Abstract
Arrow’s theorem poses limits to the translation of the different preference orders on a set of options into a single preference order. In this paper, I argue, against opinions to the contrary, that Arrow’s theorem applies fully to multi-criteria decision problems as they occur in engineering design, making solution methods to such problems subject to the theorem’s negative result. Discussing the meaning and consequences for engineering design, I review the solution methods to such problems presented in the engineering design literature in the light of the theorem. It appears that underlying such methods is a mix-up of two fundamentally different problem definitions, as the theory of multi-attribute preferences, which is often presented as an adequate approach for engineering design, in fact fails to address the Arrowian multi-criteria problem. Finally, I suggest ways how engineering design might adopt results from discussions of Arrow’s theorem elsewhere in resolving its multi-criteria decision problems.
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Notes
In this paper, I have reserved the word ‘utility’ for the cardinal value concept operationalized by introducing probabilities, as axiomatized by Von Neumann and Morgenstern or in a more fundamental way by Savage, since this is how it is usually understood in engineering design. For the broader case, irrespective of whether it concerns an ordinal value concept, i.e. measurable on an ordinal scale, or a cardinal value concept, i.e. measurable on an interval scale, I have used the term ‘preferential value’ or simply ‘value’
Note that the same approach would be conceivable in social choice as well: the votes concerning a particular issue might be considered fixed by psychological parameters, and once the votes were cast, an ad hoc solution, valid exclusively for the subset of the domain of all conceivable alternatives consisting of the alternatives at issue, might be ‘generated’, by some arbitrary means. Few would consider this a fruitful approach to the problem of social choice.
The zero points of each scale can be shifted arbitrarily and independently.
A change of scale for either the single-criterion value function or the global value function must then also change the numerical value of the corresponding weight factor.
Note that we cannot think of the concatenation of two design options as a kind of merger of the two into a new option that takes, for instance, for each attribute the preferred value from either of the two original options. In that case, the result of concatenating an option with a copy of itself would be identical to the single original option, whereas the ‘doubling character’ of concatenating two equivalent items is crucial in the construction of a ratio scale.
No two authors refer to it under the same name, however. Other designations are multi-attribute decision analysis, multi-attribute utility theory, multi-objective decision-making and also, deplorably enough, multi-criteria decision-making and multi-criteria evaluation theory.
When the method is applied to concrete cases (e.g. Thurston and Liu 1991; Locascio and Thurston 1992), the global utility function is arbitrarily assumed to be sometimes additive and sometimes multiplicative, without any empirical assessment of the conditions that decide on the applicability of either form.
Cf. the way Krantz et al. (1971, p 141) introduce the underlying theory of conjoint measurement: the measurement of the attributes on an interval scale is what the procedure is aiming for, and the existence of a global order is instrumental to this goal.
Unless, of course, like profiles turn out not to be treated alike, such that no anonymous and neutral decision function is constituted, as will be the case when preference intensities per criterion are available. But then by the same argument a decision function with interval-scale input would be constituted, which would be subject to the Sen/Osborne/Kalai and Schmeidler impossibility theorems.
I am not denying, of course, that the theory of multi-attribute preferences can be fruitfully put to work here.
This preference order is not explicitly part of Pugh’s method, but its existence must be presumed in order to understand how the designer arrives at the comparative judgments represented by the +’s, –’s and S’s. However, it could be the case that at the start of the evaluation process, the designer is not even sure about his or her single-criterion preference orders, let alone the global order that furnishes the overall best design concepts. In that case, the method of evaluation matrices additionally covers a preliminary stage of the multi-criteria evaluation process.
Kemenyi’s method uses the so-called outranking matrix (a kl) belonging to a profile P of individual orders o j over a set of options a i , where matrix element a kl is the number of orders in the profile that rank a k ≻ a l , to construct a measure for the proximity of an arbitrary order O to the profile P and to identify as the global order the order that has, by this measure, the greatest proximity to P.
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Acknowledgements
I express my gratitude to Louis Bucciarelli, Peter Kroes, Ibo van de Poel, Jeroen de Ridder, Michiel Brumsen, Sabine Roeser en Martin Peterson for their comments on the earlier versions of this paper.
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Franssen, M. Arrow’s theorem, multi-criteria decision problems and multi-attribute preferences in engineering design. Res Eng Design 16, 42–56 (2005). https://doi.org/10.1007/s00163-004-0057-5
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DOI: https://doi.org/10.1007/s00163-004-0057-5