Open questions in utility theory

Throughout this paper, our main idea is to explore different classical questions arising in Utility Theory, with a particular attention to those that lean on numerical representations of preference orderings. We intend to present a survey of open questions in that discipline, also showing the state-of-art of the corresponding literature.This work is partially supported by the research projects ECO2015-65031-R, MTM2015-63608-P (MINECO/ AEI-FEDER, UE), and TIN2016-77356-P (MINECO/ AEI-FEDER, UE)

Semiorders and continuous Scott–Suppes representations. Debreu’s Open Gap Lemma with a threshold

The problem of finding a utility function for a semiorder has been studied since 1956, when the notion of semiorder was introduced by Luce. But few results on continuity and no result like Debreu’s Open Gap Lemma, but for semiorders, was found. In the present paper, we characterize semiorders that accept a continuous representation (in the sense of Scott–Suppes). Two weaker theorems are also proved, which provide a programmable approach to Open Gap Lemma, yield a Debreu’s Lemma for semiorders, and enable us to remove the open-closed and closed-open gaps of a set of reals while keeping the threshold.Asier Estevan acknowledges financial support from the Ministry of Science and Innovation of Spain under grants PID2020-119703RB-I00 and PID2021-127799NB-I00 as well as from the UPNA, Spain under grant JIUPNA19-2022

Continuous representability of interval orders: The topological compatibility setting

In this paper, we go further on the problem of the continuous numerical representability of interval orders defined on topological spaces. A new condition of compatibility between the given topology and the indifference associated to the main trace of an interval order is introduced. Provided that this condition is fulfilled, a semiorder has a continuous interval order representation through a pair of continuous real-valued functions. Other necessary and sufficient conditions for the continuous representability of interval orders are also discussed, and, in particular, a characterization is achieved for the particular case of interval orders defined on a topological space of finite support

Preference Modelling

This paper provides the reader with a presentation of preference modelling fundamental notions as well as some recent results in this field. Preference modelling is an inevitable step in a variety of fields: economy, sociology, psychology, mathematical programming, even medicine, archaeology, and obviously decision analysis. Our notation and some basic definitions, such as those of binary relation, properties and ordered sets, are presented at the beginning of the paper. We start by discussing different reasons for constructing a model or preference. We then go through a number of issues that influence the construction of preference models. Different formalisations besides classical logic such as fuzzy sets and non-classical logics become necessary. We then present different types of preference structures reflecting the behavior of a decision-maker: classical, extended and valued ones. It is relevant to have a numerical representation of preferences: functional representations, value functions. The concepts of thresholds and minimal representation are also introduced in this section. In section 7, we briefly explore the concept of deontic logic (logic of preference) and other formalisms associated with "compact representation of preferences" introduced for special purpoes. We end the paper with some concluding remarks

Essays in organization formation and decision making

This thesis consists of three essays in microeconomic theory. The first two are about the formation of organizations, and the third is about individual or organizational decision making in ambiguous settings. In the first essay I explore the implications of costs associated with binding agreements on equilibrium agreement structures. Establishing binding agreements is often costly in real world economies. These contracting costs are usually regarded as harmful by economists as the costs decrease the gains from cooperation. They affect which agreements form by changing the incentives of agents, potentially prevent the establishment of efficient contracts. Using an alternating offers bargaining model of coalition formation I show that the presence of transaction costs can lead to an efficient outcome in situations where inefficiency arises in equilibrium without these costs. These results provide new insights for policies targeting transaction costs. There are many situations in Economics and Political Science that involve limited possibilities for firms or parties to organize themselves into groups, mostly due to regulatory restrictions. In addition, in these settings the surplus of a given group often depends on the organizational structures formed outside of the group. The second essay introduces a coalition formation model that is able to analyze markets with both restricted cooperation and externalities across coalitions. This concept allows a more realistic modeling, opening the possibility to use this framework to analyze the welfare effects of mergers. In the third essay I propose a new model of decision making under uncertainty with multiple priors that is, unlike the well-known model of Gilboa and Schmeidler (1989), able to express attitude towards ambiguity. In addition, the decision does not necessarily depend on the two extreme (worst case and best case) priors as in the model of Ghirardato et al. (2001). I use choice correspondences by lexicographic semiorders that are generalizations of the choice functions defined in Manzini and Mariotti (2012). I also provide a method constructing lexicographic semiorders for choosing from ambiguous acts