3. One Shape of Choice

Imagine any chooser that lasts: a person, a firm, a community, a learning machine. It faces a space of possible configurations—actions, plans, allocations. Some look better than others. It cannot score every option with infinite precision. It chooses again and again over time.

From those facts plus consistency, the paper derives one structure. Alternatives fail stated requirements. That is why it is called forced, not fashionable.

The pieces in ordinary words

Symbol	Plain name	Role
X	Option space	Everything you might do or arrange
V	Value	How good a configuration looks for what you care about
µ (mu)	Reference / default	What you’d expect before this round’s valuing—the “background prior”
τ (tau)	Temperature / information price	How costly careful thinking is; how sharp vs noisy choice is
ρ (rho)	Choice distribution	The actual pattern of what gets chosen (not always a single point)

The forced choice law, written once so you can recognise it later:

ρ(x) is proportional to µ(x) × e^{V(x)/τ}

In words: start from the background reference. Tilt it toward higher value. The sharpness of the tilt is set by τ. When thinking is “expensive” (high τ), the background dominates and behaviour looks habitual. When thinking is “cheap” (low τ), value dominates and behaviour looks like sharp optimisation.

Four links (the chain without the full proof wall)

Link 1 — Bounded capacity forces a distribution, not a fake certainty.
If you could evaluate everything perfectly, you would pick a single best point. With limits, locking onto one option as if you knew it was globally best smuggles certainty you did not earn. Consistent behaviour under uncertainty is a probability pattern over options—weighted by what you actually know. Rational inattention economics says the same: attention is scarce; choice is stochastic.

Link 2 — Independence forces the exponential tilt.
You do not start from nothing. You bring a reference µ. Value V tilts it. When a problem splits into independent parts with additive value, consistent choice should factor into independent sub-choices—otherwise you invent correlations the information does not contain. That requirement forces the exponential form (the same mathematical family behind multinomial logit and soft policies in AI). Temperature τ is the one free scale: value per unit of “surprise” relative to the reference.

Link 3 — The cost of leaving the reference is KL divergence.
Among all ways to achieve a given average value, the forced distribution is the one that stays closest to the reference in a precise information distance called Kullback–Leibler (KL) divergence. τ is the price of that departure in value units. Shore and Johnson’s classic axioms pick KL out among divergences by the same independence logic. So “complexity cost” is not a vague metaphor; it is the dual face of the exponential.

Link 4 — Persistence needs a way to move toward the target.
A static formula is not yet dynamics. The object that drives motion is already in hand: the score—the gradient of log ρ—which splits as:

score = (value gradient)/τ + (reference gradient)

Agents are pulled toward higher value and toward denser regions of the reference (what feels “normal”). The canonical continuous dynamics is a noisy climb of that score (Langevin relaxation) whose resting pattern is exactly ρ*. Discrete cousins include logit dynamics. Score-based generative AI models train and sample with the same kind of object on different spaces—images instead of economic plans.

Two regimes

Regime	τ	What it feels like	Where classical econ lives
Dissipative (real life)	τ > 0	Noisy, habitual + purposeful, info is costly	Behavioural econ, rational inattention, real firms
Zero temperature (ideal)	τ → 0	Choice collapses onto best options in the reference’s support	Perfect rationality, much of general equilibrium

Perfect rationality is not a rival universe. It is the boundary of bounded rationality when the price of information falls to zero. Mainstream neoclassical work often lives on that boundary. Heterodox and behavioural work often lives in the warm interior. The framework does not pick a team. It locates both on one object.

Form vs content (read this twice)

The chain forces the form: there is a reference; value tilts it exponentially; dynamics follows the score.

It does not force the content of the slots. What µ, V, and τ are in a bank, a family, or a training lab is supplied by the domain. Form claims travel everywhere. Content claims—especially “µ’s social levels are doxa”—must be earned case by case.

Next: the three costs that sum into one action—being wrong, carrying a complex model, and changing—and why AI training and economic choice share an equation.