The basic model of decision theory under risk. Theory of expected utility (Bernoulli-Principle) Introduction to Game Theory

Transcription

1 I. Introduction to decision theory II. III. IV. The basic model of decision theory under risk Classical decision rinciles Theory of exected utility (Bernoulli-Princile) V. Doubts on exected utility theory and the non-exected utility aroach VI. Introduction to Game Theory Modelling and Decision Making

2 II. The basic model of decision theory underrisk risk Deiction Probabilities biliti - Axioms of robability calculation - Interretation of robabilities - Excursion: models without robabilities Dominance rinciles - State-by-state dominance - First-order stochastic dominance Modelling and Decision Making

3 Basic model of decision theory Model arameters: Action sace (A), consisting of the set of available actions/acts (a). State sace (), consisting of every state of the world () that the decision maker considers ossible and are relevant for the decision. Outcome sace (X), consisting of the set of results/outcomes (x) considered ossible. Result function g : A Θ X which assigns to each air (a,) an exlicit result x with. x g(a, θ) Modelling and Decision Making 3

4 Basic model of decision theory θ θ θ j θ n a x x x j x n a x x x j x n a i x i x i x ij x in a m x m x m x mj x mn Modelling and Decision Making 4

5 Axioms of robability calculation Axioms of robability theory (Kolmogorov): A function, that assigns a real number to every event E Θ is called robability measure and (E) is the robability of the event E, if the following axioms hold true: Axiom : (E) 0, E Axiom : ( Θ) Axiom 3: If E,E, are mutually exclusive events with E k E j, k j then EE j j E E E j j Modelling and Decision Making 5

6 Axioms of robability calculation Calculation rules for robabilities: ( ) 0, ( ) ( E) ( E) ( E E ) (E) (E ) - (E E ) Conditional robabilities: For any two events E and E the conditional robability of E given E is (E (E E) E ) (E ) Indeendent d Events: Two events E and E are stochastically indeendent, if y (E E) (E ) or equivalently (E E ) (E ) (E ) Modelling and Decision Making 6

7 Axioms of robability calculation Bayes rule For two mutually exclusive events E and F E F E F E E F E F E E F E E F E E F E (F) Modelling and Decision Making 7

8 The hilosohy of robability logical (or objective a riori) robabilities rincile of insufficient reason frequency (or objective a osteriori) robabilities robability as the marginal value of relativ ve frequency subjective robabilities (robabilities as subject tive figures of credibility) Modelling and Decision Making 8

9 Excursion: models without robabilities Initial situation The decision maker is able to identify the different states of the world, but she cannot figure out with which robabilities they are going to occur. The aroach is methodically not convincing and will not be discussed any further. Reasons: In general, decision theory is based on the subjective robability notion. If there are no indications for individuals that the credibility (occurrence robability) of a certain state of the world is greater than the credibility of another state, the individuals can allocate the same robability bilit to every state t of the world. Princile of insufficient reason (rincile of indifference) Modelling and Decision Making 9

10 State-by-state dominance An alternative a dominates another alternative a according to state-by-state dominance, if a leads in no state of the world to aworsere re esult, but at least in one state of the world to a better result than a. x j x j j and j : x x j j Imlications: Intuitively lausible, because an individual should never choose an alternative that is dominated according to state-by-state dominance. State-by-state dominance usually only leads to an exclusion of certain alternatives, but not to the determination ti of the otimal alternative. ti Modelling and Decision Making 0

11 State-by-state dominance Examle θ θ 0,4 θ 3 θ 0,4 θ θ 0, 3 a a a Modelling and Decision Making

12 First-order stochastic dominance An alternative a dominates an alternative a according to first-order stochastic dominance, if for every outcome x the a -robability to attain a outcome greater than x, is not smaller and for at least one result greater than the a -robability. x F x x F nd : F xˆ F xˆ an xˆ Imlications First-order stochastic dominance is a less dema anding concet as state-by-state dominance. It is intuitively lausible as resulting decisions are based on robability distributions. First-order stochastic dominance also only leads to a reselection. Modelling and Decision Making

13 First-order stochastic dominance Examle θ θ 0,4 θ θ 0,4 θ 3 θ 3 0, a a a Modelling and Decision Making 3

14 First-order stochastic dominance Examle (cumulative distribution function) (x) F i 0,8 0, x Modelling and Decision Making 4

15 First-order stochastic dominance f x Probability bilit density function f x f x Cumulative distribution function F x F x 0 x 0 x Modelling and Decision Making 5