PSDA.6:4.5 - Construct preferences from comparisons of consequences

Ask the value holder to compare concrete consequence profiles, with the same people, horizon and conditions in view. Record whether one profile is no worse, strictly better, or indifferent to another, or whether the holder leaves the comparison open. Retain the profiles and the reason given. An open comparison imposes no equality. Use actual alternatives or understandable hypothetical profiles within the declared ranges; a fictitious combination is useful only if the holder can judge its meaning.

The numerical construction below concerns preferences over known consequences. Preferences over uncertain outcomes also need a suitable account of risk and uncertainty; the deterministic trade-off alone does not supply it. Start with §4.5.1 to choose the form, then use the branch whose conditions the value account supports.

PSDA.6:4.5.1 - Choose and challenge the representation

Separate protected conditions under §4.4 before considering compensation. Then ask what the preference must preserve. Choose the least burdensome form that preserves the distinctions needed by the receiving comparison. These are usable alternatives, not stages every decision must pass through:

For a proposed additive form, compare profiles that differ on a subset of criteria while holding the others equal. Repeat the comparison at materially different common levels of those other criteria. A reversal challenges preferential independence: the preference over that subset depends on its background. Ask also about complementarities, thresholds and refusals to compensate. With two criteria, unchanged directions of preference alone give little support for addition; ask about the exchanges themselves. Statistical independence of measurements answers neither question.

For an affine form, ask the stronger question: does the holder value the same physical increment equally at different positions on each scale, and does the compensating amount remain the same at different starting profiles? Within a one-to-five-hour range, compare reducing two hours to one with reducing five hours to four, and repeat a cost-for-time exchange at a different cost level. A stated constant exchange is a substantive modelling assumption over that range. Finite elicitation can challenge it and check its local fit; it does not prove a universal preference law. Preserve that qualification when the assumption is accepted for use, and return the unresolved form choice when it is not.

Keep materially different holders’ accounts separate. Combining them requires a legitimately governed aggregation or reconciliation Method. Neither averaging nor choosing the sponsor’s account follows from the construction below.

PSDA.6:4.5.2 - Construct a family for a constant exchange

Suppose lower cost c and lower waiting time t are preferred, both matter, and the holder accepts the affine conditions above on the named ranges. Represent loss by L(c,t) = a c + b t, with a > 0 and b > 0; lower loss is preferred. A common constant and positive scale do not change the ordering. Divide by a to obtain L_r(c,t) = c + r t, where r = b/a > 0. The unit of r is cost units per hour. It belongs to this holder and use, rather than being a measured market price.

For each statement that profile (c_x,t_x) is no worse than (c_y,t_y), write c_x + r t_x <= c_y + r t_y. Use equality for stated indifference and a strict inequality for stated strict preference. Keep every statement on the same r and intersect its restrictions with r > 0. The resulting set R describes every admitted exchange in this form. A point selected for a convenient spreadsheet is additional preference information unless the holder actually supplied it.

Ranges still matter when scales are normalized. If the cost range is 20–60 thousand and the waiting range is 1–5 hours, normalized losses are (c-20)/40 and (t-1)/4. A weighted sum with positive weights w_c + w_t = 1 represents the same exchange only when r = 10 w_t/w_c. The weights compare the two whole range changes. A different range changes that correspondence; a change in the holder’s marginal values can require a different function as well. Unit conversion alone must leave the represented comparisons unchanged.

PSDA.6:4.5.3 - Retain unknown marginal values when linearity is unwarranted

For a finite comparison with an admitted additive form, use the distinct consequence levels appearing in the reference profiles and live alternatives. Order each criterion’s levels from least to most preferred. Assign an unknown value contribution q_i(z) to each level z of criterion i. Set the least preferred contribution on each criterion to zero, require the contributions to be nondecreasing along that criterion, and set the sum of the most preferred contributions to one. This normalization requires at least one valued improvement across the represented ranges; it fixes an arbitrary origin and unit without fixing relative criterion values. If the holder is indifferent across every represented profile, return that relation directly. If whether any difference matters is unresolved, retain that gap rather than normalizing it away.

Define U(x) = sum_i q_i(x_i), with higher value preferred. Each no-worse statement x versus y adds U(x) >= U(y); indifference adds equality; strict preference adds U(x) > U(y). Add elicited bounds or comparisons of value differences only with their own meaning and source. All rows constrain the same contributions. Solving them jointly supplies a family of additive value functions, rather than independently chosen intervals for each criterion. Weak monotonicity permits a flat contribution; require strict increase only where the holder’s judgement supports it.

This construction leaves unevaluated levels open. Add a new level with its monotonicity and preference constraints, or use an explicitly justified interpolation. Equal spacing on an ordinal performance scale is not equal spacing in value. A family that is feasible at the represented levels establishes compatibility there; use beyond them needs the stated shape or new elicitation.

The constraints are linear in the unknown contributions. Use direct algebra for a small family or a linear-programming solver for the finite system. When strict comparisons are present, a solver can replace each strict difference by a common positive slack epsilon and maximize that slack under the normalization: a feasible result with epsilon > 0 witnesses joint strictness. A fixed arbitrary margin can exclude compatible models. Preserve weak equalities and numerical error limits when interpreting a computation. MMP.10 and MMP.11 supply the general formulation, representation and obtaining-method work; the consequence meanings and preference assumptions supplied here govern this instance.

PSDA.6:4.5.4 - Distinguish an open answer from a failed construction

First check whether at least one model satisfies the joint constraints. For a small interval family, exhibit an admitted parameter; for a larger system, obtain a feasible witness or a justified infeasibility result. A solver timeout leaves this question unanswered.

If the family is empty, locate a conflicting subset of the elicited rows and form assumptions. Check transcription, units, profile meanings and attribution with the holder. Two opposite weak comparisons may express indifference; they are not a contradiction. A strict preference for x over y together with y no worse than x under the same conditions conflicts with the claimed preference relation. Other failures can arise because the chosen form cannot express otherwise coherent judgements. Do not resolve either failure by silently deleting a row or returning the best fitting weights. Revise a judgement only with its holder, change the form with an explicit basis, or return the unresolved conflict. An approximate fit needs an admitted error interpretation before it can support advice.

For a nonempty family, distinguish what all compatible models support from what at least one supports:

• x is necessarily no worse than y when every compatible model gives that weak preference.
• x is possibly no worse than y when at least one compatible model gives it.

Test these claims on the joint family. For the loss family, necessary preference requires L_r(x) <= L_r(y) for every r in R; a possible preference needs one such r. A proof over the range establishes necessity; one feasible witness establishes possibility and one contrary witness defeats necessity. Use MATH.20 for the required whole-domain bound. Separate strict preference from equality at a boundary. An empty family warrants neither kind of recommendation.

If the question asks for a best alternative, keep the quantifiers together. A necessary best is no worse than every rival in every compatible model. A possible best is no worse than every rival in one and the same compatible model. Separate pairwise witnesses against different rivals do not establish possible best. Return a common best when supported, the condition-dependent alternatives otherwise, or the precise comparison gap. No probability distribution over the compatible models is implied.

PSDA.6:4.5.5 - Reopen only what the changed judgement affects

A further judgement by the same holder on the same profiles adds a constraint to the existing family. Intersect, check nonemptiness again, and recalculate the dependent comparisons. A correction replaces its old row and every restriction derived from it; retaining the superseded row could manufacture a conflict. Preserve consequence measurements, protected conditions and comparisons whose grounds remain valid.

A changed holder, horizon, range, consequence meaning or compensation rule can invalidate the representation itself. Reconsider that premise before intersecting parameters in the old form. State which results depend on it and which remain usable. Stop when the receiving comparison is settled to the needed strength or the missing judgement, form or authority is clear enough to obtain.