Prior-Free Information Design Framework

• The paper introduces a prior-free framework that achieves robust decision-making by learning unknown state distributions and minimizing regret through binary-search and robustification techniques.
• It unifies Bayesian persuasion, cheap talk, and robust experiment design by employing minimal information withholding, thereby ensuring implementability of optimal actions under uncertainty.
• The framework applies to both repeated games and one-shot settings, offering practical algorithms that balance exploration and exploitation to support robust causal inference and effective persuasion.

A prior-free framework for information design operates under the premise that the designer cannot rely on perfect information regarding the distribution over states of the world. Instead, the designer aims to induce optimal or robust decision-making by learning or withholding dimensions of information, relevant both in repeated games (interactive persuasion) and one-shot or static environments (robust causal inference). Two major methodologies have emerged: learning-based prior-free persuasion (Lin et al., 7 Oct 2024) and partial-identification-driven robust experiment design (Rosenthal, 23 Nov 2025).

1. Model Structures and Primitive Objects

In information design with unknown prior, the core elements are:

• State Space ($\Omega$): A finite set $\{\omega_1,...,\omega_m\}$ representing possible states of the world.
• Action Space ($A$): Either a discrete set $\{a_1,...,a_n\}$ (Lin et al., 7 Oct 2024) or a compact metric space, with (possibly mixed) action $\alpha \in \Delta(A)$ (Rosenthal, 23 Nov 2025).
• Utility Functions: For the receiver, $v(a, \omega) \in [0,1]$; for the designer, $u(a, \omega) \in [0,1]$.
• Prior ($\mu^*$ / $\mathcal{P}$): Designer may face a fixed but unknown prior $\mu^*$ or a convex set of priors $\mathcal{P} \subseteq \Delta(\Omega)$.
• Signals ($S$ / $\Sigma$): Finite signal (message) space. The signaling scheme $\pi: \Omega \to \Delta(S)$ or information structure $E: \Omega \to \Delta(\Sigma)$ implements the mapping from states to distributions over signals.

In repeated games, the designer learns $\mu^*$ sequentially through observed interaction outcomes, while in prior-free robust design the agent holds only the induced signal distribution $E\mu$ as sufficient for inference.

2. Decision-Making and Regret Benchmarks

A central performance criterion is regret, defined as the gap between the realized utility and the hypothetical optimum achievable under full prior knowledge.

• One-Period Payoff Benchmark: Given prior $\mu$ and scheme $\pi$, the receiver best-responds in equilibrium via $$a^*_s \in \arg\max_{a \in A} \mathbb{E}_{\omega \sim \mu_s}[v(a, \omega)]$$, where $\mu_s$ is the posterior induced by signal $s$.
• Designer’s Regret over $T$ Rounds:

$$\mathrm{Reg}(T) = T \cdot U^* - \mathbb{E}\left[\sum_{t=1}^T u(a^{(t)}, \omega^{(t)})\right]$$

where $U^*$ is optimal single-period payoff under $\mu^*$ and $\pi^*$ (Lin et al., 7 Oct 2024).

For robust, one-shot settings (Rosenthal, 23 Nov 2025), the decision-maker faces a set of plausible priors $$\mathcal{P}(\Sigma, E) = \{\nu \in \mathcal{P} \mid E\nu = E\mu\}$$ and ranks actions by their worst-case expected payoff:

$$V(a;\Sigma,E) = \min_{\nu \in \mathcal{P}(\Sigma,E)} \sum_{\omega \in \Omega} \nu(\omega) u(a,\omega)$$

with $\alpha \in \Delta(A)$ maximizing $$\min_{\nu\in\mathcal{P}(\Sigma,E)} \langle \alpha, \nu \rangle$$.

3. Receiver Behavioral Regimes and Designer Algorithms

3.1 Prior-Aware Receiver Model

If the receiver knows $\mu^*$ and updates by Bayes’ rule, the designer uses the following learning algorithm:

• Binary-Search Estimation: Sequential schemes estimate ratios $\mu^*(\omega_i) / \mu^*(\omega_1)$; $O(m \log T)$ rounds attain $|\hat{\mu} - \mu^*|_1 = O(1/T)$ accuracy.
• Robustifying Persuasion: The designer constructs $\hat{\mu}$, designs an optimal scheme for $\hat{\mu}$, then robustifies to ensure persuasiveness for all priors within an $\ell_1$-ball of radius $O(1/T)$.
• Regret Bound: Theorem 1 (Lin et al., 7 Oct 2024) shows $O(\log T)$ regret in the general action case.

Two-Action Case

If $|A|=2$ and designer strictly prefers $a=1$, the scheme selection reduces to a one-parameter search over $M$:

• Double-Logarithmic Search: Algorithm check-persuasiveness at geometrically shrinking intervals yields $\Theta(\log \log T)$ regret (Lin et al., 7 Oct 2024).

3.2 Learning Receiver Model

If receiver is uncertain of the prior and applies contextual multi-armed bandit algorithms:

• Exploration Phase: Designer fully reveals state across $T_0 = O((|\Omega| + \log T)/\epsilon^2)$ rounds to empirically estimate $\mu^*$.
• Strongly Persuasive Robust Scheme: Compute an $\epsilon$-optimal scheme for $\hat{\mu}$, robustify to ($\Delta,\gamma$)-persuasive variant, use for exploitation.
• Regret Bound: With receiver’s external regret $r\mathrm{Reg}(T)=o(T)$, Theorem 3 (Lin et al., 7 Oct 2024) guarantees $\mathrm{Reg}(T) = O(\sqrt{r\mathrm{Reg}(T)T})$; with $r\mathrm{Reg}(T)=O(\sqrt{T})$, designer attains $O(T^{3/4})$ regret.

4. Characterization of Implementable Actions and Information Structures

The prior-free robust framework (Rosenthal, 23 Nov 2025) introduces the following characterization:

• Supporting Prior Condition: $\alpha \in \Delta(A)$ (action/mixed-action) is implementable iff there exists $\nu \in \mathcal{P}$ such that:
  1. $$\alpha \in \arg\max_{\beta \in \Delta(A)} \langle \beta, \nu \rangle$$
  2. $$\langle \alpha, \mu \rangle \geq \langle \alpha, \nu \rangle$$

• Minimal Information Withholding: Every implementable action $\alpha$ admits an information structure $(\Sigma, E)$ whose kernel is $\mathrm{span}\{\nu - \mu\}$, yielding a one-dimensional reduction. This structure is “almost fully informative”: only one linear combination of state probabilities is concealed, everything else is revealed.

Table: Comparison of Regret and Implementability Across Frameworks

Regime / Model	Key Algorithmic Principle	Regret / Robustness Result
Prior-Aware (general action)	Binary search and robustification	$O(\log T)$ regret (Lin et al., 7 Oct 2024)
Prior-Aware (two action)	Double-log search over $M$	$\Theta(\log \log T)$ regret (Lin et al., 7 Oct 2024)
Learning receiver	Full revelation, strong persuasiveness	$O(\sqrt{r\mathrm{Reg}(T) T})$ (Lin et al., 7 Oct 2024)
Partial Identification	Withhold a one-dimensional summary	Saddle-point implementability (Rosenthal, 23 Nov 2025)

5. Applications: Robust Causal Inference and Bayesian Persuasion

Robust Causal Inference

Within the potential outcomes paradigm:

• Latent and Observed States: $(Y_1, ..., Y_k, X)$ are latent, $\omega = (Y_T, X, T)$ are observed.
• Unconfoundedness: Known assignment probabilities support identification via inverse probability weighting.
• Key Result: Proposition 5.1 and Theorem 5.2 (Rosenthal, 23 Nov 2025) establish that, under finite treatment/covariate sets and unconfoundedness, any mixed treatment rule $\alpha$ can be robustly implemented by an experiment withholding at most one linear dimension.

Bayesian Persuasion and Cheap Talk

The prior-free learning framework unifies classical Bayesian persuasion (necessarily requiring commitment to signal schemes) and cheap talk (designer does not need commitment under bandit learning receiver); both are subsumed under regret-minimizing algorithms with tight rates (Lin et al., 7 Oct 2024).

6. Extensions, Assumptions, and Limitations

Core Assumptions

• Full support of state distributions: $\mu^*(\omega) \geq p_0 > 0$.
• Bounded gap in receiver utility ($G>0$).
• States are i.i.d. across periods.

Open Directions and Generalizations

• Removing full-support or analyzing adversarial/non-i.i.d. states.
• Multi-receiver and multi-period settings, persistent agents.
• Rich signal structures in two-action cases or moment restrictions in infinite-dimensional settings.
• Embedding covariate balance, moment restrictions, or other structure into prior sets $\mathcal{P}$ for robust design.
• Multi-agent robust correlated equilibrium and dynamic generalizations remain open topics.

Limitations

• The decision maker under partial identification ranks actions by signal distribution-implied worst-case payoffs, not period-by-period Bayesian updating.
• Frameworks generally assume finite signal and state spaces; some extension possible to moment-based infinite settings.
• Strong persuasiveness and robustification require trade-offs in margin selection ($\Delta, \gamma$) to mitigate regret against bandit learners.

7. Synthesis and Perspective

Prior-free frameworks for information design demonstrate that optimal signaling and experiment design are attainable without knowledge of the underlying state distribution, subject to sublinear regret or one-dimensional information withholding. Lin & Li (Lin et al., 7 Oct 2024) provide regret-optimal learning algorithms spanning rational to learning receivers, showing information design is learnable under prior ignorance. Rosenthal (Rosenthal, 23 Nov 2025) shows all robustly optimal actions can be implemented by “almost fully informative” signal structures, which conceal at most one linear dimension. These results bridge the methodological gap between repeated-interaction persuasion and robust experiment design, unifying Bayesian persuasion, cheap talk, and partial identification. Future work will aim to extend these frameworks beyond the most restrictive assumptions, toward adversarial, dynamic, and multi-agent environments.