Abstract
Most pairwise comparison (PC) methods typically require the explicit elicitation of only half of the comparisons, and infer the rest by assuming reciprocity in the decision maker’s comparisons. However, this may imply losing useful information contained in the additional comparisons that could be made, and which might be different from the first ones. This study assesses how relevant the lack of reciprocity may be in an experimental setting, and to what extent the information included in the additional comparisons may influence results. Our experiment shows that decision makers display substantial levels of irreciprocity and inconsistency, and that they generally prefer preference vectors calculated without assuming reciprocity in their comparisons. According to our results, our main conclusion is that, in general, decision makers should be requested all the comparisons in a PC matrix.
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Notes
Following the general convention in the literature we call a pairwise comparison matrix \(\hbox {A}=[a_{ij}]\) reciprocal if \(a_{ij}=1/a_{ji}\), for every \(i,j= 1,\ldots ,n\); and transitive (or consistent) if \(a_{ij}^{{*}}a_{jk}=a_{ik}\), for every \(i,j,k=1,\ldots ,n\). Reciprocity is a necessary but not sufficient condition for transitivity.
Although of course this introduces a certain selection bias, it is not larger than the one resulting from using students as subjects, as is usual in the literature on decision theory.
Given that the respondents were familiar with AHP and knew that reciprocity is desirable in pairwise comparisons, they would probably make efforts to avoid it to the extent possible. To prevent possible biases, and in addition to elicitating the comparisons in a random order, the respondents were not informed of the goal of our experiment until their participation had been completed.
Of course, all decision problems are subjective in nature. However, some comparisons (between colors, or designs, for example) imply a more emotional component than others (for instance, fuel consumption).
This includes also the order in which the problems were presented, again, to remove any learning effects that might confound the results.
References
Barzilai, J., Cook, W. D., & Golany, B. (1987). Consistent weights for judgments matrices of the relative importance of alternatives. Operations Research Letters, 6, 131–134.
Bryson, N. (1995). A goal programming method for generating priority vectors. The Journal of the Operational Research Society, 46, 1226–1232.
Diaz-Balteiro, L., Gonzalez-Pachon, J., & Romero, C. (2009). Forest management with multiple criteria and multiple stakeholders: An application to two public forests in Spain. Scandinavian Journal of Forest Research, 24(1), 87–93.
Farkas, A., Lancaster, P., & Rozsa, P. (2003). Consistency adjustments for pairwise comparison matrices. Numerical Linear Algebra with Applications, 10, 689–700.
Finan, J. S., & Hurley, W. J. (1997). The analytic hierarchy process: Does adjusting a pairwise comparison matrix to improve the consistency ratio help? Computers and Operations Research, 24, 749–755.
Fishburn, P. C., & LaValle, I. H. (1988). Context-dependent choice with nonlinear and nontransitive preferences. Econometrica, 56, 1221–1239.
Fülöp, J., Koczkodaj, W. W., & Szarek, S. J. (2012). On some convexity properties of the least squares method for pairwise comparisons matrixes without the reciprocity condition. Journal of Global Optimization, 54, 689–706.
Gass, S. I. (1998). Tournaments, transitivity and pairwise comparison matrices. The Journal of the Operational Research Society, 49(6), 616–624.
González-Pachón, J., & Romero, C. (2004). A method for dealing with inconsistencies in pairwise comparisons. Europeal Journal of Operational Research, 158, 351–361.
Hovanov, N. V., Kolari, J. W., & Sokolov, M. V. (2008). Deriving weights from general pairwise comparison matrices. Mathematical Social Sciences, 55(2), 205–220.
Kendall, M. G., & Smith, B. B. (1940). On the method of paired comparisons. Biometrika, 31(3/4), 324–345.
Koczkodaj, W. W., & Orlowski, M. (1999). Computing a consistent approximation to a generalized pairwise comparisons matrix. Computers & Mathematics with Applications, 37(3), 79–85.
Linares, P. (2009). Are inconsistent decisions better? An experiment with pairwise comparisons. European Journal of Operational Research, 193, 492–498.
Linares, P., & Romero, C. (2002). Aggregation of preferences in an environmental economics context: A goal-programming approach. Omega, 30, 89–95.
Saaty, T. L. (1980). The analytic hierarchy process. New York: Mc Graw-Hill.
Saaty, T. L. (1994). Highlights and critical points in the theory and application of the analytic hierarchy process. European Journal of Operational Research, 74(3), 426–447.
Salo, A. A., & Hämäläinen, R. P. (1997). On the measurement of preferences in the analytic hierarchy process. Journal of Multi-Criteria Decision Analysis, 6, 309–319.
Slovic, P., & Lichtenstein, S. (1983). Preference reversals: A broader perspective. American Economic Review, 73, 596–605.
Thurstone, L. L. (1994). A law of comparative judgment. Psychological Review, 10(2), 266–270.
Tversky, A. (1969). Intransitivity of preferences. Psychological Review, 79(1), 31–48.
Tversky, A., & Kahneman, D. (1974). Judgement under uncertainty: Heuristics and biases. Science, New Series, 185(4157), 1124–1131.
von Winterfeldt, D., & Edwards, W. (1986). Decision analysis and behavioural research. New York: Cambridge University Press.
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Linares, P., Lumbreras, S., Santamaría, A. et al. How relevant is the lack of reciprocity in pairwise comparisons? An experiment with AHP. Ann Oper Res 245, 227–244 (2016). https://doi.org/10.1007/s10479-014-1767-3
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DOI: https://doi.org/10.1007/s10479-014-1767-3