© Jonathan Barzilai 2007-2008.
The questions on this page have not been asked in the classical literature. They should have been.
This page is updated periodically. Additional details, questions and answers to follow.
No, there is no such proof in the literature. Addition and multiplication are not applicable in the range of any scale representing physical or non-physical variables. The only exceptions are proper scales – see Barzilai .
Game theory is a special case of n-person decision theory. The assignment of values to objects such as outcomes and coalitions, i.e. the construction of value functions, is a fundamental concept of game theory. Value (or utility, or preference) is not a physical property of the objects being valued, that is, value is a subjective (or psychological, or personal) property. Therefore, the definition of value requires specifying both what is being valued and whose values are being measured.
Game theory’s characteristic function assigns values to coalitions so that what is being valued by this function is clear but von Neumann and Morgenstern do not specify whose values are being measured in the construction of this function. Since it is not possible to construct a value (or utility) scale of an unspecified person or a group of persons, game theory’s characteristic function is not well-defined. Likewise, all game theory solution concepts that do not specify whose values are being measured are ill-defined.
The literature of classical game theory, decision theory and measurement theory offers no insight into the fundamental problem of the construction of n-person value functions.
Yes, there are. For other game theory errors see Barzilai [1, 2].
The numbers 1-10 do not define a scale. Addition, multiplication, and common statistical operations are not applicable on numbers obtained in answer to this question. The definition of a scale requires the specification of empirical and mathematical objects and operations and possibly the relation of order – see Barzilai .
The literature of classical decision theory and measurement theory offers no insight as to why the question “On a scale of 1-10, how far is Ottawa from Toronto?” is meaningless.
The sum of two times, as opposed to time differences, is undefined because time scales are affine scales and the operation of addition is undefined for points on an affine straight line.
The literature of classical decision theory and measurement theory offers no insight as to why “2005+2007=4012″ is meaningless.
The ratio of two times is undefined because time scales are affine scales. The operation of division is undefined for points on an affine straight line. In this context the number 1.071428571 has no meaning despite its scientific appearance. For the same reason, the ratio of two potential energies is undefined.
The literature of classical decision theory and measurement theory offers no insight as to why these ratios are meaningless.
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. This construction requires a specific interpretation of the empirical operation in the context of preference measurement and although the axioms are consistent in the abstract, the interpretation of the empirical utility operation creates a contradiction. The interpretation of the utility operation in terms of lotteries constrains the values of utility scales for lotteries while the values of utility scales for prizes are unconstrained. The theory permits lotteries that are prizes and this leads to a contradiction since an object may be both a prize, which is not constrained, and a lottery which is constrained. In other words, utility theory has one rule for assigning values to prizes and a different, conflicting, rule for assigning values to lotteries. For a prize which is a lottery ticket, the conflicting rules are contradictory. See [3, §6.4.2] or [4, §4.2] for a numerical example where the ranking of the alternatives depends on their label.
 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, Game Theory Foundational Errors – Part I, Technical Report, Dept. of Industrial Engineering, Dalhousie University, pp. 1-2, 2007. Posted at www.scientificmetrics.com
 Jonathan Barzilai, Preference Modeling in Engineering Design, in Decision Making in Engineering Design, K.E. Lewis, W. Chen and L.C. Schmidt (Eds.), ASME Press, pp. 43-47, 2006.
 Jonathan Barzilai, Notes on Utility Theory, Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, pp. 1000—1005, 2004.