ẽ = (Ē + βx) + εE,
we can rewrite
$$
S \;=\; \underbrace{(\omega_A\alpha-\omega_E\beta+\omega_D
D_x)}_{\gamma}\,x\;+\;\underbrace{\omega_D\varepsilon_D-\omega_A\varepsilon_A-\omega_E\varepsilon_E}_{\eta},
$$
where Dx
is the marginal effect of effort on the metadata proxy (e.g., marginal
samples/round reported). Thus measurement noise in AoI/latency
translates into a larger $\sigma_\eta^2=\Var(\eta)$, lowering the
signal-to-noise ratio and (via Proposition~3) shrinking the optimal
incentive power α1*.
To make this concrete, fix an illustrative IIoT baseline: Ā = 10 s, Ē = 50 ms, and effort x scaled so that a one-unit increase in x reduces AoI by α = 2 s but increases latency by β = 5 ms (capturing a “freshness–responsiveness tension” due to compression, batching, or radio scheduling). Suppose the platform constructs S primarily from AoI and latency, setting ωA = 1/s and ωE = 1/ms to normalize units (and ωD = 0 for the moment). If AoI is measured with standard deviation $\sd(\varepsilon_A)=1$ s and latency with $\sd(\varepsilon_E)=3$ ms, then $\sd(\eta)\approx \sqrt{1^2+3^2}\approx 3.16$ in normalized units. If γ = ωAα − ωEβ = 2 − 5 = −3, the platform’s score is sensitive to effort (i.e., effort worsens the weighted index), so the platform would reverse weights or include metadata to ensure γ > 0 whenever increased updating is desired. This calculation is itself diagnostic: it tells the practitioner that if effort mechanically worsens measured latency enough, then a “pay-for-freshness only” scheme is fragile unless latency is also priced (or technology decouples α and β).
Once γ > 0 is ensured
(e.g., by choosing weights aligned with the platform’s marginal value),
the mutual information in the Gaussian channel is
$$
I(x;S)=\tfrac{1}{2}\log\!\Big(1+\tfrac{\gamma^2\Var(x)}{\sigma_\eta^2}\Big),
$$
so empirically plausible increases in $\Var(\varepsilon_A)$ or $\Var(\varepsilon_E)$ map directly into
decreases in I(x; O) (or I(x; S)) and thus
looser efficiency bounds of the form in Proposition~4:
$$
0\le W^*-W^*(O)\;\lesssim\; \frac{C''}{e^{2I(x;O)}-1}.
$$
Operationally, the platform can report a “telemetry SNR” and interpret
it as a in welfare units.
Privacy regulation often induces coarsening: AoI may be reported in buckets (e.g., {0–1s, 1–5s, 5–15s, 15+}), latency may be truncated (to avoid leaking network fingerprints), and metadata may be clipped (to prevent inference of local dataset size). Each of these is a Blackwell garbling of the richer telemetry and therefore weakly reduces attainable welfare (Proposition~2); the numerical task is to quantify is lost under candidate compliance regimes.
A simple plan is to simulate (or estimate) the induced distribution of the score S under effort variation, then compare (i) the “continuous” index S to (ii) a quantized index S′ = bin(S) with K bins. In Gaussian environments, coarse quantization can reduce information sharply when K is small. For example, moving from a continuous score to a 1-bit report (``good/bad’’) can reduce mutual information by an order of magnitude when the underlying SNR is moderate, which in turn can multiply the bound on W* − W*(O) by a similar factor because e2I − 1 is approximately linear in SNR for small-to-moderate I.
Practically, the platform can treat candidate telemetry policies as alternative channels P(O″ ∣ O) and compute, via Monte Carlo, the implied I(x; O″) and the induced second-best objective under the best affine monotone contract. This yields a compliance-facing : for each coarsening rule, report (a) the predicted reduction in optimal slope α1*, (b) the predicted change in induced x* (hence mean AoI/latency), and (c) the implied welfare upper bound from Proposition~4. The key sensitivity exercise is to vary: (i) bucket width, (ii) clipping thresholds, and (iii) which coordinates are reported at all (AoI-only vs. AoI+latency vs. AoI+latency+metadata). We expect complementarities: adding even a noisy latency coordinate can improve robustness to gaming and improve the effective γ in the score, but only if latency noise is not so high that it becomes pure payment variance.
A calibration is feasible without raw data access, but it does require a disciplined logging protocol. At minimum, the platform needs:
To identify how x moves telemetry, we need either randomized contracts (varying α1 across rounds) or randomized engineering “knobs” that shift effective update intensity (e.g., allowed upload frequency). This is the cleanest way to estimate γ and ση2.
The platform must log (Ã, Ẽ, D̃) at the same granularity permitted for contracting, including any coarsening/bucketing actually used. Otherwise the estimated channel will be overly optimistic.
For federated learning, a practical approach is to map each accepted update (or batch) to an offline contribution score, e.g., improvement in validation loss on a held-out dataset, a gradient-norm-based proxy, or a Shapley-style approximation computed in secure enclaves. This yields an empirical analog of v(t, x) (or at least its slope in x), sufficient to calibrate b(q) and curvature κ in the quadratic approximation.
If nodes can opt out, observed acceptance and dropout rates identify the IR constraint slackness and help pin down α0 (or selection thresholds) needed for participation.
Direct energy cost measurement may be unavailable, but device-reported battery buckets, CPU utilization, or radio-on time (all potentially coarsened) can be used to estimate the convexity of c(t, x) and the heterogeneity in σ.
The end product is not a single “true” parameter set, but a small set of stress-tested scenarios. For each scenario, we recommend reporting: (a) estimated SNR (or I(x; O)) under the current privacy policy; (b) predicted optimal incentive slope α1* and induced effort x*; (c) predicted mean AoI/latency outcomes; and (d) an information-theoretic upper bound on the welfare gap. We also recommend a table: hold the engineering environment fixed and vary only telemetry coarsening rules to quantify the marginal value of one additional bit of reporting precision. This helps avoid over-claiming: even when the bound is loose, it provides a safe, compliance-friendly statement of what efficiency is to guarantee under aggressive garbling, and what might be recovered by modestly richer (still privacy-preserving) telemetry.
Our results can be read as a design manual for : rather than asking whether the platform should “pay for performance,” the right question is which performance signals should be engineered and retained under privacy constraints, and how to translate them into contracts that are both incentive-compatible and defensible to regulators. The model does not claim that one can always recover first-best; instead, it clarifies which telemetry investments are economically meaningful and which are, at best, cosmetic.
The central requirement for incentives is that observables shift in a predictable direction with the agent’s action x. Proposition~1 formalizes this as an MLR/supermodularity condition: better observable outcomes should become more likely under higher x. This immediately yields a practical “instrumentation checklist.” AoI-like timestamps and latency-like service measures are valuable not because they are intrinsically important, but because they are plausibly . By contrast, metrics that strongly correlate with realized model improvement but are only weakly affected by x (or are confounded by unobserved environment) may be attractive for prediction yet poor for incentives: they invite paying rents for luck, topology, or hidden heterogeneity rather than for costly updating.
This distinction matters for compliance-facing governance. When auditors ask why a particular signal is collected, the strongest justification is: , and the signal has a documented causal sensitivity to the node’s controllable actions. Signals without such justification should be treated as “nice-to-have” analytics and, under privacy-by-design, excluded from the contracting surface.
When the observation channel is multidimensional, a recurring temptation is to collapse everything into a single scalar KPI. Our monotonicity result provides a disciplined alternative. Under the partial order “lower Ã, lower Ẽ, higher D̃ is better,” there exists an optimal payment p*(O) that is monotone in each coordinate. This is useful both practically and politically: monotone payment rules are interpretable (“better freshness and faster service should not pay less”) and easier to defend as non-arbitrary.
That said, the Gaussian specialization explains when a scalar score S is not merely convenient but close to sufficient: if the informative content of O for effort is essentially one-dimensional, then an index S = ωDD̃ − ωAÃ − ωEẼ captures most of what can be used for incentives. The design implication is not “always score,” but rather: In deployments, we can test this by checking whether additional coordinates materially increase I(x; O) beyond the best linear index. If not, compressing to S reduces privacy surface area without sacrificing implementable welfare.
Proposition~2 (Blackwell monotonicity) formalizes a simple but often neglected point: privacy policies that garble telemetry shrink the feasible set of contracts and weakly reduce attainable welfare. This is not an argument against privacy; it is an argument for . In practice, privacy controls commonly take the form of bucketing, clipping, and censoring. Each is a choice of a garbling kernel K, hence a move from O to O″. Our framework suggests treating candidate privacy policies as competing “measurement regimes” and evaluating them by their induced informativeness I(t; O″) (or I(x; O″)) and the corresponding implementable performance.
This yields a concrete governance proposal: before enacting a coarsening rule (e.g., moving latency from millisecond precision to coarse bins), the platform should publish an internal “efficiency impact assessment” reporting (i) the change in mutual information (or SNR in the Gaussian approximation), and (ii) the implied change in optimal incentive power. This mirrors the logic of privacy impact assessments, but it is grounded in incentive feasibility rather than merely in data minimization.
A persistent risk in procurement settings is over-promising what incentive schemes can guarantee under privacy constraints. Proposition~4 provides a disciplined way to speak cautiously: we can bound the welfare gap W* − W*(O) by an explicit decreasing function of informativeness (mutual information/SNR). The operational implication is that the platform should communicate . For example: “Under the current telemetry policy, no contract measurable in O can guarantee more than X% of first-best welfare in the worst case,” or “to reduce the worst-case efficiency loss below Y, we need at least Z additional bits of effective reporting precision in the freshness/latency score.”
Such statements are conservative by construction: they do not require believing a particular behavioral model beyond the maintained curvature and noise assumptions. They are also robust to strategic behavior: when telemetry is noisy or coarsened, stronger pay-for-performance can backfire by increasing payment variance without reliably increasing effort, a point captured by the optimal shrinkage of α1* as noise rises.
The reduced-form mapping A = Ā − αx and E = Ē + βx highlights a design hazard: if freshness effort mechanically worsens latency, then a contract that rewards lower AoI but ignores latency can induce socially costly behavior (e.g., aggressive update frequency that congests the uplink). Conversely, a contract that over-penalizes latency can suppress updating and degrade model quality. The policy implication is that , and these should be revisited as engineering changes shift α and β.
Gaming concerns reinforce the case for multi-signal design. If one coordinate is easily manipulated (e.g., metadata D̃ via misreporting), pairing it with harder-to-fake measures (e.g., server-side timestamp-based AoI) can restore approximate MLR structure and reduce the need to collect sensitive data. More generally, we should treat “what to measure” as partly an adversarial design problem: prefer signals that are (i) causally sensitive to effort, (ii) verifiable under privacy constraints, and (iii) costly to manipulate relative to their payment impact.
Finally, the privacy–telemetry constraint does not only change effort incentives; it can change . When observables are heavily garbled, the platform may need higher fixed payments α0 (or looser selection thresholds) to satisfy participation, which can disproportionately attract high-cost nodes (high σ) and deter low-cost ones if the contract cannot discriminate effectively. Thus, privacy policy can have distributional effects even when average welfare changes modestly.
A practical response is to couple monotone pay-for-performance with explicit participation tools: opt-in menus tied to coarse device classes, selection rules based on verifiable compliance events, or budgeted tournaments within privacy-permitted groups. Importantly, any such rule should be evaluated not only for average performance but also for compositional impacts (who remains in the pool), since these determine the long-run sustainability of federated procurement in heterogeneous IIoT populations.
Taken together, these implications point to a unified message: telemetry is not a neutral byproduct of systems engineering. It is an economic instrument that expands (or contracts) the space of feasible contracts. Designing “what to measure (and what not)” is therefore inseparable from designing incentives, privacy compliance, and credible performance guarantees.
We studied a procurement problem that is increasingly central to industrial federated learning: how a platform can elicit timely, high-quality updates from heterogeneous IIoT nodes when both and are private, and when privacy regulation restricts contracts to depend only on a constrained telemetry surface. The model is deliberately simple in its economic primitives—risk neutrality, convex effort cost, and a reduced-form mapping from update intensity to freshness and latency—yet it is rich in the dimension that matters for practice: the platform does not contract on the “true” contribution, but on whatever noisy and coarsened observables O = (Ã, Ẽ, D̃) can be lawfully collected and verified.
Two broad messages emerge. First, privacy constraints should be understood as that shrink the feasible set of incentive schemes. Second, within those constraints, there is substantial structure: the optimal contracting problem is not an arbitrary black box, but yields interpretable monotonicity and quantitative bounds that can be used as design guidance.
On the incentive side, our monotonicity result formalizes a principle that practitioners often apply informally but rarely justify: . Under a multidimensional monotone likelihood ratio/supermodularity condition—capturing that higher update intensity makes “better” outcomes (lower AoI, lower latency, higher metadata) more likely—there exists an optimal payment rule p*(O) that is coordinate-wise monotone. This is more than aesthetic. Monotonicity simultaneously supports implementability (it aligns marginal incentives with effort) and governance (it avoids contracts that look arbitrary or discriminatory when audited). Importantly, the result is stated at the level of observables, not latent actions: it respects the privacy/compliance premise that p(⋅) must be measurable with respect to O.
We then specialized to a quadratic-Gaussian environment to extract closed-form implications. In that specialization, the contracting problem collapses to a one-dimensional sufficient statistic S (e.g., a linear index in −Ã, −Ẽ, and D̃) and an affine incentive rule p(S) = α0 + α1S. The induced action x* responds in the expected direction to incentive power, and optimal α1* shrinks as telemetry becomes noisier. This provides a tractable bridge between economic contracting and systems measurement: noise in AoI/latency measurement is not merely a forecasting nuisance; it directly dilutes the return to pay-for-performance and, in equilibrium, worsens freshness or responsiveness. In this sense, “better instrumentation” and “stronger incentives” are complements, while “heavier coarsening” and “stronger incentives” can be substitutes (or even adversaries) because paying on noisy signals purchases variance more than effort.
The privacy contribution of the paper is to place these ideas in the language of Blackwell informativeness. When a privacy rule garbles O into a less informative O″, the platform’s attainable welfare cannot increase: W*(O″) ≤ W*(O). This statement is not normative about privacy—one may rationally accept a welfare loss to satisfy legal or ethical requirements—but it is clarifying about mechanism design: coarsening is an economically meaningful intervention because it removes contracts from the feasible set. Moreover, the strictness condition highlights when the loss is unavoidable: if first-best actions vary with type or state, then collapsing the signal generically forces pooling and leaves rents and inefficiency on the table. Put differently, privacy can be “free” only when incentives are irrelevant or when heterogeneity is immaterial; otherwise it carries a real shadow price.
To make that shadow price operational, we derived a quantitative “price of privacy” bound in the Gaussian case that links welfare loss to mutual information (equivalently signal-to-noise). The bound is conservative by construction: it does not require the platform to claim it can hit a particular performance target, only that it can bound what is impossible under a given telemetry regime. This is, we think, the right posture for privacy-compliant procurement. Rather than over-promising, the platform can translate measurement precision into a maximum attainable efficiency range, and then choose privacy and instrumentation with explicit tradeoff accounting. In environments where compliance decisions must be documented and defended, a bound-based language is often more credible than point predictions.
Our analysis also underscores an often-missed conceptual point: in federated settings, “data quality” and “timeliness” do not enter the contract symmetrically. Quality q is a type component—largely given at the node—while AoI/latency are partially effort-responsive. This asymmetry implies that the platform should focus its contracting surface on signals that move with x, using them to screen or incentivize indirectly, rather than attempting to pay directly for realized model gains that may be confounded and weakly manipulable. In our framework, the practical question is not whether to reward “accuracy improvement,” but whether the available observables satisfy something like the MLR property needed for incentives to bite.
There are, of course, limitations. We assumed risk-neutral agents, which abstracts from the insurance role of fixed payments and from variance-aversion to noisy pay-for-performance. We also treated observables as exogenous outputs of a channel P(O ∣ x, t), while in practice platforms can endogenously choose measurement frequency, secure attestation, and aggregation protocols, all of which reshape that channel. Our reduced-form mapping between effort, AoI, and latency captures a common engineering tension but cannot substitute for a full queueing/network model in congestion-heavy regimes. Finally, while we motivated a repeated environment, our main results are stationary and do not exploit dynamic incentives, reputation, or intertemporal substitution in effort.
These limitations point to natural extensions. A first direction is : allowing t or network conditions to evolve and letting O include lagged performance would connect privacy to learning and to dynamic moral hazard. A second is : when aggressive updating by one node affects others through congestion, the platform’s problem becomes one of mechanism design with externalities, and monotone contracts may need to be paired with admission control or congestion pricing. A third is : if some coordinates of O are manipulable, the relevant condition is not MLR under a benign channel but incentive compatibility under strategic signal shaping, suggesting the use of auditing, randomization, and mechanism robustness. A fourth is : differential privacy, secure enclaves, and cryptographic proofs can be modeled as alternative garblings with different information budgets, enabling a welfare-per-bit comparison of privacy tools.
The overarching conclusion is that privacy and incentives are not separate design problems in federated learning procurement. Privacy regulation determines what can be conditioned on; that, in turn, determines which actions can be induced at reasonable cost. Our framework makes this dependence explicit, yields monotone and interpretable contract forms under standard conditions, and provides quantitative bounds that translate telemetry informativeness into attainable welfare. The practical implication is a reframing: the platform’s core economic decision is not merely how much to pay, but , , and in a way that remains compliant, auditable, and robust to noise and heterogeneity.