The widespread adoption of Large Language Models (LLMs) through Application Programming Interfaces (APIs) induces a critical vulnerability: the potential for dishonest manipulation by service providers. This manipulation can manifest in various forms, such as secretly substituting a proclaimed high-performance model with a low-cost alternative, or inflating responses with meaningless tokens to increase billing. This work tackles the issue through the lens of algorithmic game theory and mechanism design. We are the first to propose a formal economic model for a realistic user-provider ecosystem, where a user can iteratively delegate T queries to multiple model providers, and providers can engage in a range of strategic behaviors. As our central contribution, we prove that for a continuous strategy space and any ε∈(0,1/2), there exists an approximate incentive-compatible mechanism with an additive approximation ratio of O(T1-ε log T), and a guaranteed quasi-linear second-best user utility. We also prove an impossibility result, stating that no mechanism can guarantee an expected user utility that is asymptotically better than our mechanism. Furthermore, we demonstrate the effectiveness of our mechanism in simulation experiments with real-world API settings.

Mechanism design on social networks has attracted extensive attention recently. The goal is to design mechanisms to incentivize participants to invite more participants via their social networks, and the challenge is that the participants are competitors. Various mechanisms have been proposed for single-/multiple-unit auctions, but it has been shown that it is challenging to design such mechanisms for more complex settings. We move this forward to investigate a double auction on a network where each trader (a buyer or a seller) can link to other buyers and sellers. Incentiving invitation is more difficult than in multi-unit one-sided auctions, because there are two different roles and a buyer (seller) seems happy to invite a seller (buyer), but again the invited seller (buyer) may invite another buyer (seller) to compete with the original buyer (seller). To combat this, we propose a solution called dynamic trade reduction (DTR), which also guarantees a non-negative revenue for the market owner. Interestingly, our solution is also applicable to the multi-unit one-sided auction when there is only one seller linking to only buyers on the network. We believe that the principle of our solution has the potential to be extended to design the multi-item one-sided auction.