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Some Remarks on Stochastic User Equilibrium
Transpn Res.-B, Vol. 32, No. 2, pp. 101±108, 1998 # 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0191-2615/98 $19.00+0.00
Pergamon PII: S0191-2615(97)00015-5
SOME REMARKS ON STOCHASTIC USER EQUILIBRIUM MARTIN L. HAZELTON Department of Statistical Science, University College London, Gower Street, London WC1E 6BT, U.K. (Received 27 April 1996; in revised form 11 February 1997) AbstractÐThe behavioural foundation of Stochastic User Equilibrium is that each traveller attempts to minimize his or her perceived travel costs, where these costs are composed of a deterministic measured cost and a random term which can be interpreted as perceptual error. In principle such a de®nition, which is in terms of random errors, should imply an equilibrium probability distribution over feasible ¯ow patterns on the transport network. Such a probability distribution could potentially allow between day variability in network ¯ows to be represented. However, traditionally a deterministic, large sample approximation has been used as the `solution' of Stochastic User Equilibrium. In this paper a representation of Stochastic User Equilibrium as a probability distribution is developed. This distribution is de®ned by the conditional route selection of each individual given the choices of all other travellers. An interpretation of the resulting assignment model as the limit (in the in®nite future) of a continuous time assignment process is discussed. The limiting behaviour as the travel demand becomes large is also investigated, and convergence to the traditional deterministic form of Stochastic User Equilibrium is proved. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: conditional probability, equilibrium distribution, stochastic route choice, trac assignment. 1. INTRODUCTION
Stochastic User Equilibrium (SUE) was introduced by Daganzo and She (1977). Their idea was to relax the unrealistic assumption of users having perfect knowledge of travel costs, which is an integral part of classical Wardrop (and user) equilibrium; see Wardrop (1952). To this end they suggested the following precept, which we term Stochastic User Behaviour. De®nition 1. Stochastic User Behaviour A traveller displaying Stochastic User (SU) Behaviour will select the route which he or she perceives to have minimum cost. Daganzo and She (1977) decided to model errors in travellers' perceptions of costs by random variables. Not only did this approach provide a natural framework for the representation of such errors; it also implied that the global ¯ow pattern resulting from universal practice of SU behaviour be a random variable drawn from an equilibrium probability distribution over the space of all feasible ¯ows (as opposed to a single deterministic ¯ow pattern). Expression of an equilibrium ¯ow pattern in terms of a probability distribution is an attractive proposition, since it allows between day variability in the transport system to be modelled. From a transport planning point of view it is all very well to have an estimate which represents an `average' ¯ow pattern in some sense, but at least as important to know whether random ¯uctuations in link ¯ows will lead to gridlock of the transport system on one day in twenty. See also the remarks of Hanson and Ho (1981, 1988), Taylor (1982), Arnott et al. (1991) and Watling and van Vuren (1993). It follows that the potential of stochastic models to estimate the variance of trac ¯ows is a strong argument in their favour in comparison with more traditional deterministic trac assignments. Nevertheless, Daganzo and She (1977) did not take full advantage of this potential, preferring instead a deterministic `large sample' de®nition of an SUE ¯ow pattern. We outline their approach in the following paragraphs, beginning by introducing the necessary notation. Let the triple N; L; K represent a ®nite transport network, where N is the node set, L the set of directed links and K a set of link cost functions. Let D be the trip matrix, whose stth element dst is 101
102
Martin L. Hazelton
the number of travellers going from origin node s to destination node t; O±D pair st as we shall call it. Let n=1TD1 be the total network demand (number of travellers) where 1 is a column vector of ones. De®ne I to be a set indexing all feasible routes (acyclic paths) and let Ij be the indexing set of feasible routes for the jth traveller. We write i k if and only if i and k are routes connecting the same O±D pair. Let R R1 ; :::; Rn be the vector of traveller route choices, so that Rj 2 Ij and i k for all i; k 2 Ij . When discussing stochastic route choice R will typically be a random vector, whose probability mass function is denoted by p r Pr R r. The ¯ow down path i 2 I is de®ned in terms of R by Fi
n X
Rj ; i;
j1
where is the Kroneker delta which is zero unless its arguments are identical, in which case it takes the value one. If F is the vector of such path ¯ows and B the network's link-path incidence matrix, then the vector of link ¯ows, X, is de®ned by X BF: Let C i be the actual (or measured) cost to a traveller taking the ith route, for i 2 I. This measured cost will usually be a function of the state of the network, and may therefore be written as C(ijR), C(ijF) or C(ijX) depending on the manner in which the trac ¯ow pattern is described. The perceived cost of the ith path for a traveller is denoted by CÄ(ijX, ") where " is a generic random error term. Usually " will be implicitly assumed and we shall write simply CÄ(ijX). Note that if " follows a suitable Gumbel distribution, and the perceptual error is additive in the sense that ~ j X C i j X "; C i then we have de®ned a logit route choice model. See Thomas (1991, p. 180) for details. She and Daganzo (1977) noted that SU behaviour dictates that traveller j will choose route i from the set Ij with probability given by ~ j X < C k ~ j X; 8k 6 i; k i: Pr C i
1
(Note that if perceptual error is modelled by continuous random variables, then equal costs will only occur with probability zero, and hence we may use a strict inequality in eqn (1) without loss of generality.) Suppose that j is a typical traveller between O±D pair st, and that dst is large. She and Daganzo argued that the Weak Law of Large Numbers (see Grimmett and Stirzaker, 1982, for example) implies that the proportion of these dst travellers taking route i will be very close to the probability of taking this route, as de®ned in eqn (1). This occurs when the stochastic variation in ¯ows becomes insigni®cant in comparison to the mean ¯ows. It is therefore at this stage of the argument that Daganzo and She's trac assignment loses its stochastic nature and becomes a deterministic model. Now, if proportion is put exactly equal to probability (the limiting case) then we get, ~ j X < C k ~ j X; 8k 6 i; k i P Pr C i
Fi ki
Fk
;
2
P where dst ki Fk since each traveller for O±D pair st must take some route k such that k i. Both sides of this equation are functions of X. Furthermore, this equation holds for all routes between all O±D pairs (equivalently, for all pairs i; k where i 2 I and k i). It therefore implicitly de®nes a ¯ow pattern, as described by de®nition 2. De®nition 2. Daganzo and She's Stochastic User Equilibrium (SUE) Let X{ be the ¯ow pattern solving eqn (2) for all routes between all O±D pairs. Then X{ is Daganzo and She's Stochastic User Equilibrium.
Stochastic user equilibrium
103
Daganzo and She's SUE is intended to de®ne the limiting case (as the number of travellers goes to in®nity) of a system in which every individual practises SU behaviour. Our aim in this paper is to consider the ®nite demand case, when stochastic variation in ¯ow patterns will not be negligible and hence equilibrium should be represented by a probability distribution. Note that X{ is de®ned as the solution to a ®xed point problem. Whilst there are any number of discrete-time stochastic process models for trac ¯ow which will converge to X{ as the travel demand becomes large (see Davis and Nihan, 1993, for example), Daganzo and She's SUE exists without need to specify such a temporal process. It is partly in this spirit of generality that we will develop our form of SU equilibrium probability distribution from a largely atemporal, process free point of view. Nevertheless, our suggested form of equilibrium assignment model does have an interpretation in terms of contemporaneous traveller interaction, as we shall discuss later. The paper is structured as follows. In Section 2 we describe problems which occur if we attempt what might be seen as the most straightforward extension of Daganzo and She's version of SUE to the ®nite demand case. We acknowledge that these diculties are largely due to our somewhat game theoretic analysis of SUE, but such a `process free' approach does have advantages as mentioned above. Our own de®nition of an SU equilibrium distribution, called Conditional Stochastic User Equilibrium (CSUE), is then presented. It is illustrated using a simple numerical example. In Section 3 we discuss the properties of CSUE, including its interpretation as the limiting case (in the in®nite future) of a continuous time assignment process. It is also proved that our assignment converges to Daganzo and She's SUE as the travel demand becomes large (under certain conditions on the link cost functions). The paper concludes with some general comments on stochastic assignment procedures. 2. CONDITIONAL STOCHASTIC USER EQUILIBRIUM
SU behaviour (and any natural variant of it) is de®ned in terms of the way in which an individual traveller chooses his or her route. It therefore seems that any SU equilibrium probability distribution will necessarily be de®ned at such a disaggregate level, in terms of probabilities of individual route selections. That is to say we must ®rst look for a distribution of R, and then derive the distribution on F and X. In order to de®ne an equilibrium probability distribution over the joint route choices of every individual, it is sucient to de®ne the conditional route choice distribution of each separate traveller. This is because a full set of conditional distributions de®nes a unique joint distribution under mild conditions, as proven by the Hammersley-Cliord theorem (see Besag, 1974). Perhaps the most straightforward approach to obtaining conditional route choice probability distributions is to use Daganzo and She's formulation of SU behaviour as represented by eqn (1). This implies that the conditional probability of the jth traveller taking route i 2 Ij is ~ j X < C k ~ j X; 8k 6 i; k i: Pr Rj i j X Pr C i This is a conditional probability since it depends (through the perceived costs) on X. However, X is in some small part determined by the actual route that the jth traveller selects. In other words, the probability that j takes route i is a function of whether or not j took route i. This is a contradiction unless the probability is zero or one. Such a diculty has occurred because of the atemporal framework within which we are working. It could, for example, be resolved by introducing a discrete time dimension to the problem and de®ning route choices at epoch t in terms of the ¯ow pattern at t ÿ 1. Then Rj at t would be dependent on Rj at t ÿ 1, but this presents no logical paradox. However, our aim is to avoid speci®cation of any particular temporal stochastic process, as mentioned earlier. We therefore propose a dierent resolution, by de®ning each traveller's behaviour conditionally upon the choices of all other individuals' route choices. The result is the following behavioural precept; cf. Hazelton et al. (1996). De®nition 3. Conditional Stochastic User Behaviour A traveller displaying Conditional Stochastic User (CSU) Behaviour will select the route which he or she perceives to have minimum cost conditional on all other travellers' route choices.
104
Martin L. Hazelton
In practice this de®nition implies that when calculating the route choice probabilities for traveller j, the path costs are computed from the ¯ow pattern determined by the journeys of all travellers except the jth. Let Rÿj be R with the jth element removed, and let F ÿj and X ÿj be the corresponding path and link ¯ow patterns. Then traveller j practising CSU behaviour will select path i 2 Ij with probability ~ j X ÿj ; 8k 6 i; k i: ~ j X ÿj < C k Pr C i
3
Since this probability no longer conditions on the route choice of traveller j there are no inherent contradictions of the type discussed above. Under the assumption that the probability of eqn (3) is non-zero for all combinations of j and i 2 Ij (that is, no feasible route is assigned probability zero by any traveller), then this set of conditional probabilities necessarily de®nes a unique joint distribution for all route choices, R1 ; :::; Rn . This is a consequence of the Hammersley-Cliord theorem. We may therefore de®ne CSUE in terms of this joint distribution in the knowledge that it exists under the mild condition just noted. De®nition 4. Conditional Stochastic User Equilibrium (CSUE) A system is in CSUE when all travellers practise CSU behaviour. The random vectors of route choices, path and link ¯ow patterns will be denoted by R*, F* and X* respectively in this state. Notice that CSUE is not a deterministic state. It is a truly stochastic assignment in which the probability of a particular route choice vector r is determined by the CSUE probability distribution, p r Pr R r. We illustrate CSUE with the aid of a simple numerical example. Example. Consider a single O±D pair connected by two parallel links with ¯ows x1 and x2 . Let the separable, monotonic and dierentiable link cost functions, c1 and c2 be given by c1 x1 x1 1 1 2 3 2 x 2 2 04x241 c2 x2 otherwise: x2 1 Notice that since each feasible path corresponds to a single link, we have C(ijX x1 ; x2 ci xi in terms of the path cost notation introduced earlier. Suppose that there are a total of two travellers so that x1 x2 2. Let route choice probabilities be determined by a logit model with logit constant set to one. Then under CSUE the probability that traveller 1 takes route 1 conditional on the fact that traveller 2 takes route 1 is given by Pr R1 1 j R2 1
exp c1 1 eÿ2 : ÿ2 exp c1 1 exp ÿc2 0 e eÿ3=2
Using all the conditional probabilities, we can calculate marginal probabilities as follows. Pr R1 1 Pr R1 1 j R2 1 Pr R2 1 Pr R1 1 j R2 2 Pr R2 2
eÿ2 eÿ1 Pr R 1 Pr R2 2: 2 eÿ2 eÿ3=2 eÿ2 eÿ1
By symmetry Pr R1 1 Pr R2 1 1 ÿ Pr R2 2 q (say). Hence, q
eÿ2 eÿ1 q 1 ÿ q eÿ2 eÿ3=2 eÿ2 eÿ1
which can be solved to give q 0:540. Joint probabilities for R1* and R2* can now be computed; for example, Pr R1 1; R2 2 Pr R1 1 Pr R2 2 j R1 1 q
eÿ3=2 0:336: eÿ3=2 eÿ2
Stochastic user equilibrium
105
Since this is equal to Pr R1 2; R2 1 by symmetry, the probability of the travellers taking dierent links is Pr X 1; 1 Pr R1 1; R2 2 Pr R1 2; R2 1 0:672: We end this example by noting that the expected (or mean) ¯ow pattern in CSUE is EX 1:08; 0:92. Daganzo and She's SUE can trivially be checked to be X{=(1,1). Whilst these ¯ow patterns are not wildly dierent, CSUE does re¯ect the fact link 1 is more attractive in the sense that c1 y4c2 y with strict inequality on the interval 04y < 1. 3. PROPERTIES OF CSUE
The development of CSUE has been from a process free perspective. Nevertheless, CSUE as an assignment model does have an interesting interpretation in terms of the limiting case of a system in which travellers interact in a contemporaneous fashion. In order to explain why, we must ®rst consider a continuous time model of the development of the network ¯ow pattern. In this model each traveller makes a route selection, from time to time, by sampling from the conditional probability distribution de®ned by eqn (3), using route costs de®ned by present link ¯ows. Two consequences of the fact that our process occurs in continuous times are worthy of note. First, route choice based on present costs is feasible in continuous time, so that travellers interact contemporaneously. (This is in contrast to discrete time models in which actions at epoch t are based upon states of network in past epochs, t ÿ 1, t ÿ 2, and so on.) Secondly, assuming that absolutely simultaneous events occur with probability zero in continuous time, we can uniquely order the travellers' route changes in time, one by one. Again, this is generally in contrast to discrete time models where groups of travellers change route together at each epoch. Consider the limiting case of this model as the ¯ow pattern evolves over an in®nitely long period, so that R (and hence F and X) is updated an in®nite number of times by sampling from conditional route choice distributions. Assume that each traveller makes an in®nite number of route choices during this time. (The speci®c sequence in which individuals make route selections is unimportant in this discussion.) Then assuming that no feasible route is assigned probability zero by any traveller, the distribution of the ¯ow pattern will converge to the CSUE probability distribution. (This holds regardless of the intial state of the network.) This result is an immediate consequence of the work of Gelfand and Smith (1990). A by-product of this argument is the development of a numerical method for simulating CSUE ¯ow patterns. This is important because actual calculation of the CSUE probability distribution, p , is not analytically feasible for anything other than tiny arti®cial networks (such as the example in the previous section). By simulating a version of the continuous time process just described, we will eventually (when convergence is achieved) be drawing samples from the CSUE probability distribution (no matter what the initial ¯ow pattern). We can then estimate properties of p , such as the variance of ¯ows on route 4 or the correlation between ¯ows on links 14 and 15, simply by computing the corresponding sample quantities from a large number of simulated realizations from p . An ecient technique for performing such simulations, using so called Markov Chain Monte Carlo methods, is described in detail in Hazelton et al. (1996). This paper also contains some numerical examples of CSUE based on real road networks. We now turn to the behaviour of CSUE trac assignment as the travel demand becomes large. Whilst our assignment process is based upon CSU behaviour, as opposed to SU behaviour, the dierence is solely in terms of the costs generated by a single traveller; an increasingly insigni®cant proportion of the total costs as the number of travellers tends to in®nity. Furthermore, since it is to be expected that the stochastic coecient of variation in ¯ows will diminish with increasing travel demand, we should expect the CSUE probability distribution to converge to the single deterministic ¯ow, X{. We formally prove that this is the case (under certain conditions) in the theorem presented below. The limiting process that we actually consider is one in which each unit of ¯ow is subdivided into m assignable parts, where m ! 1. In other words the number of travellers becomes large, but each traveller is represented by a quantum of ¯ow of weight mÿ1 only. Before proving the main result a lemma stating an alternative form of the Weak Law of Large Numbers is required.
106
Martin L. Hazelton
Lemma. Let Y1, Y2, Yn be a collection of identically distributed random variables with ®nite n P Ya ; variance and common mean . Suppose that cov (Ys, Yt)40 for all s6t. Then if Sn ÿ1
a1
p
n Sn ! as n ! 1. Proof: Let the common variance of the Ya 's be 2 . Then, were Y1 ; Y2 , ..., Yn independent, the variance of their sum would be n 2 . Now, the covariance terms can only contribute negatively to var Sn , so it follows that var Sn 4n 2 . Hence var nÿ1 Sn 4nÿ1 2 ! 0 as n ! 1: Since Enÿ1 Sn , it follows that nÿ1 Sn converges in mean square to and therefore also in probability, completing the proof; cf. Feller (1950). & Theorem. Let N = (N, L, K) be a transport network with O±D trip matrix D, and such that all link cost functions are separable, continuous and monotonic increasing. Let Nm be the corresponding network in which each unit of ¯ow is subdivided into m assignable individual parts. Then if X*(m) is the CSUE random ¯ow pattern for the network N m and X{ (m) is the corresponding Danganzo and She SUE ¯ow, p
X m ÿ Xy m ! 0 as m ! 1. Proof: Let Am be a set indexing all assignable parts for whom route i is feasible in network Nm . Then for a; b 2 Am ; cov Ra ; i; Rb ; i < 0, since conditioning on individual a taking route i increases the cost of that route (by the assumptions on the link cost functions) and hence makes it less likely that individual b will follow suit. Hence, f Ra ; i; a 2 Am g are a set of random variables satisfying the lemma. P Note that the cardinality of the set Am is given by j Am j m ki Fk . Using the lemma it follows that for network N m the proportion of ¯ow using route i in CSUE satis®es P
X Fi m p j Am jÿ1 Ra ; i ! E Ra0 ; i F m ki k a2A m
Pr Ra0
as m ! 1, where a0 2 Am is a typical traveller. Now for network N
4
i m
~ j X 0 m < C k ~ j X 0 m 8k 6 i; k i Pr Ra0 i Pr C i ÿa ÿa p ~ j X m < C k ~ j X m 8k 6 i; k i ! Pr C i
since the continuity of the measured cost functions ensures that j C k j X m ÿ C k j X ÿa0 m j O mÿ1 ! 0 for any path k"I as m ! 1. (In other words, the impact on the costs of a quantum of ¯ow of weight mÿ1 has negligible eect as m ! 1.) Combining this result with eqn (4) shows that P
Fi m p ~ ~ m ÿ Pr C i j X m < C k j X m 8k 6 i; k i ! F ki k
5
as m ! 1, and this holds for all i 2 I. But the ¯ow pattern which equates the left hand side of eqn & (5) to the zero vector is precisely X{(m), see eqn (2), completing the proof.
Stochastic user equilibrium
107
We remark that the conditions of monotonicity and separability on link costs are sucient for the theorem to hold, but not necessary. 4. DISCUSSION
Between day variation in trac ¯ow is an important feature of real trac systems. However, the vast majority of assignment modelling has been deterministic, aiming only at some estimate of an average ¯ow pattern. If an equilibrium assignment is intended to represent ¯ows over a long time interval, then the ¯ows themselves will be large and statistical laws (usually) guarantee that variation in ¯ow is small in comparison to the average. In such situations, Daganzo and She's SUE may well provide a perfectly adequate representation of the system. Nevertheless, if interest centres on modelling at a ®ner level of temporal aggregation then the trac ¯ows will not be large enough to render day-to-day ¯ow dispersion negligible. Cascetta (1989) makes the point that in many applications hourly demand ¯ows are of few units for most O±D pairs, in which case variability in link ¯ows should be a pivotal part of the modelling procedure. As an example, consider the case of Stochastic Network Loading, where measured costs are ®xed independently of the ¯ow pattern. If there is a demand of 100 units between an O±D pair and the probability of a traveller using a particular route is 0.2 (a function of the ®xed costs), then the number of travellers using this route will follow a binomial distribution. It follows that the mean ¯ow down this route will be 20 units, but the range 8; 32 is an approximate 95% con®dence interval for the route ¯ow. Hence, even for a moderate O±D demand, the range of ¯ows can be comparable to the magnitude of the average ¯ow. Our CSUE assignment is capable of representing such stochastic variation in ¯ow patterns. Whilst independent ¯ow patterns drawn from p will not provide a sensible model of consecutive days' trac ¯ows (which are highly dependent), they will give an idea of the magnitude of the between day variation that can be expected over a long period of time. The ability to model variation in trac ¯ows is a strong argument in favour of assignment procedures which give rise to a probability distribution on ¯ow patterns. A further advantage of modelling stochastically is that such an approach provides a natural framework for developing assignment procedures which are based on a richer representation of traveller behaviour than those which are currently in use. For example, models incorporating traveller learning and the impact of advanced traveller information systems would naturally be formulated in terms of stochastic processes, since the rate at which individuals gather information and the way in which they react to it varies in a highly complex manner across the population. See, for example, Watling and van Vuren (1993). We believe that stochastic assignment models based upon more realistic behavioural assumptions will provide an exciting area for future research. AcknowledgementsÐThe author thanks John Polak, Seungjae Lee and anonymous referees for their useful comments on SUE and related areas. The research in this paper was supported by grant GR/K18337 from the Engineering and Physical Sciences Research Council, U.K. REFERENCES Arnott, R., de Palma, A. and Lindsey, R. (1991) Does providing information to drivers reduce congestion? Transportation Research A 25, 309±318. Besag, J. (1974) Spatial interaction and the analysis of lattice systems (with discussion). Journal of the Royal Statistical Society, Series B 36, 192±236. Cascetta, E. (1989) A stochastic process approach to the analysis of temporal dynamics in transportation networks. Transportation Research B 23, 1±17. Daganzo, C. F. and She, Y. (1977) On stochastic models of trac assignment. Transportation Science 11, 253±274. Davis, G. A. and Nihan, N. L. (1993) Large population approximations of a general stochastic trac assignment model. Operations Research 41, 169±178. Feller, W. (1950) An Introduction to Probability Theory and its Applications. John Wiley & Sons, New York. Gelfand, A. E. and Smith, A. F. (1990) Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association 85, 398±409. Grimmett, G. and Stirzaker, D. (1982) Probability and Random Processes. Clarendon Press, Oxford. Hanson, S. and Ho, J. (1981) Assessing day-to-day variability in complex travel patterns. Transportation Research Record 891, 18±24. Hanson, S. and Ho, J. (1988) Systematic variability in repetitious travel. Transportation 13, 111±135. Hazelton, M. L., Lee, S. and Polak, J. W. (1996) Stationary states in stochastic process models of trac assignment: a Markov Chain Monte Carlo approach. In Proceedings of the 13th Symposium on Transportation and Trac Theory, ed. J-B. Lesort, pp. 341±359. Pergamon, Oxford.
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Taylor, M. A. P. (1982) Travel time variabilityÐthe case of two public modes. Transportation Science 16, 507±521. Thomas, R. (1991) Trac Assignment Techniques. Avebury Technical, Aldershot. Wardrop, J. G. (1952) Some theoretical aspects of road trac research. Proceedings of the Institute of Civil Engineering, Part II 1, 325±378. Watling, D. and van Vuren, T. (1993) The modelling of dynamic route guidance systems. Transportation Research C 1, 159± 182.
Pergamon PII: S0191-2615(97)00015-5
SOME REMARKS ON STOCHASTIC USER EQUILIBRIUM MARTIN L. HAZELTON Department of Statistical Science, University College London, Gower Street, London WC1E 6BT, U.K. (Received 27 April 1996; in revised form 11 February 1997) AbstractÐThe behavioural foundation of Stochastic User Equilibrium is that each traveller attempts to minimize his or her perceived travel costs, where these costs are composed of a deterministic measured cost and a random term which can be interpreted as perceptual error. In principle such a de®nition, which is in terms of random errors, should imply an equilibrium probability distribution over feasible ¯ow patterns on the transport network. Such a probability distribution could potentially allow between day variability in network ¯ows to be represented. However, traditionally a deterministic, large sample approximation has been used as the `solution' of Stochastic User Equilibrium. In this paper a representation of Stochastic User Equilibrium as a probability distribution is developed. This distribution is de®ned by the conditional route selection of each individual given the choices of all other travellers. An interpretation of the resulting assignment model as the limit (in the in®nite future) of a continuous time assignment process is discussed. The limiting behaviour as the travel demand becomes large is also investigated, and convergence to the traditional deterministic form of Stochastic User Equilibrium is proved. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: conditional probability, equilibrium distribution, stochastic route choice, trac assignment. 1. INTRODUCTION
Stochastic User Equilibrium (SUE) was introduced by Daganzo and She (1977). Their idea was to relax the unrealistic assumption of users having perfect knowledge of travel costs, which is an integral part of classical Wardrop (and user) equilibrium; see Wardrop (1952). To this end they suggested the following precept, which we term Stochastic User Behaviour. De®nition 1. Stochastic User Behaviour A traveller displaying Stochastic User (SU) Behaviour will select the route which he or she perceives to have minimum cost. Daganzo and She (1977) decided to model errors in travellers' perceptions of costs by random variables. Not only did this approach provide a natural framework for the representation of such errors; it also implied that the global ¯ow pattern resulting from universal practice of SU behaviour be a random variable drawn from an equilibrium probability distribution over the space of all feasible ¯ows (as opposed to a single deterministic ¯ow pattern). Expression of an equilibrium ¯ow pattern in terms of a probability distribution is an attractive proposition, since it allows between day variability in the transport system to be modelled. From a transport planning point of view it is all very well to have an estimate which represents an `average' ¯ow pattern in some sense, but at least as important to know whether random ¯uctuations in link ¯ows will lead to gridlock of the transport system on one day in twenty. See also the remarks of Hanson and Ho (1981, 1988), Taylor (1982), Arnott et al. (1991) and Watling and van Vuren (1993). It follows that the potential of stochastic models to estimate the variance of trac ¯ows is a strong argument in their favour in comparison with more traditional deterministic trac assignments. Nevertheless, Daganzo and She (1977) did not take full advantage of this potential, preferring instead a deterministic `large sample' de®nition of an SUE ¯ow pattern. We outline their approach in the following paragraphs, beginning by introducing the necessary notation. Let the triple N; L; K represent a ®nite transport network, where N is the node set, L the set of directed links and K a set of link cost functions. Let D be the trip matrix, whose stth element dst is 101
102
Martin L. Hazelton
the number of travellers going from origin node s to destination node t; O±D pair st as we shall call it. Let n=1TD1 be the total network demand (number of travellers) where 1 is a column vector of ones. De®ne I to be a set indexing all feasible routes (acyclic paths) and let Ij be the indexing set of feasible routes for the jth traveller. We write i k if and only if i and k are routes connecting the same O±D pair. Let R R1 ; :::; Rn be the vector of traveller route choices, so that Rj 2 Ij and i k for all i; k 2 Ij . When discussing stochastic route choice R will typically be a random vector, whose probability mass function is denoted by p r Pr R r. The ¯ow down path i 2 I is de®ned in terms of R by Fi
n X
Rj ; i;
j1
where is the Kroneker delta which is zero unless its arguments are identical, in which case it takes the value one. If F is the vector of such path ¯ows and B the network's link-path incidence matrix, then the vector of link ¯ows, X, is de®ned by X BF: Let C i be the actual (or measured) cost to a traveller taking the ith route, for i 2 I. This measured cost will usually be a function of the state of the network, and may therefore be written as C(ijR), C(ijF) or C(ijX) depending on the manner in which the trac ¯ow pattern is described. The perceived cost of the ith path for a traveller is denoted by CÄ(ijX, ") where " is a generic random error term. Usually " will be implicitly assumed and we shall write simply CÄ(ijX). Note that if " follows a suitable Gumbel distribution, and the perceptual error is additive in the sense that ~ j X C i j X "; C i then we have de®ned a logit route choice model. See Thomas (1991, p. 180) for details. She and Daganzo (1977) noted that SU behaviour dictates that traveller j will choose route i from the set Ij with probability given by ~ j X < C k ~ j X; 8k 6 i; k i: Pr C i
1
(Note that if perceptual error is modelled by continuous random variables, then equal costs will only occur with probability zero, and hence we may use a strict inequality in eqn (1) without loss of generality.) Suppose that j is a typical traveller between O±D pair st, and that dst is large. She and Daganzo argued that the Weak Law of Large Numbers (see Grimmett and Stirzaker, 1982, for example) implies that the proportion of these dst travellers taking route i will be very close to the probability of taking this route, as de®ned in eqn (1). This occurs when the stochastic variation in ¯ows becomes insigni®cant in comparison to the mean ¯ows. It is therefore at this stage of the argument that Daganzo and She's trac assignment loses its stochastic nature and becomes a deterministic model. Now, if proportion is put exactly equal to probability (the limiting case) then we get, ~ j X < C k ~ j X; 8k 6 i; k i P Pr C i
Fi ki
Fk
;
2
P where dst ki Fk since each traveller for O±D pair st must take some route k such that k i. Both sides of this equation are functions of X. Furthermore, this equation holds for all routes between all O±D pairs (equivalently, for all pairs i; k where i 2 I and k i). It therefore implicitly de®nes a ¯ow pattern, as described by de®nition 2. De®nition 2. Daganzo and She's Stochastic User Equilibrium (SUE) Let X{ be the ¯ow pattern solving eqn (2) for all routes between all O±D pairs. Then X{ is Daganzo and She's Stochastic User Equilibrium.
Stochastic user equilibrium
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Daganzo and She's SUE is intended to de®ne the limiting case (as the number of travellers goes to in®nity) of a system in which every individual practises SU behaviour. Our aim in this paper is to consider the ®nite demand case, when stochastic variation in ¯ow patterns will not be negligible and hence equilibrium should be represented by a probability distribution. Note that X{ is de®ned as the solution to a ®xed point problem. Whilst there are any number of discrete-time stochastic process models for trac ¯ow which will converge to X{ as the travel demand becomes large (see Davis and Nihan, 1993, for example), Daganzo and She's SUE exists without need to specify such a temporal process. It is partly in this spirit of generality that we will develop our form of SU equilibrium probability distribution from a largely atemporal, process free point of view. Nevertheless, our suggested form of equilibrium assignment model does have an interpretation in terms of contemporaneous traveller interaction, as we shall discuss later. The paper is structured as follows. In Section 2 we describe problems which occur if we attempt what might be seen as the most straightforward extension of Daganzo and She's version of SUE to the ®nite demand case. We acknowledge that these diculties are largely due to our somewhat game theoretic analysis of SUE, but such a `process free' approach does have advantages as mentioned above. Our own de®nition of an SU equilibrium distribution, called Conditional Stochastic User Equilibrium (CSUE), is then presented. It is illustrated using a simple numerical example. In Section 3 we discuss the properties of CSUE, including its interpretation as the limiting case (in the in®nite future) of a continuous time assignment process. It is also proved that our assignment converges to Daganzo and She's SUE as the travel demand becomes large (under certain conditions on the link cost functions). The paper concludes with some general comments on stochastic assignment procedures. 2. CONDITIONAL STOCHASTIC USER EQUILIBRIUM
SU behaviour (and any natural variant of it) is de®ned in terms of the way in which an individual traveller chooses his or her route. It therefore seems that any SU equilibrium probability distribution will necessarily be de®ned at such a disaggregate level, in terms of probabilities of individual route selections. That is to say we must ®rst look for a distribution of R, and then derive the distribution on F and X. In order to de®ne an equilibrium probability distribution over the joint route choices of every individual, it is sucient to de®ne the conditional route choice distribution of each separate traveller. This is because a full set of conditional distributions de®nes a unique joint distribution under mild conditions, as proven by the Hammersley-Cliord theorem (see Besag, 1974). Perhaps the most straightforward approach to obtaining conditional route choice probability distributions is to use Daganzo and She's formulation of SU behaviour as represented by eqn (1). This implies that the conditional probability of the jth traveller taking route i 2 Ij is ~ j X < C k ~ j X; 8k 6 i; k i: Pr Rj i j X Pr C i This is a conditional probability since it depends (through the perceived costs) on X. However, X is in some small part determined by the actual route that the jth traveller selects. In other words, the probability that j takes route i is a function of whether or not j took route i. This is a contradiction unless the probability is zero or one. Such a diculty has occurred because of the atemporal framework within which we are working. It could, for example, be resolved by introducing a discrete time dimension to the problem and de®ning route choices at epoch t in terms of the ¯ow pattern at t ÿ 1. Then Rj at t would be dependent on Rj at t ÿ 1, but this presents no logical paradox. However, our aim is to avoid speci®cation of any particular temporal stochastic process, as mentioned earlier. We therefore propose a dierent resolution, by de®ning each traveller's behaviour conditionally upon the choices of all other individuals' route choices. The result is the following behavioural precept; cf. Hazelton et al. (1996). De®nition 3. Conditional Stochastic User Behaviour A traveller displaying Conditional Stochastic User (CSU) Behaviour will select the route which he or she perceives to have minimum cost conditional on all other travellers' route choices.
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In practice this de®nition implies that when calculating the route choice probabilities for traveller j, the path costs are computed from the ¯ow pattern determined by the journeys of all travellers except the jth. Let Rÿj be R with the jth element removed, and let F ÿj and X ÿj be the corresponding path and link ¯ow patterns. Then traveller j practising CSU behaviour will select path i 2 Ij with probability ~ j X ÿj ; 8k 6 i; k i: ~ j X ÿj < C k Pr C i
3
Since this probability no longer conditions on the route choice of traveller j there are no inherent contradictions of the type discussed above. Under the assumption that the probability of eqn (3) is non-zero for all combinations of j and i 2 Ij (that is, no feasible route is assigned probability zero by any traveller), then this set of conditional probabilities necessarily de®nes a unique joint distribution for all route choices, R1 ; :::; Rn . This is a consequence of the Hammersley-Cliord theorem. We may therefore de®ne CSUE in terms of this joint distribution in the knowledge that it exists under the mild condition just noted. De®nition 4. Conditional Stochastic User Equilibrium (CSUE) A system is in CSUE when all travellers practise CSU behaviour. The random vectors of route choices, path and link ¯ow patterns will be denoted by R*, F* and X* respectively in this state. Notice that CSUE is not a deterministic state. It is a truly stochastic assignment in which the probability of a particular route choice vector r is determined by the CSUE probability distribution, p r Pr R r. We illustrate CSUE with the aid of a simple numerical example. Example. Consider a single O±D pair connected by two parallel links with ¯ows x1 and x2 . Let the separable, monotonic and dierentiable link cost functions, c1 and c2 be given by c1 x1 x1 1 1 2 3 2 x 2 2 04x241 c2 x2 otherwise: x2 1 Notice that since each feasible path corresponds to a single link, we have C(ijX x1 ; x2 ci xi in terms of the path cost notation introduced earlier. Suppose that there are a total of two travellers so that x1 x2 2. Let route choice probabilities be determined by a logit model with logit constant set to one. Then under CSUE the probability that traveller 1 takes route 1 conditional on the fact that traveller 2 takes route 1 is given by Pr R1 1 j R2 1
exp c1 1 eÿ2 : ÿ2 exp c1 1 exp ÿc2 0 e eÿ3=2
Using all the conditional probabilities, we can calculate marginal probabilities as follows. Pr R1 1 Pr R1 1 j R2 1 Pr R2 1 Pr R1 1 j R2 2 Pr R2 2
eÿ2 eÿ1 Pr R 1 Pr R2 2: 2 eÿ2 eÿ3=2 eÿ2 eÿ1
By symmetry Pr R1 1 Pr R2 1 1 ÿ Pr R2 2 q (say). Hence, q
eÿ2 eÿ1 q 1 ÿ q eÿ2 eÿ3=2 eÿ2 eÿ1
which can be solved to give q 0:540. Joint probabilities for R1* and R2* can now be computed; for example, Pr R1 1; R2 2 Pr R1 1 Pr R2 2 j R1 1 q
eÿ3=2 0:336: eÿ3=2 eÿ2
Stochastic user equilibrium
105
Since this is equal to Pr R1 2; R2 1 by symmetry, the probability of the travellers taking dierent links is Pr X 1; 1 Pr R1 1; R2 2 Pr R1 2; R2 1 0:672: We end this example by noting that the expected (or mean) ¯ow pattern in CSUE is EX 1:08; 0:92. Daganzo and She's SUE can trivially be checked to be X{=(1,1). Whilst these ¯ow patterns are not wildly dierent, CSUE does re¯ect the fact link 1 is more attractive in the sense that c1 y4c2 y with strict inequality on the interval 04y < 1. 3. PROPERTIES OF CSUE
The development of CSUE has been from a process free perspective. Nevertheless, CSUE as an assignment model does have an interesting interpretation in terms of the limiting case of a system in which travellers interact in a contemporaneous fashion. In order to explain why, we must ®rst consider a continuous time model of the development of the network ¯ow pattern. In this model each traveller makes a route selection, from time to time, by sampling from the conditional probability distribution de®ned by eqn (3), using route costs de®ned by present link ¯ows. Two consequences of the fact that our process occurs in continuous times are worthy of note. First, route choice based on present costs is feasible in continuous time, so that travellers interact contemporaneously. (This is in contrast to discrete time models in which actions at epoch t are based upon states of network in past epochs, t ÿ 1, t ÿ 2, and so on.) Secondly, assuming that absolutely simultaneous events occur with probability zero in continuous time, we can uniquely order the travellers' route changes in time, one by one. Again, this is generally in contrast to discrete time models where groups of travellers change route together at each epoch. Consider the limiting case of this model as the ¯ow pattern evolves over an in®nitely long period, so that R (and hence F and X) is updated an in®nite number of times by sampling from conditional route choice distributions. Assume that each traveller makes an in®nite number of route choices during this time. (The speci®c sequence in which individuals make route selections is unimportant in this discussion.) Then assuming that no feasible route is assigned probability zero by any traveller, the distribution of the ¯ow pattern will converge to the CSUE probability distribution. (This holds regardless of the intial state of the network.) This result is an immediate consequence of the work of Gelfand and Smith (1990). A by-product of this argument is the development of a numerical method for simulating CSUE ¯ow patterns. This is important because actual calculation of the CSUE probability distribution, p , is not analytically feasible for anything other than tiny arti®cial networks (such as the example in the previous section). By simulating a version of the continuous time process just described, we will eventually (when convergence is achieved) be drawing samples from the CSUE probability distribution (no matter what the initial ¯ow pattern). We can then estimate properties of p , such as the variance of ¯ows on route 4 or the correlation between ¯ows on links 14 and 15, simply by computing the corresponding sample quantities from a large number of simulated realizations from p . An ecient technique for performing such simulations, using so called Markov Chain Monte Carlo methods, is described in detail in Hazelton et al. (1996). This paper also contains some numerical examples of CSUE based on real road networks. We now turn to the behaviour of CSUE trac assignment as the travel demand becomes large. Whilst our assignment process is based upon CSU behaviour, as opposed to SU behaviour, the dierence is solely in terms of the costs generated by a single traveller; an increasingly insigni®cant proportion of the total costs as the number of travellers tends to in®nity. Furthermore, since it is to be expected that the stochastic coecient of variation in ¯ows will diminish with increasing travel demand, we should expect the CSUE probability distribution to converge to the single deterministic ¯ow, X{. We formally prove that this is the case (under certain conditions) in the theorem presented below. The limiting process that we actually consider is one in which each unit of ¯ow is subdivided into m assignable parts, where m ! 1. In other words the number of travellers becomes large, but each traveller is represented by a quantum of ¯ow of weight mÿ1 only. Before proving the main result a lemma stating an alternative form of the Weak Law of Large Numbers is required.
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Lemma. Let Y1, Y2, Yn be a collection of identically distributed random variables with ®nite n P Ya ; variance and common mean . Suppose that cov (Ys, Yt)40 for all s6t. Then if Sn ÿ1
a1
p
n Sn ! as n ! 1. Proof: Let the common variance of the Ya 's be 2 . Then, were Y1 ; Y2 , ..., Yn independent, the variance of their sum would be n 2 . Now, the covariance terms can only contribute negatively to var Sn , so it follows that var Sn 4n 2 . Hence var nÿ1 Sn 4nÿ1 2 ! 0 as n ! 1: Since Enÿ1 Sn , it follows that nÿ1 Sn converges in mean square to and therefore also in probability, completing the proof; cf. Feller (1950). & Theorem. Let N = (N, L, K) be a transport network with O±D trip matrix D, and such that all link cost functions are separable, continuous and monotonic increasing. Let Nm be the corresponding network in which each unit of ¯ow is subdivided into m assignable individual parts. Then if X*(m) is the CSUE random ¯ow pattern for the network N m and X{ (m) is the corresponding Danganzo and She SUE ¯ow, p
X m ÿ Xy m ! 0 as m ! 1. Proof: Let Am be a set indexing all assignable parts for whom route i is feasible in network Nm . Then for a; b 2 Am ; cov Ra ; i; Rb ; i < 0, since conditioning on individual a taking route i increases the cost of that route (by the assumptions on the link cost functions) and hence makes it less likely that individual b will follow suit. Hence, f Ra ; i; a 2 Am g are a set of random variables satisfying the lemma. P Note that the cardinality of the set Am is given by j Am j m ki Fk . Using the lemma it follows that for network N m the proportion of ¯ow using route i in CSUE satis®es P
X Fi m p j Am jÿ1 Ra ; i ! E Ra0 ; i F m ki k a2A m
Pr Ra0
as m ! 1, where a0 2 Am is a typical traveller. Now for network N
4
i m
~ j X 0 m < C k ~ j X 0 m 8k 6 i; k i Pr Ra0 i Pr C i ÿa ÿa p ~ j X m < C k ~ j X m 8k 6 i; k i ! Pr C i
since the continuity of the measured cost functions ensures that j C k j X m ÿ C k j X ÿa0 m j O mÿ1 ! 0 for any path k"I as m ! 1. (In other words, the impact on the costs of a quantum of ¯ow of weight mÿ1 has negligible eect as m ! 1.) Combining this result with eqn (4) shows that P
Fi m p ~ ~ m ÿ Pr C i j X m < C k j X m 8k 6 i; k i ! F ki k
5
as m ! 1, and this holds for all i 2 I. But the ¯ow pattern which equates the left hand side of eqn & (5) to the zero vector is precisely X{(m), see eqn (2), completing the proof.
Stochastic user equilibrium
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We remark that the conditions of monotonicity and separability on link costs are sucient for the theorem to hold, but not necessary. 4. DISCUSSION
Between day variation in trac ¯ow is an important feature of real trac systems. However, the vast majority of assignment modelling has been deterministic, aiming only at some estimate of an average ¯ow pattern. If an equilibrium assignment is intended to represent ¯ows over a long time interval, then the ¯ows themselves will be large and statistical laws (usually) guarantee that variation in ¯ow is small in comparison to the average. In such situations, Daganzo and She's SUE may well provide a perfectly adequate representation of the system. Nevertheless, if interest centres on modelling at a ®ner level of temporal aggregation then the trac ¯ows will not be large enough to render day-to-day ¯ow dispersion negligible. Cascetta (1989) makes the point that in many applications hourly demand ¯ows are of few units for most O±D pairs, in which case variability in link ¯ows should be a pivotal part of the modelling procedure. As an example, consider the case of Stochastic Network Loading, where measured costs are ®xed independently of the ¯ow pattern. If there is a demand of 100 units between an O±D pair and the probability of a traveller using a particular route is 0.2 (a function of the ®xed costs), then the number of travellers using this route will follow a binomial distribution. It follows that the mean ¯ow down this route will be 20 units, but the range 8; 32 is an approximate 95% con®dence interval for the route ¯ow. Hence, even for a moderate O±D demand, the range of ¯ows can be comparable to the magnitude of the average ¯ow. Our CSUE assignment is capable of representing such stochastic variation in ¯ow patterns. Whilst independent ¯ow patterns drawn from p will not provide a sensible model of consecutive days' trac ¯ows (which are highly dependent), they will give an idea of the magnitude of the between day variation that can be expected over a long period of time. The ability to model variation in trac ¯ows is a strong argument in favour of assignment procedures which give rise to a probability distribution on ¯ow patterns. A further advantage of modelling stochastically is that such an approach provides a natural framework for developing assignment procedures which are based on a richer representation of traveller behaviour than those which are currently in use. For example, models incorporating traveller learning and the impact of advanced traveller information systems would naturally be formulated in terms of stochastic processes, since the rate at which individuals gather information and the way in which they react to it varies in a highly complex manner across the population. See, for example, Watling and van Vuren (1993). We believe that stochastic assignment models based upon more realistic behavioural assumptions will provide an exciting area for future research. AcknowledgementsÐThe author thanks John Polak, Seungjae Lee and anonymous referees for their useful comments on SUE and related areas. The research in this paper was supported by grant GR/K18337 from the Engineering and Physical Sciences Research Council, U.K. REFERENCES Arnott, R., de Palma, A. and Lindsey, R. (1991) Does providing information to drivers reduce congestion? Transportation Research A 25, 309±318. Besag, J. (1974) Spatial interaction and the analysis of lattice systems (with discussion). Journal of the Royal Statistical Society, Series B 36, 192±236. Cascetta, E. (1989) A stochastic process approach to the analysis of temporal dynamics in transportation networks. Transportation Research B 23, 1±17. Daganzo, C. F. and She, Y. (1977) On stochastic models of trac assignment. Transportation Science 11, 253±274. Davis, G. A. and Nihan, N. L. (1993) Large population approximations of a general stochastic trac assignment model. Operations Research 41, 169±178. Feller, W. (1950) An Introduction to Probability Theory and its Applications. John Wiley & Sons, New York. Gelfand, A. E. and Smith, A. F. (1990) Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association 85, 398±409. Grimmett, G. and Stirzaker, D. (1982) Probability and Random Processes. Clarendon Press, Oxford. Hanson, S. and Ho, J. (1981) Assessing day-to-day variability in complex travel patterns. Transportation Research Record 891, 18±24. Hanson, S. and Ho, J. (1988) Systematic variability in repetitious travel. Transportation 13, 111±135. Hazelton, M. L., Lee, S. and Polak, J. W. (1996) Stationary states in stochastic process models of trac assignment: a Markov Chain Monte Carlo approach. In Proceedings of the 13th Symposium on Transportation and Trac Theory, ed. J-B. Lesort, pp. 341±359. Pergamon, Oxford.
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Taylor, M. A. P. (1982) Travel time variabilityÐthe case of two public modes. Transportation Science 16, 507±521. Thomas, R. (1991) Trac Assignment Techniques. Avebury Technical, Aldershot. Wardrop, J. G. (1952) Some theoretical aspects of road trac research. Proceedings of the Institute of Civil Engineering, Part II 1, 325±378. Watling, D. and van Vuren, T. (1993) The modelling of dynamic route guidance systems. Transportation Research C 1, 159± 182.