Faster responses can eliminate pure equilibria in adaptive competition

When interventions change the environment observed by competitors, reducing a response delay can alter the existence of a participation equilibrium. We study a simultaneous game of N agents in which each exits or chooses a constant adaptation rate, pays a fixed operating charge per unit time and a cost per cycle, and receives a reward for tracking a binary source whose switching rate increases with competitors’ activity. For a specified range of externality strength and participation cost, we prove three distinct joint limits as normalized cycle cost c and delay d vanish. If d ≪ c, the unique pure equilibrium has all agents participating. If c ≪ d ≪ √c, no pure equilibrium exists, including asymmetric participation and rate profiles. If √c ≪ d ≪ 1, exactly one agent participates in each of N pure equilibria. In the intermediate regime, a symmetric mixed equilibrium exists. Every such equilibrium satisfies p → 1 − b^(1/(N−1)) and 2ηda = W(K_b d/c) + o(1). The participation probability coincides with that of a classical collision game, while the continuous adaptation rate follows a separate asymptotic balance. An explicit finite-parameter certificate makes pure-equilibrium enumeration exhaustive for a twenty-agent example in which halving the delay removes every pure equilibrium. A sufficiently large finite rate cap preserves this example, whereas any fixed cap changes the joint limiting problem. The results concern equilibrium in a specified strategic model; they establish neither temporal oscillations nor a thermodynamic extremum principle. Revised research preprint prepared with disclosed AI assistance. Five GPT Astra agents conducted independent first reviews followed by cross-checks; these checks are not independent human peer review or empirical validation. The accompanying source archive contains reconstructed LaTeX sources, an independent Python implementation, generated figures and synthetic numerical outputs, with reproduction instructions. The original LaTeX and the original program announced in the supplied PDF were unavailable. Companion preprint: Endogenous information obsolescence and energetic viability in an adaptation game, https://doi.org/10.5281/zenodo.23435371.