Lecture 1: Preference Relations and Utility Functions

Transcription

1 Microeconomics I: Game Theory Lecture 1: Preference Relations and Utility Functions (see Osborne, 2009, Sect 1.2) Dr. Michael Trost Department of Applied Microeconomics October 25, 2013 Dr. Michael Trost Microeconomics I: Game Theory Introduction 1 / 21

2 The theory of rational choice A decision maker is said to act rationally if the action chosen by her is at least as good, according to her preferences, as every other available action. Dr. Michael Trost Microeconomics I: Game Theory Introduction 2 / 21

3 Features of the theory of rational choice The theory of rational choice is based on the following three features: the set A of possible (imaginable) actions the set C A of available actions a preference relation on the set A of possible actions Dr. Michael Trost Microeconomics I: Game Theory Introduction 3 / 21

4 Possibility set and constraint The set A of possible (or imaginable) actions is called the possibility set. But not any action of the possibility set is realizable by the decision maker, maybe due to technical, financial or time reasons. The set C A of actions which are realizable (or synonymously, available for the decision maker ) is called constraint. Dr. Michael Trost Microeconomics I: Game Theory Introduction 4 / 21

5 Decision problem The decision problem of a decision maker is to choose a single action from the set of available actions. Dr. Michael Trost Microeconomics I: Game Theory Introduction 5 / 21

6 Example: Vacation planning A student consulting a travel agency in Erfurt to book a vacation package. The travel agency offers four vacation packages, Havanna, Istanbul, London and Paris. The vacation package Havanna includes an expensive intercontinental flight and is not affordable for the student. QUESTION: What is the student s possibility set and what is her constraint? Describe her decision problem! Dr. Michael Trost Microeconomics I: Game Theory Introduction 6 / 21

7 Example: Vacation planning Dr. Michael Trost Microeconomics I: Game Theory Introduction 7 / 21

8 (Weak) preference relation The decision maker is assumed to be endowed with a (weak) preference relation that describes her ranking of the possible actions. Let a, b A be possible actions of the decision maker, then the expression a b means that the decision maker considers action a at least as good as action b (or synonymously, she weakly prefers a to b) Dr. Michael Trost Microeconomics I: Game Theory Introduction 8 / 21

9 Assumptions on preference relations A preference relation is assumed to satisfy the following two properties: (Completeness) For every actions a, b A, a b or b a holds. (Transitivity) For every actions a, b, c A, if a b and b c hold, then a c holds. Dr. Michael Trost Microeconomics I: Game Theory Introduction 9 / 21

10 Assumptions on preference relations The assumption of completeness postulates that the decision maker is able to rank every pair of actions. For any pair of actions a and b, the decision maker knows whether a is at least as good as b or b is at least as good as a. The assumption of transitivity postulates that the decision maker s ranking is consistent. If the decision maker considers a at least as good as b and b at least as good as c, then she must consider a at least as good as c. Dr. Michael Trost Microeconomics I: Game Theory Introduction 10 / 21

11 Indifference and strict preference The decision maker is said to be indifferent between actions a and b, in symbols a b, if both a b and b a hold. The decision maker is said to strictly prefer action a to action b, in symbols a b, if a b holds, but not b a. Dr. Michael Trost Microeconomics I: Game Theory Introduction 11 / 21

12 Rational choice Consider a decision maker who has a preference relation on set A of possible actions and who is constrained to choose between the available actions C A. The decision maker is said to be rational if he selects one of the actions which belong to set {a C : a b for every b C} Elements of this set are called rational choices of the decision maker. Dr. Michael Trost Microeconomics I: Game Theory Introduction 12 / 21

13 Example: Vacation planning The student consulting the travel agency in Erfurt reveals the following preferences. She prefers vacation in Havanna to vacation in Istanbul and vacation in Istanbul to vacations in Paris or in London. The latter two vacation packages are regarded as equivalent by her. QUESTION: Write out completely the preference relation of the student! What are her rational choices? Dr. Michael Trost Microeconomics I: Game Theory Introduction 13 / 21

14 Utility function Let A be a set of possible actions and let U be a function assigning to each action a A a real number U(a) R. A preference relation on set A of possible actions is said to be representable by a utility function U whenever, for every actions a, b A, a b holds if and only if U(a) U(b) holds. Dr. Michael Trost Microeconomics I: Game Theory Introduction 14 / 21

15 Why utility functions? A utility function is a useful device for representing preferences. It reproduces the preferences of a decision maker in a compact manner. Dr. Michael Trost Microeconomics I: Game Theory Introduction 15 / 21

16 Utility representation theorems Theorem 1.1 Let A be a finite set of possible actions and a preference relation on A. Then is representable by a utility function. Dr. Michael Trost Microeconomics I: Game Theory Introduction 16 / 21

17 Utility representation theorems Theorem 1.2 Let A be a set of possible actions and U a function assigning to each action a A a real number U(a) R. Then there exists a preference relation which is representable by utility function U. Dr. Michael Trost Microeconomics I: Game Theory Introduction 17 / 21

18 Example: Vacation planning The student s preferences is representable by utility function U, where U(Havanna) = 2, U(Istanbul) = 1, U(London) = U(Paris) = 0 hold, but also by utility function V, where hold. V (Havanna) = 100, V (Istanbul) = 0, V (London) = V (Paris) = π QUESTION: Can you find another utility function W representing her preference relation? Dr. Michael Trost Microeconomics I: Game Theory Introduction 18 / 21

19 Irrelevance of positive monotone transformations Theorem 1.3 Let be a preference relation on set A of possible actions which is representable by utility function U. Function V is another utility function for if and only if there exists a strictly increasing function f so that V (a) = f (U(a)) holds for every a A. Dr. Michael Trost Microeconomics I: Game Theory Introduction 19 / 21

20 Utility as an ordinal concept of measurement The last theorem says that utility is a concept of measurement which is unique up to positive monotone transformations. Such concepts of measurement are termed ordinal. In ordinal measurement, the attributes are ranked-ordered. Distances between attributes have no meaning. In context of utility theory, this means that utility values are only auxiliary devices to rank the possible actions. They do not convey any information about the intensity of preferences. Dr. Michael Trost Microeconomics I: Game Theory Introduction 20 / 21

21 Rational choice and utility maximization Consider a decision maker who has preference relation on set A of possible actions. Her preference relation is representable by utility function U, and she is constrained to choose a single action from set C A of available actions. Then, an action a C is a rational choice of her if and only if it maximizes her utility among all available actions. Hence, under above assumptions, making rational choices boils down to maximizing the utility under the availability constraint. Dr. Michael Trost Microeconomics I: Game Theory Introduction 21 / 21