Random Utility Theory
3.2 Basic Assumptions
Random utility theory is based on the hypothesis that every individual is a rational decision-maker, maximizing utility relative to his or her choices. Specifically, the theory is based on the following assumptions.
(a) The generic decision-makeri, in making a choice, considers mi mutually ex- clusive alternatives that constitute herchoice set Ii. The choice set may dif- fer according to the decision-maker (e.g., in the choice of transport mode, the choice set of an individual without a driver’s license or car obviously should not include the alternative “car as a driver”);
(b) Decision-makeriassigns to each alternativej in his choice set aperceived util- ityor “attractiveness”Uji and selects the alternative that maximizes this utility;
(c) The utility assigned to each choice alternative depends on a number of measur- able characteristics, orattributes, of the alternative itself and of the decision- maker:Uji=Ui(Xij), whereXij is the vector of attributes relative to alternative j and to decision-makeri;
(d) Because of various factors described later, the utility assigned by decision- maker i to alternativej is not known with certainty by an external observer (analyst) wishing to model the decision-maker’s choice behavior, thusUji must be represented in general by a random variable.
From the above assumptions, it is not usually possible to predict with certainty the alternative that the generic decision-maker will select. However, it is possible to express the probability that the decision-maker will select alternativej conditional on her choice setIi; this is the probability that the perceived utility of alternativej is greater than that of all the other available alternatives:
pi(j/Ii)=Pr
Uji> Uki ∀k=j, k∈Ii
(3.2.1)
3.2 Basic Assumptions 91 The perceived utilityUji can be expressed as the sum of two terms: a system- atic utility and a random residual. Thesystematic utilityVji represents the mean (expected value) utility perceived by all decision-makers having the same choice context (alternatives and attributes) as decision-makeri. Therandom residualεji is the (unknown) deviation of the utility perceived by userifrom this mean value; it captures the combined effects of the various factors that introduce uncertainty into choice modeling:
Uji =Vji+εji ∀j ∈Ii (3.2.2a) with
Vji=E Uji
, σi,j2 =Var Uji and therefore
E Vji
=Vji, Var Vji
=0 E
εji
=0, Var εji
=σi,j2 Replacing expression (3.2.2a) in (3.2.1) yields:
pi[j/Ii] =Pr
Vji−Vki> εki −εij ∀k=j, k∈Ii
(3.2.3a) From (3.2.3a) it follows that the choice probability of an alternative depends on the systematic utilities of all competing (available) alternatives, and on the joint probability law of the random residualsεj.
Random utility models and the variables they involve can be compactly repre- sented using vector notation. Let
pi be the vector of choice probabilities, of dimension (mi×1), with elements pi[j]
Ui be the vector of perceived utilities, of dimension(mi×1), with elements Uji
Vi be the vector of systematic utility values, of dimension(mi×1), with ele- mentsVji
εi be the vector of random residuals, of dimension(mi×1), with elementsεji f (ε) be the joint probability density function of the random residuals
F (ε) be the joint probability distribution function of the random residuals Expression (3.2.2a) can therefore be written in vector notation as:
Ui=Vi+εi (3.2.2b)
In general, the choice model (3.2.3a) can be viewed as a function, known as a choice function, that associates a vector of choice probabilities to each vectorVi of systematic utilities for a given probability law of random residuals:
pi=pi(Vi) ∀Vi ∈Emi (3.2.3b)
A random utility model is said to beinvariant (oradditive) if neither the form nor the parameters of the joint probability density function of the random residuals, f (ε), depends on the vectorV of systematic utilities:
f (ε/V)=f (ε) ∀ε∈Emi
It follows immediately from expression (3.2.3a) that, for invariant models, the choice probabilities of the alternatives do not change if a constantV0is added to the systematic utility of each of them:
pi[j/Ii] =Pr
Vji+V0−Vki−V0> εki −εji
=Pr
Vji−Vki> εki −εji
∀k=j; j, k∈Ii (3.2.4) From the previous expression it also follows that, in the case of invariant models, choice probabilities depend on the differences between the systematic utility of each alternative and that of a reference alternativeh; these differencesVj−Vhare known as relative systematic utilities.
Before describing some of the random utility models derived from particular assumptions on the random residual joint probability functions, some further general remarks on the implications of the hypotheses introduced so far should be made.
The variance–covariance matrix of random residuals. In general, a variance–
covariance matrixΣ is symmetric and positive semidefinite (see Appendix3.B).
When the variance of each random residualεk is zero,σkk=0, all the covariances must also be zero,σkh=0∀h, and therefore the variance–covariance matrix is itself zero,Σ=0; this case yields thedeterministicchoice model whose properties are described in Sect.3.4. If the variance–covariance matrix is not zero,Σ=0, anon- deterministicchoice model is obtained. In this case, it is usually assumed that the varianceσkk=σk2of each random residualεkis strictly positive,σkk>0, and that the random residuals are imperfectly correlated,(σkh)2< σk2σh2; that is, the rows (or columns) ofΣ are pairwise linearly independent. These conditions are equivalent to assuming that the variance–covariance matrix is not singular,|Σ| =0, in addi- tion to being nonzero,Σ=0. In this case the models are calledprobabilistic,3and the choice functionp=p(V)can be shown to be continuous with continuous first partial derivatives.
The set of available alternativesIi, or choice set, significantly influences the choice probabilities, as can be seen from (3.2.1) and (3.2.3a). If a particular decision-maker’s choice setIi is known, the definition of choice probability (3.2.1) can be applied directly. However, it often happens that the analyst has no exact knowledge of the generic decision-maker’s choice set. In this case, the problem can be handled with different levels of approximation, as shown in Sect.3.5.
3The case in which the variance–covariance matrix is nonzero,Σ=0, but singular,|Σ| =0, be- cause the variance of a random residual is zero and/or two random residuals are perfectly corre- lated, is of limited practical interest and is not given further attention.
3.2 Basic Assumptions 93 The expression for the systematic utility. Systematic utility is the mean perceived utility among all individuals who have the same attributes; it is expressed as a func- tionVji(Xikj)of attributesXkji relative to the alternatives and the decision-maker.
Although in principle the functionVji(Xij)may be of any type, it is usually as- sumed for analytical and statistical convenience that the systematic utilityVji is a linear function, with coefficientsβk, of the attributesXkji or of functional transfor- mationsfk(Xkji )of them:
Vji Xij
=
k
βkXkji =βTXij (3.2.5a) or
Vji Xij
=
k
βkfk Xikj
=βTf Xij
(3.2.5b) Further details on the specification of the systematic utility are given in Chap. 8.
The attributes included in the vectorXijcan be classified in different ways. Those related to the service offered by the transport system are known aslevel-of-serviceor performance attributes(times, costs, service frequency, comfort, etc.). Those related to the land-use characteristics of the study area (e.g., the number of shops or schools in each zone) are known asactivity system attributes. Those related to the decision- maker or to his household (income, holding a driver’s license, number of cars in the household, etc.) are referred to associoeconomic attributes.
Attributes of any type are calledgenericif they are included in the systematic utility of more than one alternative in the same form and with the same coeffi- cientβk. They are calledspecificif they are included with different functional forms and/or coefficients in the systematic utilities of different alternatives. A dummy vari- able is usually included in the systematic utility of the generic alternativej; its value is one for alternativej and zero for the others. This variable is usually denoted the Alternative Specific Attribute(ASA) or “modal preference” attribute,4and its co- efficientβk is known as the Alternative Specific Constant (ASC). The ASA is a kind of constant term in the systematic utility; it can be viewed as the difference between the mean utility of an alternative and the portion that is explained by its other attributesXkj.
From expression (3.2.4), it can be seen that the choice probabilities of invariant models depend in part on the differences between the ASC of each alternativej and that of a reference alternativeh. If alternative specific attributes were included in the systematic utilities of all alternatives, any combination of coefficientsβ that led to the same ASC differences between alternatives would result in the same choice probability values, so the ASCs could not be statistically estimated. For this reason, when specifying invariant models, the ASC of at least one of the alternatives must
4This term derives from early applications of random utility models to the choice among different transport modes.
Vwalking=β1twl
Vauto=β1twla+β2tba+β3mca+β4AVAIL+β5INC+β6AUTO Vbus=β1twlb+β2tbb+β3mcb+β7twb+β8BUS
Alternative Level of service Socioeconomic
specific attributes (ASA) attributes attributes
AUTO tb=Time on board (generic) AVAIL=# Auto/# licenses BUS tw=Waiting time at stop (specific) INC=Disposable
twl=Walking time (generic) household income mc=Monetary cost (generic)
Fig. 3.1 Specification of systematic utilities and classification of attributes
be set to zero; equivalently, ASAs may be included in the systematic utilities of at most all the alternatives except one.
An elementary example of systematic utilities related to transport mode choice is given in Fig.3.1. Many other examples are given in the following chapters.
Utilities are merely a way of capturing the preference ordering among alterna- tives, and so have no intrinsic units of measurement; alternatively, they can be ex- pressed in arbitrary dimensionless units, sometimes calledutils. From expression (3.2.5) it can be seen that, in order to sum attributes expressed in different units (e.g., time and cost), their respective coefficientsβk have to be expressed in units that are inverses of those of the attributes themselves (e.g., time−1 and cost−1).
The coefficientsβare sometimes called reciprocal substitution coefficients because they make it possible to evaluate reciprocal “exchange rates” (rates of substitution) between attributes. This point is developed in Chap. 4.
The randomness of perceived utilities. Various factors account for the difference between the utility perceived by an individual decision-maker and the systematic utility common to all decision-makers with equal values of the attributes. These factors are related both to the model (factors a, b, and c below) and to the decision- maker (factors d and e). They include:
(a) Errors in measuring the attributes that are included in the systematic utility. For example, level-of-service attributes are often obtained from a network model and so are subject to modeling and aggregation (zoning) errors; other attributes are intrinsically variable and only their average value can be considered.
(b) Attributes that are not included in the systematic utility because they are not directly observable or are difficult to evaluate (e.g., travel comfort or total travel time reliability).
(c) Instrumental attributes that are included in the systematic utility specification but only imperfectly represent the actual attributes that influence the alterna- tives’ perceived utility (e.g., modal preference attributes replacing variables such as the comfort, privacy, image, etc. of a certain transport mode; the to- tal number of commercial establishments in a given zone replacing the number and variety of shops).