Strategic Pricing in LLM Routing Markets

Routing systems for large language models choose among models and collaboration modes to balance answer quality against cost. They take supplier prices as given. When suppliers set prices strategically, the router defines a game with two difficulties. Its equilibrium need not be pure, and a single trained router can reward a supplier for raising its own price. We compute equilibria by fictitious play and accept a profile only after checking every unilateral deviation on the bid grid. We learn answer quality once, independently of price, and add payments through an economic layer, so a supplier's demand cannot rise with its own price. We then study two platform controls. A committed outside option gives the customer side bargaining power. Under a router that picks the best mode, it raises customer surplus whenever its committed value exceeds baseline surplus, and repricing cannot remove that gain. In simulations with model accuracies calibrated to three open-weight models, its benefit rises with its capability and falls with its price. A capability-blind near-tie rule changes $9.1\%$ of allocations at a moderate threshold and raises the least-used supplier's share in every run, at a bounded direct cost. Menu expansion buys contestability. Redistribution within the menu buys inclusion.