Pricing the incommensurable: sign, rank, and the welfare metric under pluralist objection

Beta-W orders system loss per revenue dollar without treating protected goods as saleable

SAPM narrows what beta-W claims. It signs and ranks system-welfare effects per audited annual revenue, while values that cannot be priced without loss move into constraints or flags.

• The causal metric is beta-W(a) = -dW(a) / dPi(a), with dPi defined as audited annual revenue.
• A welfare loss has positive beta-W because the metric reverses the sign of dW.
• The paper rejects strong commensurability and keeps protected values in K(a), outside the price term.

The objection targets the social meaning of pricing

The paper treats the pluralist objection as a real constraint on welfare metrics, not as discomfort with uncertainty. The problem is that pricing can change the good being measured.

• Incommensurability: friendship, citizenship, bodily integrity, sacred land, and ecological inheritance are governed by different norms.
• Corruption: payment can replace civic, moral, or relational reasons with market reasons.
• Category error: a named child, voting right, or sacred relation is misdescribed when a dollar figure is offered as full value.

SAPM survives because beta-W does not require full monetization

The paper locates the weak premise in the anti-pricing argument: beta-W uses monetary units for measurable welfare effects, but it does not require every plural good to become a price.

• The rejected premise is that a monetary measure of system welfare change requires commensurating all plural goods.
• A partial shadow price can sign and rank a measurable effect without authorizing the harm.
• Contingent valuation is not used as a universal method when asking for payment would itself distort the good.

The flawed game excludes the system from enforceable payoffs

System-Excluded Bilateral Pricing has three players: firm, counterparty, and system. The firm and counterparty transact inside private payoffs while the system supplies health, trust, public order, ecology, democratic capacity, and option value without a payoff term.

• MST states that when W is outside the payoff function, local optimization can create private gains while degrading the system.
• Pluralist goods are especially likely to be omitted because ordinary transactional machinery has no clean way to represent them.
• The Hollow Win appears when welfare pricing is added but still authorizes degradation because only the measurable residue was priced.

Every material system effect gets a decision status before authorization

Decision Accounting inserts system welfare into the record before capital, licensing, disclosure, or regulatory action proceeds. It separates measured welfare loss from protected-value constraints.

• If an effect can be measured without category error, measure it.
• If it can be signed but not measured, state the sign and evidence.
• If it can be ranked but not priced, rank it against relevant alternatives.
• If pricing would corrupt the good, carry it as K(a) or an override flag.

The decision score has a welfare term and a protected-value term

The paper gives a formula for executing Decision Accounting rather than only describing it: Decision Score(a) = Pi(a) - Cp(a) - L(a) - lambda dW_hat(a) - K(a).

• Pi is annual revenue, Cp is private cost, and L is legally internalized liability.
• lambda is the institutional weight assigned to measured system loss.
• K changes the decision class when a non-compensable site, right, institutional norm, or ecological threshold is triggered.

The paper separates description, monetization, ranking, compensation, and authorization

The Sign-Rank Theorem concerns ranking. It does not say that every harm can be monetized, that payment repairs the loss, or that payment permits the activity.

• Description asks what changes and requires a domain-specific record.
• Monetization labels any priced component as a partial shadow price.
• Ranking orders system load without making rank a moral totality.
• Compensation is separated from valuation, and authorization is governed by protected-value constraints.

Four axioms are enough for sign and rank

The theorem proves that beta-W can sign and rank system-welfare effects without assuming a single cardinal value scale for all goods.

• A1: each effect can be welfare-improving, neutral, reducing, or indeterminate under stated evidence.
• A2: audited annual revenue is positive and observable for revenue-generating activities.
• A3: relevant pairs can be ranked, tied, or declared indeterminate per revenue dollar.
• A4: values that cannot be permissibly monetized remain in K(a) or qualitative flags.

Positive revenue preserves sign, rank, and monotonic transformations

Because revenue is positive, dividing by revenue cannot change sign. The leading negative sign turns welfare loss into positive beta-W, which lets the metric order burden per revenue dollar.

• sign(beta-W(a)) = -sign(dW(a)) for every priced component.
• For equal revenues, beta-W(ai) > beta-W(aj) iff dW(ai) < dW(aj).
• For unequal revenues, beta-W(ai) > beta-W(aj) iff dW(ai)/Pi(ai) < dW(aj)/Pi(aj).
• Any strictly increasing f with f(0)=0 preserves sign and rank.

Revenue measures activity scale better than profit

The paper uses audited annual revenue because it tracks the flow through which system load is generated. Profit is an accounting residual and can be negative while system damage remains large.

• Profit can be shifted by use, transfer pricing, depreciation, and cyclical margins.
• A loss-making activity cannot escape welfare accounting by reporting no taxable income.
• Revenue makes cross-activity comparisons about system loss intensity rather than tax profitability.

Some values can be ordered only under a non-sale condition

Protected rank lets institutions compare harms inside a protected class without implying that the good can be bought. The order helps choose designs, remedies, or prohibitions while K(a) preserves non-compensability.

• A community can rank route alternatives by harm to sacred land while rejecting sale of the site.
• A constitutional court can rank burdens on speech while rejecting payment as a remedy for censorship.
• An endangered-species agency can rank habitat options while treating extinction as non-compensable.

SAPM fails if signs reverse, ranks collapse, omitted constraints reverse decisions, or adoption worsens welfare

The paper makes SAPM testable and guards against Hollow Win. The framework is wrong if the metric legitimates harms it fails to measure or if matched regimes without SAPM produce better system welfare.

• Falsification 1: beta-W signs reverse under plausible assumptions.
• Falsification 2: ordinal ranks fail under sensitivity analysis.
• Falsification 3: later inclusion of protected values reverses an approval made under beta-W alone.
• Falsification 4: SAPM adoption worsens system welfare relative to matched decision regimes.