Rationality is Temperature

"The question is not whether agents are rational. The question is: how rational, at what cost, and measurable from what data?"

Standard economic models treat agents as perfectly rational: they maximise utility exactly, choose the unique best option, and never make mistakes. This is the limit \(\beta \to \infty\) — infinite inverse temperature, zero decision noise, pure argmax.

EconIAC replaces argmax with a single parameter:

\[Z_\beta = \frac{1}{\beta} \ln \sum_i e^{\beta U_i}\]

\(\beta\) is the inverse temperature of rationality — the inverse of decision noise. High \(\beta\): agents behave nearly classically. Low \(\beta\): agents choose more randomly, weighted by utility but not determined by it.

What this one substitution buys

The classical argmax is a step function. It is not differentiable, not calibratable from data, and not continuous in its parameters. The Gibbs softmax \(e^{\beta U_i} / Z_\beta\) is all three.

Classical model	EconIAC equivalent	What changes
Argmax (perfect choice)	Gibbs softmax	Smooth, differentiable, calibratable
Hard threshold (VaR breach)	Sigmoid	Continuous tipping point; early-warning signal
Leontief minimum (production)	SoftMin	Differentiable supply chain; exact policy gradient
Nash equilibrium	Quantal response equilibrium (QRE)	Calibratable from observed choice variance
Shapley value	Thermal Shapley	One backward pass; exact attribution

The substitution is not an approximation to the classical model — it is a one-parameter family that contains the classical model as its limiting case. At \(\beta \to \infty\) you recover Nash equilibria, Leontief multipliers, and Shapley values exactly. At finite \(\beta\) you gain differentiability.

Calibrating β from data

\(\beta\) is not a free parameter to be tuned by hand. It is a physical quantity measurable from the variance of observed choices:

\[\beta^* = \frac{1}{\mathrm{Var}(\text{observed choices})}\]

More precisely: in a Gibbs ensemble, the variance of a utility-weighted observable is \(\partial^2 \ln Z_\beta / \partial \beta^2\). Inverting this gives \(\beta^*\) from any dataset of observed decisions — bids in an auction, portfolio allocations, lending choices, voting records.

This calibration has a physical interpretation: \(\beta^*\) is the point at which the system is at its empirical operating temperature. It is not the temperature at which agents are "approximately rational" — it is the temperature at which their observed behaviour is thermodynamically consistent.

The tipping point connection

Near a phase transition — a bank run, a repo freeze, a cascading default — the susceptibility

\[\chi(\beta) = \frac{\partial^2 \ln Z_\beta}{\partial \beta^2}\]

diverges. This is a computable early-warning signal: as the system approaches a tipping point, \(\chi(\beta)\) rises sharply before any individual threshold is breached.

This is the EconIAC version of the standard early-warning literature (Scheffer et al. 2009), but derived from first principles rather than fitted to historical data. The signal is exact, not heuristic, because it follows from the analytic structure of \(Z_\beta\).

Why temperature, not noise

It is tempting to read \(\beta\) as "how much noise agents have" — a behavioural friction. This misses the point.

In statistical mechanics, temperature is not noise — it is the parameter that controls the trade-off between energy (utility) and entropy (diversity of choices). A system at high temperature explores many states; a system at low temperature concentrates on the ground state. The ground state of an economic system is the Nash equilibrium or Pareto optimum; the high-temperature states are the heterogeneous, partially-coordinated behaviours actually observed in markets.

The reason EconIAC uses temperature rather than noise is that temperature is extensive and measurable: it is the same parameter that controls differentiability, calibration, tipping-point detection, and Shapley attribution. A noise parameter would give you differentiability but not the thermodynamic structure that makes \(\chi(\beta)\) an early-warning signal.

Connection to the other core ideas

Rationality-as-temperature is the foundation that makes the rest of EconIAC computable:

The three levels of risk (H⁰/H¹/H²) are measurable because \(\beta\) makes the consistency conditions differentiable.
The Pentagon identity can be tested empirically because the Gibbs lift turns hard thresholds into smooth residuals.
Clearing and netting topology changes the phase structure of the Gibbs ensemble — CCP novation changes which configurations are reachable at a given \(\beta\).