Amos Tversky (1969), Psychological Review 76(1):31–48 — DOI-10.1037/h0026750
Key takeaways
Registered, not read. The model is given as a later source states it.
- The lexicographic semiorder. In Manzini and Mariotti’s statement of it, “a preference is generated by the sequential application of numerical criteria, by declaring an alternative x better than an alternative y if the first criterion that distinguishes between x and y ranks x higher than y by an amount exceeding a fixed threshold”. With a zero threshold it is the ordinary lexicographic order; with a positive one it is Luce’s semiorder (LIT-1460) placed in front of a second criterion.
- Why it cycles. Take options x₁, …, xₙ in which each step lowers the first criterion by less than the threshold and raises the second. Each adjacent pair is then decided by the second criterion, so x₂ beats x₁, x₃ beats x₂, and so on. Across the whole chain the first-criterion difference passes the threshold, so x₁ beats xₙ. That is a cycle. The construction is the record’s, from the definition; the paper’s own experiments are reported (by search summaries, not read) to use risky gambles built in this way, small steps in probability against steps in payoff.
- Contested. Birnbaum and Gutierrez (2007) and Regenwetter, Dana and Davis-Stober (2011) are reported to find little reliable individual intransitivity on retest. Neither was read.
Standing in the record
Filed on 2026-10-23 for the correction to CLAIM-436 and CLAIM-435. Does Tversky’s lexicographic semiorder match the “threshold override”? Only in one of two readings, and the claims should keep them apart.
- A level threshold is part 3a’s construction. Above a cost c* the alarm ranks by cost alone; below it a continuous trade-off decides. That is a weak order and has a (discontinuous) representation, as part 3a shows. It is not a semiorder.
- A discrimination threshold is Tversky’s construction. The alarm decides between two options when their costs differ by more than ε, and otherwise the resource decides. That is a lexicographic semiorder: its indifference is intransitive, its strict preference can cycle, and it is not a weak order. Where it cycles it has no numerical representation even on a finite set: neither a utility (x ≻ y when u(x) > u(y)) nor Luce’s threshold kind (x ≻ y when u(x) > u(y) + δ), since either form makes strict preference acyclic. It is not even a semiorder then, a semiorder’s strict part being acyclic.
ALARM’s “100% weight” (THEORY-090) can be read either way. The two readings predict different things in the trade-off experiment CLAIM-435 proposes. A level threshold predicts a discontinuity at c* that does not move with the resource. A discrimination threshold predicts no fixed discontinuity, but cycles among options whose costs step by less than ε while their resources step up. So the test should include a Tversky-style chain as well as the indifference curve. That needs a within-subject design with repeated pairwise choices, and the contested replication record warns how hard such cycles are to establish.
No instruction for machine-learning practice; nothing here belongs in the anthology.