Mathematical Topics on Representations of Ordered Structures and Utility Theory. Essays in Honor of Professor Ghanshyam B. Mehta

We met Prof. Dr. G. B. Mehta for the first time in 1992, in a congress held in Paris, France. Since then we have fruitfully and continuously contributed with him in the publication of several papers on Utility Theory, as well as in the preparation of congresses, workshops, and seminars. He has always encouraged us to go ahead in our research on Utility Theory and, in particular, on the mathematical theory of the numerical representability of ordered structures. Before his retirement, we met him for the last time in Madrid, Spain in 2006 (August), where the International Congress of Mathematicians ICM 2006 was held. The idea of preparing a book in his honor is old, so-to-say. However, for different reasons, we were obliged to delay it until now. Professor Mehta reached the age of 75 y.o. on July 8, 2018, and just a few months before, we started to convert the idea of editing a festschrift in something real, having also in mind the possibility of this new book to be also a tribute and recognition to a whole lifetime dedicated to research, on the occasion of his 75th anniversary. Professor Ghanshyam B. Mehta (Bombay, now Mumbai, India 1943) belongs to a family with plenty of people devoted to Mathematics. His wife Meena, brother Vikram (1946–2014), and daughter Maithili (to cite only those of whom we know personally, probably there are even more) are mathematicians, too. He got the titles of Bachelor in Economics in Bombay University in 1963, Master of Arts in Economics in the University of California in Berkeley, USA. in 1965, and Doctor of Philosophy in Economics also in the University of California in Berkeley, USA. in 1971, with a thesis about the structure of the Keynesian Revolution. Then he worked as assistant professor at the University of New Brunswick, Canada in the period 1971–1972. Perhaps fascinated by billabongs, kangaroos, and coolibah trees, he arrived in Australia in 1974 where he settled, founded a home, and currently lives there. He got a Master of Science in Mathematics at the University of Queensland in 1978, and finally the title of Doctor in Philosophy in Mathematics, again in the University of Queensland, in 1987, with a thesis on Topological Methods in Equilibrium Analysis. He has been the advisor of several Ph.D. theses, both in Economics and Mathematics. Some of them are clearly devoted to Utility Theory (e.g., to put just one example, “Preference and utility in economic theory and the history of economic thought”, by K. L. Mitchener, School of Economics, University of Queensland, 2007). Thus, Prof. Mehta features the merit of having a wide formation and knowledge in Mathematics as well as in Economics, with doctorates in both disciplines. Obviously, this fact has allowed him to be a leader in those areas where Mathematics and Economics meet, and in particular in all that has to do with Utility Theory, understood as the mathematical problem of converting qualitative scales in which agents declare their preferences in order to compare things (usually, elements of a set of alternatives), into numerical or quantitative ones. That is, instead of just comparing things one another, the agents may alternatively and equivalently (when this is possible) compare numbers, in the sense of “the bigger the better”. Mathematically, this corresponds to the theory of numerical representability of ordered structures in the Euclidean real line endowed with its usual linear order. Most of the classical problems arising in Utility Theory have been analyzed by Prof. G. B. Mehta in some of his valuable papers and contributions: (i) Numerical representations of different kinds of orderings. (ii) Existence and nonexistence of utility representations. (iii) Existence of maximal elements for suitable orderings. (iv) Construction of utility functions by means of scales and separable systems. (v) Particular constructions of utility functions on classical spaces arising in Economics. (vi) Infinite-dimensional utility theory. (vii) Mathematical Intuitionism and Constructive Utility Theory. (viii) Continuous representability properties of topological spaces. (ix) Numerical representations of orderings in codomains different from the real line. (x) Utility in Economics vs. Entropy in Physics: the common mathematical background. His book (jointly written with D. S. Bridges) entitled Representations of Preference Orderings (Springer. Berlin. 1995) is undoubtedly a classical one in the theory of numerical representations of ordered structures, header book and the starting point of many past and ongoing researches in this framework. Apart from this, Prof. G. B. Mehta has also paid a relevant attention to the analysis of classical works and authors in Utility Theory, showing the necessity of understanding well, and periodically reviewing, the contributions made by the pioneers (Arrow, Birkhoff, Cooper, Debreu, Fleischer, Hahn, Keynes, Milgram, Newman, Peleg, Rader, Read, Wold, etc.). He has been a specialist in showing how many modern ideas recently issued in key papers in Utility Theory sometimes already had an implicit intuition that could be noticed in some of the papers of the most classical authors. Thus, it is crucial to (re)-read the classical papers in order to get new insights and inspiration in forthcoming research on these topics related to Utility Theory.This festschrift has been distributed in four parts. However, the distribution could perhaps be considered a bit whimsical: the reason is that they are not closed or watertight compartments, but instead, they have many connections and interactions sharing basic ideas. Anyways, we have tried to distribute the material in parts where the corresponding papers could have some key features in common. Thus, the final distribution goes as follows: (i) General Utility Theory. (ii) Particular Kinds of Preferences and Representations. (iii) Extensions of Utility. (iv) Applications into Economics. The first part, on General Utility Theory, consists of three contributions that can be considered as surveys on this framework. The first one, by Alan F. Beardon, is a really nice study on the mathematical background that supports the numerical representability of orderings, mainly leaning on General Topology. The second one, by Juan C. Candeal, analyzes the mathematical reasons that provoke the existence or the nonexistence of utility functions in several kinds of environments, as the algebraic one (looking for utility functions that, in addition, are also homomorphisms on the real line endowed with the corresponding algebraic structure), and the topological one as well (looking mainly for continuous utility functions defined on some ordered topological space). Finally, the third and last contribution in this part, by María J. Campión and Esteban Indurain, is unusual so-to-say, but, in our opinion, quite necessary for any potential researcher in the theory of the numerical representation of ordered structures. It is a survey in which the most classical theorems in Utility Theory are stated with the corresponding references, and then it is said that “beyond this point, everything is an open question.” That is, the contribution furnishes a wide list of open problems in Utility Theory, explaining in each case what is known, and what is still unknown, and paying a special attention to the “boundary of knowledge” here, that is, to the points from which the terrain is virgin, still unknown or unexplored. This will invite and encourage the potential researcher in this theory to develop some of her/his further research in the next future on these topics. The second part deals with particular kinds of preferences and representations. Basically, the most typical orderings that frame preferences are the total preorders (i.e., transitive and complete binary relations), the interval orders, and the semiorders. Therefore, in this part we have included those contributions that are directly related to the numerical representability of either a total preorder (through a suitable real-valued utility function), an interval order (through two intertwined utility functions), or a semiorder (by means of a real-valued utility function and a strictly positive real constant called threshold of discrimination). This part consists of seven contributions. The first one, by Y. Rebillé, as well as the second one, by A. Estevan, deal with particular features of the numerical representability of interval orders defined on topological spaces, paying particular attention to continuity. The third one, by D. Bouyssou and M. Pirlot deals with semiorders, analyzing in-depth a recently discovered condition that is necessary for the existence of a Scott-Suppes numerical representation. The fourth one, by D. Bouyssou and J.-P. Doignon, deals with biorders: this structure relates two given sets, generalizing the concept of an interval order (that appears when those two given sets coincide). This contribution studies different aspects related to numerical representations of nested families of biorders. The fifth one, by C. Chis, H.-P. A. Künzi and M. Sanchis, deals with the existence of utility representations on ordered topological groups, so relating order, algebra, and topology.The sixth contribution in this part, due to M. Cardin, studies preferences defined on convex structures, discussing in-depth the role played by convexity in different contexts of Utility Theory, so that is those settings, the economic agents exhibit an inclination for diversification and so they prefer a more balanced bundle to bundles with a more extreme composition. Finally, the seventh contribution, by C. Hervés and P. K. Monteiro, analyzes monotonicity properties of preferences, and how in remarkable situations of infinite dimensionality, so that the commodity space is rich enough, strictly monotonic preferences are neither representable by utility functions nor continuous in any linear topology.