Random Utility Theory
3.4 Expected Maximum Perceived Utility and Mathematical Properties of Random Utility Models
3.4 Expected Maximum Perceived Utility and Mathematical Properties of Random Utility 133
3.4 Expected Maximum Perceived Utility and Mathematical
By definition, s(V)=
∞ ε1=−∞
· · · ∞
εm=−∞
max(V +ε)f (ε) dε and becausef (ε)≥0 and max(V +ε)≥Vk+εk,∀k∈I, it follows that
s(V)= ∞
ε1=−∞
· · · ∞
εm=−∞
max(V+ε)f (ε) dε
≥ ∞
ε1=−∞
· · · ∞
εm=−∞
Vkf (ε) dε+ ∞
ε1=−∞
· · · ∞
εm=−∞
εkf (ε) dε
=Vk ∞
ε1=−∞
· · · ∞
εm=−∞
f (ε) dε+ ∞
ε1=−∞
· · · ∞
εm=−∞
εkf (ε) dε
=Vk+E[εk] =Vk ∀k∈I
Therefores(V)is greater than or equal to the largest systematic utility,s(V)≥ Vk, ∀k∈I.
In addition, the mean systematic utility, calculated by weighing the systematic utility of each alternativekby its respective choice probabilitypk(V), is less than or equal to the EMPU variable. From expression (3.4.3), it follows that
p(V )TV =
k
pk(V )Vk≤
k
pk(V )max(V )=max(V )≤s(V )
In order to analyze the EMPU variable in more detail, consider first a multino- mial logit model with constant parameterθ. For this model,s(V)can be expressed in closed form. Referring to the results reported for the maximization of Gumbel variables,16the EMPU is given by expression (3.3.5), repeated here:
s(V)=θln
j
exp(Vj/θ ) (3.4.4)
It can easily be shown that expression (3.4.4) satisfies condition (3.4.3); Fig.3.14il- lustrates this result. From expression (3.4.4) it can also be deduced that the EMPU of a multinomial logit model increases if the systematic utility of one or more alterna- tives increases because the functions ln(·)and exp(·)are both monotonic increasing.
Furthermore, because of the nonnegativity of the exponential function, the EMPU increases with the number of available alternatives. In fact, the addition of a new alternative to the choice set results in an increase in the EMPU even if the new al- ternative has a systematic utility less than that of the alternatives already available.
This is because of the randomness of perceived utilities: there is a positive probabil- ity that the new alternative will be perceived as having a utility greater than that of
16The maximum of i.i.d. Gumbel variables having scale parameterθis also a Gumbel variable with the same scale parameter. See also Appendix3.B.
3.4 Expected Maximum Perceived Utility and Mathematical Properties of Random Utility 135 Fig. 3.14 Example of
calculation of the expected maximum perceived utility (EMPU)
any other alternative. In this case, the maxj(Ui)will clearly increase, and this will lead to a general increase in the mean value of maxj(Ui), which is the EMPU.
The example in Fig.3.14also illustrates this point.
These properties of EMPU, directly derived here for the multinomial logit model, also apply to the larger class of invariant random utility models. Recall that, for these models, the density function of the random residuals does not depend onV.
f (ε/V)=f (ε) ∀ε∈Em (3.4.5) All of the random utility models described in Sect.3.3are invariant if the para- meters off (ε)do not depend on the vectorV. If the joint density function of the random residualsf (ε)is continuous with continuous first derivatives, the choice probabilitiesp(V)and the EMPUs(V)are also continuous functions ofV with continuous first derivatives. All random utility models described in Sect.3.3satisfy these continuity requirements. Under these assumptions, invariant random utility models share a number of general mathematical properties that are connected with the expected maximum perceived utility.
(1) Thepartial derivativeof the EMPU with respect to the systematic utilityVk is equal to the choice probability of alternativek:
∂s(V)
∂Vk =p[k](V) (3.4.6)
The gradient of the EMPU is thus equal to the vector of choice probabilities:
∇s(V)=p(V) (3.4.7a)
and its Hessian is equal to the Jacobian of choice probabilities:
Hess s(V)
=Jac p(V)
(3.4.7b) For a continuous function with continuous first derivatives, the integration and differentiation operators can be exchanged:
∂s(V)
∂Vk = ∂
∂Vk ∞
ε1=−∞
· · · ∞
εm=−∞
max(V+ε)f (ε) dε
= ∞
ε1=−∞
· · · ∞
εm=−∞
∂max(V +ε)
∂Vk f (ε) dε (3.4.8) Because
∂max(V +ε)
∂Vk
=
1 forksuch thatVk+εk=max(V +ε) 0 otherwise
the integral (3.4.8) is equal to the probability that the perceived utility of al- ternativek, Vk+εk, is the largest among all themalternatives available, from which expression (3.4.6) derives.
This result can be checked immediately for the multinomial logit model, for which the EMPU, expressed by (3.4.4), can be differentiated analytically:
∂
∂Vk
θln
j
exp(Vj/θ )
= exp(Vk/θ )
jexp(Vj/θ )=p[k](V ) (3.4.9) Furthermore, because the choice probabilityp[k]is always greater than or equal to zero (3.4.6) shows that the derivative of the EMPU with respect to the systematic utility is always nonnegative: the EMPU increases (or does not de- crease) as the systematic utility of each alternative increases and, by extension, as the number of available alternatives increases.17
(2) The EMPU function is convex18 with respect toV, the vector of systematic utilities.
In fact, for eachε, f (ε)≥0 and max(V +ε)is a convex function ofV; it follows that the expected maximum perceived utility functions(V), expressed by (3.4.2), is a linear combination with nonnegative coefficients of convex func- tions, and therefore is convex too.
Note that by virtue of property (2) the EMPU function has a Hessian matrix, Hess(s(V)), which is (symmetric and) positive semidefinite. Consequently, the Jacobian of choice probabilities,Jac(p(V)), is (symmetric and) positive semi- definite (see (3.4.7b)).
(3) If the EMPU function is continuous and differentiable then:
s(V′)≥s(V′′)+p(V′′)T(V′−V′′) ∀V′,V′′ (3.4.10a) and the choice probabilities are monotonic increasing functions of the system- atic utilities.
p(V′)−p(V′′)T
(V′−V′′)≥0 ∀V′,V′′ (3.4.10b)
17The availability of a new alternative can be seen, in fact, as a change in the systematic utility of that alternative from minus infinity to a finite value.
18Convexity of a scalar-valued function of a vector is defined in Appendix A.
3.4 Expected Maximum Perceived Utility and Mathematical Properties of Random Utility 137 Because the EMPU function is convex and differentiable, it follows that
s(V′)≥s(V′′)+ ∇s(V′′)T(V′−V′′) ∀V′,V′′
and its gradient must be an increasing monotonic function (see Appendix A):
∇s(V′)− ∇s(V′′)T
(V′−V′′)≥0 ∀V′,V′′
Applying (3.4.7a), the two preceding expressions can be formulated in terms of the vector of choice probabilities as in (3.4.10a) and (3.4.10b). Moreover, from (3.4.10a) it follows that:
s(V′)−s(V′′)≥p(V′′)T(V′−V′′) ∀V′,V′′
s(V′′)−s(V′)≥p(V′)T(V′′−V′) ∀V′,V′′
Summing the last two inequalities yields:
0≥p(V′′)T(V′−V′′)+p(V′)T(V′′−V′) ∀V′,V′′
from which (3.4.10b) is easily obtained.
In particular, (3.4.10b) can be expressed for a single alternative, assuming that the systematic utilities of all other choice alternatives are constant:
pk(Vk′)≥pk(Vk′′) ifVk′≥Vk′′
In other words, the choice probability of a generic alternative does not de- crease as its systematic utility increases, if all the other systematic utilities re- main unchanged. Using an analogous argument it can be demonstrated that, as Vktends to minus infinity, the choice probability of alternativektends to zero:
Vk→−∞lim p[k] =0
Mathematical properties of the deterministic choice model. The deterministic choice model19 is obtained if the random residuals are all equal to zero. In this case, the perceived utility coincides with the systematic utility and only the alternative(s) having maximum utility can be chosen:
p[k]>0 ⇒ Vk=max(V) and
Vk=max(V) ⇒ p[k] ∈ [0,1], Vk<max(V) ⇒ p[k] =0
19Deterministic utility models and their properties are mainly used in Sect. 4.3.3 on path choice models and in Chap. 5 on assignment models.
Note that the deterministic choice model satisfies condition (3.4.5) and can there- fore be considered an invariant model. If there are two or more alternatives with (equal) maximum systematic utility, there are infinitely many choice probability vectors satisfying the above conditions. In this case, the relationp(V)is not a func- tion, but a one-to-many map. LetpDET(V)be one of the possible choice probability vectors corresponding to vectorV through the deterministic choice map.
The following necessary and sufficient condition guarantees that a probability vectorp∗(withp∗≥0and1Tp∗=1) is a deterministic choice probability vector:
p∗=pDET(V) ⇔ VTp∗=max(V)1Tp∗=max(V) (3.4.11a) Given a vector of deterministic probabilities p∗ =pDET(V), it follows that VTp∗=max(V)becausep∗k can be positive only for an alternativekhaving max- imum systematic utility, and conversely. Furthermore, the condition1Tp∗=1 im- plies that max(V)1Tp∗=max(V).
In general, for any vector of choice probabilitiesp, because 1Tp=1 then, as observed earlier:
VTp≤max(V)1Tp=max(V) ∀p:p≥0, 1Tp=1
Consistent with (3.4.11a), equality holds in the above relationship only for a vec- tor of deterministic probabilities. Combining the above relationship with (3.4.11a), the following basic relationship can be obtained.
V −max(V)1T
p−pDET(V)
≤0 ∀p:p≥0, 1Tp=1 (3.4.11b) This is applied in the analysis of deterministic assignment models in Chap. 5.
The deterministic utility model has properties (2) and (3) described above for probabilistic and invariant models.20 Regarding property (2), the expected maxi- mum perceived utility of a deterministic model is a convex function of systematic utilities and is equal to the maximum systematic utility:
s(V)=max(V)=pDET(V)TV (3.4.12) This condition and result (3.4.3) imply that, for a given vector of systematic utilitiesV, the EMPU of a deterministic choice model is less than or equal to that of any probabilistic choice model involving the same systematic utility. A behavioral interpretation of this result suggests that the presence of random residuals makes the perceived utility for the chosen alternative, on average, larger than the alternative’s systematic utility, which is the perceived utility in a deterministic choice model.
Regarding property (3), the deterministic choice map is monotone nondecreasing with respect to systematic utilities, just as are invariant probabilistic choice func- tions:
s(V′)≥s(V′′)+pDET(V′′)T(V′−V′′) ∀V′,V′′ (3.4.13a)
20Property (1) requires the introduction of the concept of subgradients of a convex function.