© Jonathan Barzilai 2007-2008.
This page is updated periodically. Additional details, questions and answers to follow.
The construction of the mathematical foundations of social science disciplines, including economic theory, requires the application of mathematical operations to non-physical variables, i.e., to variables that describe psychological or subjective properties such as preference.
Whether psychological properties can be measured, and hence whether mathematical operations such as addition and multiplication can be applied to psychological variables, was debated by a Committee that was appointed in 1932 by the British Association for the Advancement of Science but the opposing views in this debate were not reconciled in the Committee’s 1940 Final Report.
In 1944, game theory was proposed by von Neumann and Morgenstern as the proper instrument with which to develop the mathematical foundations of economic theory where utility theory was to be the means for measuring preference. This goal was not achieved:
The operations of addition and multiplication are not applicable on utility scale values — see The Principle of Reflection in [1, §§3.1—3.2 and 5, §2]. Variants of utility theory implicitly assume that “interval” scale type implies the applicability of addition and multiplication on utility scale values but this implication is false — see .
The interpretation by von Neumann and Morgenstern of utility equality as a true identity precludes the possibility of indifference between a prize and a lottery which is utilized in the construction of utility scales while under the interpretation of utility equality as indifference the construction of lotteries is not single-valued and is therefore not an operation (see [5, §4.4]).
As an abstract mathematical system, von Neumann and Morgenstern’s utility axioms are consistent. However, while von Neumann and Morgenstern establish the existence and uniqueness of scales that satisfy these axioms, they do not address utility scale construction. The interpretation of the utility operation which this construction requires creates an intrinsic inconsistency — see Barzilai’s Paradox.
These shortcomings of utility theory render it unsuitable to serve as the mathematical foundation of economics, decision theory, or other scientific disciplines.
No, the AHP is not a valid methodology. The proponents of the AHP should have acknowledged that credit for most of its elements is due to J.R. Miller who published his methodology in the 1960s (see [10-12]). Miller was not a mathematician and his methodology is based on mathematical errors although some of its non-mathematical elements are valuable. The AHP is based on these mathematical errors and additional ones.
As is the case for all mathematical decision methodologies, the AHP is a method for constructing preference scales and, as is the case for other methodologies, the operations of addition and multiplication are not applicable on AHP scale values (see Barzilai ). The applicability of addition and multiplication must be established before these operations are used to compute AHP eigenvectors and the fact that eigenvectors are unique up to a multiplicative constant does not imply the applicability of addition and multiplication.
The literature of classical decision theory and classical measurement theory offers no insight into the fundamental problem of applicability of mathematical operations on scale values.
In order for addition and multiplication to be applicable on preference scale values, the alternatives must correspond to points on a straight line in an affine geometry (see Barzilai ). Since the ratio of points on an affine straight line is undefined, preference ratios, which are the building locks of AHP scales, are undefined. In addition, pairwise comparisons cannot be used to construct affine straight lines.
The fundamental mathematical error of using inapplicable operations to construct AHP scales renders the numbers generated by the AHP meaningless. For other AHP errors see Barzilai [2-9]. These include the fact that the coefficients of a linear preference function cannot correspond to weights representing relative importance and cannot be decomposed using Miller’s criteria tree; the eigenvector method is not the correct method for constructing preference scales; the assignment of the numbers 1-9 to AHP’s “verbal scales” is arbitrary, and there is no foundation for these “verbal scales”; etc. Additional details to follow.
 Jonathan Barzilai, On the Mathematical Foundations of Economic Theory, Technical Report, Dept. of Industrial Engineering, Dalhousie University, pp. 1-13, 2007. Posted at www.scientificmetrics.com
 Jonathan Barzilai, Avoiding MCDA Evaluation Pitfalls, Presented April 2007, NATO Advanced Research Workshop, Lisbon, 2007. Posted at www.scientificmetrics.com
 Jonathan Barzilai, Pairwise Comparisons, in N.J. Salkind (Ed.), Encyclopedia of Measurement and Statistics, Sage Publications (Thousand Oaks, CA), Vol. II, pp. 726-727, 2007.
 Jonathan Barzilai, On the Mathematical Modeling of Measurement, pp. 1-4, 2006. Posted at www.scientificmetrics.com
 Jonathan Barzilai, Notes on Utility Theory, Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, pp. 1000—1005, 2004.
 Jonathan Barzilai, Notes on the Analytic Hierarchy Process, Proceedings of the NSF Design and Manufacturing Research Conference, Tampa, Florida, pp. 1-6, 2001.
 Jonathan Barzilai, On the Decomposition of Value Functions, Operations Research Letters, Vol. 22, Nos. 4-5, pp. 159-170, 1998.
 Jonathan Barzilai, Understanding Hierarchical Processes, Proceedings of the 19th Annual Meeting of the American Society for Engineering Management, pp. 1-6, 1998.
 Jonathan Barzilai, Deriving Weights from Pairwise Comparison Matrices, Journal of the Operational Research Society, Vol. 48, No. 12, pp. 1226-1232, 1997.
 J.R. Miller, Professional Decision-Making, Praeger, 1970.
 J.R. Miller, Assessing Alternate Transportation Systems, Memorandum RM-5865-DOT, The RAND Corporation, 1969.
 J.R. Miller, The Assessment of Worth: A Systematic Procedure and Its Experimental Validation, doctoral dissertation, M.I.T., 1966.