Decision Models in Skiable Areas - EPFL

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3 METHODOLOGY - DISCRETE CHOICE MODELS 4 3 Methodology - Discrete Choice Models This section is an introduction in discrete choice models, and some of them are explained. Discrete choice models are used when an analyst wants to understand some human based decisions. Such models are often applied in transportation problems. 3.1 Decision Maker The decision maker is considered to be an individual, like an person or a group of persons. In the case of a group, the decision is considered to be one of the whole entire group. For example, the decision maker can be a skier in a skiable area which has to decide between a set of lifts and ski slopes to continue his trip. 3.2 Alternatives The set of alternatives is called the choice set. This set may be continuous (or not enumerable), like the time to spend in a restaurant, or discrete, like a set of lifts. In discrete choice models, only finite choice sets a considered, therefor the name. It is distinguished between the universal choice set, noted C, and the reduced choice set. The universal choice set is the set of all possible alternatives that an individual can chose. The reduced choice set is associated to an particular individual and is the set of all possible alternatives for this individual in the actual situation. 3.3 Attributes The attributes are a set of characteristics (an attribute) of the alternatives. They allow them to be different between them. Each attribute must take a measurable value. It is distinguished between generic and specific attributes. An generic attribute is one that can be measured for all alternatives while a specific attribute is one that exist only for a subset of alternatives. Example: The decision maker is a skier who has to choose his path. Suppose that the skier is at the end of a ski slope and that he has to choose the next lift of ski slope. The universal choice set is the set of all lifts and ski slopes. The reduced choice set is the set of all lifts and ski slopes that issues from his actual point. An generic attribute is the length of the lift resp. the ski slope. An specific attribute, defined for each lift, is the length of the queue of this one. An specific attribute, defined only for the ski slopes, is the level of difficulty. 3.4 Random Utility Models In discrete choice models, it exists a map which associates for each alternative (with the actual values of the attributes) and each individual a value, called utility U i a. It is a measurement of utility for the decision maker for each alternative. In discrete choice models, it is always assumed that the analyst don’t posses the complete information. It is caused to unobserved attributes and measurement errors. So the utility is considered to be a random variable composed of two components U i a = V i a + ε i a (1)
3 METHODOLOGY - DISCRETE CHOICE MODELS 5 The component V i a is the deterministic term and discussed in section 3.4.1, whereas ε i a is the random term and discussed in section 3.4.2. It should be remarked that the deterministic term is usually of the same form, while the different random terms determines the kind of Model. Suppose that for all alternatives, the utility is determined. Then it is supposed that the decision maker chooses the alternatives with the highest utility. But, because the utility is considered to be a random variable, a probability Pi C (a) can be associated to each alternative. PiC (a) is the probability that the individual i will chose alternative a between the set of alternatives C. We have accordingly the following result: P i C(a) = P[ U i a = max b ∈ C U i b ] (2) Once the random and deterministic terms fixed, we know now how to chose the alternative. 3.4.1 The Deterministic Term As already mentioned, the deterministic term is a function of the attributes, the alternative and the decision maker. Usually it is a affine function of the values of the attributes with one fixed coefficients for each attribute. E.g. the coefficients of two deterministic functions corresponding to two different alternatives are the same. The constants are depending of the alternatives, this are the alternative specific constants (ASC). It is also imageable to introduce a second constant related to an individual or a group of individuals. The deterministic term has in general the following form: V i a = V i a(x i a) = � βkx i a(k) + ASCa (3) k where x i a is the vector of all attributes for alternative a and individual i. The scalar value x i a(k) is the k-th element of the vector x i a and ASCa is the alternative specific constant related to the alternative a. 3.4.2 The Random Term This section handles about the random term ε i a and illustrates the principal properties of it. The random term ε i a is like already mentioned supposed to be a random variable. Without lost of generality it can be assumed that the mean of ε i a is zero. This property is illustrated in the following example where the mean of the random variables is supposed to be different from zero. A decision maker i has the choice between two alternatives 1 and 2 (C = {1, 2}). It is supposed to know the corresponding deterministic terms V i 1 and V i 2 . Let be m1 and m2 be the means of ε i 1 and ε i 2. Then P i C(1) = P(U i 1 ≥ U i 2) = P(V i 1 + ε i 1 ≥ V i 2 + ε i 2) = P(V i 1 − V i 2 ≥ ε i 2 − ε i 1) (4) As next, we define two new random variables εi ′ 1 and εi 2 ε i ′ 1 = ε i 1 − m1 ε i ′ 2 = ε i 2 − m2 It is evident that εi ′ i ′ i 1 resp. ε2 are of mean zero and of the same variance as ε1 resp. εi 2. Then P i C(1) = P(V i 1 −V i 2 ≥ ε i ′ 2 +m2−ε i ′ 1 −m1) = P(V i 1 +m1−V i 2 −m2 ≥ ε i ′ i ′ 2 −ε1 ) (6) ′ : (5)
Page 1 and 2: Decision Models in Skiable Areas Se
Page 3: 1 INTRODUCTION 3 1 Introduction Thi
Page 7 and 8: 3 METHODOLOGY - DISCRETE CHOICE MOD
Page 9 and 10: 6 MODEL OF DEMAND IN A SKIABLE AREA
Page 19 and 20: 7 CONCLUSIONS 19 • the alternativ
Page 21 and 22: REFERENCES 21 References [1] Bierla
Page 23 and 24: REFERENCES 23 Appendix B: Detailed
Page 25: REFERENCES 25 Numéro Nom Capacité

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