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Theory and Decision
Published in: Theory and Decision 3/2017

26-04-2017

A multiattribute decision time theory

Author: Nobuo Koida

Published in: Theory and Decision | Issue 3/2017

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Abstract

In this study, we analyze choice in the presence of some conflict that affects the decision time (response time), a subject that has been documented in the literature. We axiomatize a multiattribute decision time (MDT) representation, which is a dynamic extension of the classic multiattribute expected utility theory that allows potentially incomplete preferences. Under this framework, one alternative is preferred to another in a certain period if and only if the weighted sum of the attribute-dependent expected utility induced by the former alternative is larger than that induced by the latter for all attribute weights in a closed and convex set. MDT uniquely determines the decision time as the earliest period at which the ranking between alternatives becomes decisive. The comparative statics result indicates that the decision time provides useful information to locate indifference curves in a specific setting. MDT also explains various empirical findings in economics and other relevant fields.
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Appendix
Available only for authorised users
Footnotes
1
The decision time is also referred to as the response time, reaction time, and contemplation time.
 
2
Spiliopoulos and Ortmann (2015) provided a comprehensive survey on the economic analyses of decision time in other contexts.
 
3
Because we consider the set of all Borel probability measures over X, our framework may appear to permit a correlation among attribute values. However, the following analysis of this study discards this possibility (i.e., it identifies two alternatives that have the identical marginal probability distributions of all \(X_i\)) because we impose the axioms called independence and single-attribute regularity. Alternatively, we may consider \({\mathcal {P}}(X_1) \times \cdots \times {\mathcal {P}}(X_n)\), that is, the product set of Borel probability measures over each attribute set \(X_i\), as the domain of choice, which would simplify some of our analyses. However, for reasons of generality, we assume \({\mathcal {P}}(X)\) as our domain of choice.
 
4
We require Axiom 4(a) because the preferences are possibly incomplete. This statement trivially holds with complete preferences.
 
5
Under the consistency axiom, Definition 2(b) is equivalent to the statement that “\(\tau ^*(p, q)\) equals the smallest non-negative integer \(\tau \) such that \(p \succsim ^{\tau } q\) or \(q \succsim ^{\tau } p\).”
 
6
Chapter 2 of Keeney and Raiffa (1993) discusses multiattribute utility models in a similar form.
 
7
Consistency implies that \(\cap ^{\infty }_{\tau = 0} \varLambda ^\tau \) is nonempty.
 
8
Such alternatives p and q are indifferent once the ranking between them has become decisive, which explains the terminology of the long-run indifference curve.
 
9
A similar argument can hold for the case \(n > 2\), under the assumption that the considered alternatives have the identical marginal probability distributions in all but two attributes.
 
10
If \(\varLambda ^\tau \) comprises circular cones, that is, \(\varLambda ^\tau = \{\lambda \in \varDelta ^{n-1} : d^\tau \le \frac{\lambda \cdot \lambda ^\infty }{\Vert \lambda \Vert \Vert \lambda ^\infty \Vert } \le 1 \}\) for all \(\tau \) and some \(d^\tau \in [0, 1]\) such that \(d^\tau \le d^{\tau '}\) for all \(\tau \le \tau '\), the contraction of set \(\varLambda ^\tau \) is uniform in the sense that each \(\varLambda ^\tau \) consists of all the vectors in \(\varDelta ^{n-1}\) that form an angle with \(\lambda ^\infty \) that is less than a certain value. By assuming such a case, statements (a) and (b) in Theorem 3 hold for non-congruent alternative pairs.
 
11
For simplicity, we assume two calendar dates and exponential discounting, although the conclusion of this example can be extended to models with three or more calendar dates and hyperbolic discounting, by increasing the number of attributes (i.e., calendar dates) and setting an appropriate discounting rate for each of them.
 
12
A simple calculation yields that \(\tau ^*(x, y)\) equals the smallest integer larger than or equal to \((1-2\epsilon )/2\epsilon \) for the second group, which is strictly greater than zero if \(\epsilon < 0.5\).
 
13
They also considered strategic processes, which analyze the structure of a game, and the motor function response, which actually implements the act of choice. However, a discussion of these processes is omitted here because the former is irrelevant to our individual choice framework and the latter is a non-decision process.
 
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Metadata
Title
A multiattribute decision time theory
Author
Nobuo Koida
Publication date
26-04-2017
Publisher
Springer US
Published in
Theory and Decision / Issue 3/2017
Print ISSN: 0040-5833
Electronic ISSN: 1573-7187
DOI
https://doi.org/10.1007/s11238-017-9601-4

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