# PREFERENCE MODELING AND CHOICE THEORY

**Academic Year 2020/2021**- 1° Year

**Teaching Staff:**

**Alfio Giarlotta**

**Credit Value:**6

**Taught classes:**35 hours

**Exercise:**12 hours

**Term / Semester:**2°

## Learning Objectives

**1. Knowledge and understanding: **The formative process of the course aims at acquiring the basic theoretical principles in (1) preference modeling, (2) utility representation of preferences, and (3) choice theory, in particular rational choices and bounded rationility aprroaches. The formative process is intended to deveop students' inductive and deductive skills. Learning is tested via homework to be done at home and corrected in class, along with a final (oral and/or written) exam. During the course, the lecturer will make sure that knowledge is transmitted to students, by stimulating their direct partecipation.

**2. Applying knowledge and ****understanding: **The teaching methodology will be oriented toward the operational acquisition of the needed conceptual and technical tools. The aim is to developing students' capacity to constructively criticize the learned techiques, in a continuous process of analysys/synthesis. Special attention is devoted to students who are going to graduate, because they may be end up facing professional problems similar to those taught in this course, often treated in multidisciplinary fashion.

**3. Making judgements: **The development of an autonomous critical capacity in the context of the topics of this course is one of the main objectives. Indeed, theoretical and operational skills ought to be complemented by students' capacity to evaluate the most suited methodolodies to be applied, emphasizing their limits and problems, and, in particular, the --often hidden-- hypothesis on which the modelization is based.

**4. Communication skills: **Students will be able to explain, also to non-experts, the basic features of approaches to preference modeling, utility representations, and choice theory, by using proper technical and non-techincal language.

**5. Learning skills: **From the outset of the course, students are stimulated to partecipate to the learning process, also in order to improve their learning methodology. During classes, the lecturer will constantly verify whether knowledge is effectively transferred to students. The lecturer may also decide to change the teaching methodology, according to the composition ad the background of the class.

## Course Structure

The course is structured in 3+1 modules, taught in classes. The first 3 parts of the course are mandatory, the forth is optional. The course should bring the student to the very frontier of recent research in the fields of preference modeling, utility representations, and choice theory.

All classes are taught in presence. However, should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

## Detailed Course Content

**PART I. Preferences, utility representations, and choices: the classical "mono-approach" **

**Objectives:** Introduction to the notion of preference and its representation by means of utility functions. Introduction to the notion of choice function/correspondence, with special emphasis on rational choices and choices with bounded rationality.

**Description of content:** Binary relations. The two classical tenets of rationality: transitivity and completeness. Binary preferences, preorders, complete preorders. Interval orders and semiorders. Utility functions, representability of transitive and complete preferences, tiopological and non-topological characterizations. Debreu's Open Gap Lemma. Choice functions/correspondences and rationalizability. The theory of revealed preferences. Problems and shortcomings of the classical approach, and possibile ways to cope with them.

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**PART II. The transition toward a "multi-approach"**

**Objectives: **Presentation of some models that suggest the advantages of a multi-approach versus a mono-approach

**Description of content:** Utility representations that emply lexicoraphic products and, in particular, lexicograpohic powers. The representability number of a linear ordering. Universal semiorders and shifted lexicographic products. (m,n)-Ferrers properties, discrete degrees of transitivity, and money pump phenomena. (m,n)-rationalizable choices.

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**PART III. Preferences, utility representations, and choices: a "multi-approach"**

**Objectives: **An overview of recent approaches** **of multi-preferences, multi-utility representations, and multi-rationalizability.

**Description of content:** B-preferences and NaP-preferences. A generalization of a famous theorem by Schmeidler (1971). Multi-utility representations and modal utility representations. Bounded rationality models for choices. Sequential and simultaneous choice multi-rationalizability.

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**PART IV. Miscellanea (optional, if time allows)**

**Objectives:** Introduction to the basic principles of decision under uncertainly. Decomposition of preferences and choices: resolutions.

**Description of content:** Objective and subjective probability. Von Neumann-Morgenstern model, Savage model, maximin model of expected value, Bewley preferences, and justifiable preferences. Resolutions: topological spaces, choices, preferences, and convex geometries. Intergenerational justice and interactions with mathematical utility theory. Topics in mathematical utility theory.

## Textbook Information

[1] F. Aleskerov, D. Bouyssou, and B. Monjardet, **Utility Maximization, Choice and Preference. **Springer, 2007.

[2] G. Bosi, G., M.J. Campión, J.C. Candeal, and E. Indurain, * Mathematical Topics on Representations of Ordered Structures and Utility Theory. Essays in Honor of Professor Ghanshyam B. Mehta. *Springer, 2020.

[3] D. Bridges and G.B. Mehta, * Representations of Preference Orderings. *Springer, 1995

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[4] A.** **Giarlotta, **New trends in preference, utility, and choice: From a mono-approach to a multi-approach. **In: M. Doumpos, J.R. Figueira, S. Greco, and C. Zopounidis (eds.), *New Perspectives in Multiple Criteria Decision Making,*** **pp. 3–80. Springer, 2019.

[5]* *I. Gilboa, ** Theory of decision under uncertainty.** Cambridge University Press, 2009.

[6] D. M. Kreps, **Microeconoimic Foundations I: Choice and Competitive Markets. **Princeton University Press, 2012.

[7] S. Watson, **The Construction of Topological Spaces: Planks and Resolutions.** In: M. Husek and J. van Mill (eds.), *Recent Progress in General Topology.* North Holland, 1992.