Ranking an endless future fairly is not the same as giving it a score

An infinite future, one generation at a time

Imagine describing a possible future as an endless list of well-being levels, one for each generation. Comparing two such lists raises a question that finite accounting does not settle: can society rank every possible future consistently while respecting fairness between better-off and worse-off generations?

In an arXiv preprint, Ram Sewak Dubey and Giorgio Laguzzi investigate this question through mathematical proofs, rather than experiments or economic forecasts. Their central distinction is between a ranking, a numerical representation and an explicit construction. These are different demands, and satisfying one does not automatically satisfy the others.

A social welfare order compares every pair of infinite lists and is transitive: its comparisons cannot form contradictory preference cycles. A real-valued social welfare function assigns each list a numerical score whose comparisons reproduce the ranking. An explicit construction supplies a specified ranking rule, rather than merely proving that some suitable ranking exists through nonconstructive mathematical principles.

What fairness requires

The paper studies three redistribution principles. Strong Equity considers changes affecting only two generations: the worse-off generation gains, the better-off generation loses, and the worse-off generation remains strictly worse off afterward. Society must strictly prefer the changed future. The principle does not require the gain and loss to balance.

Hammond Equity requires only weak preference in the same comparison, allowing a tie. The paper’s strict Pigou–Dalton principle requires strict preference when the redistribution also preserves the two generations’ combined utility.

Efficiency is a separate requirement. Strong Pareto says that raising at least one generation’s well-being without lowering anyone else’s must make a future strictly preferable. Monotonicity is weaker: such a change must not make it worse.

An explicit rule built from thresholds

The authors construct a ranking satisfying Strong Equity and Strong Pareto when the allowed utility levels are well-ordered under their usual numerical order. Well-ordering means that every nonempty subset of allowed levels has a least member. Finite sets and the positive integers are examples.

Their method does not simply give the earliest generation priority. Instead, it converts each future into binary indicators recording whether each generation’s utility lies at or below a given threshold. It compares thresholds from lower to higher and, within each threshold, generations in sequence. At the first differing indicator, being above the threshold wins.

Well-ordering ensures that a first differing position exists. In a Strong Equity redistribution, the relevant first difference favors the generation lifted above its original low utility level. For a Pareto improvement, no indicator changes in the unfavorable direction.

This construction also satisfies Hammond Equity and the paper’s Pigou–Dalton principle. But an explicit comparison rule is not automatically a real-number scoring formula—or a guarantee of an executable procedure for arbitrary infinite inputs.

When numerical scores exist

For Strong Equity alone, without an added efficiency requirement, the authors give an exact existence boundary: a real-valued welfare function exists if and only if the set of allowed utility levels is countable.

Countability concerns the available levels, not the collection of infinite futures. The positive proof groups streams that eventually become identical, extends the equity comparisons within each group to a linear order, and embeds that order into the rational numbers. The construction uses nonconstructive choices in general.

For uncountably many allowed utility levels, the impossibility proof exploits the fact that every nonempty open interval of real numbers contains a rational number. The comparisons forced by Strong Equity ultimately demand more disjoint intervals than this permits.

The paper also identifies a sharper distinction for explicit numerical formulas. It provides a continuous Strong Equity welfare function on a seven-level domain. With eight levels, such a function cannot be continuous in the standard product topology, and its existence implies a non-Ramsey collection. Continuity here concerns whether fixing a sufficiently long initial segment controls the score’s sensitivity to the remaining future.

What the obstructions mean

A non-Ramsey collection classifies infinite subsets of the natural numbers so that every infinite pool contains subsets in both classes. The authors treat its existence as a marker of nonconstructiveness.

They derive such obstructions for Strong Equity plus monotonicity on domains containing an infinite strictly descending sequence. For Pigou–Dalton plus monotonicity, their theorem requires additional equal-gap structure, reflecting the conservation requirement. Strong Equity alone also yields an obstruction on domains containing an integer-like subset.

These are not blanket claims that fair rankings do not exist. They distinguish existence from explicit description under stated assumptions. The work is a theoretical arXiv preprint, not an empirical validation or a ready-made policy calculator. Its lesson is that fairness axioms, utility-domain structure and the requested form of the answer must be examined separately.