Abstract
Consistency of choice is a fundamental and recurring theme in decision theory, social choice theory, behavioral economics, and psychological sciences. The purpose of this paper is to study the consistency of choice independent of the particular decision model at hand. Consistency is viewed as an inherently logical concept that is fundamentally void of connotation and is thus disentangled from traditional rationality or consistency conditions imposed on decision models. The proposed formalization of consistency takes two forms: internal consistency, which refers to the property that a choice model does not generate contradictory statements; and semantic consistency, which refers to the idea that a theory’s predictions are valid with respect to some observed data. In addressing semantic consistency, the relationship between theory and data is analyzed in terms of so-called duality mappings, which allow a passage between the two universes in a way that consistency is preserved. The formalization of consistency concepts relies on adapting the revealed preference theory to the context-dependent setting. The paper concludes by discussing the implications of the proposed framework and how it relates to classical revealed preference theory and other formalizations of the relationship between the theory and reality of choice.
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Notes
For a discussion of a similar view, the reader is referred to Gilboa (2010).
Because this assertion is foundational in nature and its topic is, at least tangentially, tied to diverse philosophical and methodological questions regarding choice, the boundaries of the proposed definition are stressed throughout this paper. The purpose of this is not to repudiate a viewpoint, but to clarify the extent and, more importantly, the limitations of the proposed formalization of consistency.
Note that the type of deductive logic considered is formal, free of judgment, experience or perception. This coincides with Ramsey’s notion of the logic of consistency or formal logic. In his seminal work, Ramsey (1926) writes: “it seems to me, therefore, that we can divide arguments into two radically different kinds, which we can distinguish in the words of Peirce as (1) ‘explicative, analytic, or deductive’ and (2) ‘amplifiative, synthetic, or (loosely speaking) inductive’. Arguments of the second type are from an important point of view much closer to memories and perceptions than to deductive arguments. We can regard perception, memory and induction as the three fundamental ways of acquiring knowledge; deduction on the other hand is merely a method of arranging our knowledge and eliminating inconsistencies or contradictions. Logic must then fall very definitely into two parts: (excluding analytic logic, the theory of terms and propositions) we have the lesser logic, which is the logic of consistency, or formal logic; and the larger logic, which is the logic of discovery, or inductive logic”.
The terminology of semantic consistency is borrowed from formal studies of logic and model theory. Indeed, in traditional Aristotelian logic, consistency is a semantic notion: two or more statements are called consistent if they are simultaneously true in a model under some interpretation. In modern logic, internal consistency is also called syntactic consistency, and a set of statements is called syntactically consistent with respect to a certain logical calculus, if no formula and its negation are derivable from those statements by the rules of the calculus, that is, the theory is free from contradictions. If these definitions are equivalent for a logic, the equivalence amounts to the completeness of its system of rules, a notion that is far beyond the scope of this paper.
For example, in the traditional setup of mathematical finance, decision makers choose from the vector space \(\mathbb {L}^\infty \) of essentially bounded real-valued random variables on a probability space. In expected utility theory, alternatives are considered over the space of lotteries, which are formalized via probability distributions. The agent’s choice set is hence the set \(\mathcal {M}\) of all probability measures on a separable metric space. The formalization of the algebraic structure of a set relies on work by Birkhoff (1935) and is beyond the scope of this article.
Even with this distinction in mind, one may then question whether in the study of choice behavior the idea of context dependence can be used to form aggregate prediction over a collection of different menus. This is an especially relevant question in social choice theory and aggregation questions lie far beyond the scope of this paper. Note that context dependence in our framework would for example imply that weak Pareto optimality or the independence of irrelevant alternatives in social choice theory is not necessarily satisfied.
See Sen (1997) for a formalization of menu-independent preferences, menu-independent choice functions, and their relationship.
The definition of semantic consistency along with the symbol \(\models \) are inspired by the view taken in classical deductive logic and model theory, where lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e., there exists an interpretation under which all theorems in the theory are true. The syntactic definition requires that there is no theorem in the theory such that both the theorem and its negation can be derived from the theory, which corresponds to internal consistency in the choice theoretic setting.
For notational simplicity, the set \(\tau (\mathcal {C}^*)\) is overloaded to denote both the theory itself and its set of axioms.
The interested reader is referred to Denecke et al. (2004), who give an introduction to the history of Galois connections in mathematics and review some applications. Lawvere (1969) uses Galois connections to study a similar relationship between sets of axioms and classes of models in the mathematical foundations of algebraic theories and categorical logic.
A mapping \(F:A\rightarrow B\) between two partially ordered sets is a reverse-order isomorphism if it is bijective and monotonically order reversing.
Other views contend that people rely on domain-specific or content-sensitive rules of inference or on mental representations that correspond to imagined possibilities.
For example, intuition often plays an important role. An intuitive approach is not primarily based on an understanding of the assumptions and logical deduction, but on its perceived similarity to other situations, for which an appropriate behavior in known, which can be transferred to the problem.
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Mahmoud, O. On the consistency of choice. Theory Decis 83, 547–572 (2017). https://doi.org/10.1007/s11238-017-9635-7
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DOI: https://doi.org/10.1007/s11238-017-9635-7