Menu Pricing of Large Language Models

### 6.2 OpenAI

Unlike Anthropic's tier structure, which differentiates mostly through constraints on usage, OpenAI's ChatGPT pricing differentiates through both quantity and exclusive model access: higher tiers grant access to more capable models and relax usage constraints. This joint screening aligns with the multi-model monopolist of Section 5.2.

Figure 3 summarizes OpenAI's tier structure as of January 2026. The Free tier provides limited GPT-4o access with automatic downgrade to GPT-4o-mini when capacity is constrained. Plus (\$20/month) expands to approximately 80 GPT-4o messages per 3-hour window and adds reasoning models (o3, o4-mini) with a separate weekly limit of roughly 100 queries. Pro (\$200/month) removes most constraints and provides exclusive access to o1pro, OpenAI's most compute-intensive reasoning model. Thus, both Anthropic and OpenAI converge on similar price points (\$20 standard, \$200 premium), but implement very different screening strategies.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 3: OpenAI ChatGPT subscription tiers (January 2026). Higher tiers grant access to more capable models and larger usage allocations.

In Proposition 6, we showed that when models differ in cost curvature, the optimal mechanism assigns higher-capability models to higher buyer types. OpenAI's decision to reserve its o1-pro model for the highest tier is consistent with this prediction. In consumer subscriptions, the relevant cost differences across models may reflect inference-time compute intensity rather than user fine-tuning, but the qualitative implication of a monotone assignment with discrete jumps at upgrade thresholds is the same.


Menu Pricing of Large Language Models

Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.

Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.

### 6.3 Model Aggregators: GitHub and Quora

A number of AI platforms, including GitHub Copilot and Quora's Poe, do not develop their own models. Instead, they aggregate models produced by others (OpenAI, Anthropic, Google, Mistral, and others) and sell access to users through a single interface. A common contractual arrangement is one whereby subscribers pay a monthly fee, receive a budget of platform-specific credits, and allocate those credits across the available models. Selecting a more capable model typically depletes the budget faster. We now describe Quora's and GitHub's platforms in detail and then relate their pricing structures to the indirect implementation of the optimal mechanism discussed in Section 4.2.

Quora-Poe Poe offers subscribers access to over 100 AI models through a unified chat interface. A subscriber pays a fixed monthly fee and receives an allocation of "compute points," which serve as the platform's internal currency. Each model is priced in points per message at a rate that reflects its inference cost. ${ }^{13}$

Table 1 summarizes Poe's tier structure. The entry-level Basic plan provides 300,000 points per month for \$5; the top-tier Premium plan provides 12,500,000 points for \$250. Above the Basic tier, the effective price per million points is constant at \$20, so that higher tiers simply offer a proportionally larger budget at a fixed unit rate. Table 2 reports point costs for selected models, illustrating the wide dispersion across model classes.

| Tier | Monthly Price | Points/Month | Eff. \$/M Points |
| :--- | :--- | :--- | :--- |
| Basic | \$5 | 300,000 | \$16.67 |
| Standard | \$20 | 1,000,000 | \$20.00 |
| Pro | \$50 | 2,500,000 | \$20.00 |
| Advanced | \$100 | 5,000,000 | \$20.00 |
| Premium | \$250 | 12,500,000 | \$20.00 |

Table 1: Poe subscription tiers (2025).

The connection to the maximum-spend framework of Section 4.2 is immediate. The monthly subscription fee corresponds to the transfer $T_{n}$; the point allocation corresponds to the budget $B_{n}$; and the per-model point costs correspond to marginal-cost pricing of different token classes. The subscriber chooses how to allocate the budget taking the marginal prices into account. Finally, once the budget is exhausted, access to all models is suspended until the next billing cycle, and points do not roll over. Thus, Poe's menu is best captured by a maximum-spend mechanism.

[^11]| Class | Model | Points/Message |
| :--- | :--- | :--- |
| Budget | GPT-4o-mini | 9 |
|  | Gemini-2.0-Flash | 9 |
| Mid-tier | GPT-4o | 224 |
|  | Claude-3.5-Sonnet | 276 |
| Frontier | Claude-3-Opus | 1,697 |
|  | Claude-Opus-4 | 4,105 |

Table 2: Poe per-model point costs (2025).

GitHub Copilot GitHub Copilot provides AI-assisted coding within a developer's integrated environment. Like Poe, Copilot offers multiple pricing tiers, each bundling a monthly fee with a budget of "premium requests" that the developer allocates across model calls. Table 3 summarizes the plan structure.

| Plan | Monthly Fee | Premium Requests | Notes |
| :--- | :--- | :--- | :--- |
| Free | \$0 | 50 | Limited quota, basic completions |
| Pro | \$10 | 300 | Unlimited completions, chat, IDE support |
| Pro+ | \$39 | 1,500 | Larger quota, advanced models |
| Business | \$19/user | 300/user | Org controls, policy management |
| Enterprise | \$39/user | 1,000/user | Enterprise features |

Table 3: GitHub Copilot pricing tiers and premium request budgets (2026).

The mechanism by which Copilot meters usage differs slightly from Poe's point system but serves the same function. Rather than assigning each model a point cost per message, Copilot assigns each model a multiplier that determines how many premium requests a single interaction consumes. A set of baseline models, including GPT-5 mini, GPT-4.1, and GPT-4o, carry a multiplier of 0 on paid plans: interactions with these models are included at no additional budget cost. Efficient reasoning models such as Claude Haiku 4.5 and o4-mini carry multipliers of 0.25-0.33. Standard models such as Claude Sonnet 4.x and GPT-5.x carry a multiplier of 1. At the frontier, Claude Opus 4.5 carries a multiplier of 3, and GPT-4.5 carries a multiplier of 50.

This mechanism matches the framework of Section 4.2 quite closely. The plan fee corresponds to $T_{n}$; the monthly premium-request allocation corresponds to $B_{n}$; and the model multipliers play the role of heterogeneous marginal costs. A distinctive feature of Copilot is that it extends this budget mechanism with an overage option. When a developer (i.e., a buyer) exhausts her monthly allocation, Copilot allows continued use at a fixed charge
of \$0.04 per premium request. The minimum-spend framework of Section 4.2 matches this feature rather well: each user commits to a certain amount of monthly spending, there are no refunds, but the developer can increase her consumption ex post at linear prices. ${ }^{14}$

Thus, both Poe and Copilot implement the core logic of the committed-spend mechanisms: a non-bankable budget, priced up front via a subscription fee, that the user freely allocates across inputs with heterogeneous per-unit costs. The key distinction lies in how each platform handles the overages. Poe enforces a hard cap: once points are exhausted, access is suspended until the next billing cycle. This corresponds exactly to the maximumspend mechanism. Copilot, by contrast, allows the cap to be relaxed at a known marginal price (which varies across models through the multiplier system), effectively implementing a minimum-spend mechanism.


Menu Pricing of Large Language Models

### 6.4 API Pricing

Alongside consumer subscriptions, every major LLM provider operates a developer-facing API through which applications can programmatically submit prompts and receive completions. A developer who calls the API pays per token, with separate rates for input tokens (the prompt) and output tokens (the model's response). There is no subscription fee, no bundled allocation, and no minimum commitment: the developer pays only for what she uses. Table 4 reports prices for selected models across three providers as of January 2025.

| Provider | Model | Input (\$/M) | Output (\$/M) |
| :--- | :--- | :--- | :--- |
| OpenAI | GPT-4o-mini | 0.15 | 0.60 |
| OpenAI | GPT-4o | 2.50 | 10.00 |
| OpenAI | o1 (reasoning) | 15.00 | 60.00 |
| Anthropic | Claude 3.5 Haiku | 0.80 | 4.00 |
| Anthropic | Claude 3.5 Sonnet | 3.00 | 15.00 |
| Anthropic | Claude 3 Opus | 15.00 | 75.00 |
| Google | Gemini 1.5 Flash | 0.075 | 0.30 |
| Google | Gemini 1.5 Pro | 1.25 | 5.00 |

Table 4: API pricing across major providers (January 2025). Prices per million tokens.

The pricing structure is notably uniform. Output tokens are priced at $3-5 \times$ input tokens across all providers, reflecting that output generation is sequential and autoregressive (each token requires a forward pass conditioned on all preceding tokens), whereas input encod-

[^12]ing is parallelizable and therefore substantially cheaper per token. Pricing is strictly linear: there are no quantity discounts, volume commitments, or tiered rates in standard API access. Although outside our model, all providers offer batch discounts (typically 50\%) for asynchronous processing and prompt-caching discounts (up to 90\%) for repeated inputs.

This linear, pay-per-token structure stands in sharp contrast to the nonlinear menus that characterize consumer subscriptions. It corresponds instead to the constrained-efficient allocation of Corollary 1: when a provider faces capacity constraints but wishes to maximize total surplus rather than profit (for instance, to lock in consumers in anticipation of future monetization), the constrained-efficient allocation can be implemented via linear prices equal to marginal costs inflated by shadow costs. The absence of nonlinear screening in API pricing is consistent with providers currently prioritizing adoption over rent extraction.

The uniformity of the pricing format across providers is itself evidence of competitive pressure: if any single provider were to introduce nonlinear screening in the API segment, it would risk losing developers to competitors who maintain simpler, linear pricing.

Evidence from pricing dynamics is consistent with this interpretation. First, API prices have declined rapidly: GPT-4-class capability fell from \$30/\$60 per million tokens at launch (March 2023) to \$2.50/\$10 by August 2024, a 90\% reduction in 16 months. Second, providers appear to cut prices in response to competitive launches rather than cost improvements; OpenAI's August 2024 reductions followed Claude 3.5 Sonnet and Llama 3.1 releases. Third, gross margins of 50-75\% imply prices remain above marginal cost but are being compressed by competition. The entry of DeepSeek in January 2025 with reasoning-model pricing at \$0.55/\$2.19, versus OpenAI's \$15/\$60 for comparable capability, suggested that further compression is viable. See Demirer et al. (2025) for a comprehensive treatment of market share dynamics in the API segment.


Menu Pricing of Large Language Models

## 7 Conclusion

A priori, the problem of pricing LLM access appears intractable: users have high-dimensional private information, the allocation space is also high-dimensional, and the user's hidden allocation of tokens across tasks introduces moral hazard. In this paper, we have shown that homogeneity of the gain function generates a sufficient-statistic reduction that makes the problem solvable. Users' high-dimensional type profiles collapse to a scalar aggregate type; the seller's problem reduces to one-dimensional screening; and the optimal mechanism admits simple implementations (committed-spend contracts and two-part tariffs) that correspond precisely to the pricing structures observed at leading providers.

While we have phrased the analysis in terms of large language models, the framework applies more broadly to any setting in which a provider sells access to a general-purpose technology with multiple input classes and heterogeneous users. Cloud computing, where providers offer a multiplicity of services with both horizontal and vertical differentiation, is a natural example. In both settings, the core pricing problem is how to allocate and monetize costly computing resources, particularly when operating under capacity constraints.

Several extensions would enrich the analysis. First, we assumed that all tasks are homogeneous in their use of input, output, and fine-tuning tokens, and differ only vertically in how valuable a given task is to the buyer. Allowing the gain function parameters to vary across tasks would break the aggregation result. Understanding how the optimal mechanism changes when the scalar sufficient statistic no longer exists is a challenging but important open question, especially in the context of competing specialized models.

Second, we assumed that all buyer types have fine-tuning data readily available. In practice, buyers face constraints on data availability or may be unwilling to share data with the provider because of privacy, compliance, or intellectual-property concerns. A data-poor buyer may then substitute toward inference-time tokens, while a data-rich buyer substitutes toward fine-tuning tokens. When data availability is the user's private information, the provider's problem involves screening on two dimensions (aggregate type and data availability), which may require new techniques.

Third, LLM platforms exhibit network effects and data externalities that our static framework abstracts from. A provider that attracts more users may improve its model through fine-tuning, reinforcement learning from human feedback, or simply from the revenue that funds further training. These dynamic complementarities between adoption and model quality could substantially reshape optimal pricing, potentially justifying the aggressive belowaverage-cost pricing observed in the API segment not just as a means to capture market share but as investment in model improvement.

Fourth, our analysis treats each provider as either a monopolist or a leader facing a competitive fringe. A full oligopoly analysis with multiple differentiated proprietary models would capture the strategic interactions among Anthropic, OpenAI, Google, and others that increasingly shape the market. The tractability of our framework through the reduction to one-dimensional types suggests that such an extension may be feasible and would yield further predictions about equilibrium pricing and product differentiation.

The LLM industry is at an inflection point between growth-oriented pricing and profitmaximizing pricing. Our framework provides a theoretical foundation for understanding the mechanisms through which this transition will unfold, and for evaluating whether the resulting market structure efficiently allocates this critical economic input.


Menu Pricing of Large Language Models

## A Proofs and Derivations

Proof of Proposition 1 For any $z$ and $i$, the optimal allocation of inference tokens on task $i$ solves

$$
\max _{x_{i j} \geq 0} w_{i} \Psi\left(x_{i}\right) \Phi(z)-\sum_{j=1}^{J} c_{j} x_{i j} .
$$

If $w_{i}=0$ or $\Phi(z)=0$, then $x_{i}=0$. Otherwise, by the Inada assumption, the problem admits an interior solution, which satisfies the system of $J$ first-order conditions:

$$
w_{i} \nabla \Psi\left(x_{i}\right) \Phi(z)=c .
$$

By (35), for all $i, \nabla \Psi\left(x_{i}\right)$ belongs to a ray with a direction $c$. Because $\Psi$ is homogeneous and strictly concave, the ray in the space of gradients corresponds to a ray in the space of tokens. ${ }^{15}$ Thus, for each $i$, any optimal $x_{i}$ can be written as:

$$
x_{i}=d y_{i},
$$

where $d \in \mathbb{R}_{+}^{J}$ is the unique vector that solves $\nabla \Psi(d)=c$.
Therefore, by the homogeneity of $\Psi$, we have $\nabla \Psi\left(x_{i}\right)=y_{i}^{\sigma-1} \nabla \Psi(d)$, and (35) can be rewritten as:

$$
w_{i} y_{i}^{\sigma-1} \nabla \Psi(d) \Phi(z)=\nabla \Psi(d),
$$

which pins down the optimal scale as:

$$
y_{i}=w_{i}^{\frac{1}{1-\sigma}} \Phi(z)^{\frac{1}{1-\sigma}} .
$$

The resulting total surplus ignoring fine-tuning costs is:

$$
\begin{aligned}
& \int_{0}^{1} w_{i} \Psi\left(x_{i}\right) \Phi(z) d i-\sum_{j=1}^{J} c_{j} \int_{0}^{1} x_{i j} d i=\int_{0}^{1} w_{i} y_{i}^{\sigma} \Psi(d) \Phi(z)-y_{i}(d \cdot c) d i \\
& =\int_{0}^{1} w_{i}^{\frac{1}{1-\sigma}} \Phi(z)^{\frac{1}{1-\sigma}}(\Psi(d)-d \cdot c) d i=\theta(w)^{\frac{1}{1-\sigma}} \Phi(z)^{\frac{1}{1-\sigma}}(1-\sigma) \Psi(d)
\end{aligned}
$$

[^13]where $\theta(w)$ is the aggregate type defined as
$$
\theta(w) \triangleq\left(\int_{0}^{1} w_{i}^{\frac{1}{1-\sigma}} d i\right)^{1-\sigma}
$$
and the last equality in (36) follows from the definition of $d$ and Euler's theorem for homogeneous functions: $d \cdot \nabla \Psi(d)=\sigma \Psi(d)$.

Similarly, the total amount of inference tokens of class $j$ is:

$$
\int_{0}^{1} x_{i j} d i=\int_{0}^{1} d_{j} y_{i} d i=\int_{0}^{1} d_{j} w_{i}^{\frac{1}{1-\sigma}} \Phi(z)^{\frac{1}{1-\sigma}} d i=\theta(w)^{\frac{1}{1-\sigma}} d_{j} \Phi(z)^{\frac{1}{1-\sigma}}
$$

This completes the proof. $\square$

Proof of Corollary 1 The Lagrangian approach applies. Thus, associating Lagrange multipliers $\lambda_{j} \geq 0$ and $\hat{\lambda}_{k} \geq 0$ with the capacity constraints for the different token classes, the optimal solution solves

$$
\max _{\left(x_{i}\right)_{i \in[0,1], z \geq 0}} \int_{0}^{1} w_{i} g\left(x_{i}, z\right) d i-\sum_{j=1}^{J}\left(c_{j}+\lambda_{j}\right) \int_{0}^{1} x_{i j} d i-\sum_{k=1}^{K}\left(\hat{c}_{k}+\hat{\lambda}_{k}\right) z_{k} .
$$

As such, the solution is an efficient allocation given the adjusted costs $c_{j}^{\prime} \triangleq c_{j}+\lambda_{j}$ and $\hat{c}_{k}^{\prime} \triangleq \hat{c}_{k}+\hat{\lambda}_{k}$. The result follows. $\square$

Proof of Proposition 2 Consider the buyer-optimal inference token allocation across tasks for any given token budgets $(X, Z)$ such that $\Phi(Z)>0$ :

$$
\max _{x_{i j} \geq 0} \int_{0}^{1} w_{i} \Psi\left(x_{i}\right) \Phi(Z) d i, \quad \text { s.t. } \int_{0}^{1} x_{i j} d i=X_{j} \text { for } j \in[J] .
$$

Because $\Phi(Z)$ is a strictly positive constant, it can be ignored. Assume that for each $j$, $X_{j}>0$; otherwise, $x_{i j}=0$ for all $i$. Lagrangian approach applies. We associate Lagrange multipliers $\lambda=\left(\lambda_{1}, \ldots, \lambda_{J}\right)$ with the budget constraints of (38); the resulting first-order conditions are:

$$
w_{i} \nabla \Psi\left(x_{i}\right)=\lambda .
$$

By the same logic as in the efficiency analysis, these conditions imply that the relative ratios of input tokens are the same across all tasks:

$$
x_{i}=y_{i} d,
$$

for some $d \in \mathbb{R}_{+}^{J}$. The budget constraints imply that $\left(\int_{0}^{1} y_{i} d i\right) d=X$ and hence $d$ is proportional to $X$ and can be normalized to satisfy $\sum_{j=1}^{J} d_{j}=1$, leading to

$$
d=\frac{1}{\sum_{j=1}^{J} X_{j}} X
$$

The optimal scales $y_{i}$ solve:

$$
\max _{y_{i} \geq 0} \int_{0}^{1} w_{i} y_{i}^{\sigma} \Psi(d) d i, \quad \text { s.t. } \quad \int_{0}^{1} y_{i} d i=\sum_{j=1}^{J} X_{j} .
$$

At the optimum, the marginal gains $w_{i} \sigma y_{i}^{\sigma-1}$ are equalized across tasks, so:

$$
y_{i}=w_{i}^{\frac{1}{1-\sigma}} \frac{\sum_{j=1}^{J} X_{j}}{\int_{0}^{1} w_{r}^{\frac{1}{1-\sigma}} d r} .
$$

The optimal buyer's payoff is:

$$
\begin{aligned}
U & =\int_{0}^{1} w_{i}\left(w_{i}^{\frac{1}{1-\sigma}} \frac{\sum_{j=1}^{J} X_{j}}{\int_{0}^{1} w_{r}^{\frac{1}{1-\sigma}} d r}\right)^{\sigma} \Psi(d) \Phi(Z) d i \\
& =\left(\int_{0}^{1} w_{i}^{\frac{1}{1-\sigma}} d i\right)^{1-\sigma}\left(\sum_{j=1}^{J} X_{j}\right)^{\sigma} \Psi(d) \Phi(Z)=\theta(w) \Psi(X) \Phi(Z)
\end{aligned}
$$

where in the last equality we used the homogeneity of $\Psi$ and (39). $\square$

Proof of Lemma 1 The fact that $C(Q)$ is strictly increasing and strictly convex is immediate because $g(X, Z)=\Psi(X) \Phi(Z)$ is strictly increasing and strictly concave in its arguments whenever $g(X, Z)>0$.

To show $C_{+}^{\prime}(0)=0$, observe that since $C(Q)$ is convex, $C_{+}^{\prime}(0)=\inf _{Q>0} C(Q) / Q$. Pick any $X_{0}$ such that $\Psi\left(X_{0}\right)>0$ and $Z_{0}$ as in Footnote 3. For $r>0$, consider $\left(X_{r}, Z_{r}\right)=r\left(X_{0}, Z_{0}\right)$ and set $Q_{r}=g\left(X_{r}, Z_{r}\right)$. We have,

$$
\frac{C\left(Q_{r}\right)}{Q_{r}} \leq \frac{c \cdot X_{r}+\hat{c} \cdot Z_{r}}{Q_{r}}=\frac{r\left(c \cdot X_{0}+\hat{c} \cdot Z_{0}\right)}{r^{\sigma} \Psi\left(X_{0}\right) \Phi\left(r Z_{0}\right)}=\frac{c \cdot X_{0}+\hat{c} \cdot Z_{0}}{\Psi\left(X_{0}\right)} \frac{1}{\Phi\left(r Z_{0}\right) / r^{1-\sigma}} .
$$

By the choice of $Z_{0}, \Phi\left(r Z_{0}\right) / r^{1-\sigma} \rightarrow+\infty$ as $r \downarrow 0$, hence the right-hand side goes to 0 . Since $Q_{r} \downarrow 0$, it follows that $\inf _{Q>0} C(Q) / Q=0$. $\square$

Proof of Proposition 3 The result is an application of Mussa and Rosen (1978) to the buyer's payoff $\theta Q$ from Proposition 2 and the cost function $C(Q)$ from Lemma 1. The optimal quality schedule (12) and transfers (13) follow from standard arguments. $\square$


Menu Pricing of Large Language Models

Proof of Proposition 4 Fix an optimal direct mechanism $\left(X^{*}(\theta), Z^{*}(\theta), T^{*}(\theta)\right)$ with the associated aggregate quality $Q^{*}(\theta)=\Psi\left(X^{*}(\theta)\right) \Phi\left(Z^{*}(\theta)\right)$.

Maximum-spend mechanism. Consider a maximum-spend mechanism with $p=$ $(c, \hat{c}), T(\theta)=T^{*}(\theta)$, and $B(\theta)=C\left(Q^{*}(\theta)\right)$. Under this mechanism, since tokens are priced at marginal cost, each type $w$ purchasing budget $B\left(\theta^{\prime}\right)$ would optimally allocate tokens in a constrained-efficient way, thus deriving payoff $\theta(w) Q$ where $C(Q)=B\left(\theta^{\prime}\right)$, i.e., $Q=Q^{*}\left(\theta^{\prime}\right)$. Hence, the choice across items in this mechanism is equivalent to the choice in the direct mechanism, and this mechanism implements the allocation of the direct mechanism.

Two-part-tariff mechanism. When type $w$ faces any two-part tariff, his optimal token allocation problem is equivalent to the efficient allocation problem (5) with prices taking the roles of costs. Therefore, all types $w$ with the same $\theta(w)$ have the same payoff from any possible two-part tariff and can be treated as a single type.

Faced with a given item, the problem of type $\theta$ can be written as value-maximizationpayment-minimization:

$$
\max _{Q \geq 0} \theta Q-P(Q),
$$

where

$$
P(Q) \triangleq \min _{X_{1}, \ldots, X_{J}, Z_{1}, \ldots, Z_{K} \geq 0} \sum_{j=1}^{J} p_{j} X_{j}+\sum_{k=1}^{K} \hat{p}_{k} Z_{k}, \quad \text { s.t. } \Psi(X) \Phi(Z)=Q .
$$

Consider the following two-part-tariff mechanism:

$$
\begin{aligned}
& p_{j}(\theta)=m(\theta) c_{j}, \hat{p}_{k}(\theta)=m(\theta) \hat{c}_{k}, j \in[J], k \in[K], \\
& p_{0}(\theta)=t(\theta)-m(\theta) C(Q(\theta)),
\end{aligned}
$$

where $C(Q)$ is as defined in (10), $Q(\theta)$ is as defined in (12), and $t(\theta)$ is as defined in (13). Under this mechanism, $\sum_{j=1}^{J} p_{j} X_{j}+\sum_{k=1}^{K} \hat{p}_{k} Z_{k}=m(\theta)\left(\sum_{j=1}^{J} c_{j} X_{j}+\sum_{k=1}^{K} \hat{c}_{k} Z_{k}\right)$. Thus, the buyer-optimal allocation is efficient, and $P(Q)=m(\theta) C(Q)$.

The rest of the argument is standard (e.g., Tirole (1988, pp. 154-157)). A buyer-optimal $Q(\theta)$ is determined by the first-order condition:

$$
\theta=P^{\prime}(Q(\theta))=m(\theta) C^{\prime}(Q(\theta)),
$$

and thus the buyer-optimal level of quality satisfies the optimality condition:

$$
\bar{\varphi}(\theta)=C^{\prime}(Q(\theta)) .
$$

If $m(\theta)$ is decreasing, then the implied payment schedule in units of $C(Q)$ is concave. Thus, a menu of two-part tariffs with markups $m(\theta)$, constant across inputs, implements the desired quality schedule $Q(\theta)$, with each type $\theta$ consuming the optimal amount of tokens and paying the optimal total transfer.

Minimum-spend mechanism. Consider a minimum-spend mechanism that consists of $p(\theta)=r(\theta)(c, \hat{c})$ and $B(\theta)=T^{*}(\theta)$ for all types such that $Q^{*}(\theta)>0$, where $r(\theta)=$ $\frac{T^{*}(\theta)}{C\left(Q^{*}(\theta)\right)}$. By construction, since prices are proportional to costs, type $w$, when consuming aggregate quality $Q$, would optimally derive value $\theta(w) Q$. Thus, the buyer's problem of optimal spending can be formulated in terms of $Q$. The restriction of minimum spend when reporting $\theta^{\prime}$ can be written as $r\left(\theta^{\prime}\right) C(Q) \geq T^{*}\left(\theta^{\prime}\right)$, which simplifies to $Q \geq Q^{*}\left(\theta^{\prime}\right)$. Thus, the optimal payoff of type $\theta$ when reporting type $\theta^{\prime}$ is

$$
U\left(\theta, \theta^{\prime}\right)=\max _{Q \geq Q^{*}\left(\theta^{\prime}\right)} \theta Q-r\left(\theta^{\prime}\right) C(Q) .
$$

This mechanism implements the allocation of the direct mechanism if each type prefers to report truthfully and choose $Q=Q^{*}(\theta)$. Since $B(\theta)$ is increasing, for this to happen, $r(\theta)$ must be decreasing.

In fact, $r(\theta)$ being decreasing is not only necessary but also sufficient for implementability. To see this, note that $Q^{*}(\theta)$ is continuously increasing and $r(\theta) \geq 1$. Thus, the optimal deviation is always weakly below the maximum quality offered in the direct menu:

$$
\arg \max _{Q \geq Q^{*}\left(\theta^{\prime}\right)} \theta Q-r\left(\theta^{\prime}\right) C(Q) \leq \arg \max _{Q \geq Q^{*}\left(\theta^{\prime}\right)} \bar{\theta} Q-C(Q)=Q^{*}(\bar{\theta}),
$$

where $\bar{\theta}$ is the maximum point in $\operatorname{supp} F$. Moreover, if type $\theta$ deviates to type $\theta^{\prime}$ and consumes quality $Q \leq Q^{*}(\bar{\theta})$, then there exists a type $\theta^{\prime \prime} \geq \theta^{\prime}$ such that $Q=Q^{*}\left(\theta^{\prime \prime}\right)$. Since $r$ is decreasing, it is more profitable for type $\theta$ to deviate to type $\theta^{\prime \prime}$ and consume $Q^{*}\left(\theta^{\prime \prime}\right)$. But such a deviation is equivalent to the deviation to $\theta^{\prime \prime}$ in the direct menu and is therefore not profitable.

It remains to show that decreasing $m$ implies decreasing $r$. Assume that $m$ is decreasing. Since $m$ and $r$ are continuous, it suffices to show that their derivatives are negative at all
differentiable points. At those points, $m^{\prime}(\theta) \leq 0$, which implies $\bar{\varphi}(\theta)-\bar{\varphi}^{\prime}(\theta) \theta \leq 0$, and

$$
r^{\prime}(\theta)=\frac{T^{* \prime}(\theta) C\left(Q^{*}(\theta)\right)-T^{*}(\theta) C^{\prime}\left(Q^{*}(\theta)\right) Q^{* \prime}(\theta)}{C\left(Q^{*}(\theta)\right)^{2}}=\frac{Q^{\prime}(\theta)}{C\left(Q^{*}(\theta)\right)^{2}}\left(\theta C\left(Q^{*}(\theta)\right)-T^{*}(\theta) \bar{\varphi}(\theta)\right),
$$

where we used the fact that $T^{* \prime}(\theta)=\theta Q^{* \prime}(\theta)$ and $C^{\prime}\left(Q^{*}(\theta)\right)=\bar{\varphi}(\theta)$. Since $Q$ is increasing, it suffices to show that $\frac{\theta}{\bar{\varphi}(\theta)} \leq \frac{T^{*}(\theta)}{C\left(Q^{*}(\theta)\right)}$, i.e., the markup in the two-part tariff implementation is lower than the markup in the minimum-spend implementation. To show this, observe that at the maximal excluded type $\theta_{0}, \theta C\left(Q^{*}(\theta)\right)-T^{*}(\theta) \bar{\varphi}(\theta)=0$, and for all types $\theta>\theta_{0}$,


Menu Pricing of Large Language Models

$$
\left(\frac{\theta C\left(Q^{*}(\theta)\right)-T^{*}(\theta) \bar{\varphi}(\theta)}{\theta}\right)^{\prime}=\frac{T(\theta)}{\theta^{2}}\left(\bar{\varphi}(\theta)-\bar{\varphi}^{\prime}(\theta) \theta\right) \leq 0 .
$$

Therefore, $\theta C\left(Q^{*}(\theta)\right)-T^{*}(\theta) \bar{\varphi}(\theta) \leq 0$, and $r$ is decreasing. $\square$

Proof of Lemma 2 For general, not necessarily identical, $\sigma_{l}$, the marginal gain from assigning an infinitesimal task with value $w_{i}$ to model $l$ is, by (18):

$$
\delta_{l}\left(w_{i}\right)=\frac{1-\sigma_{l}}{\left(\int_{I_{l}} w_{r}^{1 /\left(1-\sigma_{l}\right)} d r\right)^{\sigma_{l}}} Q_{l} w_{i}^{1 /\left(1-\sigma_{l}\right)}
$$

Under an optimal task split, if a task with value $w_{i}$ is assigned to a model $l$, then it must be that for all $m, \delta_{l}\left(w_{i}\right) \geq \delta_{m}\left(w_{i}\right)$. Because $\delta_{m}\left(w_{i}\right) / \delta_{l}\left(w_{i}\right)$ is increasing in $w_{i}$ for all $m>l$, it follows that an optimal task-model allocation exists that forms a monotone partition, with models of higher $l$ (and thus higher $\sigma_{l}$ ) being allocated the more valuable tasks.

Denoting by $i_{l}$ the delimiting points, with $i_{0}=0$ and $i_{L}=1$, the optimal partition is determined by equalizing the marginal gains, $\delta_{l}\left(w_{i_{l}}\right)=\delta_{l+1}\left(w_{i_{l}}\right)$ for $l \in[L-1]$ :

$$
\frac{\left(1-\sigma_{l}\right) Q_{l} w_{i_{l}}^{1 /\left(1-\sigma_{l}\right)}}{\left(1-\sigma_{l+1}\right) Q_{l+1} w_{i_{l}}^{1 /\left(1-\sigma_{l+1}\right)}}=\frac{\left(\int_{I_{l}} w_{i}^{1 /\left(1-\sigma_{l}\right)} d i\right)^{\sigma_{l}}}{\left(\int_{I_{l+1}} w_{i}^{1 /\left(1-\sigma_{l+1}\right)} d i\right)^{\sigma_{l+1}}} .
$$

When $\sigma_{l} \equiv \sigma$, the optimality condition (42) simplifies to the following expression: ${ }^{16}$

$$
\frac{Q_{l}}{Q_{l+1}}=\frac{\left(\int_{I_{l}^{*}} w_{i}^{1 /(1-\sigma)} d i\right)^{\sigma}}{\left(\int_{I_{l+1}^{*}} w_{i}^{1 /(1-\sigma)} d i\right)^{\sigma}}
$$

[^14]Thus,

$$
\int_{I_{l}^{*}} w_{i}^{1 /(1-\sigma)} d i=\frac{Q_{l}^{1 / \sigma}}{\sum_{m} Q_{m}^{1 / \sigma}} \int_{0}^{1} w_{i}^{1 /(1-\sigma)} d i
$$

or, equivalently,

$$
\theta_{l}\left(I_{l}^{*}\right)=\left(\frac{Q_{l}^{1 / \sigma}}{\sum_{m} Q_{m}^{1 / \sigma}}\right)^{1-\sigma} \theta .
$$

The collection $\left(\theta_{l}\left(I_{l}^{*}\right)\right)_{l=1}^{L}$ determines an optimal (monotone) partition $\left(I_{l}^{*}\right)_{l=1}^{L} .{ }^{17}$ All models with $Q_{l}>0$ are employed at some tasks. The resulting optimal payoff is then (20). $\square$

Proof of Lemma 3 Drop the $l$ index. First, consider the case $\hat{\sigma}>0$. Define the unit-cost indices

$$
e \triangleq \min _{x \geq 0}\{c \cdot x: \Psi(x) \geq 1\}, \quad \hat{e} \triangleq \min _{z \geq 0}\{\hat{c} \cdot z: \Phi(z) \geq 1\} .
$$

For strictly concave, increasing, homogeneous $\Psi, \Phi$ these are finite and attained, and the respective minimizers $d, \hat{d}$ are unique.

By homogeneity and the definition of $e$, the minimal cost to reach $\Psi(x)=\Psi_{0}$ is

$$
\min _{x \geq 0}\left\{c \cdot x: \Psi(x) \geq \Psi_{0}\right\}=\Psi_{0}^{1 / \sigma} e,
$$

attained at $x=\Psi_{0}^{1 / \sigma} d$. Similarly, $\min _{z \geq 0}\left\{\hat{c} \cdot z: \Phi(z) \geq \Phi_{0}\right\}=\Phi_{0}^{1 / \hat{\sigma}} \hat{e}$, attained at $z=\Phi_{0}^{1 / \hat{\sigma}} \hat{d}$.
The cost minimization reduces to

$$
\min _{\Psi_{0}, \Phi_{0} \geq 0} \Psi_{0}^{1 / \sigma} e+\Phi_{0}^{1 / \hat{\sigma}} \hat{e} \quad \text { s.t. } \quad \Psi_{0} \Phi_{0}=Q
$$

Straightforward calculation gives:

$$
\Psi_{0}=Q^{\frac{\sigma}{\sigma+\tilde{\sigma}}}\left(\frac{\hat{e} \sigma}{e \hat{\sigma}}\right)^{\frac{\sigma \hat{\sigma}}{\sigma+\tilde{\sigma}}}, \quad \Phi_{0}=Q^{\frac{\hat{\sigma}}{\sigma+\hat{\sigma}}}\left(\frac{\hat{e} \sigma}{e \hat{\sigma}}\right)^{-\frac{\sigma \hat{\sigma}}{\sigma+\hat{\sigma}}},
$$

which corresponds to the optimal token allocation:

$$
x^{*}(Q)=Q^{\frac{1}{\sigma+\hat{\sigma}}}\left(\frac{\hat{e} \sigma}{e \hat{\sigma}}\right)^{\frac{\hat{\sigma}}{\sigma+\hat{\sigma}}} d, \quad z^{*}(Q)=Q^{\frac{1}{\sigma+\hat{\sigma}}}\left(\frac{\hat{e} \sigma}{e \hat{\sigma}}\right)^{-\frac{\sigma}{\sigma+\hat{\sigma}}} \hat{d}
$$

The resulting cost function is:

$$
C(Q)=e \Psi_{0}^{1 / \sigma}+\hat{e} \Phi_{0}^{1 / \hat{\sigma}}=e^{\frac{\sigma}{\sigma+\hat{\sigma}}} \hat{e}^{\frac{\hat{\sigma}}{\sigma+\hat{\sigma}}}\left[\left(\frac{\sigma}{\hat{\sigma}}\right)^{\frac{\hat{\sigma}}{\sigma+\hat{\sigma}}}+\left(\frac{\hat{\sigma}}{\sigma}\right)^{\frac{\sigma}{\sigma+\hat{\sigma}}}\right] Q^{\frac{1}{\sigma+\hat{\sigma}}}=C_{0} Q^{\frac{1}{\sigma+\hat{\sigma}}} .
$$

[^15]The case $\hat{\sigma}=0$ is analogous and simpler because in that case $\Phi(z) \equiv \Phi_{0}>0$ and fine-tuning can be ignored. The resulting cost function is

$$
C(Q)=e\left(\frac{Q}{\Phi_{0}}\right)^{\frac{1}{\sigma}}=C_{0} Q^{\frac{1}{\sigma}} .
$$

This completes the proof. $\square$

Proof of Proposition 5 By Lemma 3, each model's surplus $\theta Q-\kappa_{l} Q^{1 /\left(\sigma+\hat{\sigma}_{l}\right)}$ is concave in $Q$ and can be maximized in closed form. The efficient allocation selects the model that yields the highest surplus, yielding (22)-(23). Single-model usage follows because the cost function (21) is concave in $Q^{1 / \sigma}$. $\square$

Proof of Proposition 6 The result follows by pointwise maximization of virtual surplus with $\varphi(\theta)$ in place of $\theta$ in Proposition 5. Monotonicity of $\varphi$ ensures that the pointwise solution $Q^{\mathrm{m}}(\theta)=Q^{*}(\varphi(\theta))$ is increasing and therefore implementable. $\square$

Proof of Lemma 4 When $\hat{\sigma}_{F}=0$, (29) simplifies into:

$$
q_{L}^{\left(\sigma+\hat{\sigma}_{L}\right) / \sigma}+q_{F}=\left(\frac{\theta \sigma}{c_{F}}\right)^{1 /(1-\sigma)}
$$

The outside option corresponding to $q_{L}=0$ is

$$
\psi(\theta)=\theta^{\frac{1}{1-\sigma}}(1-\sigma)\left(\frac{\sigma}{c_{F}}\right)^{\sigma /(1-\sigma)}
$$

The formulation (31) follows. $\square$

Proof of Proposition 7 Denote the expression in the brackets of (32) by $\Pi(\theta, q)$ and observe that


Menu Pricing of Large Language Models

$$
\Pi(\theta, q(\theta))= \begin{cases}c_{F} q(\theta)^{\left(\sigma+\hat{\sigma}_{L}\right) / \sigma}-c_{L} q(\theta), & \text { if } q(\theta)<\hat{q}(\theta), \\ \varphi(\theta) q(\theta)^{\sigma+\hat{\sigma}_{L}}-c_{L} q(\theta)-\psi(\theta)+\frac{1-F(\theta)}{f(\theta)} \psi^{\prime}(\theta), & \text { if } q(\theta)>\hat{q}(\theta) .\end{cases}
$$

Consider the pointwise maximization of $\Pi(\theta, q(\theta))$. In the region $q(\theta) \in[0, \hat{q}(\theta)], \Pi(\theta, q(\theta))$ is convex in $q(\theta)$ and therefore attains its maximum at a corner, $q(\theta)=0$ or $q(\theta)=\hat{q}(\theta)$.

Direct calculation shows that $\Pi(\theta, 0)>\Pi(\theta, \hat{q}(\theta))$ if and only if $\theta<\theta_{1}$, where

$$
\theta_{1} \triangleq \frac{c_{F}}{\sigma}\left(\frac{c_{L}}{c_{F}}\right)^{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right) / \hat{\sigma}_{L}}
$$

In the region $q(\theta) \geq \hat{q}(\theta)$, if $\varphi(\theta) \leq 0$, then optimally $q(\theta)=\hat{q}(\theta)$. If $\varphi(\theta)>0$, then $\Pi(\theta, q(\theta))$ is concave in $q(\theta)$ and therefore the maximum is either at the corner, $q(\theta)=\hat{q}(\theta)$, or in the interior, in which case it equals the optimal monopoly quantity

$$
q^{\mathrm{m}}(\theta)=\left(\frac{\left(\sigma+\hat{\sigma}_{L}\right) \varphi(\theta)}{c_{L}}\right)^{1 /\left(1-\sigma-\hat{\sigma}_{L}\right)}
$$

Thus, the solution over the region $q(\theta) \geq \hat{q}(\theta)$ is interior if and only if $\Pi_{q}(\theta, \hat{q}(\theta))>0$, or

$$
\varphi(\theta)>\frac{c_{L}}{\sigma+\hat{\sigma}_{L}}\left(\frac{\theta \sigma}{c_{F}}\right)^{\frac{\sigma\left(1-\sigma-\sigma_{L}\right)}{(1-\sigma)\left(\sigma+\tilde{\sigma}_{L}\right)}} .
$$

Condition (48) can hold on disjoint intervals of $\theta$ even if $\varphi(\theta)$ is increasing, because the right-hand side is increasing in $\theta$. However, the power of $\theta$ on the right-hand side is strictly less than 1 . Thus, if $F$ satisfies monotone hazard rate, then $\varphi^{\prime}(\theta) \geq 1$, and condition (48) holds, if at all, for all $\theta>\theta_{2}$ where $\theta_{2}$ solves:

$$
\varphi\left(\theta_{2}\right)=\frac{c_{L}}{\sigma+\hat{\sigma}_{L}}\left(\frac{\theta_{2} \sigma}{c_{F}}\right)^{\frac{\sigma\left(1-\sigma-\sigma_{L}\right)}{(1-\sigma)\left(\sigma+\sigma_{L}\right)}}
$$

If (49) doesn't admit a solution, then the solution is never interior and thus for all $\theta<\theta_{1}$, $q(\theta)=0$ and for all $\theta>\theta_{1}, q(\theta)=\hat{q}(\theta)$. If (49) admits a solution at $\theta_{2}>\theta_{1}$, then: for all $\theta<\theta_{1}, q(\theta)=0$; for all $\theta \in\left(\theta_{1}, \theta_{2}\right), q(\theta)=\hat{q}(\theta)$; for all $\theta \geq \theta_{2}, q(\theta)=q^{\mathrm{m}}(\theta)$. Finally, if (49) admits a solution at $\theta_{2}<\theta_{1}$, then $\theta_{3} \in\left[\theta_{2}, \theta_{1}\right]$ exists such that for all $\theta<\theta_{3}, q(\theta)=0$ and for all $\theta \geq \theta_{3}, q(\theta)=q^{\mathrm{m}}(\theta)$. (The threshold $\theta_{3}>\theta_{2}$ is determined by the condition $\Pi\left(\theta_{3}, q^{\mathrm{m}}\left(\theta_{3}\right)\right)=0$.)

Since all these (relaxed) solutions are implementable, the result follows. $\square$


Menu Pricing of Large Language Models

## Leader-Fringe Competition with Uniform Types

Competitive Mechanism- With $\hat{\sigma}_{F}=0$ the buyer's first-order condition implies

$$
q_{L}^{\left(\sigma+\hat{\sigma}_{L}\right) / \sigma}+q_{F}=\left(\frac{\theta \sigma}{c_{F}}\right)^{1 /(1-\sigma)}
$$

On the fringe-indifference boundary $\left(q_{F}=0\right)$ the leader's quantity solves

$$
\hat{q}(\theta)=\left(\frac{\theta \sigma}{c_{F}}\right)^{\frac{\sigma}{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)}} .
$$

When the buyer single-homes on the leader and the fringe constraint is slack,

$$
q^{\mathrm{m}}(\theta)=\left(\frac{\left(\sigma+\hat{\sigma}_{L}\right) \varphi(\theta)}{c_{L}}\right)^{\frac{1}{1-\sigma-\hat{\sigma}_{L}}}=\left(\frac{\left(\sigma+\hat{\sigma}_{L}\right)(2 \theta-1)}{c_{L}}\right)^{\frac{1}{1-\sigma-\hat{\sigma}_{L}}} .
$$

We want the optimal menu to be characterized by $0<\theta_{1}<\theta_{2}<1$ such that

$$
q^{L F}(\theta)=\left\{\begin{array}{ll}
0, & \theta \leq \theta_{1}, \\
\hat{q}(\theta), & \theta_{1}<\theta<\theta_{2}, \\
q^{\mathrm{m}}(\theta), & \theta \geq \theta_{2},
\end{array} \quad q_{F}^{L F}(\theta)= \begin{cases}\left(\frac{\theta \sigma}{c_{F}}\right)^{\frac{1}{1-\sigma}}, & \theta \leq \theta_{1}, \\
0, & \theta>\theta_{1} .\end{cases}\right.
$$

The entry cutoff is

$$
\theta_{1}=\frac{c_{F}}{\sigma}\left(\frac{c_{L}}{c_{F}}\right)^{\frac{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)}{\hat{\sigma}_{L}}} .
$$

The boundary-interior switch $\theta_{2}$ is the unique solution to

$$
\varphi(\theta)=\frac{c_{L}}{\sigma+\hat{\sigma}_{L}}\left(\frac{\theta \sigma}{c_{F}}\right)^{\frac{\sigma\left(1-\sigma-\hat{\sigma}_{L}\right)}{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)}},
$$

which exists in (1/2, 1) if and only if $\frac{c_{L}}{\sigma+\hat{\sigma}_{L}}\left(\frac{\sigma}{c_{F}}\right)^{\frac{\sigma\left(1-\sigma-\hat{\sigma}_{L}\right)}{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)}}<1$. Furthermore, $\theta_{2}>\theta_{1}$ if and only if $\theta_{1}<\left(\sigma+\hat{\sigma}_{L}\right) /\left(\sigma+2 \hat{\sigma}_{L}\right)$. Therefore, for $0<\theta_{1}<\theta_{2}<1$ to take place, the necessary and sufficient conditions are:

$$
\frac{c_{L}}{\sigma+\hat{\sigma}_{L}}\left(\frac{\sigma}{c_{F}}\right)^{\frac{\sigma\left(1-\sigma-\hat{\sigma}_{L}\right)}{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)}}<1, \quad \frac{c_{F}}{\sigma}\left(\frac{c_{L}}{c_{F}}\right)^{\frac{(1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)}{\hat{\sigma}_{L}}}<\frac{\sigma+\hat{\sigma}_{L}}{\sigma+2 \hat{\sigma}_{L}} .
$$

Efficient Allocation - Efficiency features single-homing with a model-switch cutoff $\hat{\theta}$ :

$$
\left(q_{L}^{*}(\theta), q_{F}^{*}(\theta)\right)= \begin{cases}\left(0,\left(\frac{\theta \sigma}{c_{F}}\right)^{\frac{1}{1-\sigma}}\right), & \theta<\hat{\theta} \\ \left(\left(\frac{\theta\left(\sigma+\hat{\sigma}_{L}\right)}{c_{L}}\right)^{\frac{1}{1-\sigma-\hat{\sigma}_{L}}}, 0\right), & \theta \geq \hat{\theta}\end{cases}
$$

with

$$
\hat{\theta}=\left(\frac{1-\sigma}{1-\sigma-\hat{\sigma}_{L}}\right)^{\frac{\left(1-\sigma-\hat{\sigma}_{L}\right)(1-\sigma)}{\hat{\sigma}_{L}}}\left(\frac{\sigma}{c_{F}}\right)^{\frac{\sigma\left(1-\sigma-\hat{\sigma}_{L}\right)}{\hat{\sigma}_{L}}}\left(\frac{c_{L}}{\sigma+\hat{\sigma}_{L}}\right)^{\frac{\left(\sigma+\hat{\sigma}_{L}\right)(1-\sigma)}{\hat{\sigma}_{L}}} .
$$

Multi-Model Monopoly- Virtual type is $\varphi(\theta)=2 \theta-1$. Types $\theta<1 / 2$ are excluded. Among
served types the monopolist assigns a single model, switching at $\theta^{\mathrm{m}}=(1+\hat{\theta}) / 2$. Optimal quantities are

$$
\left(q_{L}^{\mathrm{m}}(\theta), q_{F}^{\mathrm{m}}(\theta)\right)= \begin{cases}(0,0), & \theta<\frac{1}{2}, \\ \left(0,\left(\frac{\sigma \varphi(\theta)}{c_{F}}\right)^{\frac{1}{1-\sigma}}\right), & \frac{1}{2} \leq \theta<\theta^{\mathrm{m}}, \\ \left(\left(\frac{\left(\sigma+\hat{\sigma}_{L}\right) \varphi(\theta)}{c_{L}}\right)^{\frac{1}{1-\sigma-\hat{\sigma}_{L}}}, 0\right), & \theta \geq \theta^{\mathrm{m}} .\end{cases}
$$


Menu Pricing of Large Language Models

## B Contractible Tasks

Throughout the manuscript, we focus on the more realistic case in which the LLM provider cannot contract on the allocation of tokens across the buyer's tasks. However, the fullcontracting setting provides a natural benchmark and we study it in this section. Thus, the seller designs a direct menu that specifies a token allocation across different tasks:

$$
\left\{\left(\left(x_{i 1}(w), \ldots, x_{i J}(w)\right)_{i \in[0,1]}, z_{1}(w), \ldots, z_{K}(w), t(w)\right)\right\}_{w} .
$$

The allocation being contractible means the buyer has no freedom to reallocate tokens across tasks. This naturally increases the scope for screening. This also makes the problem intractable in full generality, because the reduction to a one-dimensional aggregate type and quality as in Proposition 2 does not apply. Nevertheless, in this section we obtain complete characterizations in two special settings: the case of separable type distributions (Section B.1) and the case of two types (Section B.2). In the former case, the optimal solution can be obtained via a menu of token budgets. In the latter case, the optimal solution can be obtained by identifying the structure of binding incentive constraints.


Menu Pricing of Large Language Models

## B. 1 Separable Type Distribution and Cost-Based Pricing

In this section, we provide a sufficient condition on type distribution under which contracting on tasks is not profitable. In fact, under that condition an even less contractually restrictive class of mechanisms, cost-based tariffs, is optimal:

Definition 4 (Cost-Based Tariff). A cost-based tariff is a menu of monetary budgets and transfers $\left\{\left(B_{n}, T_{n}\right)\right\}_{n}$ such that, upon purchasing item $n$, the buyer pays $T_{n}$ for access to budget $B_{n}$, which he can freely spend on tokens priced at their marginal costs.

We build on the analysis of Armstrong (1996) and introduce the conditions that are jointly sufficient for the optimality of cost-based tariffs: demand separability and type separability. We show that the former is always satisfied in our setting, whereas the latter is equivalent to the aggregate type not being informative about the relative task weights.

To this end, consider a problem of optimal token allocation by type $w$ given a monetary budget $B$ and token prices equal to marginal costs:

$$
V(w, B)=\max _{\left\{\left(x_{i}\right)_{i \in[0,1], z \geq 0\}}\right.} \int_{0}^{1} w_{i} g\left(x_{i}, z\right) d i, \quad \text { s.t. } \int_{0}^{1} \sum_{j=1}^{J} c_{j} x_{i j} d i+\sum_{k=1}^{K} \hat{c}_{k} z_{k}=B,
$$

Definition 5 (Demand Separability). Demand is separable if there exist functions $V_{1}(w)$ and $V_{2}(B)$ such that

$$
V(w, B)=V_{1}(w) V_{2}(B) .
$$

If demand is separable, then faced with a cost-based tariff, all types $w$ with the same $V_{1}(w)$ purchase the same item.

Demand separability always holds in our setting. Indeed, by Proposition 2, if type $w$ purchases a total amount of inference tokens $X=\left(X_{1}, \ldots, X_{J}\right)$ and fine-tuning tokens $Z=$ $\left(Z_{1}, \ldots, Z_{K}\right)$, then his optimal payoff is $\theta(w) g(X, Z)$. Therefore, (51) holds with $V_{1}(w)=$ $\theta(w)$ and

$$
V_{2}(B)=\max _{X \geq 0, Z \geq 0} g(X, Z), \quad \text { s.t. } \sum_{j=1}^{J} c_{j} X_{j}+\sum_{k=1}^{K} \hat{c}_{k} Z_{k}=B .
$$

Definition 6 (Type Separability). Type distribution is separable if $f_{1}, f_{2}$ exist such that $f(w)=f_{1}(\theta(w)) f_{2}(w)$ for all $w$ and $f_{2}$ is homogeneous of degree zero.

A separable type distribution means that knowing $\theta(w)$ provides no information about which ray from the origin $w$ lies on, that is, about the relative weights the buyer assigns to different tasks. Equivalently, denoting by $\|\cdot\|_{p}$ a standard $L_{p}$ norm, $\theta(w)=\|w\|_{1 /(1-\sigma)}$ and the type separability is equivalent to $\|w\|_{1 /(1-\sigma)}$ and $w /\|w\|_{1 /(1-\sigma)}$ to be independent. Thus, any type distribution generated by a draw of a "total size" $\|w\|_{1 /(1-\sigma)}$ according to an arbitrary distribution together with an independent draw of "relative weights" $w /\|w\|_{1 /(1-\sigma)}$ according to any other distribution, is separable.

Proposition 8 (Cost-Based Optimality). If the type distribution is separable, then a cost-based tariff is optimal.

Proof. Since in our setting demand is always separable, the result follows from the arguments presented by Armstrong (1996). Specifically, we can relax the IC constraints between types located on different rays from the origin. Then, applying the Envelope Theorem ray by ray, we can find a solution to the relaxed problem as follows:

$$
\left(x^{*}, z^{*}\right) \in \arg \max _{\left\{\left(x_{i}\right)_{i \in[0,1],} z \geq 0\right\}}\left(1-\frac{\int_{1}^{\infty} r^{n-1} f(r w) d r}{f(w)}\right) u(w, x, z)-c(x, z) .
$$

Given the demand separability, this condition can be rewritten as:

$$
\left(x^{*}, z^{*}\right) \in \arg \max _{\left\{\left(x_{i}\right)_{i \in[0,1]}, z \geq 0\right\}} u(w, x, z), \quad \text { s.t. } c(x, z) \leq B^{*}(w),
$$

and

$$
B^{*}(w) \in \arg \max _{B \geq 0}\left(1-\frac{\int_{1}^{\infty} r^{n-1} f(r w) d r}{f(w)}\right) \theta(w) V_{2}(B)-B .
$$

At the same time, the optimal cost-based tariff implements an allocation

$$
\left(x^{*}, z^{*}\right) \in \arg \max _{\left\{\left(x_{i}\right)_{i \in[0,1]}, z \geq 0\right\}} u(w, x, z), \quad \text { s.t. } c(x, z) \leq B^{*}(w),
$$

where

$$
B^{*}(w) \in \arg \max _{B}\left(\theta(w)-\frac{1-F(\theta(w))}{f(\theta(w))}\right) V_{2}(B)-B .
$$

If the type distribution is separable, then

$$
\left(1-\frac{\int_{1}^{\infty} r^{n-1} f(r w) d r}{f(w)}\right) \theta(w)=\theta(w)-\frac{1-F(\theta(w))}{f(\theta(w))},
$$

and the cost-based tariff implements the solution to the relaxed problem. Therefore, it is (indirectly) optimal. $\square$


Menu Pricing of Large Language Models

## B. 2 Binary Types

Alternatively, let there be only two types, $w_{1}$ and $w_{2}$, which occur with prior strictly positive probabilities $f_{1}$ and $f_{2}$, respectively. In this case, the optimal mechanism depends on what happens if the seller attempts to extract the first-best level of surplus, that is, if she offers a menu containing the efficient amounts of tokens for each type with prices equal to their respective added values. If this menu is incentive compatible, then it is clearly optimal, and we call the type with the higher payment "high" and associate the label $H$ with it. If this menu is not incentive compatible, then we call "high" the type whose incentive constraint is violated. We call the other type "low" and associate the label $L$ with it.

To design an optimal menu, it is important to determine which type out of $w_{1}$ and $w_{2}$ is high and which one is low. If $w_{2}$ dominates $w_{1}$ for every task, then it is clearly high; alternatively, the types are in some sense horizontally differentiated and the distinction is less clear. However, it turns out that the ranking can be derived from the aggregate types, and the optimal menu admits a simple characterization.

Proposition 9 (Binary Types). The high type is the one with the higher aggregate type, i.e., $\theta\left(w_{H}\right) \geq \theta\left(w_{L}\right)$. In the optimal menu, the token allocation of $w_{H}$ is always efficient. If

$$
\int_{0}^{1}\left(w_{H i}-w_{L i}\right) w_{L i}^{\frac{\sigma}{1-\sigma}} d i \leq 0
$$

then the token allocation of $w_{L}$ is also efficient and the seller extracts full surplus. Otherwise, the token allocation of $w_{L}$ is efficient with respect to a virtual type $w_{L}-\left(w_{H}-w_{L}\right) f_{H} / f_{L}$.

Proof. With a small abuse of notation relative to previous sections, denote by $q=\left(q_{i}\right)_{i \in[0,1]}$ the profile of qualities delivered to each task, $q_{i} \triangleq g\left(x_{i}, z\right)$, and denote by $C(q)$ the minimal total cost of generating a given profile $q$ :

$$
C(q) \triangleq \min _{x_{i}, z \geq 0} \int_{0}^{1} \sum_{j=1}^{J} c_{j} x_{i j} d i+\sum_{k=1}^{K} \hat{c}_{k} z_{k}, \quad \text { s.t. } g\left(x_{i}, z\right)=q_{i}, \forall i \in[0,1] .
$$

Because the set of feasible profiles $q$ is convex, it follows from the analysis of Haghpanah and Siegel (2024) on general screening problems with two buyer types that in our setting either (i) the seller extracts full surplus; or (ii) the incentive constraint of type $w_{H}$ and the individual rationality constraint of type $w_{L}$ bind.

It follows that if the seller cannot extract full surplus, then the seller's problem can be written as

$$
\begin{aligned}
\max _{q_{L}, q_{H}, t_{L}, t_{H}} & f_{L}\left(t_{L}-C\left(q_{L}\right)\right)+f_{H}\left(t_{H}-C\left(q_{H}\right)\right) \\
\text { s.t. } & \int_{0}^{1} w_{H i} q_{H i} d i-t_{H}=\int_{0}^{1} w_{H i} q_{L i} d i-t_{L}, \int_{0}^{1} w_{L i} q_{L i} d i-t_{L}=0
\end{aligned}
$$

Solving for transfers from the constraints, the problem can be restated as:

$$
\max _{q_{L}, q_{H}} f_{L}\left(\int_{0}^{1}\left(w_{L i}-\frac{f_{H}}{f_{L}}\left(w_{H i}-w_{L i}\right)\right) q_{L i} d i-C\left(q_{L}\right)\right)+f_{H}\left(\int_{0}^{1} w_{H i} q_{H i} d i-C\left(q_{H}\right)\right),
$$

which is solved by the allocation efficient relative to the virtual types.
It is left to determine which type is high and provide conditions for full surplus extraction. Consider a mechanism that attempts full surplus extraction. By the efficiency analysis behind Proposition 1, under this mechanism type $w$, when reporting type $\tilde{w}$, obtains added value

$$
u(w, \tilde{w})=\int_{0}^{1} w_{i} \Psi\left(x_{i}(\tilde{w})\right) \Phi(z(\tilde{w})) d i=\int_{0}^{1} w_{i} \tilde{w}_{i}^{\frac{\sigma}{1-\sigma}} \Psi(d) \Phi(z(\tilde{w}))^{\frac{1}{1-\sigma}} d i
$$

Because under truth-telling each type obtains zero rents, the corresponding payment is
$t(w)=u(w, w)$, and the incentive constraint $u(w, w)-t(w) \geq u(w, \tilde{w})-t(\tilde{w})$ is violated if and only if:

$$
\Psi(d) \Phi(z(\tilde{w}))^{\frac{1}{1-\sigma}} \int_{0}^{1}\left(w_{i}-\tilde{w}_{i}\right) \tilde{w}_{i}^{\frac{\sigma}{1-\sigma}} d i>0
$$

By Jensen's inequality and the concavity of the logarithm:

$$
w_{i} \tilde{w}_{i}^{\frac{\sigma}{1-\sigma}}=w_{i}^{\frac{1-\sigma}{1-\sigma}} \tilde{w}_{i}^{\frac{\sigma}{1-\sigma}} \leq(1-\sigma) w_{i}^{\frac{1}{1-\sigma}}+\sigma \tilde{w}_{i}^{\frac{1}{1-\sigma}} .
$$

Therefore, inequality (53) implies

$$
\int_{0}^{1}\left(w_{i}^{\frac{1}{1-\sigma}}-\tilde{w}_{i}^{\frac{1}{1-\sigma}}\right) d i>0
$$

and the incentive violation is possible only if $\theta(w)>\theta(\tilde{w})$, i.e., from high to low aggregate type. Therefore, if full surplus extraction is not incentive compatible, then the high type is the one with the higher aggregate type. If full surplus extraction is incentive compatible, then the high type is the one with the higher aggregate type directly by the analysis behind Proposition 1. The result follows. $\square$

Note that even though the virtual types in Proposition 9 follow the standard formula of the single-dimensional case, each type there is infinite-dimensional.

One might wonder whether the correspondence between the incentive order and aggregate types extends beyond the binary-type case. This is not the case. In Section B.3, we study a special case of our model in which the buyer's type can be effectively parameterized by two variables: the number of ex ante homogeneous tasks to which the buyer attaches positive value, and the value he attaches to each of those tasks. We show that the incentive constraints that bind in the optimal mechanism do not admit a one-dimensional structure. Instead, they form an infinite collection of one-dimensional segments. In that setting the optimal mechanisms also admit a natural implementation via a menu of two-part tariffs. However, in contrast to the indirect implementations described in Section 4.2, each item in the menu must be accompanied by task caps, that is, restrictions on the number of tasks the buyer can process.


Menu Pricing of Large Language Models

## B. 3 Value-Scale Heterogeneity

In this section, we characterize an optimal menu with contractible token allocations across tasks in the case of value-scale heterogeneity. In this case, each multidimensional type $w$ is
characterized (with a small abuse of notation) by two parameters $(w, s)$ such that

$$
w_{i}=\left\{\begin{array}{l}
w, \text { if } i \leq s \\
0, \text { if } i>s
\end{array}\right.
$$

in which $w$ and $s$ are independently distributed according to CDFs $F_{w}$ and $F_{s}$, with $F_{w}$ featuring increasing virtual values. We will argue that in this case, under additional assumptions, the binding incentive constraints are those within each scale, and the seller is able to generate the same profit as if the scale were observable.

For this section, we drop the product structure and the homogeneity requirements of (1) and instead require that $g\left(x_{i}, z\right)$ is (i) positive and continuous on $\mathbb{R}_{+}^{J+K}$, (ii) strictly monotone, twice continuously differentiable, strictly concave with negative definite Hessian, and with all cross-partial derivatives strictly positive on $\mathbb{R}_{++}^{J+K}$, (iii) $g(0)=0$, and (iv) Inada at zero, i.e., $\lim _{y_{m} \downarrow 0} g_{y_{m}}\left(y_{m}, y_{-m}\right)=+\infty$ for all $y_{-m} \in \mathbb{R}_{+}^{J+K-1}$ such that $g\left(0, y_{-m}\right)>0 .{ }^{18}$

Consider the problem in which the scale is commonly known to be $s>0$. The buyer's payoff from any given item on the menu (50) is

$$
w \int_{i=0}^{s} g\left(x_{i}, z\right) d i-t
$$

For any reported $w$ the seller should optimize token allocation to deliver a promised level of (total) quality $q$,

$$
w q-t,
$$

with the minimal cost function $C(q, s)$ of delivering a given quality being:

$$
C(q, s)=\min _{\left(x_{i}\right)_{i \in[0,1], z \geq 0}} \int_{i=0}^{s} \sum_{j=1}^{J} c_{j} x_{i j} d i+\sum_{k=1}^{K} \hat{c}_{k} z_{k}, \quad \text { s.t. } \int_{i=0}^{s} g\left(x_{i}, z\right) d i=q \text {. }
$$

Since $g$ is strictly concave, the solution to this problem is achieved by allocating the inference tokens uniformly across the $s$ tasks. The problem can be equivalently stated as:

$$
C(q, s)=\min _{x, z \geq 0} s \sum_{j=1}^{J} c_{j} x_{j}+\sum_{k=1}^{K} \hat{c}_{k} z_{k}, \quad \text { s.t. } s g(x, z)=q .
$$

The resulting cost function $C(q, s)$ satisfies, over the domain of admissible $(q, s)$, the following properties:

[^16]


Menu Pricing of Large Language Models

## Lemma 5 (Cost Function).

1. $C(q, s)$ is strictly increasing and strictly convex in $q$ with $C_{q}(0, s)=0$.
2. $C(q, s)$ is strictly decreasing in $s$ for $q>0$.
3. $C(q, s)$ is submodular, i.e., $C_{q}(q, s)$ is decreasing in $s$ for all $q$.

Proof. The first two statements follow directly from our assumptions on $g$. To establish the third property, posit the Lagrangian for the cost minimization problem:

$$
L=s \sum_{j=1}^{J} c_{j} x_{j}+\sum_{k=1}^{K} \hat{c}_{k} z_{k}+\lambda(q-s g(x, z)) .
$$

By the Envelope Theorem, $C_{q}(q, s)=\lambda(q, s)$. Thus, it suffices to show that $\lambda(q, s)$ is decreasing in $s$ for a fixed $q$.

To this end, for any $q>0$, by the Inada condition, the solution must be interior, $(x, z) \gg$ 0 . The constraint binds, so $\lambda>0$. Denoting $(x, z)$ by $y$, the first-order conditions are:

$$
g_{x_{j}}(y)=c_{j} / \lambda, \quad g_{z_{k}}(y)=\hat{c}_{k} /(\lambda s) .
$$

Define the Hessian function $H(y) \triangleq \nabla^{2} g(y)$. Denote by $A, B \in \mathbb{R}^{J+K}$ vectors such that $A_{r}=c_{r} / \lambda^{2}$ if $r \leq J$ and $=\hat{c}_{r-J} /\left(\lambda^{2} s\right)$ if $r>J, B_{r}=0$ if $r \leq J$ and $=\hat{c}_{r-J} /\left(\lambda s^{2}\right)$ if $r>J$. Then, differentiating (57) with respect to $s$, we obtain:

$$
H \frac{d y}{d s}=-A \frac{d \lambda}{d s}-B .
$$

At the same time, differentiating the constraint $s g(y(s))=q$ with respect to $s$, we obtain:

$$
g(y)+s \nabla g(y)^{\top} \frac{d y}{d s}=0 .
$$

Solving for $d y / d s$ in (58) and substituting it into (59), we obtain, omitting the dependence on $y$ :

$$
\frac{d \lambda}{d s} s \nabla g^{\top}\left(-H^{-1} A\right)=-g-s \nabla g^{\top}\left(-H^{-1} B\right) .
$$

Now, observe that for all $y \in \mathbb{R}_{++}^{J+K}$, by assumption on $g, H(y)$ is symmetric negative definite; moreover, by the positivity of cross-derivative, all off-diagonal entries of $H(y)$ are strictly positive. Then, $-H(y)$ is a Stieltjes matrix and, consequently, all elements of $H(y)^{-1}$ are negative. It immediately follows from (60) that whenever $q>0$ and $s>0, d \lambda / d s<0$. □

As such, the seller's problem for any given $s$ is analogous to Mussa and Rosen (1978). Since $w$ and $s$ are independently distributed, we can drop the dependence on $s$ and define the virtual value as

$$
\varphi(w) \triangleq w-\frac{1-F_{w}(w)}{f_{w}(w)} .
$$

Since $\varphi(w)$ is increasing, all $w$ with $\varphi(w) \leq 0$ are excluded and all other $w$ receive the quality level $q(w, s)$ that solves:

$$
\varphi(w)=C_{q}(q(w, s), s) .
$$

The corresponding optimal transfers are

$$
t(w, s)=w q(w, s)-\int_{0}^{w} q(r, s) d r
$$

Lemma 5 shows that it is cheaper to generate an extra unit of (total) quality when you have more tasks. This property is intuitive given that the returns on each task are diminishing. Thus, the optimal quality, and hence the buyer's rent, increase in scale for any given value $w$.

Assumption 2 (Bounded Rent Increase). For all $w, s$, the function $q(w, s)$ defined in (62) satisfies $\int_{0}^{w} s q_{s}(r, s) d r \leq w q(w, s)$.

Assumption 2 requires that the buyer's rent does not grow too quickly and, specifically, that the marginal increase of buyer rent from having an additional task is smaller than the average equilibrium value generated by LLM across existing tasks.

Proposition 10 (Optimal Menu of Token Allocations). Under Assumption 2, an optimal menu is

$$
\left(\left(x_{i}(w, s)\right)_{i \in[0,1]}, z(w, s), t(w, s)\right)_{(w, s)},
$$

where for each $(w, s),\left(\left(x_{i}(w, s)\right)_{i \in[0,1]}, z(w, s)\right)$ are cost-minimizing tokens from (55) that deliver quality $q(w, s)$ as defined in (62), and $t(w, s)$ is as defined in (63).

Proof. If each type reports truthfully, then the menu attains the profits of the observablescale benchmark and is thus optimal.

If type $(w, s)$ deviates to $(w, \tilde{s})$ with $\tilde{s} \leq s$, then, under the proposed menu, he obtains exactly the same payoff as type $(w, \tilde{s})$, because he processes the same number of tasks with the same willingness to pay for quality. By Lemma $5, C_{q}(q, s)$ is decreasing in $s$ for all $q$, and thus $q(w, s)$ is increasing in $s$ for all $w$. Therefore, the rents accrued by type $(w, s)$ under truth-telling,

$$
U(w, s)=\int_{0}^{w} q(r, s) d r
$$

are increasing in $s$ for all $w$. Therefore, $(w, s)$ does not want to deviate to $(w, \tilde{s})$ with $\tilde{s} \leq s$. Furthermore, by incentive compatibility within a given $\tilde{s},(w, \tilde{s})$ does not want to deviate to $(\tilde{w}, \tilde{s})$. Therefore, $(w, s)$ does not want to deviate to any $(\tilde{w}, \tilde{s})$ with $\tilde{s} \leq s$.

If type $(w, s)$ deviates to $(\tilde{w}, \tilde{s})$ with $\tilde{s}>s$, then he obtains gross payoff $w q(\tilde{w}, \tilde{s}) s / \tilde{s}$ and pays the transfer $t(\tilde{w}, \tilde{s})$. Therefore, the optimal double deviation strategy for a misreporting type solves

$$
\max _{\tilde{w} \geq 0}\left[w q(\tilde{w}, \tilde{s}) \frac{s}{\tilde{s}}-\tilde{w} q(\tilde{w}, \tilde{s})+\int_{0}^{\tilde{w}} q(r, \tilde{s}) d r\right] .
$$

Because the mechanism incentivizes truthful reporting by any $\tilde{s}$-truthtelling type, including the type ( $w s / \tilde{s}, \tilde{s}$ ), it follows that

$$
\tilde{w}^{*}=\frac{w s}{\tilde{s}}, \quad U(w ; s, \tilde{s})=\int_{0}^{\frac{w s}{\tilde{s}}} q(r, \tilde{s}) d r
$$

The condition that discourages local deviations to $\tilde{s}>s$ is:

$$
s \int_{0}^{w} q_{s}(r, s) d r \leq w q(w, s)
$$

for all $(w, s)$, which is precisely Assumption 2. Furthermore, observe that for $\tilde{s}>s$ :

$$
\begin{aligned}
U_{\tilde{s}}(w ; s, \tilde{s}) & =\int_{0}^{\frac{w s}{\tilde{s}}} q_{s}(r, \tilde{s}) d r-\frac{w s}{\tilde{s}^{2}} q\left(\frac{w s}{\tilde{s}}, \tilde{s}\right) \\
& =\frac{1}{\tilde{s}}\left(\tilde{s} \int_{0}^{\frac{w s}{\tilde{s}}} q_{s}(r, \tilde{s}) d r-\frac{w s}{\tilde{s}} q\left(\frac{w s}{\tilde{s}}, \tilde{s}\right)\right) \leq 0
\end{aligned}
$$

where the inequality holds by Assumption 2 with $(w, s)$ replaced by $(s w / \tilde{s}, \tilde{s})$. Therefore, the upward global deviations in $\tilde{s}$ are suboptimal and the result follows. $\square$


Menu Pricing of Large Language Models

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Kaplan, J., S. McCandlish, T. Henighan, T. B. Brown, B. Chess, R. Child, S. Gray, A. Radford, J. Wu, and D. Amodei (2020): "Scaling Laws for Neural Language Models," arXiv preprint arXiv:2001.08361.

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[^0]:    *Bergemann: Department of Economics, Yale University, dirk.bergemann@yale.edu. Bonatti: MIT Sloan, bonatti@mit.edu. Smolin: Toulouse School of Economics, alexey.v.smolin@gmail.com. We thank Mark Armstrong, Mert Demirer, Scott Kominers, Antonio Russo, Ron Siegel, and Frank Yang for valuable comments and discussions, as well as audiences at the Triangle Conference, USC, Caltech, Virginia, the Chicago workshop, the Sciences Po workshop, the NBER Market Design workshop, VSET, Brown, LSE, the TSE Digital Economics Conference, the IESE Economics of AI Conference, and Cambridge. An extended abstract of an earlier version of this paper appeared as "The Economics of Large Language Models: Token Allocation, Fine-Tuning, and Optimal Pricing" in the proceedings of EC'25. Dirk Bergemann gratefully acknowledges financial support from NSF SES 2049754 and ONR MURI. Alessandro Bonatti gratefully acknowledges financial support from NSF SES 2519401. Alex Smolin gratefully acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (grant ANR-17-EURE-0010) and through the AI Interdisciplinary Institute ANITI (grant ANR-23-IACL-0002).

[^1]:    ${ }^{1}$ For the practical challenges associated with LLM pricing, see https://www.wsj.com/articles/no-one-knows-how-to-price-ai-tools-f346ea8a.

[^2]:    ${ }^{2}$ That is, for all $j$ and $x_{i,-j} \in \mathbb{R}_{+}^{J-1}$ such that $\Psi\left(0, x_{i,-j}\right)>0, \lim _{x_{i j} \downarrow 0} \Psi_{j}\left(x_{i j}, x_{i,-j}\right)=+\infty$.
    ${ }^{3}$ That is, there exists $z_{0} \geq 0, z_{0} \neq 0$, such that $\lim _{r \downarrow 0} \Phi\left(r z_{0}\right) / r^{1-\sigma}=+\infty$. Sufficient conditions for this property are (i) $\Phi(0)>0$ or (ii) $\Phi$ is homogeneous of degree $\hat{\sigma} \in(0,1)$ with $\sigma+\hat{\sigma}<1$.

[^3]:    ${ }^{4}$ For tractability, we will often study the case $g(x, 0)=0$. This can be viewed as a normalization that doesn't affect the economics of the problem.

[^4]:    ${ }^{5}$ The parameter counts of leading models are estimated to be in the several-billion range; see https: //codingscape.com/blog/most-powerful-llms-large-language-models.
    ${ }^{6}$ Recent progress has leveraged this channel; see, for instance, https://arcprize.org/blog/oai-o3-pub-breakthrough.

[^5]:    ${ }^{7}$ Note that fine-tuning typically does not change the model's architecture or size and therefore should not directly affect marginal inference costs.

[^6]:    ${ }^{8}$ In Section 5, we show that if $\Phi(Z)$ is also a homogeneous function, then $C(Q)$ is simply a power function.

[^7]:    ${ }^{9}$ Equivalently, we could assume that the buyer can combine tokens from two models on any task, with a performance given by the sum of individual gain functions.

[^8]:    ${ }^{10}$ See Calzolari and Denicolò (2013) for a related analysis of nonlinear competition between a dominant firm and a competitive fringe under private information about demand.


Menu Pricing of Large Language Models

[^9]:    ${ }^{11}$ Indeed, $\underline{\hat{\theta}}=\left[\left(\frac{1-\sigma}{1-\sigma-\hat{\sigma}_{L}}\right)^{1-\sigma-\hat{\sigma}_{L}}\left(\frac{\sigma}{\sigma+\hat{\sigma}_{L}}\right)^{\sigma+\hat{\sigma}_{L}}\right]^{(1-\sigma) / \hat{\sigma}_{L}}$, and thus, by the strict concavity of the logarithm and Jensen's inequality, $\ln \left(\frac{\hat{\theta}}{\underline{\theta}}\right)=\frac{1-\sigma}{\hat{\sigma}_{L}}\left[\left(1-\sigma-\hat{\sigma}_{L}\right) \ln \left(\frac{1-\sigma}{1-\sigma-\hat{\sigma}_{L}}\right)+\left(\sigma+\hat{\sigma}_{L}\right) \ln \left(\frac{\sigma}{\sigma+\hat{\sigma}_{L}}\right)\right]<0$.

[^10]:    ${ }^{12}$ Calzolari and Denicolò (2015) show that a dominant firm with a competitive advantage can profitably impose exclusive dealing on privately informed buyers; the mechanism here is analogous, though exclusivity arises from the optimal nonlinear tariff rather than from contractual restrictions.

[^11]:    ${ }^{13}$ For instance, a message to GPT-4o-mini costs 9 points, while a message to Claude Opus 4 costs 4,105 points, a ratio of roughly 450×.

[^12]:    ${ }^{14}$ As in the theory, the overage charge interacts with the multiplier system: a query to a 1 × model costs \$0.04 in the overage region, while a query to a $3 \times$ model (Claude Opus 4.5) costs \$0.12.

[^13]:    ${ }^{15}$ By homogeneity, for all $x \in \mathbb{R}_{+}^{J}$ and $r>0, \nabla \Psi(r x)=r^{\sigma-1} \nabla \Psi(x)$, so any token ray corresponds to a gradient ray. By strict concavity, if $x_{1} \neq x_{2}$, then $\nabla \Psi\left(x_{1}\right) \neq \nabla \Psi\left(x_{2}\right)$, so distinct token rays map to distinct gradient rays.

[^14]:    ${ }^{16}$ In the general case of heterogeneous $\sigma_{l}$, the optimal task split and the resulting buyer's payoff do not admit tractable closed-form solutions. In particular, the fine details of the profile $w$ could matter.

[^15]:    ${ }^{17}$ Non-monotone partitions may also be optimal, but $\theta_{l}\left(I_{l}^{*}\right)$ are uniquely determined.

[^16]:    ${ }^{18}$ The Inada condition is imposed only for simplicity of arguments in Lemma 5.


Menu Pricing of Large Language Models

## Citation and provenance

**Authors:** Dirk Bergemann; Alessandro Bonatti; Alex Smolin

**Canonical citation:** Bergemann, Dirk, Alessandro Bonatti, and Alex Smolin. “Menu Pricing of Large Language Models.” Working paper, 2026.

**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/menu-pricing-of-large-language-models.md

**Source record:** https://arxiv.org/abs/2502.07736

**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.

Provenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt


Calibrated Mechanism Design

> Machine-readable author manuscript.
> Authors: Laura Doval; Alex Smolin.
> Canonical citation: Doval, Laura, and Alex Smolin. “Calibrated Mechanism Design.” TSE Working Paper 26-1718, 2026.
> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt


Calibrated Mechanism Design

# Calibrated Mechanism Design

**Authors:** Laura Doval; Alex Smolin

**Manuscript date:** 2026-02-18

#### Abstract

We study mechanism design when a designer repeatedly uses a fixed mechanism to interact with strategic agents who learn from observing their allocations. We introduce a static framework, calibrated mechanism design, requiring mechanisms to remain incentive compatible given the information they reveal about an underlying state through repeated use. In single-agent settings, we prove implementable outcomes correspond to two-stage mechanisms: the designer discloses information about the state, then commits to a state-independent allocation rule. This yields a tractable procedure to characterize calibrated mechanisms, combining information design and mechanism design. In private values environments, full transparency is optimal and correlationbased surplus extraction fails. We provide a microfoundation by showing calibrated mechanisms characterize exactly what is implementable when an infinitely patient agent repeatedly interacts with the same mechanism. Dynamic mechanisms that condition on histories expand implementable outcomes only by weakening incentive constraints, but not by enriching the designer's ability to obfuscate learning.

[^0]


Calibrated Mechanism Design

## 1 Introduction

Many economic institutions rely on mechanisms that remain fixed while agents interact with them repeatedly. Online platforms commit to stable auction formats for advertising slots, lenders use persistent scoring algorithms for loan decisions, and regulators establish durable rules for market participants. When the mechanism's operation depends on information known only to the designer-such as the platform's data about match values, the lender's assessment of credit market conditions, or the regulator's understanding of market fundamentals-participants may infer this information by observing their outcomes across repeated interactions. This learning creates a fundamental constraint: the information a mechanism reveals through repeated use limits what outcomes it can implement in the long run. Participants can use the information gleaned from past interactions when deciding whether and how to participate, tightening the designer's incentive constraints. A lender whose approval decisions depend on unobserved credit market conditions will gradually reveal these conditions to borrowers through his lending decisions, constraining the lender's ability to provide credit efficiently. We study how this endogenous information leakage shapes the set of implementable outcomes in mechanism design.

A simple example illustrates how learning prevents the designer from exploiting his information. Consider a seller who repeatedly offers a good whose demand depends on an unobserved state, which can be either low $(L)$ or high $(H)$. Each state is equally likely. The seller faces a buyer whose value for the good can take one of two values, 1/2 or 1. The probability that the buyer's value is 1 is higher when the demand state is high. Table 1 summarizes the value distribution conditional on the demand state:

|  | $v=1 / 2$ | $v=1$ |
| :--- | :--- | :--- |
| $L$ | 2/3 | 1/3 |
| $H$ | 1/3 | 2/3 |

Table 1: Value distribution conditional on demand state.

Suppose the seller can design the terms of trade, that is, the probability with which he allocates the good to the buyer $(q \in[0,1])$ and the payment the buyer makes to the seller $(t \in \mathbb{R})$. The buyer's payoff is $v q-t$, and the seller's is $t$. The buyer can always choose to not trade with the seller and ensure a payoff of 0.

Suppose first the buyer and the seller interact only once. Table 2 depicts an optimal mechanism for the seller in this case:

|  | $v=1 / 2$ | $v=1$ |
| :--- | :--- | :--- |
| $L$ | $(1,0)$ | $(1,0)$ |
| $H$ | (1,3/2) | (1,3/2) |

Table 2: Trade probabilities and payments as a function of buyer's value and demand state.

In this mechanism, the buyer gets the good for free when the demand state is $L$ and pays a price of 3/2 when it is $H$. If this mechanism were offered once without the buyer observing the demand state, the buyer obtains a payoff of 0 from participating and truthfully reporting her type. Unsurprisingly,
the seller extracts the buyer's surplus: the seller knows the demand state, which is correlated with the buyer's type, and exploits this information in the design of his mechanism (cf. Crémer and McLean, 1988).

Suppose now the buyer interacts repeatedly with the mechanism, but the state remains fixed. If the buyer observes nothing from her interaction with the mechanism, the buyer is willing to participate and truthfully report her value into the mechanism, no matter how many times it is offered: In each period, she anticipates getting a (continuation) payoff of 0 from engaging with the mechanism. Suppose, instead, the buyer observes her allocation in the mechanism. If the demand state is $L$, the buyer gets the good for free at the end of the first period, and from now on knows this is what she will get in the mechanism. If the demand state is $H$, the buyer gets the good and pays a price of 3/2 as she agreed to when she decided to participate in period 1, but anticipating a price of 3/2 from then onwards, never again participates in the mechanism. Thus, whereas the seller can implement the outcomes in Table 2 when the buyer does not observe her allocations, this is no longer the case when she can.

This paper develops a framework for mechanism design in which agents' ability to learn about the designer's information from repeatedly playing a mechanism constrains implementable outcomes. In our framework, allocations depend on agents' reports and on a state known only to the designer. Through repeated participation, agents observe their allocations and gradually learn about this state. A mechanism therefore serves a dual role: it determines allocations based on reports, and it acts as an information structure that reveals the underlying state. The more the mechanism conditions on the state, the more information it leaks, and the tighter the constraints on implementable outcomes.

We approach our analysis in two steps. First, we introduce a static solution concept for mechanism design that directly models the feedback between the mechanism, the information it reveals, and participants' behavior. This solution concept allows us to tractably capture the limits on the set of implementable outcomes implied by agents' learning, while abstracting from the dynamics of experimentation. Second, we provide a dynamic microfoundation showing this static solution concept precisely captures the implementable outcomes when an infinitely patient agent repeatedly interacts with the same mechanism.


Calibrated Mechanism Design

In Section 2, we introduce a static solution concept-calibrated mechanism design-requiring that mechanisms remain incentive compatible and individually rational given the information they reveal about the state through their allocations. We formalize this requirement through the notion of a calibrated mechanism. We couple each mechanism with an information structure that describes what participants learn about the state from the mechanism. The information structure reveals to each agent an interim allocation rule-the mapping from her type reports to lotteries over her allocations-capturing what she would learn from repeatedly observing her outcomes in the mechanism. We require the information structure to be calibrated in the sense of Foster and Vohra (1997): the interim allocation rule each agent observes must accurately describe the allocation probabilities she faces. Throughout the paper, we study calibrated mechanism design: the designer chooses a mechanism that remains incentive compatible and individually rational when participants have access to the mechanism's calibrated information structure before playing. Calibration imposes a constraint
on the designer relative to standard mechanism design: the more the allocation rule depends on the state, the more informative the calibrated information structure becomes, and hence the more incentive and participation constraints the designer must satisfy.

In private values environments, the constraint that the mechanism must remain incentive compatible and individually rational given the information it reveals about the state pushes the designer to full transparency. We show in Theorem 1 that, under the calibration constraint, the designer can do no better than inducing in each state the optimal direct mechanism when there is common knowledge of that state. In particular, in settings with transferable utility in which the designer has statistical information about the agents' types, Theorem 1 implies the designer cannot extract full surplus.

In Section 3, we characterize optimal calibrated mechanisms through a tractable class we dub twostage mechanisms. In a two-stage mechanism, the designer first discloses information about the state to the agent-inducing a belief about the state-then commits to an allocation rule that depends only on the agent's report, not the state itself. Theorem 2 shows that in single-agent settings, calibrated mechanisms and two-stage mechanisms implement exactly the same outcome distributions. This equivalence yields a practical algorithm for finding optimal calibrated mechanisms, combining tools from information design and mechanism design: for each possible belief the designer might induce, solve a standard mechanism design problem given that belief; then choose the optimal information disclosure by concavifying the resulting value function.

In the case of multiple agents, Proposition 1 shows calibrated mechanisms admit a similar representation via generalized two-stage mechanisms: Like two-stage mechanisms, the designer individually discloses to each agent a belief about the state and offers an incentive compatible and individually rational interim allocation rule that no longer conditions on the state. Whereas the designer observes the disclosed belief profile, each agent only observes the belief disclosed to her. ${ }^{1}$ Moreover, each agent learns only her own interim allocation rule-how her reports map to her allocations-rather than the complete mapping from type profiles to allocations. This partial observability requires additional consistency conditions to ensure agents' interim allocation rules are mutually compatible. In contrast to the single-agent case, not every generalized two-stage mechanism induces a calibrated mechanism, as generalized two-stage mechanisms may reveal strictly less information than calibrated mechanisms.

In Section 4, we study optimal calibrated mechanism design in the canonical setting of quasilinear utilities, single-dimensional types and allocations. In Section 4.1, we study the single-agent case. We show that if the order of types is state independent, then optimal two-stage mechanisms fully reveal the state, whereas this conclusion can be reversed when the order of types is state-dependent. In Section 4.2, we compare optimal calibrated mechanism design against the Myersonian benchmark. We provide sufficient conditions under which the designer realizes the payoff of the Myersonian benchmark under the calibration constraint; under these conditions, the optimal Myersonian mechanism satisfies the agent's incentive constraints state-by-state. Building on that result, we analyze multi-agent applications in Section 4.3.

[^1]Section 5 provides a microfoundation for calibrated mechanism design. We analyze an infinitehorizon game where an infinitely patient agent repeatedly plays the same mechanism. ${ }^{2}$ The state remains fixed, but the agent's type is redrawn each period independently of the state. ${ }^{3}$ Our notion of implementation is based on the long-run expected frequency of allocation-type-state tuples when the agent best responds to the mechanism. Theorem 3 shows that the implementable outcome distributions are precisely those induced by incentive compatible two-stage mechanisms. This result validates our static framework: calibrated mechanism design captures exactly what is implementable through repeated play.


Calibrated Mechanism Design

We then ask whether giving the designer additional flexibility helps. In a dynamic mechanism, the designer can condition each period's allocation on the complete history of past reports and allocations, rather than using the same mechanism repeatedly. Theorem 4 shows that dynamic mechanisms expand implementable outcomes in a specific way: they correspond to two-stage mechanisms with weaker incentive compatibility and individual rationality conditions. The designer can now exploit the ability to monitor the frequency of type reports over time, which allows him to punish detectable deviations-reporting strategies whose frequency distribution differs from the true type distribution. Instead, the mechanism must be robust to undetectable ones. Importantly, in environments with transferable utility, this distinction vanishes: As shown in Rahman (2024), eliminating profitable undetectable deviations is equivalent to incentive compatibility, so dynamic mechanisms implement exactly the same distributions over physical allocations, types, and states as our static calibrated mechanisms.

Related Literature The paper lies at the intersection of four literatures: rational expectations equilibria, (public) information disclosure in mechanism design, the computer science literature on learning in repeated auctions, and dynamic implementation.

The definition of a calibrated mechanism is in the spirit of rational expectations equilibria (Radner, 1979; Green, 1977; Kreps, 1977). Indeed, requiring a mechanism to remain incentive compatible given the information it reveals about the state mirrors the rational-expectations requirement that prices clear markets given the information they convey. Unlike rational expectations equilibrium, where the only role of prices is to clear the market, calibrated mechanisms are chosen by a designer who understands the incentive implications of the mechanism's information leakage and trades this off against the value of conditioning the mechanism on the state. Similar to our analysis in Section 5, some papers in the literature have studied the question of whether rational expectations equilibria emerge from learning dynamics (see, for instance, Milgrom, 1981; Blume et al., 1982).

Following Milgrom and Weber (1982), a literature has studied whether a designer should publicly disclose information he knows before a mechanism is played. Ottaviani and Prat (2001) show revealing a signal affiliated with the buyer's value is optimal in a single-agent screening problem. When considering the case of an informed principal, they consider what we call two-stage mechanisms to bound the monopolist's profits. Szabadi (2018) and Yamashita (2018) study the optimal release of

[^2]public information followed by an optimal mechanism conditional on that disclosure, while Fu et al. (2012) study this question in the context of a second price auction. In those papers, the restriction to public disclosure and the independence of the mechanism on information other than the disclosed one is a constraint on the class of mechanisms the designer can use. Instead, we show this class of mechanisms is without loss when the designer faces our calibration constraint in the single-agent case, but it may not be in the multi-agent case. Note, however, that when full or no disclosure are optimal in the Myersonian benchmark the distinction between private and public disclosure is immaterial. For that reason, the results on the achievability of the Myersonian benchmark are similar across their and our work. Daskalakis et al. (2016) lift the restriction to public disclosure and study the Myersonian benchmark in an auction setting, showing that the complexity of that problem is the same as that of a multi-product monopolist (cf. Guesnerie and Laffont, 1984). ${ }^{4}$

Motivated by the prevalence of fixed auction formats with which bidders interact repeatedly, a literature in computer science studies the properties of bidder learning algorithms and the implications for the auctioneer (see, for instance, Golrezaei et al., 2019; Nedelec et al., 2019; Kanoria and Nazerzadeh, 2020, and Nedelec et al., 2022 for a survey treatment). A common finding is that learning bidders can take advantage of "naive" auction formats which are no longer incentive compatible when bidders learn. Inspired by this literature, we develop a framework which allows us to systematically study the question of optimal mechanism design in the presence of learning agents.

Our dynamic implementation results relate to the literature that studies whether a mechanism can be implemented either by linking decisions (Jackson and Sonnenschein, 2007; Ball and Kattwinkel, 2023) or in the patient limit of a repeated interaction (Renou and Tomala, 2015; Margaria and Smolin, 2018; Meng, 2021). Both strands identify cyclical monotonicity as the condition for implementation (cf. Rochet, 1987). Rahman (2024) shows that cyclical monotonicity is equivalent to the absence of profitable undetectable deviations.

By focusing on what agents learn from the designer's information, our paper is distinct from the literature on mechanism design with interdependent payoffs which focuses on agents' learning about others' types through their actions in the mechanism (Green and Laffont, 1987; Niemeyer, 2022; Häfner et al., 2025). Moreover, by focusing in the case of a designer with commitment, we are distinct from the literature on the informed principal (Myerson, 1983; Maskin and Tirole, 1990).

Lastly, our paper contributes to two literatures. First, by studying the informational role of the mechanism, we contribute to the literature on feedback in auctions, which analyzes how different feedback rules affect bidders' information about other agents, and ultimately behavior in first price auctions (see, for instance, Esponda, 2008; Bergemann and Hörner, 2018; Cesa-Bianchi et al., 2024). Second, by showing the designer's problem involves solving information and mechanism design problems, our paper joins a recent literature that highlights the dual role of the mechanism as an information structure and an allocation rule (Calzolari and Pavan, 2006; Dworczak, 2020; Doval and Skreta, 2022).

[^3]


Calibrated Mechanism Design

## 2 Calibrated Mechanism Design

In this section, we introduce the static setting and solution concept that captures the impact of agents' learning from the mechanism on the set of implementable outcomes. We defer to Section 5 the analysis of the dynamic game whose outcomes our static solution concept captures.

Primitives A designer (he) interacts with $N$ privately informed agents (she) to determine an allocation. Let $\Theta_{i}$ denote the set of types of agent $i$, and $\Theta \equiv \times_{i=1}^{N} \Theta_{i}$. Each agent knows her type, but not those of other agents. The allocation space is given by $A \equiv \times_{i=1}^{N} A_{i} .{ }^{5}$ Finally, let $\Omega$ denote a set of states, which are known to the designer, but not to the agents. The sets $\Theta_{i}, A_{i}$, and $\Omega$ are assumed to be finite throughout. ${ }^{6}$ Agent $i$ 's payoffs are given by $u_{i}: A_{i} \times \Theta_{i} \times \Omega \rightarrow \mathbb{R}$. That is, agent $i$ cares about her dimension of the allocation, her type, and the state, and not about other agents' allocations or types.

Denote by $\mu_{0}$ the distribution over $\Omega$. For each $\omega \in \Omega$, let $f(\cdot \mid \omega) \in \Delta(\Theta)$ denote the type distribution. We assume throughout the types are independently distributed conditional on the state, that is,

$$
f(\theta \mid \omega)=\prod_{i=1}^{N} f_{i}\left(\theta_{i} \mid \omega\right),
$$

for all $\theta \in \Theta$ and $\omega \in \Omega$. Together with the assumption on agents' payoffs, the assumption on $f(\cdot \mid \omega)$ allows us to isolate the effect of learning about the state from that of learning about others' types (perhaps because others' types provide additional information about the state).

Mechanisms We model mechanisms as mappings

$$
\phi: \Theta \times \Omega \times[0,1] \rightarrow \Delta(A),
$$

where $\varepsilon \in[0,1]$ is a uniformly distributed random variable, which we refer to as the randomization device.

Several comments are in order. First, to understand how a mechanism works, the timing of when the different random variables is drawn is important. In particular, we assume that both the state $\omega$ and the realization of the randomization device $\varepsilon$ are independently drawn at the beginning, but not observed by the agents. This determines the direct mechanism $\phi(\cdot, \omega, \varepsilon): \Theta \rightarrow \Delta(A)$ to which the agents send type reports, which in turn determines the lottery from which the allocation is drawn. Thus, the allocation is random in our setting for two reasons: on the one hand, the agents do not know the realization of $(\omega, \varepsilon)$, and hence the direct mechanism $\phi(\cdot, \omega, \varepsilon)$ they face. Second, conditional on $(\omega, \varepsilon)$, the allocation may be drawn at random. Mathematically, we could have subsumed all sources of randomness in the allocation into the randomization device. However, as we explain next, the definition in Equation 2 allows us to distinguish the source of randomness in the allocation that is informative about the state from that which is not.

[^4]Second, it is useful to consider the reason for the randomization device in the definition of a mechanism. For simplicity, consider the case of the designer facing a single agent. If the agent had repeated access to the mechanism, the agent would stand to learn the mapping $\phi(\cdot, \omega, \varepsilon): \Theta \rightarrow \Delta(A)$ by experimenting with different reports into the mechanism and observing the resulting allocations. ${ }^{7}$ Without the randomization device, the agent would stand to learn a partition of the set of states, where states in the same cell of the partition induce the same direct mechanism $\phi(\cdot, \omega, \varepsilon)$. By allowing the designer to rely on the randomization device, we allow him to obfuscate the agent's learning beyond a simple partitional structure. Contrast this with the Myersonian benchmark in which without loss of generality the designer would offer mechanisms that do not rely on such devices, that is, $\phi_{\mathrm{My}}: \Theta \times \Omega \rightarrow \Delta(A)$. Indeed, the Myersonian designer is not concerned with the agents' learning: without loss of generality, he does not disclose anything about the state to the agents, so that the question of how to optimally release information about the state is moot.

Lastly, note that we assume the mechanism asks the agents for type reports. In Appendix D, we show that the revelation principle holds in the setting of this section: it is without loss of generality to focus on direct and incentive compatible mechanisms that induce full participation.

Calibrated information structures We now describe how a mechanism induces an information structure, which we define using the language in Green and Stokey (2022) and Gentzkow and Kamenica (2017). An information structure is a mapping ${ }^{8}$

$$
\pi: \Omega \times[0,1] \rightarrow S_{1}^{*} \times \cdots \times S_{N}^{*}
$$

where $\varepsilon \in[0,1]$ is a uniformly distributed random variable-in fact, it is the same as in the definition of a mechanism-and

$$
S_{i}^{*}=\Delta\left(A_{i}\right)^{\Theta_{i}}
$$

is the set of agent $i$ 's interim allocation rules. ${ }^{9}$ We choose this language for the information structure to capture the idea that if agent $i$ plays the mechanism repeatedly, she stands to learn how her reports influence her allocation probabilities, i.e., her interim allocation rule. The interim allocation rule, in turn, depends on the mechanism and the strategies of others. Below, we require the interim allocation rule is well-calibrated with the mechanism and others' strategies:

Definition 1 (Calibrated information structures). We say that the information structure is calibrated to mechanism $\phi$ if for all $(\omega, \varepsilon) \in \Omega \times[0,1]$ such that $\pi(\omega, \varepsilon)=\left(s_{1}^{*}, \ldots, s_{N}^{*}\right)$ we have that for all $i \in\{1, \ldots, N\}$,