Decision-Theoretic Ecological Optimization of the Curzon-Ahlborn Heat Engine

Finite-time heat engines must balance useful power output against irreversibility. We develop a decision-theoretic framework for ecological optimization of the endoreversible Curzon-Ahlborn engine based on its normalized power-loss Pareto frontier. We first examine bargaining-based selection rules. For Newtonian heat transfer, a symmetric Nash compromise between retained power and reduced exergy destruction selects a unique interior operating point with efficiency ηN=1-τ3/4, where τ=Tc/Th is the cold-to-hot reservoir temperature ratio. The same Newtonian state is recovered by the Arias-Hernández-Angulo-Brown prescription, which fixes the ecological weight from the maximum-power reference. Our contribution is therefore not a new efficiency law but a decision-theoretic interpretation of the established 3/4 law as a distinguished compromise between maximum-power operation and the reversible Carnot limit. Under the same normalization, the Kalai-Smorodinsky and egalitarian selectors also identify the same balance point. We then extend the bargaining analysis beyond Newton's law of cooling and compare the Nash continuation with the Arias-Hernández-Angulo-Brown Eϵ - Cϵ prescription. Although coinciding in the Newtonian limit, the two approaches separate at first order when the heat-transfer law is perturbed, thereby distinguishing their underlying selection principles. This comparison raises a further question: how should an operating state be chosen when the relative priority assigned to power and irreversibility reduction is itself uncertain? We address this question through minimax-regret optimization, which selects the operating point that minimizes the largest loss relative to the optimum that would have been chosen had the priority been known. Under complete normalized uncertainty in the ecological priority, the regret-robust selector again yields η=1-τ3/4, giving the same operating state a complementary interpretation in terms of robustness to priority uncertainty.