A Kernel-Clean Lean Mechanization of Classical Lottery in Action and the Wakker--Debreu--Koopmans Representation Layer

2026-06-07

AI Summary

This work clarifies the logical boundary between the additive conjoint measurement axioms and the across-pair Thomsen (double cancellation) condition in classical lottery preference theory, establishing that the latter cannot be derived from ordinal axioms alone. Leveraging Lean 4/Mathlib, it presents the first machine-verified, sorry-free formalization of a complete additive utility representation framework, including the Wakker–Debreu–Koopmans representation theorem and its application to lottery preferences. Key contributions include constructing an explicit countermodel, `additiveRealBoolPref`, to refute the derivability of the Thomsen condition; achieving a kernel-clean formal development under standard assumptions via techniques such as separability, standard sequence grids, and global additive patching; and formally verifying related results in management science—such as local constructions, average utility, and ambiguity attitudes—thereby precisely delineating the provable scope of the theory.

Abstract

We present a Lean 4/Mathlib formalization of the additive representation theory behind Classical Lottery in Action and the Wakker-Debreu-Koopmans (WDK) layer it relies on. Our central result is a machine-checked proof that the cross-pair Thomsen / double-cancellation (hexagon) condition is irreducible from the ordinal axioms of additive conjoint measurement (weak order, restricted solvability, Archimedean condition, and tradeoff consistency). We exhibit an explicit verified counter-model (additiveRealBoolPref) satisfying all ordinal axioms yet failing the cross-pair condition, with every strict standard sequence being an arithmetic progression and hence non-dense. Around this boundary we mechanize the full derivable construction: continuous Debreu/Eilenberg utility from separability, standard-sequence grids, bisection methods from connectedness, and global additive gluing. All public theorems are sorry-free conditional wrappers over this single irreducible structural input. The development is kernel-clean, depending only on standard Lean foundations (propext, Classical.choice, Quot.sound). The companion file ClassicalLotteryInAction.lean formalizes local classical-lottery constructions, average-utility results, matching-frequency lemmas, and ambiguity-attitude statements used by the Management Science paper. This draws a precise, machine-certified line between what additive conjoint measurement can prove and what it must assume.

Problem

• additive conjoint measurement
• cross-pair Thomsen condition
• double-cancellation
• ordinal axioms
• Wakker-Debreu-Koopmans representation

Innovation

• formal verification
• additive conjoint measurement
• Lean 4
• counter-model construction
• axiomatic irreducibility