Agentic Information Acquisition

• Agentic information acquisition is a dynamic learning process where agents strategically design information gathering under strict entropy and capacity constraints to shape decision timing.
• It employs rigorous mathematical models using jump processes, Bregman divergences, and the HJB framework to characterize belief dynamics and time-risk profiles.
• Applications span R&D, clinical trials, and digital marketing, highlighting optimal trade-offs between fast, risky decisions and slow, deterministic learning outcomes.

Agentic information acquisition denotes the design and implementation of information-gathering processes in which decision makers (agents) dynamically and strategically determine both the structure and timing of their own learning activities, typically under explicit resource constraints and with objectives that reflect their risk preferences regarding learning outcomes, especially with respect to the timing and predictability of decision resolution (Chen et al., 2018). The field is characterized by a rigorous mathematical treatment of belief dynamics, learning constraints, optimal policy design, and their economic or operational implications. Recent research demonstrates how agents can choose among fundamentally different information acquisition strategies to shape not only the precision of their eventual decisions but also the risk profile of when critical information will be available.

1. Formal Model and Capacity Constraints

The canonical framework for agentic information acquisition is the dynamic learning model for a binary state, where the agent’s evolving belief $\mu_t$ about the state reaches an upper ($\bar{\mu}$) or lower ($\underline{\mu}$) threshold at a random stopping time $\tau$. The agent may design any learning process—i.e., any signal structure—so long as it satisfies a hard constraint on the instantaneous reduction of entropy: $$\mathbb{E}\left[ \frac{d}{dt}\, H(\mu_t) \;\big|\; \mathcal{F}_t \right] \leq I,$$ where $H(\cdot)$ is a strictly convex entropy function, $I$ is the learning capacity (normalized to $1$ without loss of generality), and $\mathcal{F}_t$ is the natural filtration. This constraint governs the maximum instantaneous mutual information that may be extracted from the environment per unit time.

A crucial consequence is that all admissible, “exhaustive” strategies that saturate this constraint yield the same expected stopping time, namely the initial entropy $H(\mu_0)$, as formalized by the optional stopping theorem: $$\mathbb{E}[\tau - t \mid \mathcal{F}_t ] = -H(\mu_t).$$ Thus, the agent’s degree of control is fundamentally about the distributional properties (dispersion, risk) of $\tau$, not the mean time to threshold.

2. Time-Risk Preferences and Strategic Design

With the mean stopping time fixed for all exhaustive strategies, the key dimension of control is the “time risk”: the shape of the threshold-hitting time distribution. Agents are ordered by their preference over lotteries of $\tau$. Two polar strategies emerge, corresponding to opposite attitudes toward time dispersion.

• Greedy Exploitation (time-risk seeking/convex time preference):

The agent designs the signal process to maximize the probability of immediate resolution. Formally, belief dynamics are dictated by jumps toward the nearest threshold in terms of Bregman divergence:

$$d_H(\nu, \mu) = H(\nu) - H(\mu) - H'(\mu)(\nu - \mu).$$

If $\mu_t > \mu^*$ (the Bregman-median), jumps target $\bar{\mu}$ with rate $\lambda_t = I / d_H(\bar{\mu}, \mu_t)$. Otherwise, the lower threshold is targeted. Between jumps, the compensator forces a slow drift away from the “closer” threshold. This yields highly dispersed (risky) hitting times.

• Pure Accumulation (time-risk averse/concave time preference):

The process structures the learning so that any jump event keeps the entropy constant—jumps are “entropic” but not informative. The entire reduction in uncertainty is achieved deterministically by drift, resulting in a degenerate (deterministic) stopping time:

$$d\mu_t^P = [\mu^H(\mu_t^P) - \mu_t^P] dJ_t(\lambda_t) - \lambda_t [\mu^H(\mu_t^P) - \mu_t^P] dt,$$

where $\mu^H(\mu)$ solves $H(\mu^H(\mu)) = H(\mu)$. All progress is via deterministic entropy reduction.

The dispersion-minimizing strategy eliminates time risk entirely, while the dispersion-maximizing one generates maximal mean-preserving spread.

3. Jump Processes and Belief Dynamics

In both cases, the evolution of beliefs is governed by compensated Poisson jump processes with controlled intensity. The general jump dynamics take the form: $$d\mu_t = \sum_{i} [\nu^{i}(t, \mu_t) - \mu_t][dJ_t^i(\lambda^i(t, \mu_t)) - \lambda^i(t, \mu_t) dt] + \text{(diffusion terms)}$$ The diffusion component (continuous increments) is strictly suboptimal except in the degenerate risk-neutral case; all optimal strategies are pure-jump. Jumps may be interpreted as large, instantaneous leaps in belief, with rate and direction modulated by risk preferences and the current belief’s position relative to Bregman-divergence midpoints.

For Greedy Exploitation, jumps target the “nearest” threshold energetically: $$d\mu_t^G = (\bar{\mu} - \mu_t^G)[dJ_t^1(\lambda_t) - \lambda_t dt] \quad \text{with} \quad \lambda_t = 1/d_H(\bar{\mu}, \mu_t^G).$$ For Pure Accumulation, the process is constructed to maintain invariant entropy across jumps, strictly restricting all uncertainty reduction to the drift.

4. Mathematical Structure and Verification

The analysis is supported by a dynamic programming characterization using the Hamilton–Jacobi–BeLLMan (HJB) equation. The Bregman divergence, rather than Euclidean or Kullback–Leibler divergence, is essential for the geometry of the mean-preserving spread, and for defining the “closest” threshold. The key characterization is that, with the entropy constraint binding,

$\mathbb{E}[\tau - t | \mathcal{F}_t] = -H(\mu_t),$

so the agent’s entire effect on the learning process is encapsulated in higher moments (the risk profile) of $\tau$. The strategies given above are shown to be optimal for, respectively, convex and concave valuations of the time-to-resolution, fully characterizing the mean-preserving spread ordering for $\tau$.

5. Trade-Offs, Applications, and Extensions

• Trade-off:

The pivotal insight is that in strategic/agentic information acquisition, the speed of learning (mean time) and the risk profile (variance/spread) can and should be disentangled. The choice of experiment, signal structure, and belief jump mechanism is not dictated uniquely by speed, but by how much temporal risk is tolerable or desirable.

• Strategy selection:

Time-risk loving agents (willing to gamble on quick—but possibly indefinitely delayed—resolution) adopt the aggressive, stochastic Greedy Exploitation approach. Time-risk averse agents (requiring predictability and suppressing suspense or delay spread) must use Pure Accumulation, accepting slowly delivered, but perfectly predictable, learning.

• Applications:
  • R&D and experimental design, where impatience or regulatory predictability affect the desired learning mechanism.
  • Clinical trials, where sponsors may prefer determinism or faster potential breakthroughs depending on regulatory and market pressures.
  • Digital marketing and platform experimentation, where time-risk loving managers may “bet” on fast wins to maximize early impact.
• Model-guided caution:

The paper emphasizes that optimal signal structures for agentic acquisition are not generally of Brownian or reduced-form Gaussian type—risk preferences induce qualitative differences in how entropy reduction is distributed between jump and drift.

6. Broader Theoretical Implications

The results illuminate the fundamental separation between information acquisition as the reduction of uncertainty and the management of risk in the time required to reach a decisive state. By focusing on agentic flexibility under a binding entropy-reduction (learning-capacity) constraint, the framework generalizes across a variety of learning environments. The use of Bregman geometry to order strategies by risk profile is central, yielding an explicit toolkit for the design and analysis of dynamic information-gathering mechanisms with heterogeneous temporal preferences.

Further, the results extend to multi-agent settings and organizational experimentation strategies, implying that the “shape” of efficient information acquisition cannot be ascertained from mean time criteria alone—its higher-order risk profile and the underlying convexity/concavity of agent valuations are fundamental. Decisions about whether to maximize “suspense,” exploit early risky wins, or ensure perfectly deterministic learning need to be formally anchored in the preference structure for time lotteries.

---

Agentic information acquisition, as defined and characterized by (Chen et al., 2018), is thus an inherently dynamic, risk-sensitive process, where agents design not only the total speed at which they consume learning resources, but also the entire distributional contour of when actionable information will crystallize. The core mathematical architecture—centered on constraints for entropy reduction, Bregman divergences for risk geometry, compensated Poisson jump processes for belief evolution, and HJB formalism for strategic solution—provides a generalizable and robust framework for understanding and optimizing real-world learning systems under resource and temporal risk constraints.