Imagine you’re deciding between three lunch options: a burger, a salad, or pasta. You prefer the burger to the salad, the salad to the pasta, but somehow the pasta seems better than the burger when you compare them directly. Something doesn’t add up, right? This puzzling situation where preferences form a circle rather than a clear ranking is exactly what economist William Gorman identified in welfare economics-a problem that shook the foundations of how economists evaluate social welfare.
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The pursuit of a perfect welfare test
In the early twentieth century, welfare economists were searching for an objective way to determine whether economic policy changes would improve society’s wellbeing. The challenge was thorny: how do you compare situations where some people gain while others lose? Nicholas Kaldor and John Hicks proposed compensation criteria suggesting that if winners could theoretically compensate losers and still be better off, the change would increase social welfare.
But economist Tibor Scitovsky quickly spotted a fatal flaw in this approach. He demonstrated what became known as the Scitovsky Paradox: it was possible for allocation A to be deemed better than allocation B using the Kaldor-Hicks test, but once society moved to allocation B, the same test could prove that moving back to A was also an improvement. This created a logical impossibility where both forward and backward movements could simultaneously increase welfare.
To fix this contradiction, Scitovsky developed his double criterion, which required both that gainers could compensate losers in moving from state A to state B, and that losers could not bribe gainers to prevent the change. This seemed to solve the reversal problem. Yet William Gorman would reveal an even deeper issue lurking beneath.
The emergence of intransitive chains
In his groundbreaking work, William Gorman demonstrated that even Scitovsky’s refined double criteria couldn’t prevent intransitive rankings. Intransitivity means that if you prefer option G to option D, and D to F, logic would dictate you should prefer G to F. But Gorman showed this wasn’t guaranteed under compensation principles.
Here’s how this plays out in practice. Consider three different economic allocations: G, D, and F. Using the compensation tests, society might determine that allocation G is preferred to D because winners could compensate losers. Similarly, D might be preferred to F. Common sense suggests G should therefore be preferred to F. However, Gorman proved that when applying the double criteria, it’s entirely possible that F could be preferred to G, creating an illogical chain where G beats D, D beats F, but F beats G.
This isn’t just an abstract mathematical curiosity. Imagine a government evaluating three tax policy proposals. Using compensation criteria, they might find Policy One better than Policy Two, and Policy Two better than Policy Three. But when they directly compare Policy One and Policy Three, they discover Policy Three appears superior. How can policymakers make rational decisions when their evaluation method produces circular preferences?
Why intransitivity matters for real decisions
The intransitivity problem means that the sequence in which alternatives are compared can determine which one is ultimately selected. If you evaluate proposals in the order G-D-F, you might end up choosing G. But evaluate them in the order F-G-D, and F might win instead. The final choice depends not on which option is genuinely best for society, but on the arbitrary order of comparison.
This creates opportunities for manipulation. A skilled policymaker could engineer which proposal gets adopted simply by controlling the agenda-the sequence in which options are voted on or evaluated. The “best” policy becomes whatever option benefits from favorable sequencing, not what actually maximizes social welfare.
Visualizing intransitivity with multiple production possibilities
To understand why intransitivity emerges, we need to visualize how economists map society’s choices. The Production Possibility Frontier represents all the combinations of goods an economy can produce given its resources and technology. Think of it as the menu of options available to society-if you produce more cars, you must produce fewer computers given fixed resources.
Community Indifference Curves, on the other hand, represent society’s preferences-which combinations of goods provide equal satisfaction to the community as a whole. When economists evaluate whether a policy change improves welfare, they examine whether society can move to a higher indifference curve.
With just two Production Possibility Frontiers and their associated Community Indifference Curves, Scitovsky’s double criteria could identify welfare improvements without reversals. But Gorman showed that once you introduce three or more PPFs with their corresponding indifference curves-representing different economic states or policy scenarios-the double criteria breaks down completely.
The geometry of confusion
Picture three Production Possibility Frontiers representing three different technological states or policy environments, each with its own set of Community Indifference Curves. When you compare points on the first two frontiers using Scitovsky’s double test, you might find a clear ranking. Compare points on the second and third frontiers, and again you get a definite ordering. But when you compare points on the first and third frontiers directly, the relationship may contradict the transitive ordering you’d expect.
The problem arises because the compensation test only compares pairs of points at a time. Point G on the first frontier might dominate point D on the second frontier by the double criteria. Point D might dominate point F on the third frontier. But the double criteria don’t guarantee that G dominates F. The pairwise comparisons can be consistent individually while producing an inconsistent overall ranking.
This geometric reality exposes a fundamental limitation: pairwise comparison methods cannot reliably construct a complete social ordering across multiple alternatives. You can successfully compare pairs, but assembling those pairwise judgments into a coherent ranking of all options may be impossible.
The limitation of existing criteria
Gorman’s intransitivity problem revealed a deeper flaw in the entire compensation principle framework. Economists had hoped that Scitovsky’s double criteria, by eliminating the reversal paradox, would provide a value-free, objective way to rank social states based purely on potential compensation. But intransitivity demonstrated that this approach was insufficient for guaranteeing a rational, transitive social welfare ordering across all possible economic states.
The implications were profound. If even the sophisticated double criteria couldn’t ensure consistent rankings, what hope was there for an objective welfare economics? The search for a mechanical rule that could rank social alternatives without value judgments appeared doomed. Every criterion devised seemed to harbor internal contradictions or produce nonsensical results.
Moreover, the intransitivity problem highlighted that the compensation principle-focusing on whether winners could hypothetically compensate losers-might be asking the wrong question entirely. Perhaps the issue wasn’t finding a more clever compensation test, but recognizing that welfare comparisons involving gainers and losers inevitably require explicit ethical judgments about distribution and fairness, judgments that can’t be avoided through technical wizardry.
Moving beyond mechanical rules
Gorman’s critique helped shift welfare economics away from the hope of finding purely technical, value-free criteria for welfare improvements. Economists increasingly recognized that evaluating policies affecting different people differently requires incorporating explicit social welfare functions-mathematical representations of society’s ethical priorities regarding distribution and equity.
Rather than trying to avoid value judgments, modern welfare economics embraces them transparently. When comparing policies, economists now typically specify assumptions about society’s willingness to trade off gains for some people against losses for others. These value judgments are made explicit rather than buried in seemingly neutral technical criteria.
The lesson extends beyond economics. In any collective decision involving tradeoffs among different people or groups, purely procedural rules without substantive ethical content will likely produce inconsistencies or manipulable results. Gorman’s intransitivity problem stands as a warning: there’s no escape from making moral choices about distribution and fairness.
What do you think? Can you think of real-world policy debates where different sequences of comparison might lead to different outcomes? How should societies handle situations where no clear, transitive ranking of alternatives exists?
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