Thermodynamics of Cooperation: Necessary Constraints

We derive the constraints that necessarily govern resource allocation among multiple self-maintaining open thermodynamic systems sharing a finite-resource environment. Starting from the Second Law of Thermodynamics and the condition that each system actively maintains its structural boundary, we prove that mutual boundary constraints are necessary for any feasible multi-agent allocation to exist. Each constraint carries a computable shadow price quantifying its energetic cost. Using the Generalized Nash Equilibrium Problem (GNEP) framework with thermodynamic cost functions, we show that the constrained system admits a variational equilibrium with uniform shadow prices. We prove that removing constraints under resource collision admits no stable allocation, that the constrained equilibrium is welfare-superior to unconstrained alternatives, and that constraint value scales monotonically with scarcity. Worked examples with asymmetric agents demonstrate that the variational equilibrium outperforms both greedy and equal-split allocations, with welfare gains scaling with agent heterogeneity and resource scarcity.