Introduction to Formal Preference Spaces

In the article the formal characterization of preference spaces [1] is given. As the preference relation is one of the very basic notions of mathematical economics [9], it prepares some ground for a more thorough formalization of consumer theory (although some work has already been done - see [17]). There was an attempt to formalize similar results in Mizar, but this work seems still unfinished [18]. There are many approaches to preferences in literature. We modelled them in a rather illustrative way (similar structures were considered in [8]): either the consumer (strictly) prefers an alternative, or they are of equal interest; he/she could also have no opinion of the choice. Then our structures are based on three relations on the (arbitrary, not necessarily finite) set of alternatives. The completeness property can however also be modelled, although we rather follow [2] which is more general [12]. Additionally we assume all three relations are disjoint and their set-theoretic union gives a whole universe of alternatives. We constructed some positive and negative examples of preference structures; the main aim of the article however is to give the characterization of consumer preference structures in terms of a binary relation, called characteristic relation [10], and to show the way the corresponding structure can be obtained only using this relation. Finally, we show the connection between tournament and total spaces and usual properties of the ordering relations.Niewiadomska Eliza - Institute of Mathematics University of Białystok Akademicka 2, 15-267 Białystok PolandGrabowski Adam - Institute of Informatics University of Białystok Akademicka 2, 15-267 Białystok PolandKenneth J. Arrow. Social Choice and Individual Values. Yale University Press, 1963.Robert J. Aumann. Utility theory without the completeness axiom. Econometrica, 30(3): 445-462, 1962.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Klaus E. Grue and Artur Korniłowicz. Basic operations on preordered coherent spaces. Formalized Mathematics, 15(4):213-230, 2007. doi:10.2478/v10037-007-0025-4.Sören Halldén. On the Logic of Better. Lund: Library of Theoria, 1957.Emil Panek. Podstawy ekonomii matematycznej. Uniwersytet Ekonomiczny w Poznaniu, 2005. In Polish.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.George F. Schumm. Transitivity, preference, and indifference. Philosophical Studies, 52: 435-437, 1987.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313-319, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Freek Wiedijk. Arrow’s impossibility theorem. Formalized Mathematics, 15(4):171-174, 2007. doi:10.2478/v10037-007-0020-9.Krzysztof Wojszko and Artur Kuzyka. Formalization of commodity space and preference relation in Mizar. Mechanized Mathematics and Its Applications, 4:67-74, 2005.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990

Interdependent Preferences and Strategic Distinguishability

A universal type space of interdependent expected utility preference types is constructed from higher-order preference hierarchies describing (i) an agent's (unconditional) preferences over a lottery space; (ii) the agent's preference over Anscombe-Aumann acts conditional on the unconditional preferences; and so on. Two types are said to be strategically indistinguishable if they have an equilibrium action in common in any mechanism that they play. We show that two types are strategically indistinguishable if and only if they have the same preference hierarchy. We examine how this result extends to alternative solution concepts and strategic relations between types.Interdependent preferences, Higher-order preference hierarchy, Universal type space, Strategic distinguishability

'First Portal in a Storm': A Virtual Space for Transition Students

The lives of millennial students are epitomised by ubiquitous information, merged technologies, blurred social-study-work boundaries, multitasking and hyperlinked online interactions (Oblinger & Oblinger, 2005). These characteristics have implications for the design of online spaces that aim to provide virtual access to course materials, administrative processes and support information, all of which is required by students to steer a course through the storm of their transition university experience. Previously we summarised the challenges facing first year students (Kift & Nelson, 2005) and investigated their current online engagement patterns, which revealed three issues for consideration when designing virtual spaces (Nelson, Kift & Harper, 2005). In this paper we continue our examination of students’ interactions with online spaces by considering the perceptions and use of technology by millennial students as well as projections for managing the virtual learning environments of the future. The findings from this analysis are informed by our previous work to conceptualise and describe the architecture of a transition portal

The Topology-Free Construction of the Universal Type Structure for Conditional Probability Systems

We construct the universal type structure for conditional probability systems without any topological assumption, namely a type structure that is terminal, belief-complete, and non-redundant. In particular, in order to obtain the belief-completeness in a constructive way, we extend the work of Meier [An Infinitary Probability Logic for Type Spaces. Israel Journal of Mathematics, 192, 1-58] by proving strong soundness and strong completeness of an infinitary conditional probability logic with truthful and non-epistemic conditioning events.Comment: In Proceedings TARK 2017, arXiv:1707.0825

Integrating multicriteria decision analysis and scenario planning : review and extension

Scenario planning and multiple criteria decision analysis (MCDA) are two key management science tools used in strategic planning. In this paper, we explore the integration of these two approaches in a coherent manner, recognizing that each adds value to the implementation of the other. Various approaches that have been adopted for such integration are reviewed, with a primary focus on the process of constructing preferences both within and between scenarios. Biases that may be introduced by inappropriate assumptions during such processes are identified, and used to motivate a framework for integrating MCDA and scenario thinking, based on applying MCDA concepts across a range of "metacriteria" (combinations of scenarios and primary criteria). Within this framework, preferences according to each primary criterion can be expressed in the context of different scenarios. The paper concludes with a hypothetical but non-trivial example of agricultural policy planning in a developing country

E-Learning and support tools for Information and Computer Sciences

The availability of Web 2.0 tools together with associated Open Educational Resources (OER) enables the creation of new social and collaborative learning spaces. This paper investigates student preferences (across three cohorts) in terms of openly and freely accessible Web 2.0 tools to provide a space where students can interact with each other and their tutor to discuss concerns that arise within their final year project-based learning. This intervention was planned since existing arrangements that support communication between tutor and distance learning students appeared insufficient to facilitate the necessarily intense episodes of interaction required for productive supervision. The findings suggest that different student cohorts are interested in using a variety of Web 2.0 tools. This paper gives initial feedback about intended usage of Web 2.0 tools for co-operative and collaborative learning for final year project work