Disaggregated inference architectures physically separate prefill and decode phases onto distinct GPU pools, creating competing "agents" that share a fixed hardware budget. We provide, to our knowledge, the first formal game-theoretic analysis of this architecture, using NVIDIA Dynamo as a concrete case study. We model disaggregated serving as three coupled games: a two-player resource game between prefill and decode pools, a selfish caching game over the hierarchical KV cache, and a congestion game with positive externalities for request routing. We empirically validate the latter two; the P/D resource game is treated analytically (Section 9.2). We characterize how GPU saturation induces regime transitions that shift the game's payoff structure: below saturation, selfish behavior has bounded Price of Anarchy (PoA); at saturation, superlinear latency and cache externalities drive our empirical estimator PoA-hat (defined in Section 6.4) upward. Based on this analysis, we design an adaptive controller that detects saturation transitions in real time and adjusts routing parameters accordingly, shifting from cache-affinity exploitation to load-balanced congestion avoidance. We instantiate our framework on a 3-node NVIDIA B200 cluster running Dynamo with two models, Nemotron-4-340B (TP=8, full-node workers with cross-InfiniBand KV transfers) and Llama-3.1-70B (TP=4), and find the same three-regime PoA-hat structure with the same first post-knee grid point (C=128) on both models. Adaptive routing shifts each model to a better operating point. Our strongest result is on the 70B 1P/5D topology, where PoA-hat drops 3.1x (66.4 to 21.5) in the saturated phase at a 13% throughput cost. On the 70B 1P/2D, PoA-hat drops 2.2x and TTFT P99 drops 7.6x (see Section 8.5).