Abstract
Choosing among alternatives by proceeding lexicographically through a sequence of criteria is a common description of practical decision-making. This paper uses these sequences as a theoretical tool. First, we show that there are sequences of criteria that form reasonable decision-making procedures but that lead to preferences whose utility representations are monotone injections from ℝⁿ into ℝ and that enjoy a fractal or self-similarity property. Second, sequences of binary criteria provide a uniform measure of how concisely a preference can be represented. This measure leads to plausible conclusions about which preferences are easy to represent that differ from the longstanding tradition of simply checking if a preference has a utility representation. Finally, we generalize the classical result that preferences with a countable order-dense subset have utility representations and provide simple proofs of the extension theorems that show that strict partial orders can be extended to complete or linear orders.
Highlights
When decision-making criteria are ordered lexicographically, the first criterion that orders a pair of alternatives determines a choice or preference for that pair
I will argue that criteria provide a better measure of concision than the economics test of checking whether a preference has a utility representation
If it seems counterintuitive that there are preferences with utility representations, such as ≿ in Example 5, that we classify as hard to represent, keep in mind that length-cardinality gaps are defined relative to the number of indifference classes
Summary
When decision-making criteria are ordered lexicographically, the first criterion that orders a pair of alternatives determines a choice or preference for that pair. This paper uses minimal sets of criteria to measure how concisely preferences can be represented and how efficiently an agent can make decisions. Well-ordered sets of binary criteria eliminate this difficulty They vindicate Chipman’s position: Debreuvian lexicographic preferences show a sizable length-cardinality gap and qualify as concise. Since the utilities that arise discriminate between every pair of bundles, they define one-to-one mappings (injections) of Rn into R and generalize the injections defined by Cantor, the discoverer of the first such mappings This class of utility functions can represent ‘fractal’ preferences where the pattern of indifference classes when they are grouped together coarsely matches their pattern when grouped together finely (Mandler 2020c). I segregate the abstract applications of ordinals to the last two sections of the paper
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