📈 economics

Private Languages

This paper demonstrates that in sender-receiver games, costly deviations from privately observed anchors can enhance information transmission by generating continuous messages and enabling uninformative anchors to convey strategic distortions, potentially allowing organizations to achieve near-full revelation even with minimal reporting costs.

Original authors: Jeremy Bertomeu

Published 2026-05-26

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jeremy Bertomeu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Core Idea: Everyone Has Their Own "Internal Ruler"

Imagine you are trying to explain how tall a building is to a friend.

The Standard Model (Cheap Talk): You just say, "It's about 50 stories high." You have no reason to lie, but you also have no reason to be perfectly precise. You might round up or down.
The "Private Language" Model: You have a specific, internal ruler in your head that you use to measure things. Maybe your ruler is slightly warped, or maybe you measure from the roof instead of the ground. Your friend doesn't see your ruler; they only hear your final number.

The paper argues that in the real world, people (senders) often report numbers based on their own private internal standards (anchors), like a proprietary financial model, an internal audit score, or a personal grading rubric. The person listening (the receiver) sees the final number but doesn't see the internal ruler used to create it.

The Big Discovery: "No Gaps" in the Conversation

In traditional economic theories, communication is often like a staircase. If you are a "low" type, you say "Low." If you are a "high" type, you say "High." There are gaps in between where no one speaks.

Bertomeu's finding is different: Because everyone has a slightly different internal ruler, the conversation becomes a smooth, continuous slide rather than a staircase.

The Analogy: Imagine a room full of people trying to guess the temperature. In the old model, they would shout "Hot," "Warm," or "Cold." In this new model, because everyone has a slightly different thermometer (some are broken, some are calibrated differently), they end up shouting every single temperature between freezing and boiling.
Why? If you have a "high" internal ruler, you might report "80 degrees" for a state that is actually "70." If someone else has a "low" ruler, they might report "60" for that same "70" state. Because there are so many different internal rulers, every possible number gets used. There are no "gaps" in the conversation.

The Surprising Twist: Small Costs Create Big Truth

Usually, economists think that if it costs nothing to lie (or if it's easy to stretch the truth), people will lie a lot, and communication breaks down.

The Paper's Counter-Intuitive Result:
If there is even a tiny cost to deviating from your own internal ruler, communication actually becomes more truthful, almost perfectly so.

The Analogy: Imagine you are painting a wall. If it costs you nothing to paint it the wrong color, you might paint it whatever color you want. But if it costs you a tiny bit of effort to paint it a color that doesn't match your internal sketch, you will try very hard to match your sketch.
The Mechanism: To get the listener to understand the true state of the world, the sender will "inflate" their report. They will stretch the numbers out so much that the "noise" of their private ruler gets washed out. It's like turning up the volume on a radio so much that the static (the private ruler's error) becomes irrelevant compared to the music (the true state).
The Catch: This "truth" comes at a price. The sender has to work very hard to stretch their reports, which costs them energy (or money). As the cost of lying gets smaller, the sender has to stretch the numbers infinitely to be perfectly understood.

The "Uninformative" Anchor Trick

What if the sender's internal ruler is completely random and tells them nothing about the truth? (e.g., a manager uses a random number generator as their "standard").

The Paper's Finding: Even a useless ruler can help transmit information!

The Analogy: Imagine two people trying to guess the weight of a rock. One person uses a broken scale that always says "10 lbs" regardless of the rock. The other uses a broken scale that always says "20 lbs."
How it works: If the rock is actually heavy, the first person has to lie a lot to make it look like 10 lbs. The second person has to lie a little to make it look like 20 lbs. The amount they have to lie reveals the truth. The receiver learns not from the number itself, but from how much the sender had to distort their own private standard to say it.

Hard Numbers vs. Soft Words (The Hybrid System)

The paper also looks at systems where people give both a number (like a stock price target) and a label (like "Buy" or "Sell").

The Finding: When people are very aligned (they want the same thing), just using words (Cheap Talk) is actually better than using numbers + words.
Why? Adding a number forces the "words" (the labels) to become coarser. It's like trying to sort books. If you just say "Good" or "Bad," you can make very fine distinctions. But if you are forced to attach a specific price to "Good," you might have to group "Good" and "Okay" together just to make the math work.
The Sweet Spot: However, if people are somewhat misaligned (they have different goals), the hybrid system (Number + Label) is the best. The label sorts the big categories, and the number refines the details inside those categories.

Summary of the "Rules vs. Discretion" Debate

The paper answers a practical question for bosses and regulators: Should we force everyone to use a strict, standardized checklist (an anchor), or let them use their own judgment?

If everyone agrees on the goal: Let them use their own judgment (Discretion). Forcing a standard actually makes communication worse because it forces them to use "coarse" categories.
If people have different goals: Use the Standardized Checklist (Anchors). It helps discipline the conversation and prevents people from hiding behind vague language.

The Bottom Line

This paper shows that the "language" people use isn't just about the words they say; it's about the private tools they use to measure the world. When people have different private tools, they end up saying a continuous stream of numbers rather than distinct categories. Surprisingly, a tiny cost to being "faithful" to your own tool can make the whole system incredibly transparent, even if the tool itself is flawed.

Technical Summary: Private Languages in Strategic Communication

Problem and Motivation
Strategic communication is frequently modeled under the assumption that senders and receivers share a common language mapping states to messages. However, in many institutional settings—such as financial analysts using proprietary valuation models, auditors applying internal checklists, or managers relying on accounting estimates—the "anchor" used to judge the fidelity of a report is privately observed by the sender. The sender faces a cost for deviating from this private anchor, distinct from the cost of lying about the objective state. This paper investigates how such private languages, where the cost structure of reporting depends on a privately observed reference point, alter equilibrium communication compared to standard cheap talk or deterministic lying-cost models.

Methodology
The paper employs a sender-receiver game framework. The state $\theta$ is payoff-relevant and observed by both parties, but the sender also observes a private anchor $b$ . The sender chooses a report $r$ , incurring a direct cost $D(r, b)$ for deviating from $b$ , while the receiver chooses an action $a$ to maximize utility based on the posterior belief about $\theta$ .

The analysis focuses on regular equilibria, defined as those where the receiver's action varies continuously and monotonically with the report. The core methodological innovation is a geometric characterization of the reporting strategy. Unlike deterministic lying-cost models where a fixed state maps to a single report, the presence of a private anchor implies that for a fixed state $\theta$ , the set of optimal reports is a continuum indexed by the realization of $b$ .

The author specializes the model to additive anchors ( $b = b_0(\theta) + \sigma x$ ) and translation-invariant reporting costs ( $D(r, b) = c\phi(r-b)$ ). This structure allows the equilibrium conditions to be reduced to a differential equation (ODE) governing the receiver's action rule, derived from the sender's first-order condition and Bayes' rule. The analysis further explores hybrid equilibria combining qualitative labels (cheap talk) with quantitative anchored reports.

Key Results

Geometry of Communication (Continuity and No-Holes):
- In regular equilibria, the map from the private anchor to the report is continuous and non-decreasing. Consequently, for any fixed state, the set of used reports forms a connected interval.
- This leads to a "no-holes" property: unlike standard lying-cost models which often feature pooling at boundaries and gaps in the interior, private anchoring generates continuous variation in messages. If the anchor support is unbounded, every report is used by every state (full-use property).
Discontinuity at Zero Cost:
- There is a sharp discontinuity between zero and positive reporting costs. At zero cost, the model collapses to standard cheap talk with partitioned equilibria.
- However, for any arbitrarily small positive cost $c > 0$ , a branch of regular equilibria exists where communication approaches full revelation as $c \to 0$ . The sender utilizes "message inflation" (expanding the report scale) to filter out the noise of the private anchor. While the reports become asymptotically fully informative, the expected reporting cost converges to a finite cumulative incentive term (or diverges if the state space is unbounded with non-vanishing conflict).
Information from Uninformative Anchors:
- Even if the anchor is entirely uninformative (independent of the state), it can support informative communication. Different states have different marginal incentives to move the receiver's action, leading them to distort their reports by different amounts away from the same uninformative anchor. The receiver learns from the pattern of these strategic distortions.
- In a Gaussian-Quadratic setting with uninformative anchors, a specific calibration of the reporting cost and anchor noise allows for a family of informative equilibria. A designer selecting the most informative equilibrium can approach full revelation, though this imposes infinite welfare loss on the sender if the bias is non-zero.
Hybrid vs. Pure Communication Channels:
- The paper analyzes a hybrid equilibrium where a qualitative label (cheap talk) partitions the state space, and an anchored numerical report refines information within each label.
- Moderate Misalignment: The hybrid channel strictly dominates both pure anchored communication and pure cheap talk. The label sorts states, and the anchor refines beliefs within the sort.
- Low Misalignment: As the conflict of interest vanishes ( $d \to 0$ ), pure cheap talk becomes strictly preferred over the hybrid. While the anchor provides within-label refinement, it forces a coarsening of the qualitative partition to maintain incentive compatibility. The welfare loss from this partition widening (a second-order effect) outweighs the second-order gain from within-label refinement. Thus, organizations with highly aligned interests may optimally rely exclusively on informal, unanchored channels.

Significance and Contribution
The paper claims to contribute to the literature on strategic communication by treating the "language" of communication as a private variable rather than a common primitive.

Institutional Design: It reframes compliance regimes (e.g., accounting standards, audit checklists) not merely as devices to align reports with truth, but as mechanisms that coordinate senders on a common internal language. The optimal design depends on the alignment between sender and receiver; strict anchors may be detrimental when interests are closely aligned.
Equilibrium Selection: The model provides a mechanism for equilibrium selection in cheap talk environments. The private anchor acts as a perturbation that selects a specific informative equilibrium branch, resolving the multiplicity of cheap-talk equilibria.
Form of Communication: It predicts continuous cross-sectional variation in reports tied to private references, contrasting with the partition-valued reports of cheap talk or the single-report-per-state predictions of deterministic lying-cost models. This offers a theoretical basis for observing continuous variation in reports (e.g., price targets) even when the underlying state is fixed, and for the coexistence of qualitative and quantitative disclosure.

The analysis remains within the theoretical bounds of the model, noting that these mechanisms apply to settings where formal verification and informal persuasion coexist, such as financial forecasting, auditing, and performance ratings.