Stochastic user equilibrium in the presence of state dependence

Abstract

We consider the following two state-dependent effects at the level of route choice: inertia to change and, as a consequence of experience, lower perception variance for the currently used route. A heteroscedastic extreme value model embodying heterogeneity across alternatives in the mean of the random terms is used. Estimations based on stated preference data confirm the presence of both state-dependent effects. We introduce a new class of stochastic user equilibrium (SUE) models that take state-dependent effects into account. The class includes conventional SUE as special case. The equilibrium conditions are formulated as fixed-point states of deterministic day-to-day assignment processes. At the equilibrium (1) no user can improve her/his utility by unilaterally changing route, and (2) if each user shifts from her/his current route to her/his newly chosen route the observed route flows do not change. The existence of the equilibrium is guaranteed under usually satisfied conditions. A modified method of successive averages is proposed for solution. Examples related to a two arc network and to the Nguyen-Dupuis network illustrate the model.

Appendix

1.1 Notation

The following notation (in alphabetical order) is used. The symbols are, wherever possible, self-explaining because based on the initials of the associated quantity. Quantities (flows and travel times) that are related to arcs are lowercase, quantities that are related to routes are uppercase.

a:

Arc

A:

Arc set

C_a:

Capacity of arc

d^w:

Demand of OD pair $w$

$f_{a}$:

Flow of arc $a$

$F_{r}^{w}$:

Flow of route $r$ of OD pair w

${\mathbf{F}}$:

Vector of route flows

$k$:

Iteration counter

$I_{r\left| j \right.}^{w}$:

Indicator function related to the identity of route $r$ at day $n$ and route $j$ the day before for OD pair w

$j$:

Route

$M_{r}^{w}$:

Monetary cost of route $r$ of OD pair $w$

$n$:

Day

$P_{r\left| j \right.}^{w(n)}$:

Conditional probability of choosing route $r$ at day $n$ having chosen route $j$ the day before for OD pair $w$

$r$:

Route

$R^{w}$:

Route set of OD pair $w$

$t_{a}^{(n)}$:

Travel time of arc $a$ at day $n$

$t_{a}^{0}$:

Free-flow travel time of arc $a$

$T_{r}^{w(n)}$:

Travel time of route $r$ of OD pair $w$ at day $n$

$V_{r\left| j \right.}^{w(n)}$:

Conditional systematic utility of route $r$ of OD pair $w$ at day $n$ having chosen route $j$ the day before

$w$:

OD pair

$W$:

OD pair set

$\beta_{T}$:

Estimation coefficient of route travel time

$\beta_{M}$:

Estimation coefficient of route monetary cost

$\gamma$:

Algorithm convergence tolerance

$\delta_{a,r}^{w}$:

Entry of the arc-route incidence matrix

$\eta$:

Estimation coefficient of the inertia term

$\theta$:

Scale parameter of the route chosen the day before

$\lambda$:

Eigenvalue of the Jacobian, computed in the fixed point, of the transition functions

${\varvec{\Psi}}$:

Route-flow based fixed-point map