LLM agents are known to deviate from Nash equilibria in strategic interactions, but nobody has looked inside the model to understand why, or asked whether the deviation can be reversed. We do both. Working with four open-source models (Llama-3 and Qwen2.5, 8B to 72B parameters) playing four canonical two-player games, we establish the behavioral picture through self-play and cross-play experiments, then open up the 32-layer Llama-3-8B model and examine what actually happens during a strategic decision. The mechanistic findings are clear. Opponent history is encoded with near-perfect fidelity at the first layer (96% probe accuracy) and consumed progressively, while Nash action encoding is weak throughout, never exceeding 56%. There is no dedicated Nash module. Instead, the model privately favors the Nash action through most of its forward pass, but a prosocial override rooted in pretraining on human text concentrated in the final layers reverses this, reaching 84% probability of cooperation at layer 30. Injecting a learned Nash direction into the residual stream shifts behavior bidirectionally and causally, confirmed through concept clamping. The behavioral experiments surface six scale- and architecture-dependent findings, the most notable being that chain-of-thought reasoning worsens Nash play in small models but achieves near-perfect Nash play above 70B parameters. The cross-play experiments reveal three phenomena invisible in self-play: a small model can unravel any partner's cooperation by defecting early; two large models reinforce each other's cooperative instincts indefinitely; and who moves first determines which Nash equilibrium the system reaches. LLMs do not lack Nash-playing competence. They compute it, then suppress it.