Fuzzy constraints, choice, and utility

Abstract

We revisit the notion of fuzzy revealed preference, with the intention of using it in decision making. A decision making agent manages resources that are priced, and it has a certain wealth. We review proposed fuzzifications of preference relations. Although they are well founded, they cannot deal well with decision making under fuzzy constraints. We introduce the fuzzy budget set as a general model of fuzzy constraints on resources. We then show that a general condition for rationality, the fuzzy axiom of revealed preference should not be an axiom anymore, but just a statement that is true to a certain degree. We show how to calculate this degree of truth. We define a fuzzy utility function, and show how it can be maximised, subject to fuzzy budget constraints. Keywords— fuzzy choice, fuzzy utility, fuzzy budget, qualitative reasoning I. Fuzzy choice and intelligent agents ELECTRONIC commerce allows buyers and sellers to use increasingly complicated decision procedures. Large amounts of information are available, and computers can process this to advise the economic agent in the choice of an alternative. The information gathered, over the web for example, is often inconsistent. If it is to be used in decision making, the decision making process will have to be able to deal with uncertainty. Humans deal with uncertainty in a natural way, via generalization. Automatic procedures use either probability theory or fuzzy logic. We prefer fuzzy logic because it deals with linguistic variables in a more intuitive way that probability theory. The linguistic variables are classes that have evolved over time, in a certain application domain, to be effective in generalization. Linguistic variables are the method adopted by humans to indicate choice and preference. Many intelligent agents aim to capture the preferences of their human owner. If decisions are to be made by a computer agent in e-commerce, it is essential that the computer agent agrees with the human owner. Should we study psychology before implementing a shop-bot? This would create problems, as a psychological analysis of economical behaviour returns results that are difficult to implement in a computer algorithm. Most economists hold that their theory, micro-economics based on game theory, gives an accurate description of human economical behaviour. They even have applied the microeconomic paradigm to areas such as social interactions, and irrational behaviour in households and firms [1]. For the management of resources, a core economic activity, the micro-economic approach is prevailing. This is what e-commerce is mostly about: buying and selling quantifiable resources. If we can allow the quantities to be fuzzy, e-commerce and e-management of resources will be even more widely applied than it is now. E-commerce and e-management of resources can operate automatically using intelligent software agents. To achieve this, we need to re-formulate micro-economy so that it can deal with fuzzy choice and preferences. The Orlovsky choice function is often used as the basis for fuzzy choice [2], [3]. We will start from an entirely different starting point, immediately taking into account prices of resources that affect the choice among alternatives. Another approach, ranking based on pairwise comparisons is described in [4]. Choice among attributes that have multiple attributes is reviewed in [5]. The attributes of our alternatives will be the prices of goods in the consumption bundle. This will allow us to have more specific procedures for ranking than in [5], [4]. Once a basic concept, such as the Orlovsky choice function is proposed and adopted, scientists usually start refining and generalizing it. This happened to fuzzy choice functions, just as it happened to Nash equilibrium, expert systems, etc. Much of the current theory about fuzzy choice has become so abstract that it is impossible to implement in an e-commerce agent. The refined theory of choice can certainly be used to model particular user’s decisions very accurately, but this matching of theory and user requires extensive human intervention. If the e-commerce agent has to implement fuzzy choice automatically for a large class of users, we have to turn back, and use a more intuitive theory. Kulshreshtha and Shekar [6] have recently attempted to present an intuitive perspective on fuzzy preference. It becomes clear from this paper that there is an array of possible choice functions, with no clear criteria as to which ones to prefer. There are even some intuitive contradictions. The authors point out the need to conduct experiments to find the most appropriate fuzzy preference relations for real life situations. We will not conduct experiments, but consider the crisp theory of preference closer to the application (resource management), before fuzzifying it. II. Two weak axioms of fuzzy revealed