Thermal Economics: Implicit Differentiation Through Fixed Points as a Unifying Schema for Differentiable Economics

Economics is populated by combinatorial objects — the Nash equilibrium, the Leontief inverse, the Shapley value, the Gini coefficient, the value-at-risk quantile — that are exact but discontinuous. Each resists differentiation: you cannot take the gradient of argmax, and the derivative of a quantile is a Dirac mass.

This paper argues that each of these objects has a natural thermal relaxation: a one-parameter family parametrised by an inverse temperature $\beta \geq 0$ that (1) recovers the original combinatorial object in the tropical limit $\beta \to \infty$, (2) is smooth in $\beta$ and in all model parameters $\theta$ for finite $\beta$, and (3) admits an exact gradient via the Implicit Function Theorem (IFT), enabling end-to-end calibration by gradient descent. The unifying mechanism is always the same: the relaxed object is the fixed point $x^(\theta, \beta)$ of a smooth equation $F(x^, \theta, \beta) = 0$, and its gradient is $\partial x^* / \partial \theta = -(\partial F / \partial x)^{-1} \partial F / \partial \theta$, computed at the fixed point. We call this programme Thermal Economics.

Seven principal instances are developed. (1) Thermal Shapley Values: the Shapley value is replaced by a Gibbs-weighted marginal contribution, differentiable in all coalition utility parameters. (2) SoftLeontief: the Leontief input-output inverse $(I - A)^{-1}$ is replaced by a temperature-regularised resolvent, enabling gradient-based supply-chain optimisation. (3) Differentiable Nash / Quantal Response Equilibrium: the Nash fixed point is the $\beta \to \infty$ limit of the McKelvey-Palfrey quantal response equilibrium, which is smooth and IFT-differentiable at finite $\beta$. (4) SoftGini: hard rank sorting is replaced by the NeuralSort doubly-stochastic relaxation, giving a differentiable inequality index. (5) SoftVaR / SoftCVaR: the entropic risk measure $\rho_\beta(X) = \beta^{-1} \ln \mathbb{E}[e^{\beta X}]$ recovers CVaR at $\beta \to \infty$ by Varadhan's lemma. (6) Sinkhorn / Entropic Optimal Transport: the entropic regularisation of the earth-mover distance gives a differentiable market-clearing mechanism. (7) Gibbs Social Choice: the Gibbs softmax escapes Arrow's Impossibility Theorem by outputting a probability measure rather than a ranking.

The Tropical Recovery Theorem establishes that every Gibbs relaxation converges to its combinatorial limit at rate $O(\beta^{-1} \ln n)$, uniformly over the parameter space. The tropical gap quantifies the price paid for differentiability; calibrating $\beta^*$ to match empirical noise implicitly sets the gap to be of the same order as measurement error.

The inverse temperature $\beta$ is identified as the Lagrange multiplier on the mean-utility constraint in the entropy-maximisation dual of the Gibbs distribution — the shadow price of relaxing the constraint that binds in the combinatorial limit. The research programme connecting Thermal Economics to the Adelic Simplicial Architecture (Papers 289, 291, 293, 294, 300, 315, 316) is outlined. This paper is part of Portfolio G of the Adelic Simplicial Architecture research programme.