Coordinating extreme-heat adaptation at mega-sporting events: early warning and cost sharing in a multi-actor differential game

Introduction Coordinating extreme-heat adaptation at mega-sporting events is a time-dependent public health challenge because public authorities, event organizers, and participating teams control different protective measures while heat risk evolves over a finite competition horizon. Existing sport-heat research describes exposure and prevention yet provides limited guidance on how adaptation should be timed, coordinated, and financed across responsible institutions. Methods We develop a three-actor linear-quadratic differential game with gross event heat-risk pressure and accumulated adaptive capacity as state variables. Open-loop Nash, government-led Stackelberg, and cooperative solutions are evaluated under four normalized heat-shock structures. Early-warning lead time enters through a piecewise information structure, and a budget-balanced cost-sharing rule supports decentralized implementation of the cooperative path. Results In the baseline late-peak scenario with a lead time of three normalized units, cooperation reduces modeled social loss by 21.9% relative to Nash and 23.1% relative to government-led Stackelberg; peak residual heat-risk pressure falls by 28.0% and 28.7%, respectively. Under the baseline parameterization, social loss in the government-led regime exceeds the Nash value by 1.6%. Earlier warning lowers loss, residual pressure, and peak effort, although its marginal value declines once a usable preparation window is available. Across the tested combinations of warning lead time and organizer mitigation efficiency, cooperative social gains range from 16.7 to 24.4%. Within the model, the effective preparation-window threshold—defined as the first normalized lead time capturing 80% of the cooperative warning benefit available between L = 0 and L = 6—ranges from 2.2 to 2.6 and responds mainly to heat-shock duration. Cooperation also shifts effort toward organizer-controlled measures, while cost sharing compensates the organizer for carrying greater responsibility despite receiving the smallest uncompensated private gain. Discussion The Nash—Stackelberg ordering is conditional on the government-first sequence, actor-specific objectives, and parameter values used in the analysis. Within the normalized model, these findings indicate that coordinated extreme-heat adaptation can benefit from aligning actor objectives and linking warning information to actionable time, institutional capability, and financing. Event-specific application requires empirical calibration.