Information Design With Large Language Models

Imagine you are a Chef (the Sender) trying to convince a Food Critic (the Receiver) to order your most expensive dish.

In the old way of doing things (called Bayesian Persuasion), the Chef's only tool was the menu description. If the dish was actually "Spicy," the Chef could choose to say "Mildly Zesty" or "Fiery." The Critic would then update their belief based on the facts provided. The Chef's strategy was purely about what facts to reveal.

But this paper argues that in the real world, people don't just react to facts; they react to how the facts are wrapped up. This is called Framing.

The Core Idea: The "Wrapper" vs. The "Gift"

Think of information design as a two-step process:

The Wrapper (Framing): This is the slogan, the tone, the font, or the story you tell before you even show the product. It's like wrapping a gift in bright, festive paper versus plain brown paper. Even if the gift inside is the same, the wrapper changes how the Critic feels about it before they even open it.
- Example: A clothing brand doesn't just sell a jacket; they sell a "Mountain-Ready, City-Chic" lifestyle. This slogan (the frame) makes the Critic think the jacket is stylish and durable, even before they see the price tag.
The Gift (Signaling): This is the specific data or signal you give them after the frame is set. In our example, this is the discount. If the jacket is high quality, the brand might offer a small discount. If it's low quality, a huge discount. The Critic uses this signal to update their belief about the specific item.

The Big Question: Should the Chef just focus on the menu description (Signaling), or should they also spend time designing the perfect wrapper (Framing)? And can they do both at once?

The Problem: The "Language Maze"

The authors realized that while we know how to calculate the best menu description (Signaling), figuring out the best wrapper (Framing) is a nightmare.

The Space is Huge: There are infinite ways to phrase a slogan. It's like trying to find the perfect sentence in a library with infinite books.
The Reaction is Weird: A tiny change in wording (e.g., "90% fat-free" vs. "10% fat") can completely flip a person's decision. It's like a light switch: a tiny nudge turns the light from off to on, or vice versa. This makes it mathematically very hard to predict what will work.

The Solution: Using AI as a "Human Simulator"

This is where Large Language Models (LLMs) like the ones you chat with come in. The authors propose a brilliant shortcut:

Instead of running expensive focus groups with thousands of humans to see how they react to slogans, ask an AI.
The AI acts as a "proxy" for the human mind. You show it the slogan, and it predicts: "If a human saw this, they would believe there is a 70% chance this jacket is stylish."

The Two Strategies: What the Paper Found

The authors tested two approaches using this AI-assisted method:

1. The "Fixed Wrapper" Strategy (Hard Mode)

Imagine the Chef is stuck with a specific menu layout (Signaling) and can only change the wrapper (Framing).

The Result: This is mathematically broken. Because human reactions to wording are so sensitive (like that light switch), a tiny error in the AI's prediction can lead to a massive failure. It's like trying to balance a pencil on its tip; one tiny wobble and it falls. The paper proves this is computationally impossible to solve perfectly.

2. The "Master Chef" Strategy (Easy Mode)

Imagine the Chef can change both the wrapper (Framing) and the menu description (Signaling) together.

The Result: This is much more stable. When you have the freedom to adjust the signal (the discount) to match the frame (the slogan), the system becomes smooth and forgiving. If the AI predicts the wrapper makes people think the jacket is "stylish," the Chef can immediately adjust the discount to match that new belief.
The Analogy: It's like driving a car with power steering. If you turn the wheel slightly (change the frame), the car adjusts smoothly. You don't crash.

The Experiment: The "Himalaya" Jacket

To prove this works, the authors ran a real-world simulation:

The Setup: A fictional brand called "Himalaya" wanted to sell jackets to "fashionable mall shoppers" (who care more about style than technical durability).
The Process:
1. They started with a real-world slogan from a famous brand (Patagonia).
2. They asked the AI to generate new slogans and descriptions.
3. The AI predicted how the "mall shoppers" would interpret these new slogans.
4. A computer solver calculated the best discount strategy for those new beliefs.
5. The AI used that score to write an even better slogan for the next round.
The Outcome: The AI-generated slogans were significantly better at persuading the target audience than the original real-world slogan. The AI found a "sweet spot" in the language that humans hadn't thought of, perfectly aligning the "wrapper" with the "discount."

The Takeaway

This paper is a roadmap for the future of persuasion. It tells us:

Don't just rely on facts. How you say something (Framing) is just as important as what you say (Signaling).
Don't try to optimize them separately. If you can't change your facts, optimizing your words is a dangerous, unstable game.
Do it together. If you can change both your story and your offer, you can use AI to find the perfect combination that guides people to the decision you want, smoothly and effectively.

In short: Use AI to write the perfect story, and then use math to back it up with the perfect offer. That's the winning combination.

Here is a detailed technical summary of the paper "Information Design with Large Language Models".

1. Problem Statement

The paper addresses a gap in classical Information Design (specifically Bayesian Persuasion) by integrating insights from behavioral economics regarding framing effects.

Classical Model: In standard Bayesian Persuasion, a Sender influences a Receiver's decision by designing a signaling scheme (a correlation between the true state of the world and a signal). The Receiver updates their belief purely via Bayes' rule. The only variable the Sender controls is the signal structure.
The Gap: Behavioral economics (e.g., Tversky & Kahneman) demonstrates that human decision-making is heavily influenced by how information is framed (wording, context, tone), which can alter the Receiver's prior belief in non-Bayesian ways.
The Challenge: Optimizing over "framing" is difficult because:
1. The space of possible framings is vast and linguistic (natural language).
2. The mapping from a specific framing to a specific prior belief is complex, personalized, and often non-Bayesian.
3. Systematically optimizing this space is computationally intractable without a proxy for human behavior.

Core Question: How can a Sender jointly optimize a linguistic framing (to shape the Receiver's prior) and a signaling scheme (to refine that prior) to maximize utility, and what are the computational and theoretical properties of this problem?

2. Methodology

The authors propose a hybrid framework combining game-theoretic modeling with Large Language Models (LLMs) as proxies for human behavior.

A. Theoretical Framework

The model extends Bayesian Persuasion:

Framing ( $c$ ): A state-independent message (e.g., a slogan) chosen by the Sender before observing the state. It induces a subjective prior belief $\mu_c$ in the Receiver via a mapping $\ell: c \to \mu_c$ . This mapping is determined by "nature" or societal norms, not the Sender.
Signaling ( $\pi$ ): A state-dependent mapping from states to signals, designed by the Sender after observing the state.
Interaction:
- Sender commits to $(c, \pi)$ .
- Nature draws state $\omega$ .
- Receiver observes framing $c$ (updating prior to $\mu_c$ ) and signal $s$ (updating to posterior $\mu_{c,s}$ via Bayes' rule).
- Receiver takes a best-response action.
Optimization Variants:
- Framing-Only: Optimize $c$ given a fixed $\pi$ .
- Joint Optimization: Optimize both $c$ and $\pi$ .

B. Empirical Approach (LLM Integration)

To handle the linguistic space and the unknown mapping $\ell$ :

LLM as Proxy: The authors use LLMs (GPT-5.2 and Claude Sonnet 4) to simulate the Receiver. The LLM is prompted to estimate the probability distribution (belief) over states induced by a specific text framing.
Iterative Prompt Optimization: They employ a "textual gradient" approach. An LLM generates a framing candidate.
- A second LLM estimates the induced belief $\mu_c$ .
- An analytical solver computes the optimal signaling scheme $\pi^*$ for that belief and the resulting Sender utility.
- A third LLM verifies the "soundness" (factual correctness) of the framing.
- The combined score (Utility $\times$ Soundness) and textual feedback are fed back to the generator LLM to refine the next iteration.

3. Key Contributions & Theoretical Results

1. Computational Complexity of Framing-Only Optimization

Result: Optimizing framing for a fixed signaling scheme is NP-hard.
Reasoning: The authors prove that the Sender's utility function $U_\pi(\mu)$ is generally discontinuous with respect to the Receiver's prior belief $\mu$ . Small changes in framing (and thus $\mu$ ) can cause the Receiver to switch actions abruptly, leading to large jumps in utility.
Implication: Even with a perfect mapping $\ell$ , finding the optimal framing is computationally intractable. Approximation is difficult because errors in belief estimation can lead to arbitrarily large utility losses due to discontinuities.

2. Tractability of Joint Optimization

Result: Jointly optimizing framing and signaling is significantly more tractable.
Continuity: When the Sender can also design the signaling scheme $\pi$ , the optimal utility function $U^*(\mu)$ becomes continuous (specifically, locally Lipschitz) with respect to the prior belief $\mu$ .
Robustness: This continuity implies that errors in the LLM's belief estimation (simulated $\ell_\epsilon$ ) only result in small, bounded losses in utility ( $O(\epsilon)$ ).
Algorithms:
- For unconstrained belief spaces, there exists a polynomial-time approximation scheme (PTAS) and exact polynomial-time solutions for state-independent utilities.
- For constrained belief spaces, the authors provide a Quasi-Polynomial Time Approximation Scheme (QPTAS).

3. Empirical Validation

Case Study: An advertising scenario for a clothing brand ("Himalaya") targeting a specific demographic.
Belief Estimation: LLMs (specifically GPT-5.2) were shown to estimate induced beliefs that closely matched aggregate human judgments (derived from Prolific surveys using Bradley-Terry models).
Optimization: The iterative framework successfully generated new slogans and descriptions that outperformed the baseline (Patagonia's actual slogan) in terms of the Sender's utility for the target demographic, demonstrating the ability to co-design text and strategy.

4. Key Results Summary

Feature	Framing-Only Optimization	Joint Optimization (Framing + Signaling)
Complexity	NP-Hard (even with perfect belief mapping)	Tractable (Admits QPTAS/PTAS)
Utility Function	Discontinuous (Highly sensitive to errors)	Continuous (Robust to errors)
LLM Utility	Low (High risk of catastrophic failure due to discontinuity)	High (Robust to approximation errors)
Optimal Strategy	Randomization over framing offers no benefit	Deterministic framing is sufficient

5. Significance

Theoretical Bridge: The paper formally bridges the gap between classical game-theoretic information design and behavioral economics, providing a rigorous mathematical model for "framing effects."
Algorithmic Insight: It identifies a crucial structural difference: while optimizing only the frame is hard, optimizing the combination of frame and signal is easy. This suggests that in real-world applications (like marketing or policy), the ability to adjust the "hard" signal (e.g., price, discount, recommendation) makes the "soft" framing (text) much easier to optimize.
LLM as a Tool: It validates the use of LLMs not just as text generators, but as computational proxies for human belief formation, enabling the systematic optimization of linguistic variables that were previously only accessible through expensive human studies (focus groups).
Practical Application: The proposed framework offers a blueprint for automated, data-driven design of persuasive communication campaigns that are tailored to specific audience psychologies.

In conclusion, the paper argues that while framing is a powerful but computationally difficult lever on its own, it becomes a highly effective and tractable tool when combined with the strategic design of information signals, a synergy that can be effectively exploited using Large Language Models.