Robust Trust

## A. 6 Proof of Proposition 1

By Lemma 1 and Corollary 2, the choice of an optimal strategy for the agent reduces to optimization over the extreme points $\underline{\mu}, \bar{\mu}$ of the trust interval with the corresponding payoff:

$$
\begin{aligned}
& V(\underline{\mu}, \bar{\mu}) \triangleq \\
& \alpha\left(\int_{0}^{\underline{\mu}}\left(U(\underline{\mu})+U^{\prime}(\underline{\mu})(\mu-\underline{\mu})\right) \tau(\mu) d \mu+\int_{\underline{\mu}}^{\bar{\mu}} U(\mu) \tau(\mu) d \mu+\int_{\bar{\mu}}^{1}\left(U(\bar{\mu})+U^{\prime}(\bar{\mu})(\mu-\bar{\mu})\right) \tau(\mu) d \mu\right) \\
& \quad+(1-\alpha)\left(\int_{0}^{b(\underline{\mu}, \bar{\mu})}\left(U(\bar{\mu})+U^{\prime}(\bar{\mu})(\mu-\bar{\mu})\right) \tau(\mu) d \mu+\int_{b(\underline{\mu}, \bar{\mu})}^{1}\left(U(\underline{\mu})+U^{\prime}(\underline{\mu})(\mu-\underline{\mu})\right) \tau(\mu) d \mu\right)
\end{aligned}
$$

The function $V(\underline{\mu}, \bar{\mu})$ is continuously differentiable with partial derivatives (whenever

$$
\underline{\mu}<\bar{\mu}):
$$

$$
\begin{aligned}
& \frac{\partial V}{\partial \underline{\mu}}=U^{\prime \prime}(\underline{\mu})\left(\alpha \int_{0}^{\underline{\mu}}(\mu-\underline{\mu}) \tau(\mu) d \mu+(1-\alpha) \int_{b(\underline{\mu}, \bar{\mu})}^{1}(\mu-\underline{\mu}) \tau(\mu) d \mu\right) \\
& \frac{\partial V}{\partial \bar{\mu}}=U^{\prime \prime}(\bar{\mu})\left(\alpha \int_{\bar{\mu}}^{1}(\mu-\bar{\mu}) \tau(\mu) d \mu+(1-\alpha) \int_{0}^{b(\underline{\mu}, \bar{\mu})}(\mu-\bar{\mu}) \tau(\mu) d \mu\right)
\end{aligned}
$$

Intuitively, the first-order impact of a change in the trust boundary equals the change in the action played at that boundary, measured by $U^{\prime \prime}(\cdot)$, integrated over the belief regions in which the aligned and misaligned advisers induce that action, weighted by the alignment parameter. (Terms involving $\partial b / \partial \underline{\mu}$ and $\partial b / \partial \bar{\mu}$ vanish because at $\mu=b(\underline{\mu}, \bar{\mu})$ the misaligned adviser is indifferent between the two messages.)

Whenever the trust region is non-singleton, $\underline{\mu}<\bar{\mu}$, at the optimal choice of $\underline{\mu}$ and $\bar{\mu}$ these partial derivatives must equal zero, $\partial V / \partial \underline{\mu}=0$ and $\partial V / \partial \bar{\mu}=0$. Since $U^{\prime \prime}(\cdot)>0$, these first-order conditions can be rearranged as follows. Define functions $\Psi_{1}$ and $\Psi_{2}$ as

$$
\begin{aligned}
& \Psi_{1}(\underline{\mu}, \bar{\mu}, \alpha) \triangleq \alpha \int_{0}^{\underline{\mu}}(\mu-\underline{\mu}) \tau(\mu) d \mu+(1-\alpha) \int_{b(\underline{\mu}, \bar{\mu})}^{1}(\mu-\underline{\mu}) \tau(\mu) d \mu \\
& \Psi_{2}(\underline{\mu}, \bar{\mu}, \alpha) \triangleq \alpha \int_{\bar{\mu}}^{1}(\mu-\bar{\mu}) \tau(\mu) d \mu+(1-\alpha) \int_{0}^{b(\underline{\mu}, \bar{\mu})}(\mu-\bar{\mu}) \tau(\mu) d \mu
\end{aligned}
$$

Then, $\Psi_{1}(\underline{\mu}, \bar{\mu}, \alpha)=\Psi_{2}(\underline{\mu}, \bar{\mu}, \alpha)=0$ is equivalent to conditions (6) and (7).
First, we show that conditions (6) and (7) are incompatible with $\alpha<1 / 2$. (If $M$ were finite, this would follow directly from Theorem 3.) Indeed, if those conditions hold then (for the rest of the proof, we will often omit the arguments of the function $b$ for brevity):

$$
\begin{aligned}
\alpha\left(\int_{0}^{b}(b-\mu) \tau(\mu) d \mu+\int_{b}^{1}(\mu-b) \tau(\mu) d \mu\right) & \geq \alpha\left(\int_{0}^{\underline{\mu}}(\underline{\mu}-\mu) \tau(\mu) d \mu+\int_{\bar{\mu}}^{1}(\mu-\bar{\mu}) \tau(\mu) d \mu\right) \\
& =(1-\alpha)\left(\int_{0}^{b}(\bar{\mu}-\mu) \tau(\mu) d \mu+\int_{b}^{1}(\mu-\underline{\mu}) \tau(\mu) d \mu\right) \\
& \geq(1-\alpha)\left(\int_{0}^{b}(b-\mu) \tau(\mu) d \mu+\int_{b}^{1}(\mu-b) \tau(\mu) d \mu\right)
\end{aligned}
$$

where the inequalities hold because $\underline{\mu} \leq b(\underline{\mu}, \bar{\mu}) \leq \bar{\mu}$ and the equality is a consequence of (6) and (7). Because $\tau$ has full support, the multipliers on both sides of the inequality are
strictly positive, and thus $\alpha \geq 1-\alpha$, i.e., $\alpha \geq 1 / 2$.
Now we argue that for $\alpha \geq 1 / 2$ the solution to (6) and (7) such that $\underline{\mu} \leq \bar{\mu}$ exists. Note that at $\alpha=1 / 2,[\underline{\mu}, \bar{\mu}]=\left[\mu_{0}, \mu_{0}\right]$ is a solution. For the rest of this proof, we omit the dependence of $\Psi_{i}$ on $\alpha$. By Lemma 1 and direct inspection, $b(\underline{\mu}, \bar{\mu})$ is strictly and continuously increasing in its arguments, so $\Psi_{1}(\underline{\mu}, \bar{\mu})$ is strictly and continuously decreasing in $\underline{\mu}$ for each $\bar{\mu}$. Furthermore,

$$
\begin{aligned}
& \Psi_{1}(0, \bar{\mu})=(1-\alpha) \int_{b(0, \bar{\mu})}^{1} \mu \tau(\mu) d \mu \geq 0 \\
& \Psi_{1}(1, \bar{\mu})=\alpha \int_{0}^{1}(\mu-1) \tau(\mu) d \mu<0
\end{aligned}
$$

Therefore, for each $\bar{\mu}$, a best-response $b_{1}(\bar{\mu})$ such that $\Psi_{1}\left(b_{1}(\bar{\mu}), \bar{\mu}\right)=0$ exists and is unique. Since $\Psi_{1}(\underline{\mu}, \bar{\mu})$ strictly decreases in $\bar{\mu}, b_{1}(\bar{\mu})$ strictly decreases in $\bar{\mu}$. Finally, for any $\bar{\mu}$,

$$
\int_{b_{1}(\bar{\mu})}^{1}\left(\mu-b_{1}(\bar{\mu})\right) \tau(\mu) d \mu \geq \int_{b\left(b_{1}(\bar{\mu}), \bar{\mu}\right)}^{1}\left(\mu-b_{1}(\bar{\mu})\right) \tau(\mu) d \mu \geq \int_{0}^{b_{1}(\bar{\mu})}\left(b_{1}(\bar{\mu})-\mu\right) \tau(\mu) d \mu
$$

where the second inequality holds because $\alpha \geq 1 / 2$ and $\Psi_{1}\left(b_{1}(\bar{\mu}), \bar{\mu}\right)=0$. Thus, for any $\bar{\mu}$, $b_{1}(\bar{\mu}) \leq \mu_{0}$.

Analogously, for each $\underline{\mu}$, a best-response $b_{2}(\underline{\mu})$ such that $\Psi_{2}\left(\underline{\mu}, b_{2}\right)=0$, exists, is unique, strictly decreases in $\underline{\mu}$, and is everywhere greater than $\mu_{0}$.


Robust Trust

Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.

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

Therefore, a solution to (6) and (7) is any $\underline{\mu} \in\left[0, \mu_{0}\right]$ and $\bar{\mu}=b_{2}(\underline{\mu}) \in\left[\mu_{0}, 1\right]$ such that $b_{1}\left(b_{2}(\underline{\mu})\right)=\underline{\mu}$. By the established properties of $b_{1}$ and $b_{2}, b_{1}\left(b_{2}(\underline{\mu})\right)$ is continuous in $\underline{\mu}$ with $b_{1}\left(b_{2}(\underline{\mu})\right) \in\left[0, \mu_{0}\right]$ for all $\underline{\mu} \in\left[0, \mu_{0}\right]$; hence, $b_{1}\left(b_{2}(0)\right)-0 \geq 0$ and $b_{1}\left(b_{2}\left(\mu_{0}\right)\right)-\mu_{0} \leq 0$. By the intermediate value theorem, there exists $\underline{\mu} \in\left[0, \mu_{0}\right]$ such that $b_{1}\left(b_{2}(\underline{\mu})\right)=\underline{\mu}$.

So far, we showed that for $\alpha \geq 1 / 2$, a solution exists and belongs to a closed rectangular set $D=\left\{(\underline{\mu}, \bar{\mu}): \underline{\mu} \in\left[0, \mu_{0}\right], \bar{\mu} \in\left[\mu_{0}, 1\right]\right\}$. To establish uniqueness, consider the function $-\Psi=\left(-\Psi_{1},-\Psi_{2}\right)$ on $D$. Any solution must satisfy $-\Psi(\underline{\mu}, \bar{\mu})=(0,0)$. Observe that for any
$(\underline{\mu}, \bar{\mu}) \in D$,

$$
\begin{aligned}
& \frac{\partial\left[-\Psi_{1}\right]}{\partial \underline{\mu}}=\alpha \int_{0}^{\underline{\mu}} \tau(\mu) d \mu+(1-\alpha) \frac{\partial b}{\partial \underline{\mu}} \tau(b)(b-\underline{\mu})+(1-\alpha) \int_{b}^{1} \tau(\mu) d \mu>0, \\
& \frac{\partial\left[-\Psi_{1}\right]}{\partial \bar{\mu}}=(1-\alpha) \frac{\partial b}{\partial \bar{\mu}} \tau(b)(b-\underline{\mu}) \geq 0, \\
& \frac{\partial\left[-\Psi_{2}\right]}{\partial \underline{\mu}}=-(1-\alpha) \frac{\partial b}{\partial \underline{\mu}} \tau(b)(b-\bar{\mu}) \geq 0, \\
& \frac{\partial\left[-\Psi_{2}\right]}{\partial \bar{\mu}}=\alpha \int_{\bar{\mu}}^{1} \tau(\mu) d \mu+(1-\alpha) \frac{\partial b}{\partial \bar{\mu}} \tau(b)(\bar{\mu}-b)+(1-\alpha) \int_{0}^{b} \tau(\mu) d \mu>0 .
\end{aligned}
$$

Moreover, for all $(\underline{\mu}, \bar{\mu}) \in D$, the Jacobian of $[-\Psi]$ is a P-matrix, i.e., it has strictly positive principal minors:

$$
\frac{\partial\left[-\Psi_{1}\right]}{\partial \underline{\mu}}>0, \quad \frac{\partial\left[-\Psi_{1}\right]}{\partial \underline{\mu}} \frac{\partial\left[-\Psi_{2}\right]}{\partial \bar{\mu}}-\frac{\partial\left[-\Psi_{1}\right]}{\partial \bar{\mu}} \frac{\partial\left[-\Psi_{2}\right]}{\partial \underline{\mu}}>0 .
$$

By the Gale-Nikaido Theorem (Gale and Nikaido (1965), Theorem 4), it follows that $[-\Psi]$ is injective on $D$, and thus there exists at most one solution to the equation $-\Psi(\underline{\mu}, \bar{\mu})=(0,0)$.

Finally, we show that the proposed trust region strategy is robustly rationalizable by explicitly constructing a TRE. For $\alpha \geq 1 / 2$, we need to construct a measurable strategy of the misaligned adviser $\beta:[0, b] \rightarrow[\bar{\mu}, 1]$ such that for every set $X \subseteq[\bar{\mu}, 1]$ with $\alpha \tau(X)+(1-$ $\alpha) \tau\left(\beta^{-1}(X)\right)>0$,

$$
\frac{\alpha \int_{X} \mu \tau(\mu) d \mu+(1-\alpha) \int_{\beta^{-1}(X)} \mu \tau(\mu) d \mu}{\alpha \int_{X} \tau(\mu) d \mu+(1-\alpha) \int_{\beta^{-1}(X)} \tau(\mu) d \mu}=\bar{\mu}
$$

(The construction of $\beta:(b, 1] \rightarrow[0, \underline{\mu}]$ is analogous.) To this end, define two finite atomless nonnegative measures:

$$
\begin{aligned}
& \nu(Y) \triangleq(1-\alpha) \int_{Y}(\bar{\mu}-\mu) \tau(\mu) d \mu, \quad Y \subseteq[0, b] \\
& \eta(X) \triangleq \alpha \int_{X}(\mu-\bar{\mu}) \tau(\mu) d \mu, \quad X \subseteq[\bar{\mu}, 1]
\end{aligned}
$$

Observe that condition (7) is precisely $\eta([\bar{\mu}, 1])=\nu([0, b])$ whereas condition (27) is the
pushforward identity:

$$
\eta(X)=\nu\left(\beta^{-1}(X)\right), \quad X \subseteq[\bar{\mu}, 1] .
$$

In other words, we need to find $\beta$ that transports $\nu$ to $\eta$. It is always possible. For a canonical quantile construction, define the cumulative mass functions $F_{\nu}(\mu) \triangleq \nu([0, \mu])$ for $\mu \in[0, b]$ and $F_{\eta}(\mu) \triangleq \eta([\bar{\mu}, \mu])$ for $\mu \in[\bar{\mu}, 1]$. The transport map can then be set:

$$
\beta(\mu)=F_{\eta}^{-1}\left(F_{\nu}(\mu)\right), \quad \mu \in[0, b],
$$

where $F_{\eta}^{-1}(\cdot)$ is the generalized inverse: $F_{\eta}^{-1}(q)=\inf \left\{\mu \in[\bar{\mu}, 1]: F_{\eta}(\mu) \geq q\right\}$.
For $\alpha<1 / 2, T=\left\{\mu_{0}\right\}$, so the misaligned adviser is indifferent between all messages and it suffices to construct a strategy $\beta:[0,1] \rightarrow[0,1]$ such that for all $X \subseteq[0,1]$ with $\alpha \int_{X} \tau(\mu) d \mu+(1-\alpha) \int_{\beta^{-1}(X)} \tau(\mu) d \mu>0$,

$$
\frac{\alpha \int_{X} \mu \tau(\mu) d \mu+(1-\alpha) \int_{\beta^{-1}(X)} \mu \tau(\mu) d \mu}{\alpha \int_{X} \tau(\mu) d \mu+(1-\alpha) \int_{\beta^{-1}(X)} \tau(\mu) d \mu}=\mu_{0}
$$

which is equivalent to:

$$
\alpha \int_{X}\left(\mu-\mu_{0}\right) \tau(\mu) d \mu+(1-\alpha) \int_{\beta^{-1}(X)}\left(\mu-\mu_{0}\right) \tau(\mu) d \mu=0
$$

To do that, observe that $\int_{0}^{1}\left(\mu-\mu_{0}\right) \tau(\mu) d \mu=0$ and $\int_{0}^{\mu_{0}}\left(\mu_{0}-\mu\right) \tau(\mu) d \mu=\int_{\mu_{0}}^{1}(\mu-$ $\left.\mu_{0}\right) \tau(\mu) d \mu>0$. Since $\alpha \in(0,1 / 2), \alpha /(1-\alpha) \in(0,1)$ and by the intermediate value theorem, there exist $\mu_{L} \in\left(0, \mu_{0}\right)$ and $\mu_{H} \in\left(\mu_{0}, 1\right)$ such that

$$
\begin{aligned}
& \int_{0}^{\mu_{L}}\left(\mu_{0}-\mu\right) \tau(\mu) d \mu=\frac{\alpha}{1-\alpha} \int_{0}^{\mu_{0}}\left(\mu_{0}-\mu\right) \tau(\mu) d \mu \\
& \int_{\mu_{H}}^{1}\left(\mu-\mu_{0}\right) \tau(\mu) d \mu=\frac{\alpha}{1-\alpha} \int_{\mu_{0}}^{1}\left(\mu-\mu_{0}\right) \tau(\mu) d \mu
\end{aligned}
$$

By construction,

$$
\begin{aligned}
& \int_{0}^{\mu_{L}}\left(\mu_{0}-\mu\right) \tau(\mu) d \mu=\frac{\alpha}{1-\alpha} \int_{\mu_{0}}^{1}\left(\mu-\mu_{0}\right) \tau(\mu) d \mu \\
& \int_{\mu_{H}}^{1}\left(\mu-\mu_{0}\right) \tau(\mu) d \mu=\frac{\alpha}{1-\alpha} \int_{0}^{\mu_{0}}\left(\mu_{0}-\mu\right) \tau(\mu) d \mu \\
& \int_{\mu_{L}}^{\mu_{H}}\left(\mu-\mu_{0}\right) \tau(\mu) d \mu=0
\end{aligned}
$$


Robust Trust

We can set $\beta(\mu)=\beta_{L}(\mu)$ when $\mu \in\left[0, \mu_{L}\right], \beta(\mu)=\mu_{0}$, when $\mu \in\left(\mu_{L}, \mu_{H}\right)$, and $\beta(\mu)=\beta_{H}(\mu)$, when $\mu \in\left[\mu_{H}, 1\right]$. Here, $\beta_{L}$ is a quantile transport map that transports measure $\nu_{L}(Y)=(1-\alpha) \int_{Y}\left(\mu_{0}-\mu\right) \tau(\mu) d \mu$ on $\left[0, \mu_{L}\right]$ to measure $\eta_{L}(X)=\alpha \int_{X}\left(\mu-\mu_{0}\right) \tau(\mu) d \mu$ on [ $\mu_{0}, 1$ ], just like in the case of $\alpha \geq 1 / 2$; it ensures that (28) holds for all $X \subseteq\left(\mu_{0}, 1\right]$. Similarly, $\beta_{H}$ is a quantile transport map that transports measure $\nu_{H}(Y)=(1-\alpha) \int_{Y}\left(\mu-\mu_{0}\right) \tau(\mu) d \mu$ on $\left[\mu_{H}, 1\right]$ to measure $\eta_{H}(X)=\alpha \int_{X}\left(\mu_{0}-\mu\right) \tau(\mu) d \mu$ on $\left[0, \mu_{0}\right]$; it ensures that (28) holds for all $X \subseteq\left[0, \mu_{0}\right)$. (The transported masses match the targets by equations (29) and (30).) Finally, by equation (31), condition (28) holds for $\mu=\mu_{0}$. The result follows.


Robust Trust

## A. 7 Proof of Proposition 2

At $\alpha=1 / 2,[\underline{\mu}, \bar{\mu}]=\left[\mu_{0}, \mu_{0}\right]$ satisfies conditions (6) and (7). At $\alpha=1,[\underline{\mu}, \bar{\mu}]=[0,1]$ satisfies conditions (6) and (7).

For $\alpha \in(1 / 2,1)$, denote by $\Psi_{i 1}, \Psi_{i 2}$, and $\Psi_{i \alpha}$ the partial derivatives of $\Psi_{i}$ with respect to $\underline{\mu}, \bar{\mu}$, and $\alpha$ respectively. Define the Jacobian:

$$
J(\underline{\mu}, \bar{\mu}, \alpha) \triangleq\left(\begin{array}{ll}
\Psi_{11} & \Psi_{12} \\
\Psi_{21} & \Psi_{22}
\end{array}\right)
$$

As we argued in the proof of Proposition 1, for all $\underline{\mu}, \bar{\mu}, \alpha>1 / 2, \operatorname{det} J(\underline{\mu}, \bar{\mu}, \alpha)>0$, and therefore, by the implicit function theorem, optimal $\underline{\mu}(\alpha)$ and $\bar{\mu}(\alpha)$ are continuously
differentiable and ${ }^{21}$

$$
\binom{d \underline{\mu} / d \alpha}{d \bar{\mu} / d \alpha}=-J(\underline{\mu}, \bar{\mu}, \alpha)^{-1}\binom{\Psi_{1 \alpha}}{\Psi_{2 \alpha}} .
$$

Consequently,

$$
\begin{aligned}
& \frac{d \underline{\mu}}{d \alpha}=-\frac{\Psi_{2 \bar{\mu}} \Psi_{1 \alpha}-\Psi_{1 \bar{\mu}} \Psi_{2 \alpha}}{\Psi_{1 \underline{\mu}} \Psi_{2 \bar{\mu}}-\Psi_{1 \bar{\mu}} \Psi_{2 \underline{\mu}}}<0, \\
& \frac{d \bar{\mu}}{d \alpha}=\frac{\Psi_{2 \underline{\mu}} \Psi_{1 \alpha}-\Psi_{1 \underline{\mu}} \Psi_{2 \alpha}}{\Psi_{1 \underline{\mu}} \Psi_{2 \bar{\mu}}-\Psi_{1 \bar{\mu}} \Psi_{2 \underline{\mu}}}>0,
\end{aligned}
$$

where the inequalities hold because, as we already showed, $\Psi_{1 \underline{\mu}}<0, \Psi_{1 \bar{\mu}} \leq 0, \Psi_{2 \underline{\mu}} \leq 0$, $\Psi_{2 \bar{\mu}}<0$, and

$$
\begin{aligned}
& \Psi_{1 \alpha}=\int_{0}^{\underline{\mu}}(\mu-\underline{\mu}) \tau(\mu) d \mu-\int_{b}^{1}(\mu-\underline{\mu}) \tau(\mu) d \mu<0 \\
& \Psi_{2 \alpha}=\int_{\bar{\mu}}^{1}(\mu-\bar{\mu}) \tau(\mu) d \mu-\int_{0}^{b}(\mu-\bar{\mu}) \tau(\mu) d \mu>0
\end{aligned}
$$

The result follows.


Robust Trust

## A. 8 Proof of Proposition 3

Throughout, fix $\alpha>1 / 2$; the proposition is trivially true otherwise. We begin with a simple lemma.

Lemma 7. Let $U_{1}, U_{2}$ be twice differentiable and strictly convex on $[0,1]$. Assume that the ratio $U_{1}^{\prime \prime}(\mu) / U_{2}^{\prime \prime}(\mu)$ is decreasing. Then, with $b_{i}$ defined for each $U_{i}, i \in\{1,2\}$, by equation (5), we have that $b_{1}(\underline{\mu}, \bar{\mu}) \leq b_{2}(\underline{\mu}, \bar{\mu})$ for all $\underline{\mu} \leq \bar{\mu}$.

Proof. By equation (5), for $i \in\{1,2\}, b_{i}(\underline{\mu}, \bar{\mu})=\mathbb{E}_{\mu \sim f_{i}}[\mu]$, where $f_{i}(\mu)$ is a density of a

[^17]probability measure on $[\underline{\mu}, \bar{\mu}]$ defined as
$$
f_{i}(\mu) \triangleq \frac{U_{i}^{\prime \prime}(\mu)}{\int_{\underline{\mu}}^{\bar{\mu}} U_{i}^{\prime \prime}(\mu) d \mu} .
$$
By assumption, $f_{1}(\mu) / f_{2}(\mu)$ is decreasing, and hence $f_{2}$ likelihood-ratio dominates $f_{1}$. This implies that the probability distribution $f_{1}$ is first-order stochastically dominated by $f_{2}$; in particular, it has a lower mean. The result follows. $\square$

We now take the second step by showing a monotone relationship between the cutoff function $b$ and the trust region.

Lemma 8. Consider two decision problems $U_{i}, i \in\{1,2\}$, and let $\left(\underline{\mu}_{i}, \bar{\mu}_{i}\right) \in D$ denote the unique solution to the system

$$
\Psi_{1}^{i}(\underline{\mu}, \bar{\mu}, \alpha)=0, \quad \Psi_{2}^{i}(\underline{\mu}, \bar{\mu}, \alpha)=0,
$$

where $\Psi_{1}^{i}, \Psi_{2}^{i}$ and $D$ are defined as in the proof of Proposition 1 for each $U_{i}$. If $b_{1}(\underline{\mu}, \bar{\mu}) \leq$ $b_{2}(\underline{\mu}, \bar{\mu})$ for all $\underline{\mu} \leq \bar{\mu}$, then $\underline{\mu}_{2} \leq \underline{\mu}_{1}$ and $\bar{\mu}_{2} \leq \bar{\mu}_{1}$.

Proof. For any $h \in[0,1]$, define the auxiliary functions

$$
\begin{aligned}
& \widetilde{\Psi}_{1}(\underline{\mu}, h) \triangleq \alpha \int_{0}^{\underline{\mu}}(\mu-\underline{\mu}) \tau(\mu) d \mu+(1-\alpha) \int_{h}^{1}(\mu-\underline{\mu}) \tau(\mu) d \mu \\
& \widetilde{\Psi}_{2}(\bar{\mu}, h) \triangleq \alpha \int_{\bar{\mu}}^{1}(\mu-\bar{\mu}) \tau(\mu) d \mu+(1-\alpha) \int_{0}^{h}(\mu-\bar{\mu}) \tau(\mu) d \mu
\end{aligned}
$$

For each $h$, let $\underline{\mu}(h)$ be the unique solution to $\widetilde{\Psi}_{1}(\underline{\mu}, h)=0$ in $\left[0, \mu_{0}\right]$, and let $\bar{\mu}(h)$ be the unique solution to $\widetilde{\Psi}_{2}(\bar{\mu}, h)=0$ in $\left[\mu_{0}, 1\right]$, with the existence and uniqueness following from an argument analogous to that used in the proof of Proposition 1.

For each $i \in\{1,2\}$ define the scalar map

$$
\varphi_{i}(h) \triangleq b_{i}(\underline{\mu}(h), \bar{\mu}(h)) .
$$

Let $h_{i} \triangleq b_{i}\left(\underline{\mu}_{i}, \bar{\mu}_{i}\right)$ be the cutoff evaluated at the optimal endpoints of problem $i$. Then $\left(\underline{\mu}_{i}, \bar{\mu}_{i}\right)$ solves $\Psi_{1}^{i}=\Psi_{2}^{i}=0$ if and only if $\underline{\mu}_{i}=\underline{\mu}\left(h_{i}\right), \bar{\mu}_{i}=\bar{\mu}\left(h_{i}\right)$, and $h_{i}=\varphi_{i}\left(h_{i}\right)$. By the assumption
$b_{1} \leq b_{2}$ pointwise, for every $h$,

$$
\varphi_{1}(h)=b_{1}(\underline{\mu}(h), \bar{\mu}(h)) \leq b_{2}(\underline{\mu}(h), \bar{\mu}(h))=\varphi_{2}(h) .
$$

Define a region $H \triangleq\{h \in[0,1]: \underline{\mu}(h) \leq h \leq \bar{\mu}(h)\}$. Note that $H$ is an interval and $h_{1}, h_{2} \in H$. For $h \in H$, implicit differentiation yields

$$
\underline{\mu}^{\prime}(h)=-\frac{\partial \widetilde{\Psi}_{1} / \partial h}{\partial \widetilde{\Psi}_{1} / \partial \underline{\mu}} \leq 0, \quad \bar{\mu}^{\prime}(h)=-\frac{\partial \widetilde{\Psi}_{2} / \partial h}{\partial \widetilde{\Psi}_{2} / \partial \bar{\mu}} \leq 0 .
$$

Thus, $\varphi_{i}$ is weakly decreasing in $h$ on $H$ : both $\underline{\mu}(h)$ and $\bar{\mu}(h)$ are weakly decreasing in $h$, while $b_{i}(\underline{\mu}, \bar{\mu})$ is weakly increasing in each endpoint, so the composition $h \mapsto \varphi_{i}(h)$ is weakly decreasing.

It follows that $h_{2} \geq h_{1}$ : if $h_{2}<h_{1}$, then, since $\varphi_{2}$ is decreasing on $H$,

$$
h_{2}=\varphi_{2}\left(h_{2}\right) \geq \varphi_{2}\left(h_{1}\right) \geq \varphi_{1}\left(h_{1}\right)=h_{1},
$$

which is a contradiction. Since $\underline{\mu}(\cdot)$ and $\bar{\mu}(\cdot)$ are weakly decreasing,

$$
\underline{\mu}_{2}=\underline{\mu}\left(h_{2}\right) \leq \underline{\mu}\left(h_{1}\right)=\underline{\mu}_{1}, \quad \bar{\mu}_{2}=\bar{\mu}\left(h_{2}\right) \leq \bar{\mu}\left(h_{1}\right)=\bar{\mu}_{1},
$$

completing the proof. $\square$

By Lemma 7, $U_{1}^{\prime \prime} / U_{2}^{\prime \prime}$ decreasing implies $b_{1}(\underline{\mu}, \bar{\mu}) \leq b_{2}(\underline{\mu}, \bar{\mu})$ for all $\underline{\mu} \leq \bar{\mu}$. Lemma 8 then yields $\underline{\mu}_{2} \leq \underline{\mu}_{1}$ and $\bar{\mu}_{2} \leq \bar{\mu}_{1}$. This proves Proposition 3.


Robust Trust

## A. 9 Proof of Proposition 4

With a small abuse of notation, we can parameterize each private strategy by $\hat{\sigma}=\operatorname{Pr}\left(a=a_{2}\right)$. We also drop the dependence of $G$ and $L$ on $\hat{\tau}$ in the notation. Then, by the arguments behind Theorem 1, if the agent employs the set of private strategies $\hat{\Sigma}_{0}=\{\hat{\sigma}(m)\}_{m \in \Delta(\Omega)}$, the payoffs coming from both aligned and misaligned adviser depend only on $\hat{\sigma}_{L} \triangleq \inf \hat{\Sigma}_{0}$ and $\hat{\sigma}_{H} \triangleq \sup \hat{\Sigma}_{0}$, and the optimal payoffs from using $\hat{\Sigma}_{0}$ are the same as if the agent plays
$\hat{\sigma}(m)=\hat{\sigma}_{L}$ when $v(m)<0$ and $\hat{\sigma}(m)=\hat{\sigma}_{H}$ when $v(m) \geq 0$. This payoff is:

$$
\begin{aligned}
& \int_{-\infty}^{0}\left(\alpha \hat{\sigma}_{L}+(1-\alpha) \hat{\sigma}_{H}\right) v \hat{\tau}(d v)+\int_{0}^{+\infty}\left(\alpha \hat{\sigma}_{H}+(1-\alpha) \hat{\sigma}_{L}\right) v \hat{\tau}(d v) \\
& =\hat{\sigma}_{L}((1-\alpha) G-\alpha L)+\hat{\sigma}_{H}(\alpha G-(1-\alpha) L)
\end{aligned}
$$

The optimal choice of $\hat{\sigma}_{L}$ and $\hat{\sigma}_{H}$ must maximize (32) subject to $\hat{\sigma}_{L}, \hat{\sigma}_{H} \in[0,1]$ and $\hat{\sigma}_{L} \leq \hat{\sigma}_{H}$. This is a linear optimization subject to $\left(\hat{\sigma}_{L}, \hat{\sigma}_{H}\right)$ being in a triangle with vertices (0, 0), (0, 1), and $(1,1)$. A straightforward calculation gives the following solution:

If $G=L:$ if $\alpha<\hat{\alpha} \triangleq 1 / 2$, then any $\hat{\sigma}_{L}=\hat{\sigma}_{H}$ is optimal; if $\alpha>\hat{\alpha}$, then $\hat{\sigma}_{L}=0$ and $\hat{\sigma}_{H}=1$; if $\alpha=\hat{\alpha}$, then any $\left(\hat{\sigma}_{L}, \hat{\sigma}_{H}\right)$ is optimal. If $G>L$ : if $\alpha<\hat{\alpha}$, then $\hat{\sigma}_{L}=\hat{\sigma}_{H}=1$; if $\alpha>\hat{\alpha}$, then $\hat{\sigma}_{L}=0$ and $\hat{\sigma}_{H}=1$; if $\alpha=\hat{\alpha}$, then $\hat{\sigma}_{H}=1$ and any $\hat{\sigma}_{L}$ is optimal. If $G<L$ : if $\alpha<\hat{\alpha}$, then $\hat{\sigma}_{L}=\hat{\sigma}_{H}=0$; if $\alpha>\hat{\alpha}$, then $\hat{\sigma}_{L}=0$ and $\hat{\sigma}_{H}=1$; if $\alpha=\hat{\alpha}$, then $\hat{\sigma}_{L}=0$ and any $\hat{\sigma}_{H}$ is optimal.

The cases $\hat{\sigma}_{L}=\hat{\sigma}_{H}$ correspond to not trusting any message and always acting in the same way, optimal at the prior belief, so $T=\left\{\mu_{0}\right\}$. The cases $\hat{\sigma}_{L}=0$ and $\hat{\sigma}_{H}=1$ correspond to trusting all messages, so $T=\Delta(\Omega)$. Since we assumed that the probability of $v(\mu)=0$ is 0 and $G \neq L$, the corresponding optimal strategy is uniquely determined.

It is left to show that those strategies are robustly rationalizable. Define $M_{0}=\{\mu:$ $v(\mu)=0\}, M_{-}=\{\mu: v(\mu)<0\}$, and $M_{+}=\{\mu: v(\mu)>0\}$. Define probability measures $q_{+}(X)=\int_{X} v(\mu) \tau(d \mu) / G$ for $X \subseteq M_{+}, q_{-}(Y)=\int_{Y}(-v(\mu)) \tau(d \mu) / L$ for $Y \subseteq M_{-}$.

For $\alpha>\hat{\alpha}$, the agent fully trusts the adviser. Consider the following strategy of the misaligned adviser. If $\mu \in M_{0}$, then $\beta(\mu)=\mu$. If $\mu \in M_{-}$, then $\beta$ randomizes over messages $m \in M_{+}$according to $q_{+}$. If $\mu \in M_{+}$, then $\beta$ randomizes over messages $m \in M_{-}$according to $q_{-}$. This strategy is clearly adversarial. Furthermore, since $\alpha>\hat{\alpha}$, after any message $m \in M_{+}$, the posterior expected payoff from action $a_{2}$ is strictly positive: for any $X \subseteq M_{+}$ with $\tau(X)>0$,

$$
\alpha \int_{X} v(m) \tau(d m)+(1-\alpha) \int_{\Delta(\Omega)} v(\mu) \beta(X \mid \mu) \tau(d \mu)=\alpha G q_{+}(X)-(1-\alpha) L q_{+}(X)>0 .
$$

Analogously, after any message $m \in M_{-}$, the posterior expected payoff from action $a_{2}$ is
strictly negative: for any $Y \subseteq M_{-}$with $\tau(Y)>0$,

$$
\alpha \int_{Y} v(m) \tau(d m)+(1-\alpha) \int_{\Delta(\Omega)} v(\mu) \beta(Y \mid \mu) \tau(d \mu)=-\alpha L q_{-}(Y)+(1-\alpha) G q_{-}(Y)<0 .
$$

And by construction, after any message $m \in M_{0}$, the posterior expected payoff from action $a_{2}$ is zero.

For $\alpha<\hat{\alpha}$, the agent doesn't trust the adviser so any adviser's strategy is adversarial. Consider the case $G>L$; the complementary case is analogous. We need to construct the misaligned adviser strategy that makes communication not valuable. To do so, define $\gamma=\alpha L /((1-\alpha) G) \in[0,1]$. Consider the following strategy of the misaligned adviser. If $\mu \in M_{-}$, then $\beta$ randomizes over messages $m \in M_{+}$according to $q_{+}$. If $\mu \in M_{+}$, then with probability $\gamma, \beta$ randomizes over messages $m \in M_{-}$according to $q_{-}$and with probability $(1-\gamma), \beta$ randomizes over messages $m \in M_{+}$according to $q_{+}$. This strategy makes the posterior expected payoff from action $a_{2}$ zero after every message $m \in M_{-}$: for any $Y \subseteq M_{-}$ with $\tau(Y)>0$,

$$
\alpha \int_{Y} v(m) \tau(d m)+(1-\alpha) \int_{\Delta(\Omega)} v(\mu) \beta(Y \mid \mu) \tau(d \mu)=-\alpha L q_{-}(Y)+(1-\alpha) \gamma G q_{-}(Y)=0 .
$$

After any message $m \in M_{+}$, the posterior expected payoff from action $a_{2}$ is strictly positive: for any $X \subseteq M_{+}$with $\tau(X)>0$,

$$
\alpha \int_{X} v(m) \tau(d m)+(1-\alpha) \int_{\Delta(\Omega)} v(\mu) \beta(X \mid \mu) \tau(d \mu)=(G-L) q_{+}(X)>0 .
$$

Thus, this $\beta$ robustly rationalizes the strategy.
Finally, at $\alpha=\hat{\alpha}$, both full trust and no trust are optimal, along a continuum of other strategies, and robustly rationalizable by the same strategies $\beta$ as above.


Robust Trust

## A. 10 Supporting calculations for Example 1

The key step in the construction of the optimal trust region in Example 1 comes from the following simple lemma.

Lemma 9 (Spherical U). Let $U(\mu)=\tilde{U}(\|\mu-b\|)$ for some vector $b$ and function $\tilde{U}$. Let $\mu^{\prime}(r, n)=b+r n$, where $n \in \mathbb{R}^{N}$ with $\|n\|=1$ and $r \in \mathbb{R}$. Then, (i) $D_{U}\left(\mu, \mu^{\prime}(r, n)\right)$ is strictly increasing in $n \cdot(b-\mu)$ whenever $r \neq 0$, and (ii) $D_{U}\left(\mu, \mu^{\prime}(r, n)\right)$ is a unimodal function in $r$ with a minimum at $r=n \cdot(\mu-b)$.

Proof. In this case, $\nabla U\left(\mu^{\prime}(r, n)\right)=\tilde{U}^{\prime}(r) n$, and thus

$$
D_{U}\left(\mu, \mu^{\prime}(r, n)\right)=\tilde{U}(\|\mu-b\|)-\tilde{U}(r)-\tilde{U}^{\prime}(r)(n \cdot(\mu-b)-r) .
$$

Since $U(\mu)$ is strictly convex, $\tilde{U}^{\prime}(r)>0$ whenever $r \neq 0$ and $\tilde{U}^{\prime \prime}(r)>0$. Thus $D_{U}\left(\mu, \mu^{\prime}(r, n)\right)$ is strictly increasing in $n \cdot(b-\mu)$. Furthermore,

$$
\frac{\partial D_{U}\left(\mu, \mu^{\prime}(r, n)\right)}{\partial r}=-\tilde{U}^{\prime}(r)-\tilde{U}^{\prime \prime}(r)(n \cdot(\mu-b)-r)+\tilde{U}^{\prime}(r)=\tilde{U}^{\prime \prime}(r)(r-n \cdot(\mu-b)) .
$$

Because $\tilde{U}^{\prime \prime}(r)>0$, the unimodality follows. $\square$

Suppose that the misaligned adviser holds belief $\mu$. By Corollary 3, the adviser will try to maximize $D_{U}\left(\mu, \mu^{\prime}\right)$ over $\mu^{\prime}$ on the boundary of the trust region. Parameterizing $\mu^{\prime}=b+r n$ and applying Lemma 9 yields that $n$ is optimally chosen to be $(b-\mu) /\|b-\mu\|$, whereas $r$ is chosen to be maximal within the trust region, $r=r^{*}(\alpha)$. Thus, the optimal $\mu^{\prime}$ is given uniquely by $b+r^{*}(\alpha)(b-\mu) /\|b-\mu\|$.

[^0]:    *Dworczak: Department of Economics, Northwestern University and Group for Research in Applied Economics, piotr.dworczak@northwestern.edu. Smolin: Toulouse School of Economics, alexey.v.smolin@gmail.com. We thank Nageeb Ali, Ricardo Alonso, Ben Brooks, Laura Doval, Tan Gan, Alexis Ghersengorin, Marina Halac, Jason Hartline, Nicole Immorlica, Emir Kamenica, David Levine, Annie Liang, Stephen Morris, Jacopo Perego, Balázs Szentes, and Mark Whitmeyer for helpful conversations. 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 the AI Interdisciplinary Institute ANITI (grant ANR-23-IACL-0002). Part of the analysis in this paper was conducted while Alex Smolin was visiting Northwestern University and Columbia Business School, and we thank both institutions for their hospitality.

[^1]:    ${ }^{1}$ Strictly speaking, we assume that with probability $\alpha$ the Sender reveals his signal truthfully, but this can be shown to be optimal for the Receiver.

[^2]:    ${ }^{2}$ See also Levy and Szentes (2025) for a related model of AI under imperfect recall.

[^3]:    ${ }^{3}$ We allow $S$ to be infinite, which is useful for constructing tractable examples. Whenever we work with an infinite space, we endow it with the Borel $\sigma$-algebra, and require all sets and functions that we define to be measurable; statements involving "for all" should be interpreted as "for almost all" with respect to the underlying distributions.

[^4]:    ${ }^{4}$ It can be shown that the assumption of truthful reporting of the belief is equivalent (in terms of equilibrium payoff consequences) to assuming that the aligned adviser is attempting to maximize the agent's expected payoff. However, the assumption of truthful reporting is natural for an aligned AI system and useful, as it provides a natural meaning to each message (see Sobel (2020)).
    ${ }^{5}$ Formally, $\mathbb{E}_{\beta, \sigma}[u(a, \omega, \theta)]=\sum_{\omega \in \Omega} \mu_{0}(\omega) \int_{S} \int_{\Delta(\Omega)} \int_{\Theta} \int_{A} u(a, \omega, \theta) \sigma(d a \mid m, \theta) f(d \theta \mid \omega) \beta(d m \mid s) \pi(d s \mid \omega)$.

[^5]:    ${ }^{6}$ Formally, $\mathbb{E}_{\omega \sim \mu, \hat{\sigma}}[u(a, \omega, \theta)]=\sum_{\omega \in \Omega} \mu(\omega) \int_{\Theta} \int_{A} u(a, \omega, \theta) \hat{\sigma}(d a \mid \theta) f(d \theta \mid \omega)$.

[^6]:    ${ }^{7}$ Formally, to define the gradient, we extend the function $U$ beyond the probability simplex by assuming that, for any non-negative measure $\mu, U(\mu)=\mu(\Omega) U(\mu / \mu(\Omega))$.
    ${ }^{8}$ Point $m^{\prime} \in T$ is visible from $m$ if the line segment connecting $m^{\prime}$ and $m$ does not intersect $T \backslash\left\{m^{\prime}\right\}$.

[^7]:    ${ }^{9}$ However, it is not without loss of generality to assume that $T \subseteq M$.

[^8]:    ${ }^{10}$ Without loss of generality, we assume that $\beta$ uses only messages in $M$; any message $m \notin M$ cannot be sent by an aligned adviser and hence reveals that the adviser is misaligned.
    ${ }^{11}$ The assumption of finite $M$ and $\Theta$ is made for technical reasons; verifying the assumptions of Sion (1958)'s minimax theorem (in particular, its continuity requirements) is difficult for a cheap-talk-like game with infinite-dimensional strategy spaces since the impact of messages on payoffs is endogenous.

[^9]:    ${ }^{12}$ The strictly convex indirect payoff function can be a result of the agent having a continuum of actions or, as we show in Appendix A.4, finitely many actions and a continuum of private types.

[^10]:    ${ }^{13}$ One way to see that is to use our observation that in a TRE, the mapping from messages to belief defined by Bayes' rule must agree with the mapping defined by minimizing the Bregman distance to the trust region.

[^11]:    ${ }^{14}$ A recent systematic review of real-world AI deployment in medical imaging finds that AI most commonly serves as a secondary reader or a triage tool rather than as a fully autonomous reader; see Wenderott et al. (2024). For a regulatory example, the FDA-cleared DrAid chest X-ray system is indicated as a triage and prioritization aid and is not intended for stand-alone clinical decision-making; see FDA 510(k) summary K221241, released in 2024.
    ${ }^{15}$ See Dratsch et al. (2023). An RSNA summary of the study reports that incorrect AI suggestions reduced reader accuracy even for experienced radiologists.


Robust Trust

[^12]:    ${ }^{16}$ This conclusion is driven by the coarseness of the strategy space; we know from Theorem 3 that, for any $\alpha>0$, the agent would optimally use a non-trivial trust region if the state space and her action space were sufficiently rich.

[^13]:    ${ }^{17}$ To see why, note that any trust region $T$ can be convexified by replacing it with the intersection of sets $T_{\mu}$ over all $\mu \in \operatorname{supp}(\tau)$, where $T_{\mu}$ is the (convex) set of all points that are not further away from $\mu$ than any point in $T$.

[^14]:    ${ }^{18}$ It is equivalent to $\sigma^{*}$ because it is weakly better than $\sigma^{*}$ and $\sigma^{*}$ was optimal.

[^15]:    ${ }^{19}$ A matrix of adviser's posteriors can be computed by Bayes' rule as $(\mu(s))_{s \in S}=$ $\left(\operatorname{diag}\left(\mu_{0}(\omega)\right)_{\omega \in \Omega}\right) \Pi\left(\operatorname{diag}(\tau(s))_{s \in S}\right)^{-1}$. The diagonal matrices are invertible and the multiplication by an invertible matrix preserves the rank.

[^16]:    ${ }^{20}$ By the Auerbach theorem, there exist $v_{1}, \ldots, v_{R-1} \in \mathbb{W}$ and $\phi_{1}, \ldots, \phi_{R-1} \in \mathbb{W}^{*}$ such that $\left\|v_{i}\right\|_{1}=1$, $\left\|\phi_{i}\right\|_{\mathbb{W}^{*}}=1$, and $\phi_{i}\left(v_{j}\right)=\delta_{i j}$ (Section II.E, Lemma 11 in Wojtaszczyk (1991); see also Gershkov et al. (2025) for another recent application). By Hahn-Banach theorem, these $\phi_{i}$, operating on $\mathbb{W}$, can be extended to $\tilde{\phi}_{i}$, operating on $\mathbb{R}^{K}$, without a change in their norm. By the duality between spaces $l_{1}$ and $l_{\infty}$, for each $i$, there exists $x_{i} \in \mathbb{R}^{K}$ such that $\tilde{\phi}_{i}(z)=z^{\top} x_{i}$ and $\left\|x_{i}\right\|_{\infty}=\left\|\tilde{\phi}_{i}\right\|=1$. Then, for $w \in \mathbb{W}$, $w_{i}^{\top} x_{j}=\tilde{\phi}_{j}\left(w_{i}\right)=\phi_{j}\left(w_{i}\right)=\delta_{i j}$.

[^17]:    ${ }^{21}$ Differentiating the optimality conditions with respect to $\alpha$ we obtain $\Psi_{11} \frac{d \underline{\underline{\mu}}}{d \alpha}+\Psi_{12} \frac{d \bar{\mu}}{d \alpha}+\Psi_{1 \alpha}=0, \Psi_{21} \frac{d \underline{\underline{\mu}}}{d \alpha}+$ $\Psi_{22} \frac{d \bar{\mu}}{d \alpha}+\Psi_{2 \alpha}=0$.


Robust Trust

## Citation and provenance

**Authors:** Piotr Dworczak; Alex Smolin

**Canonical citation:** Dworczak, Piotr, and Alex Smolin. “Robust Trust.” Working paper, 2026.

**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/robust-trust.md

**Source record:** https://arxiv.org/abs/2602.09490

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

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


Menu Pricing of Large Language Models

> Machine-readable author manuscript.
> 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.
> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt


Menu Pricing of Large Language Models

# Menu Pricing of Large Language Models

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

**Manuscript date:** 2026-03-10

#### Abstract

We develop a framework for the optimal pricing and product design of LLMs in which a provider sells menus of token budgets to users who differ in their valuations across a continuum of tasks. Under a homogeneous production technology, we show that users' high-dimensional type profiles are summarized by a scalar index, reducing the seller's problem to one-dimensional screening. The optimal mechanism takes the form of committed-spend contracts: buyers pay for a budget that they allocate across token classes priced at marginal cost. We extend the analysis to environments with multiple differentiated models and to competition between a proprietary leader and an open-source fringe, showing that competitive pressure reshapes both the intensive and extensive margins of compute provision. Each element of our theory (token-budget menus, maximum- and minimum-spend plans, multi-model versioning, and linear API pricing) has a direct counterpart in the observed pricing practices of providers such as Anthropic, OpenAI, and GitHub.

Keywords: Large Language Models, Optimal Pricing, Menu Pricing, Fine-Tuning. JEL Codes: D47, D82, D83.

[^0]


Menu Pricing of Large Language Models

## 1 Introduction

Access to large language models is fast becoming a major input to economic activity. Businesses use LLMs for coding, customer support, legal research, and content generation; individual users rely on them for writing, analysis, and decision-making. The leading providers, Anthropic, OpenAI, and Google, collectively serve hundreds of millions of users and generate billions of dollars in annualized revenue. Yet the pricing of these services remains strikingly ad hoc: subscription tiers, token-based metering, credit systems, and volume commitments coexist with no clear theoretical foundation. Understanding how a profit-maximizing provider should price LLM access, and quantifying the distortions that arise when it does, is the goal of this paper. ${ }^{1}$

Pricing access to LLMs is fundamentally a multidimensional screening problem. An LLM user faces a continuum of tasks, each with a different value. The provider offers token budgets-bundles of input, output, and fine-tuning tokens-that the user allocates across tasks at her discretion. The provider can meter total token usage but cannot observe or contract on the user's task-by-task allocation, creating a combined adverse-selection and moral-hazard problem. User types are infinite-dimensional (a function from tasks to values), the allocation space is high-dimensional (tokens of multiple classes across many tasks), and the user's hidden action (token allocation across tasks) further compounds the difficulty. A priori, this problem seems intractable.

Our central result is that it is not. Under a homogeneous production technology-meaning that the gain function is homogeneous, so the optimal composition of inference tokens is scale-invariant-each user's entire type profile collapses to a scalar index, the $a g$ gregate type. Users with the same aggregate type make the same total token demands, receive the same total surplus, and choose the same menu item, regardless of the fine structure of their task valuations. This aggregation property reduces the provider's problem to one-dimensional screening à la Mussa and Rosen (1978), despite the underlying complexity of the environment. Building on this reduction, we develop the analysis in four steps.

First, we characterize the efficient allocation (Section 3). Efficiency requires that all tasks employ inference token classes in common proportions; only the scale of token usage varies with a task's marginal value. When the provider faces capacity constraints in each token class, the efficient allocation can be implemented through linear prices equal to inflated shadow costs (Corollary 1), a result that rationalizes the linear, pay-per-token pricing universally observed in developer-facing API markets.

[^1]Second, we characterize the optimal mechanism for a single-model monopolist (Section 4). By the aggregation result, the buyer's indirect utility from a token-budget bundle takes a multiplicative form: aggregate type times aggregate quality. The optimal menu therefore excludes low types and distorts quality downward for (almost) all others, exactly as in the standard one-dimensional framework. Crucially, the optimal mechanism admits intuitive indirect implementations (Proposition 4): it can be realized as a maximum-spend mechanism (a budget of credits allocated across tokens priced at marginal cost), as a minimum-spend mechanism (volume commitments that unlock lower per-token prices), or as a menu of twopart tariffs. These implementations are not merely theoretical curiosities; they correspond precisely to the pricing structures observed at leading platforms.

Third, we extend the framework to multiple differentiated models (Section 5). When models share a common returns-to-scale parameter in inference but differ in returns to finetuning, the buyer's payoff has a constant elasticity of substitution over model qualities. Cost minimization implies that each type uses a single model for all tasks (Proposition 6). This generates a natural versioning structure: higher-tier plans grant access to more capable models, not merely larger usage allocations. Anthropic's pricing illustrates the single-model benchmark of Section 4: all paid tiers access the same models, with differentiation occurring through compute budgets. OpenAI, by contrast, screens jointly on usage and model access, reserving its most compute-intensive reasoning model for the highest tier, consistent with the multi-model menu of Section 5.

Fourth, we study competition between a proprietary leader and an open-source fringe that sells tokens at marginal cost (Section 5). The leader designs an optimal menu subject to the buyer's ability to supplement usage from the fringe. Three regions emerge: low types purchase exclusively from the fringe; intermediate types buy from the leader at quantities that exactly deter fringe top-up; and high types are served as under pure monopoly, with standard downward distortions (Proposition 7). Competition thus reshapes both the intensive margin (how much compute is sold to each buyer) and the extensive margin (which buyers adopt the proprietary model at all).

We conclude by connecting each theoretical construct to observed pricing (Section 6). Consumer subscriptions at Anthropic and OpenAI implement the nonlinear menus of Sections 4 and 5. Model aggregators (e.g., Quora's Poe and GitHub Copilot) implement the committed-spend mechanisms of Section 4.2, differing in whether they enforce a hard budget cap (maximum spend) or allow overages at linear prices (minimum spend). API pricing is linear in tokens with no volume discounts, consistent with providers prioritizing adoption over rent extraction. Thus, pricing practices observed in the industry suggest that these mechanisms capture fundamental economic forces rather than incidental design choices.


Menu Pricing of Large Language Models

Related Literature Our paper contributes to the rapidly growing literature on the economics of AI. Existing work has examined the impact of LLMs on price competition (Fish, Gonczarowski, and Shorrer, 2024), token auctions (Duetting, Mirrokni, Paes Leme, Xu, and Zuo, 2024), and sponsored search (Bergemann, Bojko, Dütting, Paes Leme, Xu, and Zuo, 2024), and Demirer, Fradkin, Tadelis, and Peng (2025) document pricing and market-share patterns. By contrast, our focus is on the provider's own design problem, how to price and version LLM access, which has received surprisingly little theoretical attention. For example, Mahmood (2024) studies competition among producers of generative AI who can specialize in different tasks, but restricts attention to linear token pricing. We go further and characterize the fully optimal nonlinear mechanism, leveraging the sufficient-statistic reduction generated by the homogeneity of the gain function, which renders the joint screening-and-moral-hazard problem solvable and yields sharp, implementable predictions.

At their core, LLMs are an information technology, connecting our work to the literature on selling information (Babaioff, Kleinberg, and Paes Leme, 2012; Bergemann, Bonatti, and Smolin, 2018; Yang, 2022). However, LLMs possess distinctive features: they are generalpurpose technologies deployed across many tasks, usage is metered in tokens, and precision is improved by combining inference and fine-tuning, which give rise to a structurally different design problem.

Methodologically, our aggregation result relates to the literature on "1.5-dimensional" mechanism design (Fiat, Goldner, Karlin, and Koutsoupias, 2016; Devanur, Goldner, Saxena, Schvartzman, and Weinberg, 2020). Unlike most of that literature, however, we must address potential incentive clashes across the continuum of subproblems that arise from the buyer's hidden allocation of tokens, requiring an additional argument based on the production technology. Even in simpler settings, such as one-dimensional screening with moral hazard (Castro-Pires, Chade, and Swinkels, 2024) or multidimensional screening and bundling without moral hazard (Armstrong, 1996; Rochet and Stole, 2003; Daskalakis, Deckelbaum, and Tzamos, 2017), tractability issues arise. Our environment combines both, and our solution exhibits a constrained efficiency property as in Laffont and Tirole (1990) and Doligalski, Dworczak, Krysta, and Tokarski (2025).


Menu Pricing of Large Language Models

## 2 Model


Menu Pricing of Large Language Models

### 2.1 Task Environment

A buyer of LLM services faces a unit measure of tasks indexed by $i \in[0,1]$. In our baseline setting, an LLM provider offers one type of model only. The buyer can decide on (i) how
many inference tokens to use for each task $i, x_{i} \in \mathbb{R}_{+}^{J}$ and (ii) how many fine-tuning tokens to use to improve the model's performance (e.g., precision) on all tasks, $z \in \mathbb{R}_{+}^{K}$. In language models, a token is a unit of text, typically a word.

The model's performance on task $i$ is given by the gain function $g\left(x_{i}, z\right)$. The function $g$ is common across tasks and takes the following form:

$$
g\left(x_{i}, z\right)=\Psi\left(x_{i}\right) \Phi(z) .
$$

Thus, the gain function is multiplicatively separable in inference and fine-tuning tokens.
Diminishing returns to inference tokens on a single task are a fundamental feature of LLM performance, consistent with empirically observed scaling laws. Thus the function $\Psi$ is positive, increasing, strictly concave, differentiable, Inada at 0, ${ }^{2}$ and homogeneous of degree $\sigma \in(0,1)$, i.e., $\Psi\left(r x_{i}\right)=r^{\sigma} \Psi\left(x_{i}\right)$ for all $r>0$. This function captures the returns to inference tokens. A canonical example of such a function is a CES function, $\Psi\left(x_{i}\right)=\left(\sum_{j=1}^{J} \alpha_{j} x_{i j}^{\rho}\right)^{\sigma / \rho}$, where $\alpha_{j}>0, \sum_{j} \alpha_{j}=1, \rho<1$. The function $\Phi(z)$ is positive, increasing, such that $g$ is strictly concave (e.g., $\Phi(z)^{1 /(1-\sigma)}$ is strictly concave) and $(1-\sigma)$-Inada at 0. ${ }^{3}$ This function captures the returns to fine-tuning.

These properties are similar in shape to the scaling laws for training LLMs. Concavity captures the fundamental assumption that time and computing resources spent on a single task exhibit diminishing marginal returns. Homogeneity is the key tractability assumption. It ensures that the optimal mix of token classes is the same for every task; only the scale of token usage varies. This property drives the aggregation result in Section 3, which reduces the seller's infinite-dimensional screening problem to a one-dimensional problem. The parameter $\sigma$ captures the returns to scale.

The buyer's marginal value of performance on each task is captured by $w=\left(w_{i}\right)_{i \in[0,1]}$,

$$
w:[0,1] \rightarrow[0,1],
$$

which we refer to as the buyer's type. Using a $z$-fine-tuned model with a profile of $\left(x_{i}\right)_{i \in[0,1]}$ inference tokens delivers the following total payoff for buyer type $w$ :

$$
\int_{0}^{1} w_{i} g\left(x_{i}, z\right) d i .
$$

The buyer's type is distributed according to a commonly known distribution $F_{w}$. The buyer

[^2]knows his type, while the provider does not.
The provider bears the cost of processing tokens. We assume that the marginal processing costs are constant but can vary across different token classes. The cost of inference tokens is $c_{j}>0, j \in[J]$, and the cost of fine-tuning tokens is $\hat{c}_{k}>0, k \in[K]$. We let $c \in \mathbb{R}_{++}^{J}$ and $\hat{c} \in \mathbb{R}_{++}^{K}$ denote the corresponding cost profiles.

Notation Throughout the text, we use the following notation. " $\triangleq$ " indicates a definition. $[K] \triangleq\{1, \ldots, K\} . \mathbb{R}_{+}^{K} \triangleq\left\{x \in \mathbb{R}^{K}: x_{k} \geq 0, k \in[K]\right\}$ and $\mathbb{R}_{++}^{K} \triangleq\left\{x \in \mathbb{R}^{K}: x_{k}>\right.$ $0, k \in[K]\}$. All stand-alone qualifiers such as "positive," "increasing," and "concave" are understood in the weak sense, that is, as "non-negative," "non-decreasing," and "weakly concave," respectively. For constants, subscripts refer to variables or tasks; for functions, they indicate partial derivatives. All proofs are in Appendix A.


Menu Pricing of Large Language Models

### 2.2 Mechanism Design

The provider contracts on the total number of tokens of each class used by the buyer. In other words, the provider sells budgets of various kinds of tokens. A mechanism consists of an arbitrary menu of the form

$$
\left\{\left(X_{1}(w), \ldots, X_{J}(w), Z_{1}(w), \ldots, Z_{K}(w), t(w)\right)\right\}_{w},
$$

where $X_{j}$ is the total number of inference tokens of class $j$. Upon purchase, the buyer can freely distribute those tokens across tasks. (We relax this restriction in Appendix B, where we allow the provider to contract on the usage of all tokens by the buyer.) $Z_{k}$ is the number of fine-tuning tokens of class $k$, which is used for fine-tuning and cannot be distributed.

The seller's problem is an optimal mechanism design problem with infinite-dimensional private information and moral hazard. However, we show that the homogeneity of the gain function renders the problem tractable.


Menu Pricing of Large Language Models

### 2.3 Mapping the Setting to Practice

Before turning to the analysis, we describe how the primitives of our setting relate to the design and operation of contemporary large language models (LLMs).

Fine-tuning tokens, $\boldsymbol{z} \in \mathbb{R}_{+}^{\boldsymbol{K}}$ The case $z=0$ corresponds to a baseline model that has not been fine-tuned. ${ }^{4}$ In this case, the model can be interpreted as a foundation model. Its

[^3]quality depends on its size (i.e., the number of parameters) ${ }^{5}$, its architecture, the quantity and quality of training data, and the details of the training procedure. Scaling up model size typically improves capability and accuracy, but state-of-the-art performance generally requires commensurate scaling of training data as well (Kaplan et al., 2020). In our setting, we take pretraining as given. Accordingly, we focus on inference-time compute rather than training-time compute.

When $z>0$, this variable captures the number of tokens used to fine-tune the foundation model. Fine-tuning resembles pretraining in that it updates the model's parameters, without changing its size or architecture, but it is much smaller in scale. It is performed on a dataset directly relevant to a class of tasks (e.g., labeled X-ray scans, code examples, or conversational transcripts) and can be interpreted as injecting task-specific knowledge and behavior into the model.

Inference tokens, $\boldsymbol{x}_{\boldsymbol{i}} \in \mathbb{R}_{+}^{\boldsymbol{J}}$ These are the tokens used directly to process a given task. The two standard classes of inference tokens are input and output tokens. Input tokens are those provided by the user. Increasing the number of input tokens can improve predictive quality for two reasons. First, richer prompts provide more context, enabling more tailored and appropriate responses. Second, additional input may supply the data on which the model is meant to operate. For example, retrieval-augmented generation (RAG) uses the prompt to retrieve relevant passages from an external corpus and appends them to the prompt.

Output tokens are those generated by the model. A larger number of output tokens can also improve quality. First, it allows the model to provide more detailed and nuanced answers. Second, it provides opportunities for multi-step reasoning, often described as chainof-thought (CoT) computation. This additional reasoning channel can enable the model to solve more complex tasks, at the cost of generating many intermediate tokens, often hidden from the user. This inference-time reasoning is qualitatively distinct from pretraining, remains only partially understood, and is an active frontier for improving LLM performance. ${ }^{6}$

Gain function, $\boldsymbol{g}$ Tokens of different classes enter the large language model as distinct inputs. They are neither perfect substitutes nor perfect complements, but each can contribute to higher predictive quality. This motivates our gain-function formulation in (1), which is also in line with the empirically observed scaling laws for inference-time computation (e.g., Wu, Sun, Li, Welleck, and Yang, 2025).

[^4]Furthermore, one can envision other contractible token categories, either by refining the distinctions above (e.g., separating reasoning tokens from response tokens) or by introducing new modalities and resources (e.g., media files, external databases, etc.). Accordingly, our model accommodates an arbitrary number of inference and fine-tuning token classes.

Costs, $\boldsymbol{c}_{\boldsymbol{j}}, \hat{\boldsymbol{c}}_{\boldsymbol{k}}$ Processing tokens is costly because it requires executing the neural network and, for fine-tuning, computing parameter updates, which in turn requires energy and specialized hardware. Because tokens within a given class are processed symmetrically from a computational perspective, we assume that the marginal cost per token within a class is constant across tasks. ${ }^{7}$ However, because different token classes correspond to different computational routines, we allow marginal costs to differ across classes.


Menu Pricing of Large Language Models

## 3 Efficient Solution

In this section, we begin by analyzing the socially efficient allocation of tokens to buyers and tasks. This solution provides a useful benchmark and also coincides with the optimal monopoly solution when the buyer's type is known. We then turn to the problem of a social planner who faces capacity constraints for each token class, and we use the constrainedefficient solution to characterize the buyer's utility from purchasing a fixed token budget.

Unconstrained problem Given a type $w=\left(w_{i}\right)_{i \in[0,1]}$, the efficient allocation solves the following problem:

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

Consider the allocation of inference tokens first. A key implication of the homogeneity of the production function $\Psi$ is that inference tokens are optimally employed in the same proportions: the efficient allocation of tokens across tasks differs solely by the scale at which these inputs are employed. To see this, consider the system of $J$ first-order conditions for each $x_{i}=\left(x_{i 1}, \ldots, x_{i J}\right)$ :

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

By equation (6), 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

[^5]space of tokens. Thus, for each $i$, any optimal $x_{i}$ can be written as:
$$
x_{i}=r_{i} d,
$$
where $d \in \mathbb{R}_{+}^{J}$ is the unique vector that solves $\nabla \Psi(d)=c$, i.e., the cost-minimizing input shares. Furthermore, because $\Psi$ is homogeneous of degree $\sigma, \Psi_{j}$ is homogeneous of degree $\sigma-1$, so that the optimal scale is given by
$$
r_{i}=w_{i}^{\frac{1}{1-\sigma}} \Phi(z)^{\frac{1}{1-\sigma}} .
$$
Substituting back in the objective function, both the total surplus and the total consumption of each token class depend on the buyer's type $w$ only through $\int_{0}^{1} w_{i}^{1 /(1-\sigma)} d i$. It is then convenient to define the aggregate type $\theta(w)$ as
$$
\theta(w) \triangleq\left(\int_{0}^{1} w_{i}^{\frac{1}{1-\sigma}} d i\right)^{1-\sigma}
$$

Our first result establishes that the total surplus and the total amount of inference tokens depend only on the aggregate type and not on the finer details of the type profile $w$.

Proposition 1 (Efficient Allocation). Under the efficient allocation, all buyer types $w$ with the same aggregate type $\theta(w)$ consume the same number of fine-tuning tokens, consume the same total number of inference tokens in each class, and obtain the same total payoff. The number of inference tokens allocated to task $i$ is proportional to $w_{i}^{\frac{1}{1-\sigma}}$.

Proposition 1 has a key implication: two buyers with very different task profiles-one who values a few tasks intensely and many tasks little, and another who values all tasks moderately-consume the same total resources and obtain the same total surplus, provided they share the same aggregate type. The seller therefore cannot distinguish between them on the basis of total token consumption, nor would it want to. This indistinguishability is not an assumption but a consequence of the technology: homogeneity of $\Psi$ ensures that the efficient mix of token classes is task-independent, so only the scale of usage varies, and the aggregator (7) is the natural summary of how much total scale a buyer demands.

We now show that a similar logic can be used to characterize the socially efficient solution under capacity constraints in each class of tokens (training, computation, input, output).

Constrained problem We now introduce capacity constraints on each token class. This detour is not merely a generalization: the special case in which all constraints bind is precisely the problem faced by a buyer who has purchased a fixed bundle of token budgets. Thus,
the constrained-efficient allocation provides the foundation for the seller's mechanism design problem in Section 4.

Fix a type $w$ and capacity constraints $X_{j}>0, Z_{k}>0$ for all $j, k$. The capacityconstrained planner's problem is

$$
\begin{aligned}
& \max _{\left(x_{i}\right)_{i \in[0,1], z \geq 0}} \int_{0}^{1} w_{i} \Psi\left(x_{i}\right) \Phi(z) d i-\sum_{j=1}^{J} c_{j} \int_{0}^{1} x_{i j} d i-\sum_{k=1}^{K} \hat{c}_{k} z_{k} \\
& \text { s.t. } \quad \int_{0}^{1} x_{i j} d i \leq X_{j}, z_{k} \leq Z_{k} \text { for } j \in[J], k \in[K]
\end{aligned}
$$

Our next result establishes that the capacity-constrained efficient allocation coincides with the (unconstrained) efficient allocation for marginal costs inflated by the shadow costs of the capacity constraints, $\left(c^{\prime}, \hat{c}^{\prime}\right) \geq(c, \hat{c})$. This observation yields an immediate implementation of the efficient solution.


Menu Pricing of Large Language Models

## Corollary 1 (Constrained Efficient Allocation).

1. In the capacity-constrained efficient allocation, all buyer types $w$ with the same aggregate type $\theta(w)$ consume the same total number of fine-tuning tokens and inference tokens in each class, and they obtain the same total payoff.
2. The capacity-constrained efficient allocation can be implemented via linear prices equal to inflated marginal costs.

Thus, capacity constraints do not change the qualitative properties of the solution, but lead to an inefficiency in the relative allocation of token types, and in the split between those buyer types who choose to fine-tune $(z>0)$ and those who do not $(z=0)$.

The special case where all capacity constraints bind is of particular interest, because it is instrumental in the characterization of the buyer's demand for token budgets in Section 4. When all constraints bind, the total production cost is pinned down. The planner's and the buyer's solutions then coincide, i.e., the constrained efficient allocation solves the problem of a buyer who has access to inference and fine-tuning token budgets $(X, Z)$.

The optimal allocation of a fixed budget $(X, Z)$ solves the following problem:

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

Applying the homogeneity argument again, $\Phi(Z)$ factors out, and the buyer optimally sets $x_{i}=r_{i} d$ with $r_{i} \propto w_{i}^{1 /(1-\sigma)}$. Since the budget constraint for each token class $j$ requires $d_{j} \int_{0}^{1} r_{i} d i=X_{j}$, the common direction is $d=X / \int_{0}^{1} r_{i} d i$, which yields a simple expression for the buyer's optimal payoff.

The following result is the key step in our analysis. It shows that the buyer's indirect utility from any token bundle is multiplicatively separable in the aggregate type and an aggregate quality index, placing the seller's problem squarely in the Mussa-Rosen framework.

Proposition 2 (Buyer Indirect Utility). For any inference and fine-tuning token budgets $X=\left(X_{1}, \ldots, X_{J}\right) \geq 0$ and $Z=\left(Z_{1}, \ldots, Z_{K}\right) \geq 0$, the indirect utility of buyer type $w$ is

$$
U(w, X, Z)=\theta(w) \Psi(X) \Phi(Z),
$$

where $\theta(w)$ is the aggregate type defined in (7).
We therefore obtain a tractable expression for the buyer's payoff by means of a representative task, whose value is given by the aggregate type $\theta$, to which the buyer assigns the entire token budget. Heterogeneity across tasks washes out, and only the aggregate matters. A notable implication is that the buyer's unobserved allocation of tokens across tasks generates no additional incentive constraints beyond those of the standard screening problem. Under homogeneity, the cost-minimizing input mix is independent of scale, so the buyer's within-bundle allocation is pinned down regardless of type. This product form is the key input to the seller's problem, which we turn to next.


Menu Pricing of Large Language Models

## 4 Menus of Token Budgets

The main result of this section is that the seller's optimal mechanism takes the form of a menu of committed-spend contracts: each buyer pays an upfront fee for a spending budget that she allocates freely across token classes priced at the provider's marginal cost. Higher types purchase larger budgets at higher fees, with quantity discounts that are consistent with observed industry pricing. The formal analysis proceeds in two steps: we first characterize the optimal direct mechanism (Section 4.1) and then show that it admits intuitive indirect implementations (Section 4.2).

We now characterize the provider's profit-maximizing menu of token budgets. By Proposition 2, the buyer's indirect utility from a bundle $(X, Z)$ is $\theta(w) \Psi(X) \Phi(Z)$. Two features of this expression are worth emphasizing. First, private information enters only through the scalar $\theta$ : the seller's multidimensional screening problem reduces to a one-dimensional problem. Second, the payoff is multiplicatively separable in type and allocation, placing us in the framework of Mussa and Rosen (1978).


Menu Pricing of Large Language Models

### 4.1 Optimal Budget Mechanism

Define the effective type as $\theta$ and the effective allocation as the aggregate quality given by

$$
Q(X, Z) \triangleq \Psi(X) \Phi(Z) .
$$

The effective production costs of producing the aggregate quality $Q$ can be found via a cost-minimization problem:

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

The solution to the cost-minimization problem has the following properties.
Lemma 1 (Cost Function). The cost function $C(Q)$ is strictly increasing and strictly convex and satisfies $C_{+}^{\prime}(0)=0 .{ }^{8}$

Convexity follows from the strict concavity of the gain function $g$ : producing higher aggregate quality requires disproportionately more tokens. The property $C_{+}^{\prime}(0)=0$ follows from the Inada conditions. Economically, $C_{+}^{\prime}(0)=0$ means that the first unit of aggregate quality is arbitrarily cheap to produce. Combined with convexity, this implies that it is never efficient to exclude any buyer type entirely. Thus, in the monopoly problem, exclusion is driven by information rents rather than production costs.

By Lemma 1, the analysis of Mussa and Rosen (1978) applies. Denote the prior distribution of $\theta$ by $F$, which is derived from $F_{w}$. The seller chooses an aggregate quality schedule $Q(\theta)$ and a transfer schedule $t(\theta)$ to solve

$$
\max _{Q(\cdot), t(\cdot)} \mathbb{E}_{F}[t(\theta)-C(Q(\theta))],
$$

subject to incentive compatibility and individual rationality.
As usual, let

$$
\varphi(\theta) \triangleq \theta-\frac{1-F(\theta)}{f(\theta)},
$$

denote the virtual (aggregate) type. If the virtual value $\varphi$ is not increasing, let $\bar{\varphi}$ denote its Myerson-ironed version. Then, all $\theta$ with $\bar{\varphi}(\theta) \leq 0$ are excluded. For all other $\theta$, the optimal allocation is uniquely pinned down by:

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

[^6]Denoting the corresponding optimal transfers by

$$
t(\theta)=\theta Q(\theta)-\int_{0}^{\theta} Q(r) d r
$$

we can then fully characterize the optimal mechanism.
Proposition 3 (Optimal Menu). The optimal menu of token budgets is $\{(X(\theta), Z(\theta), t(\theta))\}_{\theta}$, where $(X(\theta), Z(\theta))$ are the cost-minimizing token budgets that deliver quality $Q=Q(\theta)$ at prices $t(\theta)$ as defined in (10)-(13).

The optimal menu offers a continuum of plans. All types $w$ with the same aggregate type $\theta(w)$ choose the same item. Low-type buyers are excluded. All served buyers receive distorted-downward quality (fewer tokens than the efficient allocation would prescribe) except at the very top. Buyers with higher aggregate types purchase strictly larger token budgets and pay strictly higher transfers, but enjoy quantity discounts: the average price per unit of quality falls with $\theta$.


Menu Pricing of Large Language Models

### 4.2 Indirect Mechanisms

The optimal direct mechanism specifies token quantities for each type. In practice, providers do not announce menus of token quantities; they post prices and spending limits. We now show that three natural pricing formats implement the optimum.

We first consider committed-spend mechanisms, where the terms of the contract with a buyer are related to a maximum or minimum total expenditure.

Definition 1 (Maximum-Spend Mechanism). A maximum-spend mechanism is a collection of token prices $p$ and a menu of monetary budgets and transfers $\left\{\left(B_{n}, T_{n}\right)\right\}_{n}$ such that, upon selecting item $n$, the buyer pays $T_{n}$ for access to budget $B_{n}$, which he can freely spend on tokens priced at $p$.

While the optimal budget (direct) mechanism is a cap on quantities, the maximumspend mechanism limits quality by capping expenditures at fixed prices. Those expenditures are virtual in that they do not contribute to payment beyond $T_{n}$, but they govern token allocation. As we will see in Section 6, the maximum-spend mechanism is used in practice, for example, by Quora's Poe (see Section 6), and it is closely related to the Cost-Based tariffs (where $p=c$ ) introduced in Armstrong (1996). Indeed, Proposition 8 in Appendix B derives analogous conditions to those in Armstrong (1996) under which a maximum-spend mechanism is optimal across all mechanisms, including those that specify a task-by-task token allocation.

An alternative implementation does not impose a cap on spending, but exposes the buyer to variable marginal prices, much like committed-spend mechanisms in cloud computing.

Definition 2 (Minimum-Spend Mechanism). A minimum-spend mechanism is a menu of monetary budgets and transfers $\left\{\left(p_{n}, T_{n}\right)\right\}_{n}$ such that, upon selecting item $n$, the buyer commits to spending at least $T_{n}$ on tokens priced at $p_{n}$.

A minimum-spend mechanism allows buyers to commit to larger budgets (i.e., total expenditures) to unlock variable-price discounts. In contrast to a maximum-spend mechanism, payment comes from actual token consumption rather than from an upfront payment.

Definition 3 (Two-Part-Tariff Mechanism). A two-part-tariff mechanism is a menu of prices and transfers $\left\{\left(p_{n}, T_{n}\right)\right\}_{n}$ such that, upon selecting item $n$, the buyer pays $T_{n}$ and can buy any number of tokens priced at $p_{n}$.

A two-part tariff is a classical mechanism that combines upfront and consumption payments but does not impose any token consumption limits.

Define a type-dependent markup $m(\theta)$ as:

$$
m(\theta) \triangleq \frac{\theta}{\bar{\varphi}(\theta)} .
$$

The markup $m(\theta)$ is the ratio of the true type to the virtual type; it captures the informationrent wedge that the seller imposes. Higher markups on lower types reflect greater information rents extracted from higher types in the menu.

Proposition 4 (Indirect Implementation). An optimal menu of token budgets can be implemented via a maximum-spend mechanism. If $m(\theta)$ is decreasing (e.g., if $F$ has an increasing hazard rate), then an optimal menu of token budgets can be implemented via a minimum-spend mechanism and a two-part tariff mechanism.

Implementability by a maximum-spend mechanism is straightforward and relies on the fact that the optimal token allocation is constrained-efficient, as in Doligalski et al. (2025): it suffices to set prices equal to marginal costs and to choose token expenditures and transfers so as to mimic those under the direct mechanism.

Implementability by minimum-spend and two-part tariff mechanisms requires more care. Because $\Psi$ is homogeneous, the cost-minimizing input mix is the same for every quality level $Q$; only the scale changes. A two-part tariff that preserves this input mix must therefore set token prices proportional to marginal costs; any other price vector would distort the buyer's allocation away from cost minimization. The proportionality factor is the markup $m(\theta)=$
$\theta / \bar{\varphi}(\theta)$, which inflates all marginal costs equally. The condition that $m(\theta)$ is decreasing ensures that higher types, who select larger budgets, face lower per-token prices, so the menu is incentive-compatible. In the minimum-spend implementation, the markup equals the ratio between total revenue and total cost for type $\theta$. Section 6 shows that each of these three implementations corresponds to an observed pricing format at leading platforms.


Menu Pricing of Large Language Models

## 5 Multiple Models and Competition

In practice, every major LLM provider offers multiple models that differ in capability and cost (e.g., Anthropic's Haiku, Sonnet, and Opus, or OpenAI's GPT-4o-mini and o1). A buyer can assign different tasks to different models, and the provider can screen on both usage quantity and model access. This section extends our framework to this richer environment. Two new questions arise: How does a buyer optimally allocate tasks across models? And how does a profit-maximizing provider design menus that screen on both dimensions?

Specifically, we extend our setting and allow there to be $L$ models, each with a modelspecific gain function:

$$
g_{l}\left(x_{i}, z\right)=\Psi_{l}\left(x_{i}\right) \Phi_{l}(z),
$$

where $\Psi_{l}$ and $\Phi_{l}$ are assumed to have the same properties as in the baseline model, with $\Psi_{l}$ being homogeneous of degree $\sigma_{l}$. We order the models so that $\sigma_{1} \leq \sigma_{2} \leq \cdots \leq \sigma_{L}$. The token costs are model-specific $\left(c_{l}, \hat{c}_{l}\right)$.

The buyer can process different tasks with different models; however, he cannot use two models for the same task. ${ }^{9}$ If the buyer of type $w$ processes tasks $i \in I_{l}$ with model $l$, his total payoff ignoring the payment is:

$$
\sum_{l=1}^{L} \int_{I_{l}} w_{i} g_{l}\left(x_{l i}, z_{l}\right) d i
$$

In this setting, a bundle specifies token budgets for all available models $\left(X_{l}, Z_{l}\right)_{l=1}^{L}=$ $\left(X_{l 1}, \ldots, X_{l J}, Z_{l 1}, \ldots, Z_{l K}\right)_{l=1}^{L}$.

As a preliminary step for the subsequent analysis, we show that the buyer's value for a bundle $\left(X_{l}, Z_{l}\right)_{l=1}^{L}$ depends only on the aggregate quality of each model,

$$
Q_{l} \triangleq g_{l}\left(X_{l}, Z_{l}\right),
$$

[^7]and admits a tractable closed-form expression. To this end, we fix $w$ and order the tasks so that $w_{i}$ is increasing. We assume that $Q_{l}>0$ for all $l$ (if $Q_{l}=0$ then the $l$-th model can be ignored). If the buyer allocates tasks in the set $I_{l}$ to model $l$, then his payoff from model $l$ is, by the arguments of Proposition 2,
$$
U_{l}\left(I_{l}, Q_{l}\right)=\theta_{l}\left(I_{l}\right) Q_{l} \text {, }
$$
where $\theta_{l}\left(I_{l}\right) \triangleq\left(\int_{I_{l}} w_{i}^{1 /\left(1-\sigma_{l}\right)} d i\right)^{1-\sigma_{l}}$. The total payoff from the task-model allocation $\left(I_{l}\right)_{l=1}^{L}$ is
$$
\sum_{l=1}^{L} U_{l}\left(I_{l}, Q_{l}\right)=\sum_{l=1}^{L} \theta_{l}\left(I_{l}\right) Q_{l} .
$$
The optimal payoff $U^{*}\left(Q_{1}, \ldots, Q_{L}\right)=\sum_{l=1}^{L} U_{l}\left(I_{l}^{*}, Q_{l}\right)$ is evaluated at the optimal task-model allocation $\left(I_{l}^{*}\right)_{l=1}^{L}$.

Lemma 2 (Buyer-Optimal Payoff). If $\sigma_{l}=\sigma$ for all $l \in[L]$, then

$$
U^{*}\left(Q_{1}, \ldots, Q_{L}\right)=\theta\left(\sum_{l=1}^{L} Q_{l}^{1 / \sigma}\right)^{\sigma} .
$$

By Lemma 2, despite the combinatorial complexity of assigning a continuum of tasks to multiple models, the buyer's payoff from a bundle of tokens across different models admits a simple product decomposition into an aggregate type and a CES-style aggregate quality over models. The elasticity of substitution across models is pinned down by the common returns-to-scale parameter $\sigma$, and the single-model tractability of Sections 3-4 extends to the multi-model environment.


Menu Pricing of Large Language Models

### 5.1 Efficient Solution

We first compute the efficient allocation in this setting. Since the buyer's value takes a simple form (20) as a function of aggregate qualities of each model, the efficient allocation delivers the efficient amount of these qualities in a cost-efficient way.

Assumption 1 (Homogeneous Fine-Tuning). For each $l=1, \ldots, L$ :

1. $\Psi_{l}$ is homogeneous of degree $\sigma \in(0,1)$.
2. $\Phi_{l}$ is homogeneous of degree $\hat{\sigma}_{l}<1-\sigma$.

All subsequent analysis holds under Assumption 1, so we omit it from the statements. Assumption 1 allows us to obtain closed-form expressions because of the following result:

Lemma 3 (Model-Specific Cost Function). The cost function of delivering aggregate quality $Q_{l}$ through model $l$ is $C_{l}\left(Q_{l}\right)=\kappa_{l} Q_{l}^{1 /\left(\sigma+\hat{\sigma}_{l}\right)}$ for some constant $\kappa_{l}>0$.

By Lemma 3, the minimal cost of obtaining quality $Q$ from $L$ models is

$$
C(Q)=\min _{Q_{1}, \ldots, Q_{L} \geq 0} \sum_{l=1}^{L} \kappa_{l} Q_{l}^{1 /\left(\sigma+\hat{\sigma}_{l}\right)}, \quad \text { s.t. } \sum_{l=1}^{L} Q_{l}^{1 / \sigma}=Q^{1 / \sigma} .
$$

The optimal solution utilizes a single model, resulting in

$$
C(Q)=\min _{l \in[L]} \kappa_{l} Q^{1 /\left(\sigma+\hat{\sigma}_{l}\right)} .
$$

Thus, the rich functional model heterogeneity (15) reduces to heterogeneity in two parameters $\left(\kappa_{l}, \hat{\sigma}_{l}\right)$ that roughly correspond to "cost-effectiveness" and "propensity to fine-tune." Heterogeneity in $\kappa_{l}$ is vertical: higher $\kappa_{l}$ correspond to overall costlier models. Heterogeneity in $\hat{\sigma}_{l}$ is horizontal: models with high $\hat{\sigma}_{l}$ are better at generating high $Q$ whereas models with low $\hat{\sigma}_{l}$ are better at generating low $Q$. As a result, higher values of $Q$ are optimally produced by models with higher $\hat{\sigma}_{l}$. In other words, models with higher returns to fine-tuning have flatter cost curves and are cheaper for producing high qualities, introducing a natural form of horizontal differentiation among models.

Notably, the cost function $C(Q)$ is not convex but piecewise convex, with kinks at the model switches, which translates into unusual properties of the efficient and profitmaximizing allocations. Specifically, the efficient quality allocation for type $\theta$ when using model $l$ solves:

$$
\max _{Q \geq 0} \theta Q-\kappa_{l} Q^{1 /\left(\sigma+\hat{\sigma}_{l}\right)},
$$

leading to the following characterization:
Proposition 5 (Efficient Allocation). In the efficient allocation, each type $\theta$ processes all tasks with a single model:

$$
l^{*} \in \arg \max _{l}\left(1-\sigma-\hat{\sigma}_{l}\right)\left(\frac{\left(\sigma+\hat{\sigma}_{l}\right)}{\kappa_{l}}\right)^{\left(\sigma+\hat{\sigma}_{l}\right) /\left(1-\sigma-\hat{\sigma}_{l}\right)} \theta^{1 /\left(1-\sigma-\hat{\sigma}_{l}\right)},
$$

with the aggregate quality:

$$
Q^{*}(\theta)=\left(\frac{\theta\left(\sigma+\hat{\sigma}_{l^{*}}\right)}{\kappa_{l^{*}}}\right)^{\left(\sigma+\hat{\sigma}_{l^{*}}\right) /\left(1-\sigma-\hat{\sigma}_{l^{*}}\right)}
$$

The efficient token allocation is given by Lemma 3 evaluated at $Q^{*}$ from (23).

Every type uses a single model. Higher aggregate types $\theta$ consume higher aggregate qualities by using models with higher fine-tuning capabilities $\hat{\sigma}_{l}$, i.e., $Q^{*}(\theta)$ is increasing.

Because $C(Q)$ is not continuously differentiable, the allocation is discontinuous at the model switch. Indeed, it follows from (23) and (22) that if the efficient allocation at type $\theta$ is indifferent between $Q_{l}$ of model $l$ and $Q_{l^{\prime}}$ of model $l^{\prime}$ such that $\hat{\sigma}_{l^{\prime}}>\hat{\sigma}_{l}$, then

$$
\frac{Q_{l^{\prime}}}{Q_{l}}=\frac{1-\sigma-\hat{\sigma}_{l}}{1-\sigma-\hat{\sigma}_{l^{\prime}}}>1 .
$$


Menu Pricing of Large Language Models

### 5.2 Multi-Model Monopolist

We now assume that all $L$ models are sold by a monopolist who offers a menu. Each item specifies a transfer and a collection of token budgets for different models $\left(X_{l}, Z_{l}\right)_{l=1}^{L}$.

By Lemma 2, the seller's problem is equivalent to the optimal screening problem in Mussa and Rosen (1978), in which the buyer's type is $\theta \in \mathbb{R}$, the seller chooses a menu of $(Q, t)$, the buyer's payoff is $U(\theta, Q)=\theta Q$, and the cost of obtaining a quality $Q$ is equal to $C(Q)$ as in (21). Although $C(Q)$ is not convex, the Envelope theorem applies and this problem is equivalent to a maximization of virtual surplus:

$$
\max _{Q(\cdot): \text { increasing }} \int_{0}^{1}(\varphi(\theta) Q(\theta)-C(Q(\theta))) d F(\theta)
$$

where $\varphi(\theta)=\theta-(1-F(\theta)) / f(\theta)$.
Assuming $\varphi(\theta)$ is increasing, the solution can be obtained pointwise to equal an efficient allocation with respect to the virtual type: $Q^{\mathrm{m}}(\theta)=Q^{*}(\varphi(\theta))$. By Proposition 5, if $\varphi(\theta)<0$, then the buyer is excluded, $Q^{\mathrm{m}}=0$; if $\varphi(\theta) \geq 0$,

$$
\begin{aligned}
& l^{\mathrm{m}}(\theta) \in \arg \max _{l}\left(1-\sigma-\hat{\sigma}_{l}\right)\left(\frac{\left(\sigma+\hat{\sigma}_{l}\right)}{\kappa_{l}}\right)^{\left(\sigma+\hat{\sigma}_{l}\right) /\left(1-\sigma-\hat{\sigma}_{l}\right)} \varphi(\theta)^{1 /\left(1-\sigma-\hat{\sigma}_{l}\right)}, \\
& Q^{\mathrm{m}}(\theta)=\left(\frac{\varphi(\theta)\left(\sigma+\hat{\sigma}_{l^{\mathrm{m}}}\right)}{\kappa_{l^{\mathrm{m}}}}\right)^{\left(\sigma+\hat{\sigma}_{l^{\mathrm{m}}}\right) /\left(1-\sigma-\hat{\sigma}_{l^{\mathrm{m}}}\right)} .
\end{aligned}
$$

Since $Q^{*}$ is increasing, the pointwise solution solves the problem. Optimal transfers are

$$
t^{\mathrm{m}}(\theta)=\theta Q^{\mathrm{m}}(\theta)-\int_{0}^{\theta} Q^{\mathrm{m}}(r) d r
$$

As under the efficient allocation, because $C(Q)$ is not continuously differentiable, the quality and transfer schedules are discontinuous.

Proposition 6 (Multi-Model Monopoly). If $\varphi(\theta)$ is increasing, then the optimal token-
budget menu is given by $\left\{\left(\left(X_{l}(\theta), Z_{l}(\theta)\right)_{l=1}^{L}, t(\theta)\right)\right\}_{\theta}$, where $\left(X_{l}(\theta), Z_{l}(\theta)\right)_{l=1}^{L}$ are efficient token budgets that deliver quality $Q^{\mathrm{m}}(\theta)$ at prices $t(\theta)$ as defined in (26) and (27).

The monopolist assigns each buyer type to a single model, with higher types using more capable models. All types $w$ with the same aggregate type $\theta(w)$ choose the same item and use one model, $l^{\mathrm{m}}(\theta)$ as defined in (25), for all tasks. The quality schedule exhibits discrete jumps at model-switching thresholds: a buyer at the margin between two models jumps to a strictly higher quality when upgrading. OpenAI's tier structure, which reserves its most compute-intensive reasoning model for the highest-paying subscribers, provides a direct empirical counterpart; see the discussion in Section 6.


Menu Pricing of Large Language Models

### 5.3 Leader-Fringe Competition

We now study competition between a proprietary leader and an open-source competitive fringe. The key economic forces are: (i) the fringe provides an outside option whose value depends on the buyer's type; (ii) for intermediate types, the leader must provide enough tokens to deter fringe top-up; (iii) for high types, the leader acts as an unconstrained monopolist. The resulting allocation has three distinct regions, generating a richer pattern of distortions than either the single-model monopoly or the multi-model monopoly.

Specifically, we assume that there is a single leader, corresponding to the highest-capability model $(l=L)$ in the previous section, and a continuum of firms in a competitive fringe that sell their tokens at fixed per-token prices equal to marginal costs. We assume that the leader possesses a proprietary model characterized by the aggregate cost parameter $c_{L}>0$, returns to intensity $\sigma_{L}=\sigma>0$, and returns to fine-tuning $\hat{\sigma}_{L}>0$, such that $\sigma+\hat{\sigma}_{L}<1$. The fringe possesses an open-source model characterized by the lower aggregate cost parameter $c_{F} \in\left[0, c_{L}\right)$, the same returns to intensity $\sigma_{F}=\sigma$, and lower returns to fine-tuning $\hat{\sigma}_{F} \in\left[0, \hat{\sigma}_{L}\right)$.

The buyer has a private type $w$. By Lemma 2, only the aggregate type $\theta(w)$ matters, so we take $\theta$ as a primitive and assume that $\theta$ is distributed according to $F$ with strictly positive density $f$ everywhere on [0, 1]. The buyer can multi-home and can buy at most one item from the leader.

We want to solve the leader's problem, which can be viewed as a monopolistic design of an optimal menu $(q, t)$ subject to the competitive pressure from the fringe. For notational simplicity, define the quantities:

$$
\begin{aligned}
& q_{L} \triangleq Q_{L}^{1 /\left(\sigma+\hat{\sigma}_{L}\right)}=g_{L}\left(X_{L 1}, \ldots, X_{L J}, Z_{L 1}, \ldots, Z_{L K}\right)^{1 /\left(\sigma+\hat{\sigma}_{L}\right)} \\
& q_{F} \triangleq Q_{F}^{1 /\left(\sigma+\hat{\sigma}_{F}\right)}=g_{F}\left(X_{F 1}, \ldots, X_{F J}, Z_{F 1}, \ldots, Z_{F K}\right)^{1 /\left(\sigma+\hat{\sigma}_{F}\right)}
\end{aligned}
$$

The payoff of type $\theta$ from having purchased quantity $q_{L}$ from the leader is

$$
\max _{q_{F} \geq 0} \theta\left(q_{L}^{\left(\sigma+\hat{\sigma}_{L}\right) / \sigma}+q_{F}^{\left(\sigma+\hat{\sigma}_{F}\right) / \sigma}\right)^{\sigma}-c_{F} q_{F},
$$

with a (type-dependent) outside option corresponding to $q_{L}=0$. The first-order condition for the buyer's problem of purchasing fringe quantity is

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

In general, (29) does not admit a closed-form solution. It does so in the case of $\hat{\sigma}_{F}=0$. Specifically, define

$$
\hat{q}(\theta) \triangleq\left(\frac{\theta \sigma}{c_{F}}\right)^{\sigma /\left((1-\sigma)\left(\sigma+\hat{\sigma}_{L}\right)\right)}, \quad \psi(\theta) \triangleq \theta^{\frac{1}{1-\sigma}}(1-\sigma)\left(\frac{\sigma}{c_{F}}\right)^{\sigma /(1-\sigma)} .
$$

If the quantity purchased from the leader is $q_{L}<\hat{q}(\theta)$, the buyer will purchase fringe tokens to achieve the optimal total quantity $\hat{q}(\theta)$; if $q_{L}>\hat{q}(\theta)$, the buyer will single-home with the leader. The buyer's outside option utility corresponding to $q=0$ is $\psi(\theta)$. We then obtain the following characterization of the buyer's demand for the leader's quantity.

Lemma 4 (Buyer's Payoff). The leader's problem is equivalent to one in which the buyer has a zero outside option and the following payoff:

$$
u(\theta, q)= \begin{cases}c_{F} q^{\left(\sigma+\hat{\sigma}_{L}\right) / \sigma}, & \text { if } q<\hat{q}(\theta), \\ \theta q^{\sigma+\hat{\sigma}_{L}}-\psi(\theta), & \text { if } q \geq \hat{q}(\theta) .\end{cases}
$$

The payoff $u(\theta, q)$ in Lemma 4 is continuous, differentiable, and increasing in both parameters. In the region $q<\hat{q}(\theta)$, it is convex in $q$ and independent of $\theta$. In the region $q>\hat{q}(\theta)$, it is concave in $q$ and supermodular in $q$ and $\theta$.

The two-regime structure has a clear economic interpretation. In the first regime ( $q<$ $\hat{q}(\theta))$, the buyer supplements leader tokens with fringe tokens to achieve the optimal total quantity, so her marginal value of additional leader tokens is pinned down by the fringe price $c_{F}$; the leader faces a perfectly elastic residual demand. In the second regime $(q \geq \hat{q}(\theta))$, the buyer single-homes with the leader, and her value is the standard concave payoff minus the outside option $\psi(\theta)$ she forgoes by not using the fringe. The transition between regimes is where the leader's competitive constraint binds.

The leader's problem is an instance of screening with a type-dependent outside option (Jullien, 2000); the reformulation (31) absorbs this outside option into the buyer's payoff.

An optimal menu can then be characterized by standard methods. ${ }^{10}$ Indeed, $u(\theta, q)$ satisfies the (weak) single-crossing constraints: $u_{\theta q}(\theta, q)=0$ if $q<\hat{q}(\theta)$ and $u_{\theta q}(\theta, q)>0$ if $q \geq \hat{q}(\theta)$. Allocation $q(\theta)$ is implementable only if it is increasing whenever $q \geq \hat{q}(\theta)$, and the profit associated with allocation $q$ is:

$$
\Pi(q)=\int_{0}^{1}\left(u(\theta, q(\theta))-c_{L} q(\theta)-\frac{1-F(\theta)}{f(\theta)} u_{\theta}(\theta, q(\theta))\right) f(\theta) d \theta
$$

In regular environments, this profit can be maximized pointwise. To this end, denote by $q^{\mathrm{m}}(\theta)$ the optimal monopoly quantity in the absence of the competitive fringe

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


Menu Pricing of Large Language Models

Proposition 7 (Leader-Fringe Competition). Let $F$ satisfy a monotone hazard rate and $\hat{\sigma}_{F}=0$. Then, there exist $\underline{\theta}$ and $\bar{\theta}, \underline{\theta} \leq \bar{\theta}$, such that (i) for $\theta \leq \underline{\theta}, q(\theta)=0$, (ii) for $\theta \in(\underline{\theta}, \bar{\theta})$, $q(\theta)=\hat{q}(\theta)$ given by (30), and (iii) for $\theta \geq \bar{\theta}, q(\theta)=q^{\mathrm{m}}(\theta)$ given by (33).

Under the optimal mechanism, low types purchase exclusively from the fringe. All higher types purchase exclusively from the leader. Out of those, the midline types are on the margin of whether to buy from the fringe, whereas the highline types strictly prefer to purchase from the leader. Depending on parameters, the midline region may not exist, $\underline{\theta}=\bar{\theta}$, or the highline region may not exist, $\bar{\theta}>1$.

The three-region structure reflects three distinct economic regimes. In the fringe-only region $(\theta \leq \underline{\theta})$, the buyer's willingness to pay for the leader's superior fine-tuning capability is too low to justify adoption. In the deterrence region $(\underline{\theta}<\theta<\bar{\theta})$, the leader offers exactly enough tokens to make the buyer indifferent between single-homing with the leader and supplementing with the fringe; this pins the leader's quantity to $\hat{q}(\theta)$, which rises with type. In the monopoly region $(\theta \geq \bar{\theta})$, the buyer's value is high enough that fringe competition no longer constrains the leader, who reverts to the standard downward-distorted monopoly quantity $q^{\mathrm{m}}(\theta)$. The novel region is the deterrence region, which has no analog in either the single-model monopoly or the multi-model monopoly.

[^8]Comparison with Efficient Allocation It is instructive to compare the allocation under leader-fringe competition with the efficient benchmark. By Proposition 5, defining

$$
u_{l}^{*}(\theta)=\left(1-\sigma_{l}-\hat{\sigma}_{l}\right) \theta^{1 /\left(1-\sigma_{l}-\hat{\sigma}_{l}\right)}\left(\frac{\sigma_{l}+\hat{\sigma}_{l}}{c_{l}}\right)^{\left(\sigma_{l}+\hat{\sigma}_{l}\right) /\left(1-\sigma_{l}-\hat{\sigma}_{l}\right)}
$$

type $\theta$ uses the fringe model if and only if $u_{F}^{*}(\theta)>u_{L}^{*}(\theta)$. The indifference type at which $u_{F}^{*}(\theta)=u_{L}^{*}(\theta)$ is (recall that $\sigma_{L}=\sigma_{F}=\sigma$ and $\hat{\sigma}_{F}=0$ )

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

If $\theta<\hat{\theta}$, then the type uses the fringe model: $q_{F}(\theta)=\left(\theta \sigma / c_{F}\right)^{1 /(1-\sigma)}$ and $q_{L}(\theta)=0$. If $\theta \geq \hat{\theta}$, then the type uses the leader model: $q_{F}(\theta)=0$ and $q_{L}(\theta)=\left(\theta\left(\sigma+\hat{\sigma}_{L}\right) / c_{L}\right)^{1 /\left(1-\sigma-\hat{\sigma}_{L}\right)}$.

Comparing this with the fringe-competition allocation characterized in Proposition 7, we see two main differences: First, we see that the "highline" leader quantity $q^{\mathrm{m}}(\theta)$ is distorted downward from efficiency because $\varphi(\theta)$ is smaller than $\theta$. This is natural, since in that range the leader is unconstrained by the fringe and acts as a monopolist. Second, there is an additional extensive margin distortion, because $\hat{\theta}<\underline{\theta} .{ }^{11}$

Comparison with Multi-Model Monopoly Another natural benchmark is the case of a monopolist possessing the leader and fringe models. By Proposition 6, the monopolist effectively faces a Mussa-Rosen problem with the buyer's payoff:

$$
u\left(\theta, q_{L}, q_{F}\right)=\theta\left(q_{F}+q_{L}^{\left(\sigma+\hat{\sigma}_{L}\right) / \sigma}\right)^{\sigma} .
$$

and will be supplying an efficient allocation with respect to the virtual type. If $\theta<\theta^{\mathrm{m}}$, where $\varphi\left(\theta^{\mathrm{m}}\right) \triangleq \hat{\theta}$ and $\hat{\theta}$ is defined at (34), then the type uses the fringe model and $q_{F}(\theta)=$ $\left(\varphi(\theta) \sigma / c_{F}\right)^{1 /(1-\sigma)}$ and $q_{L}(\theta)=0$. If $\theta \geq \theta^{\mathrm{m}}$, then the type uses the leader model with $q_{F}(\theta)=0$ and $q_{L}(\theta)=\left(\varphi(\theta)\left(\sigma+\hat{\sigma}_{L}\right) / c_{L}\right)^{1 /\left(1-\sigma-\hat{\sigma}_{L}\right)}$.

Relative to the efficiency benchmark, the allocation is distorted downward within each model, except at the very top. In addition, fewer types consume the leader model.

Relative to the leader-fringe environment of Proposition 7, the integrated multi-model monopolist internalizes the fringe technology and therefore does not face a competitive outside option. This has two implications. First, the fringe allocation is no longer pinned down by marginal-cost pricing: types that use the fringe model receive $q_{F}(\theta)=\left(\varphi(\theta) \sigma / c_{F}\right)^{1 /(1-\sigma)}$

[^9](and are excluded whenever $\varphi(\theta)<0$ ), whereas under leader-fringe competition the corresponding types purchase the efficient quantity $\left(\theta \sigma / c_{F}\right)^{1 /(1-\sigma)}$ from the fringe. Second, the "midline" region $\theta \in(\underline{\theta}, \bar{\theta})$ in which the leader supplies $q=\hat{q}(\theta)$ to deter top-up purchases from the fringe disappears. Because the monopolist controls both models, she uses the fringe model directly as the low-quality product in the Mussa-Rosen menu and assigns each type to a single model with a single cutoff $\theta^{\mathrm{m}}$. Finally, once the fringe constraint is slack, the intensive-margin provision coincides and equals the monopoly quantity $q^{\mathrm{m}}(\theta)$; differences between the two environments are therefore concentrated among low and intermediate types. ${ }^{12}$

Figure 1 illustrates the optimal allocation and distortions in a uniform example. All calculations for this example are in Appendix A.


Menu Pricing of Large Language Models

The top panel plots the leader quantity $q(\theta)$, and the bottom panel plots the fringe quantity $q_{F}(\theta)$. Four cutoffs, ordered $\hat{\theta}<\underline{\theta}<\theta^{\mathrm{m}}<\bar{\theta}$, partition the type space. The efficient allocation (solid) features a single switch at $\hat{\theta}$ : types below $\hat{\theta}$ use only the fringe, types above use only the leader. The leader-fringe allocation (dotted) shifts this switch rightward to $\underline{\theta}$, reflecting the extensive-margin distortion. Between $\underline{\theta}$ and $\bar{\theta}$, the leader supplies the deterrence quantity $\hat{q}(\theta)$, which lies strictly below the efficient leader quantity. Above $\bar{\theta}$, the fringe constraint is slack and the leader reverts to the monopoly quantity $q^{\mathrm{m}}(\theta)$, which coincides with the integrated monopolist's allocation (dashed). In the bottom panel, the efficient and leader-fringe curves for $q_{F}$ coincide over their respective ranges, since the fringe prices at marginal cost in both cases; the difference is that fringe usage persists up to $\underline{\theta}$ rather than $\hat{\theta}$. The integrated monopoly distorts fringe provision downward and switches to the leader at the later cutoff $\theta^{\mathrm{m}}$.


Menu Pricing of Large Language Models

## 6 LLM Pricing in Practice

A striking feature of current LLM pricing is that different market segments implement each of the optimal mechanisms in our theoretical framework: consumer subscriptions implement the nonlinear menus of Sections 4 and 5; model aggregators implement the committed-spend mechanisms of Section 4.2; and developer APIs implement the constrained-efficient linear pricing of Corollary 1.

Our main findings are as follows. The two largest proprietary providers, Anthropic and OpenAI, offer remarkably similar price points (\$20 and \$200) but implement fundamentally different screening mechanisms. Anthropic screens through quantity alone, holding model ac-

[^10]

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 1: Leader-Fringe allocations across different regimes. Example with $\theta \sim U[0,1]$, $\sigma=1 / 2, \hat{\sigma}_{L}=1 / 4, c_{F}=1 / 10, c_{L}=1 / 8$.

cess constant across paid tiers, matching the token-budget mechanism of Proposition 3. OpenAI screens through both quantity and model access, reserving its most compute-intensive reasoning model for the highest tier, matching the multi-model menu of Proposition 6. This contrast illustrates the distinction between the single-model screening of Section 4 and the multi-model versioning of Section 5. Model aggregators, which resell access to upstream providers through a single interface, implement the committed-spend mechanisms of Section 4.2: Quora's Poe enforces a hard budget cap (maximum spend), while GitHub Copilot allows overage purchasing at linear prices (minimum spend). Finally, API pricing across all major providers is linear in tokens with no volume discounts, resembling the constrainedefficient benchmark of Corollary 1 rather than any profit-maximizing mechanism, consistent with providers prioritizing market share over rent extraction in the developer segment.


Menu Pricing of Large Language Models

### 6.1 Anthropic

Anthropic's consumer pricing for Claude best illustrates the token-budget mechanism of Section 4. Anthropic offers the same model family across all paid tiers, with differentiation occurring through usage allocations. As in the optimal menu characterized in Proposition 3, users with higher aggregate types $\theta$ select items with larger token budgets while consuming the same underlying technology.

Figure 2 summarizes the structure of paid tiers as of January 2026. All paid tiers, namely Pro (\$20/month), Max 5x (\$100/month), and Max 20x (\$200/month), grant access to Haiku 4.5, Sonnet 4.5, and Opus 4.5. The key distinction is the quantity of compute: Anthropic measures usage limits in computational intensity. For example, Opus queries consume resources approximately $5 \times$ faster than Sonnet. This aligns with Proposition 3, where the optimal mechanism defines users' budgets in total tokens consumed across classes. Although Anthropic offers multiple models, all paid tiers grant access to the same set, so the relevant screening dimension is quantity rather than model access. The different depletion rates across models (e.g., Opus consuming resources $5 \times$ faster than Sonnet) function as heterogeneous per-unit costs within a single compute-budget mechanism, analogous to the token-class costs $c_{j}$ in Proposition 3.

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 2: Anthropic subscription tiers (January 2026). Model access is constant across paid tiers; differentiation occurs through usage allocations.