Keep, Customize, or Exit: Default Design and Token Pricing in LLM Reasoning Services

We study a large language model (LLM) service in which a provider chooses a per-token price and a default reasoning-token allocation, while a user may accept the default, customize the allocation, or exit. Larger allocations can improve accuracy but increase token cost and latency. We model this interaction as a Stackelberg game and derive the user's unique optimal customized allocation in closed form. For any price, the acceptable defaults form either an empty set or a compact interval. We characterize the provider's optimal default through a three-regime rule, reduce equilibrium computation to a one-dimensional price optimization, and prove the existence of the equilibrium. We further show that defaults affect the implemented reasoning allocation only when users value the convenience of avoiding customization; otherwise, every service-providing outcome implements the user's optimal customized allocation. Experiments with two compact open-weight reasoning models on five mathematics and science benchmarks support the accuracy-token model and show how model and task characteristics determine equilibrium prices, defaults, and reasoning allocations.

Strategic Persuasion Through Information Timeliness

We study a dynamic strategic communication problem in which a sender controls the timing of truthful updates from binary continuous-time Markov sources. The receiver chooses between a zero-order-hold estimator that follows the sender's updates and a prior-only default estimator, aiming to maximize a weighted correct-estimation utility. In contrast, the sender seeks to persuade the receiver to estimate the state as 1, regardless of the true state. This misalignment leads to a Stackelberg game in which the sender, as the leader, commits to state-dependent Poisson update rates, and the receiver, as the follower, decides whether to follow the sender's messages. The sender maximizes the long-term average time that the receiver's estimate equals 1, subject to a conditional intensity budget and a participation constraint (PC) ensuring that following the sender's messages does not degrade the receiver's average utility relative to its prior information. For a single source, we show that the sender's optimal policy allocates a minimum state-0 update intensity to the undesired state-0, just enough to satisfy the PC, and the remaining budget to the desired state-1. For multiple sources with heterogeneous minimum state-0 update intensities, we develop a branch-and-bound algorithm that typically avoids exhaustive search. Finally, we extend the solution to multiple receivers over dedicated channels. Our results show that controlling timeliness alone enables the sender to persuade the receiver and increase its utility.