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A network equilibrium model for oligopolistic competition in city bus services1
Transpn Res.-B, Vol. 32, No. 6, pp. 413±422, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0191-2615/98 $19.00+0.00
Pergamon PII: S0191-2615(98)00009-5
A NETWORK EQUILIBRIUM MODEL FOR OLIGOPOLISTIC COMPETITION IN CITY BUS SERVICESy LOURDES ZUBIETA*
Bishop's University, Lennoxville, QueÂbec, Canada, J1M 1Z7 and Centre de recherche sur les transports, C.P. 6128 Succ. `Centre-Ville', MontreÂal, QueÂbec, Canada, H3C 3J3 (Received 26 December 1996; in revised form 11 December 1997) AbstractÐThis paper presents a new model for a deregulated transportation system with full representation of the city network. We assume the case in which a few private bus companies provide the totality of the urban transportation services. Each private company is assumed to have exclusive rights to operate a particular transit line. Competition among companies is given only in terms of the frequency of service as demand and transit fares are considered exogenous. The bus operators seek pro®t maximization whereas passengers look for the travel strategy that minimizes expected travel time. At equilibrium, marginal revenue should equal marginal cost for each operating company and, for each origin±destination pair, travel `strategies' for passengers should be optimal. To obtain an equilibrium solution, a heuristic procedure is outlined and tested on a small network. # 1998 Elsevier Science Ltd. All rights reserved Keywords: network equilibrium, deregulation, urban transportation, transit frequencies 1. INTRODUCTION
The deregulation of urban mass transportation systems is an appealing alternative to centralized municipal transit commissions. Among the reasons invoked in favor of descentralization are, increasing discrepancies between subsidies and operating costs, suburbanization, growth in car ownership, seasoned with ideological policy shifts. In Europe, the most signi®cant example is the bus deregulation introduced in Great Britain with the Transport Act of 1985. Urban bus competition has also been introduced in several South American cities, and various experiments are being carried out in southeast Asia (Kang, 1995). Experience is now available in several cities where transit deregulation has been implemented. It suggests that, with the exception of very few short periods of price wars, the competitors matched fares by adopting either the distance-related general fare as in the U.K. (Evans, 1990), or by applying a ¯at fare throughout the city, as in Santiago de Chile (DarbeÂra, 1993). Deregulation has not change the industry structure in Chile, where the average ¯eet size went from 1.33 in 1985 to 1.67 in 1990 (DarbeÂra, 1993). However, all bus operators, with the exception of one or two, belong to a route association. These are non-pro®t organizations whose main role is to dispatch buses and provide services like insurance, legal protection, etc. Fierce competition exists between associations in terms of frequencies of services because their respective routes largely overlap. The diversity of deregulation schemes has motivated the international research community to develop analytic tools capable of providing a better insight into, and understanding of the mechanisms arising in such diverse forms of competition (Evans, 1987). The purpose of this paper is to present a network equilibrium model and a solution procedure for a deregulated transit system inspired from the experience in Santiago. The model is characterized by a full representation of the transit network with a small number of private transit ®rms providing the totality of the urban mass transportation services. Competition among the ®rms is based solely on the frequency of services as the model considers a ®xed origin±destination matrix of demand and fares are assumed constant. The solution procedure seeks an overall Nash-equilibrium state *Fax: 001 819 822 9720; e-mail: [email protected] y This work was partially funded by the National Science and Engineering Research Council of Canada (NSERC).
413
414
L. Zubieta
where passengers minimize their expected total travel time and transit ®rms frequencies maximize their pro®ts. In the transit deregulation literature, much work is devoted to providing descriptive studies of some particular city or country, and to issues such as whether or not private participation ought to be allowed. Also we ®nd some assessments of the bene®ts and/or drawbacks observed or expected in private vs public urban mass transit systems (Beesley, 1990; Mackie et al., 1995; White, 1995). The prediction of market prices, market structure and service levels in a deregulated scenario has also received some attention. Viton (1982) developed equilibrium models for the analysis of a duopoly which consisted of two ®rms competing on a simpli®ed corridor between a residential area and a central business district connected by a limited-access highway. Kanemoto (1984) provided another theoretical model for railway ®rms competing on a radial network. An important extension of Viton's model was proposed by Harker (1988). His representation of corridors is more general in that it allows passengers to embark/disembark at points other than the extremities of the corridor. Each corridor represents a dierent transit service (automobile, subway, private mini-bus, public bus, etc.) with no transfers between services; competition is present among the dierent types of services oered. FernaÁndez and Marcotte (1992) proposed the ®rst full network equilibrium model within a deregulated context. The two main advantages of this model are the full representation of the city network and the modeling of a free market competition between operators within and among transit services. When an operator chooses a line to run competition exists with other bus operators running on the same line but it also exists with other operators running on lines `attractive' to passengers. In this model, car drivers and passengers are assumed to behave according to Wardrop's First Principle (user equilibrium) while bus operators seek to maximize their individual pro®t under conditions of perfect competition. Marcotte et al. (1990) proposed an iterative scheme to solve this model, which was successfully applied to the transit network of the city of Santiago de Chile. In this paper, we modify the perfect competition assumption of the above model to re¯ect a speci®c case of market imperfection: the situation where a few companies have control of the transit market. A private company may own a ¯eet of city buses, or various members may form a cooperative in which they share garages and mechanics to decrease their operating costs. In both cases, we assume that each company has exclusive rights to run a particular bus line. In contrast to the perfect competition case, the decisions of one company aect the market as a whole, as the frequencies of service will determine, for each origin±destination (O±D) pair, the set of `attractive lines' for passengers. 2. MODEL FORMULATION
2.1. Notation and assumptions We consider a transit network where nodes represent bus stops and links represent the distance between two consecutive bus stops. Each line is operated by one private company whose only decision variable is the number of buses to run per unit of time (frequency) in order to maximize its pro®t. There is no restriction on the frequency levels, so neither a minimum nor a maximum number of buses is required. We assume that demand is exogenous, represented by a ®xed O±D matrix, and that bus fares for each company are constant. For every O±D pair, competition among bus lines is generated by the frequency of services, as passengers have several possibilities (strategies) for reaching their destination. As the bus itineraries are not disjoint, transfers between lines are allowed at bus stops. However, every time a passenger boards a vehicle the fare must be paid. Congestion is implicitly taken into account as the variable operating costs for the companies and the round trip travel times are assumed to depend on vehicle ¯ows (cars and buses). In the sections below we use the following notation: R N ; A L 1; 2; . . . ; L S
Transportation network with node set N and arc set A Set of transit lines (sequence of adjacent arcs) Strategy vector (passengers' decision vector)
Network equilibrium in city bus services
``"L V a S; a"A fir C ; V S; ` ; V S; `"L ` fij `
415
Bus frequency vector Passenger volume per time unit Passenger ¯ow boarding at node i and with ®nal destination r Passengers total transit time (travel+waiting time), (a continuous function of S and ) Operators' pro®t function (vector) Bus fare on line ` Number of passengers at node i with destination j Total operating costs per time unit for line `
2.2. The passengers sub model As stated before, passengers strive to minimize their total transit time or generalized transit cost, which includes waiting time and in-vehicle time. Given a vector of frequencies and the O±D matrix, the passenger assignment to the bus lines can be easily calculated by the state-of-the-art transit assignment models (Spiess and Florian, 1989; Nguyen and Pallottino, 1988). A passenger's strategy is a set of lines and transfer points he has in mind when going from a point of origin to a destination. At each bus stop, a strategy is characterized by a subset of available lines called the set of attractive lines. At each bus stop, a passenger boards the ®rst incoming bus belonging to this set. If bus arrivals follow a Poisson distribution (exponential headways), a passenger will choose the set of attractive lines by minimizing his expected transit time to his destination, taking into account transfers, waiting and in-vehicle travel times. For further details, see Spiess (1981) or Nguyen and Pallottino (1988). An equilibrium state is attained when passengers travel on shortest routes, according to a frequency vector , and thus the optimal strategies can be written as the solution of the following linear program parameterized in (Marcotte and Blain, 1990): min FC ; V S; s"
X a"A
ca a S;
X
fir S; ir S;
1
i"N ;r"N
where denotes the set of admissible strategies, ca is the travel time function on arc a, and ir is the waiting time for passengers boarding at node i having ®nal destination r. This is a well known transit assignment problem (Smith, 1979; Dafermos, 1980) that may be formulated as a variational inequality in terms of strategies rather than ¯ows: C ; V S; V S ; ÿ V S; 40
8S"
2
2.3. Formulation of the bus operators sub model A solution to the passengers sub model (1) can provide two useful sets of information for each node i: the number of boardings at node i with destination j; fij , and the set of `attractive lines', Uij , which compete through their frequencies to transport these passengers. If we consider fij and Uij ®xed, then the total number of boarding passengers for a given line ` can be written as a function of the frequency vector by assigning the demand at each node to the attractive lines, in proportion to their frequencies: XX ` fij `ij 3 f` k k"U ij i"N j"N where `ij
1; if line ` stops at node i 0; otherwise
We notice here that if a line is not attractive for any destination then the value of f` will be zero. The pro®t functions are then de®ned as the dierence between revenues and costs:
416
L. Zubieta
` ` f` ÿ `
4
An equilibrium state is obtained when no operator can increase his pro®t by modifying his frequency ÿ of service ` , given the present frequencies of the rest of the operators. Let ` 1 ; 2 ; . . . ; ` ; `1 ; . . . ; L 50 for each ` 1; . . . ; L. Then ` is an optimal solution of the following problem: ÿ 5 max ` ` ` 50
It can be shown that if ` "C1 ; ` 1; . . . ; L then each pro®t function ` is concave in ` and the ®rst order optimality conditions for the set of problems de®ned by (5) are: r` ` ` 0;
` 1; . . . ; L
` ` 0 ` ; ` 5 0:
6
Conditions (6) constitute a nonlinear complementarity problem, which is equivalent (Karamardian, 1971) to the following variational inequality problem: ; V S ; ÿ 40;
850
7
Given the passengers behaviour observed after a transit assignment, the bus operators can be seen as players of an L-person, noncooperative game for which we seek a Nash equilibrium. The connection between Nash equilibrium and variational inequalities was shown by Gabay and Moulin (1980) and Harker (1984). 2.4. The global model An equilibrium state for both passengers and bus, companies consists of a vector of strategies S and a vector of frequencies satisfying (2) and (7), which results in the following system of variational inequalities: C ; V S ; V S ; ÿ V S; 40;
; V S ; ÿ 40; 8 S; "ÿ
8
where ÿ denotes the set of feasible S and . The economic interpretation of such an equilibrium state follows the classical micro economic model of an oligopoly (Moulin, 1986; Friedman, 1977). For every bus operator, marginal pro®t equal marginal cost [see eqn (6)], and no ®rm can increase its pro®t by providing a dierent output if the other ®rms' output remains constant. This model does not take into account any kind of collusion among bus companies and so, the equilibrium in (8) need not be Pareto optimal. For the passengers, the equilibrium state means that no strategy other than the optimal one can reduce the expected transit time for every O±D pair. 2.5. Existence and uniqueness of solutions Given a frequency vector , the existence of at least one solution for the passenger sub model follows from the fact that C is a linear function with a bounded feasible set. Although the set of feasible ¯ows is not technically bounded in the usual formulation of the problem, it is easy to show that an optimal solution cannot occur at in®nity, given the assumption of ®xed (exogenous) demand. For the bus operators sub model, standard arguments would ensure the existence of a solution provided that the feasible set is compact and the functions are continuous. The set of feasible frequencies (50) can be limited to a compact set since the ®xed demand and the concavity of the pro®t functions imply that large frequencies would yield negative pro®ts, clearly a nonequilibrium solution. In fact, the total number of passengers boarding a bus line is obviously limited by the total number of passengers, so the revenue is bounded by a constant depending only on the problem data. As a consequence of the above, the Brouwer±Kakutani ®xed point theorem (Ortega and Rheinbolt, 1970) ensures the existence of at least one solution to the global model (8).
Network equilibrium in city bus services
417
Uniqueness of a solution can be proven for the bus operator sub model if the pro®t functions are strictly concave. For the global model, however, it would be quite surprising to ®nd reasonable conditions to the uniqueness of a solution, given that the pro®t functions are not monotone and that the passengers sub model is linear. Reasonable conditions should not restrict the topology of the bus lines to situations such as disjoint paths or non-intersecting lines. These cases are not observed in city networks. To ®nd an equilibrium (S ; ) for the global model (8) is not straightforward but we can exploit two characteristics. First, for given frequency vector , state-of-the-art transit assignment models can provide the reaction of passengers: the set of `attractive lines' Uij and the number of passengers boarding at each node of the network fij . Now, for a given Uij and fij , there is a wide body of algorithms for variational inequality problems that can be used to obtain a new vector of frequencies that maximize operators pro®t (e.g. Glowinski et al., 1976). The new frequency vector may produce a change in the passengers behaviour by revealing a dierent set of attractive lines and thus a dierent number of passengers boarding at each node of the network. A solution procedure based on these characteristics is presented next. 3. HEURISTIC PROCEDURE
In order to solve the global model (8), the following iterative scheme is proposed: start with an initial frequency vector; solve the passengers sub model to obtain the passengers reaction. Solve the operators sub model to obtain a new frequency vector. If no major dierences exist between the new and the current frequency vector, stop. Otherwise solve again the passengers sub model. Step 0. Step 1. Step 2. Step 3.
ÿ Let 0 50; 0 6 0 ; " > 0; 0 < 41; k : 0. Solve the passengers sub model (1) using k Solve the bus operators sub model (6) given the optimal strategies and passenger ¯ows obtained fromÿ Step 1. Let 0 be the solution. Let k1 k 0 ÿ k . If jk1 ÿ k j < " then STOP Otherwise k : k 1 and return to Step 1.
The above procedure is a Gauss±Seidel approach easy to implement since both Step 1 and Step 2 can be eciently solved by well-known algorithms. Solution to Step 1 is easily obtained by the state-of-the-art transit assignment software EMME/2 (1994) and Step 2 can be obtained by an iterative scheme such as Newton±Jacobi. The introduction of the relaxation parameter a is required in order to avoid abrupt changes in the optimal strategies which may prevent the procedure from converging. 4. NUMERICAL EXAMPLE
To illustrate the heuristic method proposed, consider the transit system shown in Fig. 1. It consists of 29 nodes, 86 arcs. (transit segments) and 24 O±D pairs distributed over the network. Eight bus lines serve this city network and we assume the bus fares to be $1 for all lines. The assignment results were obtained from the EMME/2 transit assignment model. The operating cost functions and the travel time functions considered are adapted from FernaÂndez and Marcotte (1992) by multiplying their cost functions per vehicle by the number of vehicles running on each line. Let FC` fixed operating cost per unit of time on line `; G` variable operating cost on line `; per unit of time : X a a a Ba 2 `a ; ` 1; . . . ; L a"A
T` round trip travel time on line ` : X a a fa Ba 2 `a ; ` 1; . . . ; L a"A
418
L. Zubieta
Fig. 1. Small network.
with `a
1; 0;
if arc a is on line ` itinerary otherwise
is the bus±car equivalence, fa is the car ¯ow on arc a; Ba ` B` `a is the total number of buses and are given in the Appendix ( 3). The operrunning on arc a, and the parameters ; ; , ating cost function is de®ned as ` ` G` T` FC` ;
` 1; . . . ; L
A particular. line may experience very little or no competition at all if it is the only `attractive line' for a subset of O±D pairs. In such cases, the revenue may become independent of the frequency vector and the latter may be driven to zero by the heuristic. In order to avoid this situation, we added one dummy `walk line' per regular line in the passengers sub model, with a ®xed frequency of 0.001 and running at 3 Km/h. These walk lines introduce a type of modal split as they represent the possibility of passengers walking instead of taking a bus if the frequency of service becomes too low. In Fig. 2 the initial value for the bus frequencies was set to 0.1 buses per min or equivalently, headways are set to one bus every 10 min. Each iteration represents the solution to the bus operators sub model (k1 ), given the transit assignment results obtained with the previous frequency vector (k ). After a few iterations it is clear that the four bottom lines rapidly converge to an equilibrium point. A second group of bus lines, with headways ranging between 6 and 9 mins, do not converge to a value but stay around a range of values. The upper bus line is a clear case of a monopoly for which there is a built-in barrier to the highest headway value passengers accept before turning away public transportation: the walk lines. In this case the revenue function is independent of frequencies and therefore pro®t maximization pulls costs to zero by increasing headway to a point where only a fraction of the clientele will continue to ride. This in turn forces the line to reduce the headway in order to increase pro®t and eventually the cycle restarts. Similar results were obtained in Fig. 3, where the initial solution was set to 1.0 min instead of 10.0 min. Again, the heuristic
Network equilibrium in city bus services
419
Fig. 2. Evolution of headways (initial value=10.0 min).
Fig. 3. Evolution of headways (initial value=1.0 min).
found the same pattern of behaviour although we have no reason to believe that this `pseudo equilibrium' is unique. The three groups of lines shown in Figs 2 and 3 can be related to the amount of competition. Table 1 shows headways, market shares and the interaction among bus lines, at the last iteration of the heuristic. We can observe that lines 102, 104, 107 and 108, having the lowest headway values, face the strongest competition: directly, by sharing passengers with another line, or indirectly, by inheriting clientele coming from other bus companies. The second group of lines (101, 103, and 105) are more aected by the transfers they receive from other bus lines than by direct competition with other lines, although line 101 shares a few passengers with line 108. Finally the monopoly situation of line 106 is clear given that this line neither shares or
420
L. Zubieta Table 1. Competition and market shares
Line
Boardings (%)
Headway (min)
Compete with
Transfers from
102 104 107 108 101 103 105 106
7.55 4.35 21.96 8.01 14.45 15.21 13.50 15.00
1.97 2.75 2.91 2.48 7.15 4.89 7.69 17.55
104,108 102,107 104 101,102 108 ± ± ±
106 101 102,103 102 103,104,106 104,107 101,107 ±
inherits any passengers. We must point out that the solution procedure proposed in this paper is memoryless in terms of all the previous headway values experienced, only the present value is considered to get the next one. Hence, the evolution patterns observed should not be interpreted as dynamic behaviour of the public transportation market. This paper presents a model and a solution procedure of a deregulated transit system where bus lines are assumed to compete for passengers in a city network. The results obtained with a small example provided a better insight into the mechanisms of such forms of competition faced by the bus companies and to explicitly relate those levels to frequencies of service. A natural extension of the model would be to allow for variable demand in the passengers sub model, an analysis which will be presented in a forthcoming paper. AcknowledgementsÐThe author wishes to thank Patrice Marcotte and an anonimous referee for their comments and suggestions. REFERENCES Beesley, M. E. (1990) Collision, prediction, and merger in the UK bus industry. Journal of Transport Economics and Policy 24, 295±310. Dafermos, S. (1980) Trac equilibria and variational inequalities. Transportation Science 14, 42±54. DarbeÂra, R. (1993) Deregulation of urban transport in Chile: what have we learned in the decade 1979±1989? Transport Reviews 13, 45±59. INRO Consultants (1995) EMME/2, User Reference Manual. Evans, A. (1987) A theoretical comparison of competition with other economic regimes for bus services. Journal of Transport Economics and Policy 21, 7±36. Evans, A. (1990) Competition and the structure of local bus markets. Journal of Transport Economics and Policy 24, 255± 281. FernaÂndez, J. E. and Marcotte, P. (1992) Operators-users equilibrium model in a partially regulated transit system. Transportation Science 26, 95±105. Gabay, D. and Moulin, H. (1980) On the uniqueness and stability of Nash-equilibria in noncooperative games. In Applied Stochastic Control in Econometrics and Management Science, eds A. Bensoussan, P. Kleindorfer and C. S. Tapiero. North Holland, Amsterdam. Glowinski, R., Lions, J. L. and TremolieÁrs, R. (1976) Analyse NumeÂrique des IneÂquations Variastionnelles, Dunod, Paris. Harker, P. T. (1984) A variational inequality approach for the determination of oligopolistic market equilibrium. Mathematical Programming 30, 105±111. Harker, P. T. (1988) Private market participation in urban mass transportation: application of computable equilibrium models of network competition. Transportation Science 22, 96±111. Kanemoto, Y. (1984) Pricing and investment policies in a system of competitive commuter railways. Review of Economic Studies 51, 665±681. Kang, Y. (1995) The development of urban bus deregulatory policy in Korea revisited. Transport Reviews 15, 357±370. Karamardian, S. (1971) Generalized complementarity problem. Journal of Optimization Theory and Applications 8, 161±168. Mackie, P., Preston, J. and Nash, C. (1995) Bus deregulation: ten years on. Transport Reviews 15(3), 229±251. Marcotte, P. and Blain, M. (1990) A Stackelberg±Nash model for the design of deregulated transit system. Fourth International Symposium of Dierential Games and Applications, eds Raimo P. HaÈmaÈlaÈinen and Harri Ehfamo, Pre prints Vol. 1. Helsinki University of Technology, Systems Analysis Laboratory, June. Marcotte, P., Zubieta, L. and Drissi-Kaitouni, O. (1990) Iterative methods for solving and equilibrium problem arising in transit deregulation. Transportation Research-B 24, 45±55. Moulin, H. (1986) Game Theory for the Social Sciences, 2nd edn. New York University Press, New York. Nguyen, S. and Pallottino, S. (1988) Equilibrium trac assignment for large scale transit networks. EJOR 37, 176±186. Ortega, J. and Rheinboldt, W. (1970) Iterative Solutions of Nonlinear Equations in Several Variables. Academic Press, New York. Smith, M. J. (1979) Existence, uniqueness and stability of trac equilibria. Transportation Research-B 13, 295±304. Spiess, H. (1981) Contributions aÁ la theÂorie et aux outils de plani®cation des reÂseaux de transport urban, Ph. D. thesis, Universite de MontreÂal, Canada.
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Spiess, H. and Florian, M. (1989) Optimal strategies: a new assignment model for transit networks. Transportation Research-B 23, 83±102. Viton, P. A. (1982) Privately-provided urban transport services. Journal of Transportation Economics and Policy 16, 85±94. White, P. (1995) Deregulation of local bus services in Great Britain: an introductory review. Transport Reviews 15(2), 185± 209.
APPENDIX
Table A1. Arc parameters ARC 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
ORIG
DEST
Car flow
1001 1002 1003 1004 1005 1006 1005 1004 1003 1002 1007 1024 1008 1025 1009 1026 1010 1011 1010 1026 1009 1025 1008 1024 1019 1002 1008 1012 1013 1012 1008 1002 1002 1017 1018 1005 1010 1015 1016 1015 1010 1005 1018 1017 1014 1013 1009 1004 1018 1020 1018
1002 1003 1004 1005 1006 1005 1004 1003 1002 1001 1024 1008 1025 1009 1026 1010 1011 1010 1026 1009 1025 1008 1024 1007 1002 1008 1012 1013 1012 1008 1002 1019 1017 1018 1005 1010 1015 1016 1015 1010 1005 1018 1017 1002 1013 1009 1004 1018 1020 1018 1004
15.0 15.0 10.0 9.8 9.4 24.4 10.0 10.0 10.0 10.0 11.0 9.8 10.0 20.0 9.4 9.4 8.0 10.0 20.0 10.2 22.0 10.8 8.0 9.4 5.0 9.8 9.4 14.0 9.8 5.0 8.0 11.0 14.0 23.4 10.6 7.0 17.0 9.4 23.4 20.0 9.4 9.0 6.0 10.2 20.0 10.2 8.0 6.8 9.8 11.0 15.6
7.5 7.5 5.0 4.9 4.7 12.2 5.0 5.0 5.0 5.0 5.5 4.9 5.0 10.0 4.7 4.7 4.0 5.0 10.0 5.1 11.0 5.4 4.0 4.7 2.5 4.9 4.7 7.0 4.9 2.5 4.0 5.5 7.0 11.7 7.8 3.5 8.5 4.7 11.7 10.0 4.7 4.5 3.0 5.1 1.0 5.1 4.0 3.4 4.9 5.5 7.8
0.390E-06 0.390E-06 0.270E-06 0.260E-06 0.260E-06 0.660E-06 0.2700E-06 0.270E-06 0.270E-06 0.270E-06 0.300E-06 0.260E-06 0.270E-06 0.540E-06 0.260E-06 0.260E-06 0.220E-06 0.270E-06 0.540E-06 0.280E-06 0.600E-06 0.290E-06 0.220E-06 0.260E-06 0.140E-06 0.260E-06 0.260E-06 0.360E-06 0.260E-06 0.140E-06 0.220E-06 0.300E-06 0.360E-06 0.630E-06 0.420E-06 0.189E-06 0.450E-06 0.260E-06 0.630E-06 0.540E-06 0.260E-06 0.240E-06 0.160E-06 0.276E-06 0.540E-06 0.276E-06 0.216E-06 0.183E-06 0.264E-06 0.297E-06 0.420E-06
0.130E-06 0.130E-06 0.900E-07 0.880E-07 0.850E-07 0.220-06 0.900E-07 0.900E-07 0.900E-07 0.900E-07 0.130E-07 0.130E-07 0.900E-07 0.180E-06 0.850E-07 0.850E-07 0.720E-07 0.900E-07 0.180E-06 0.920E-07 0.200E-06 0.970E-07 0.720E-07 0.850E-07 0.450E-07 0.880E-07 0.850E-07 0.120E-06 0.880E-07 0.450E-07 0.720E-07 0.990E-07 0.120E-06 0.210E-06 0.140E-06 0.630E-07 0.150E-06 0.850E-07 0.210E-06 0.180E-06 0.850E-07 0.800E-07 0.540E-07 0.920E-07 0.180E-06 0.920E-07 0.720E-07 0.613E-07 0.880E-07 0.990E-07 0.140E-06
40 40 50 40 40 40 70 40 40 90 40 40 40 40 50 40 40 30 40 50 40 40 40 86 40 40 40 40 49 40 40 48 40 40 57 40 40 40 40 35 40 40 40 59 40 40 40 40 40 40 40 (continued)
422
L. Zubieta Table A1Ðcontd
ARC 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87
ORIG
DEST
Car flow
1004 1009 1013 1016 1028 1011 1029 1006 1020 1021 1006 1029 1011 1028 1007 1008 1009 1010 1027 1015 1027 1010 1009 1008 1016 1011 1006 1023 1005 1022 1018 1022 1005 1023 1006 1011
1009 1013 1014 1028 1011 1029 1006 1020 1021 1006 1029 1011 1028 1016 1008 1009 1010 1027 1015 1027 1010 1009 1008 1007 1011 1006 1023 1005 1022 1018 1022 1005 1023 1006 1011 1016
11.0 11.0 11.0 9.4 13.4 13.2 10.2 13.4 14.8 9.4 6.6 9.3 9.4 6.6 17.8 6.8 24.4 17.8 11.4 14.8 8.0 6.8 1.0 10.4 5.0 5.0 8.0 7.0 16.0 9.0 6.0 6.6 8.0 6.8 8.0 6.8
5.5 5.5 15.5 4.7 6.7 6.6 5.1 6.7 7.4 4.7 3.3 4.9 4.7 3.3 8.9 3.4 12.2 8.9 5.7 7.4 4.0 3.4 5.0 5.4 2.5 2.5 4.0 3.5 8.0 4.5 3.0 3.3 4.0 3.4 4.0 3.4
0.300E-06 0.300E-06 0.300E-06 0.260E-06 0.360E-06 0.330E-06 0.280E-06 0.360E-06 0.390E-06 0.260E-06 0.180E-06 0.260E-06 0.260E-06 0.180E-06 0.480E-06 0.180E-06 0.660E-06 0.480E-06 0.300E-06 0.390E-06 0.220E-06 0.180E-06 0.270E-06 0.280E-06 0.140E-06 0.140E-06 0.220E-06 0.190E-06 0.420E-06 0.240E-06 0.160E-06 0.180E-06 0.220E-06 0.180E-06 0.220E-06 0.180E-06
0.990E-07 0.990E-07 0.990E-07 0.850E-07 0.120E-06 0.110E-06 0.920E-07 0.120E-06 0.130E-06 0.850E-07 0.600E-07 0.880E-07 0.850E-07 0.600E-07 0.160E-06 0.610E-07 0.220E-06 0.160E-06 0.100E-06 0.130E-06 0.720E-07 0.610E-07 0.900E-07 0.940E-07 0.450E-07 0.450E-07 0.720E-07 0.680E-07 0.140E-06 0.810E-07 0.540E-07 0.600E-07 0.720E-07 0.610E-07 0.720E-07 0.610E-07
40 40 40 40 46 40 40 40 57 40 40 40 49 40 40 40 40 5 40 40 40 40 40 40 46 40 40 40 40 43 40 40 40 49 40 40
Pergamon PII: S0191-2615(98)00009-5
A NETWORK EQUILIBRIUM MODEL FOR OLIGOPOLISTIC COMPETITION IN CITY BUS SERVICESy LOURDES ZUBIETA*
Bishop's University, Lennoxville, QueÂbec, Canada, J1M 1Z7 and Centre de recherche sur les transports, C.P. 6128 Succ. `Centre-Ville', MontreÂal, QueÂbec, Canada, H3C 3J3 (Received 26 December 1996; in revised form 11 December 1997) AbstractÐThis paper presents a new model for a deregulated transportation system with full representation of the city network. We assume the case in which a few private bus companies provide the totality of the urban transportation services. Each private company is assumed to have exclusive rights to operate a particular transit line. Competition among companies is given only in terms of the frequency of service as demand and transit fares are considered exogenous. The bus operators seek pro®t maximization whereas passengers look for the travel strategy that minimizes expected travel time. At equilibrium, marginal revenue should equal marginal cost for each operating company and, for each origin±destination pair, travel `strategies' for passengers should be optimal. To obtain an equilibrium solution, a heuristic procedure is outlined and tested on a small network. # 1998 Elsevier Science Ltd. All rights reserved Keywords: network equilibrium, deregulation, urban transportation, transit frequencies 1. INTRODUCTION
The deregulation of urban mass transportation systems is an appealing alternative to centralized municipal transit commissions. Among the reasons invoked in favor of descentralization are, increasing discrepancies between subsidies and operating costs, suburbanization, growth in car ownership, seasoned with ideological policy shifts. In Europe, the most signi®cant example is the bus deregulation introduced in Great Britain with the Transport Act of 1985. Urban bus competition has also been introduced in several South American cities, and various experiments are being carried out in southeast Asia (Kang, 1995). Experience is now available in several cities where transit deregulation has been implemented. It suggests that, with the exception of very few short periods of price wars, the competitors matched fares by adopting either the distance-related general fare as in the U.K. (Evans, 1990), or by applying a ¯at fare throughout the city, as in Santiago de Chile (DarbeÂra, 1993). Deregulation has not change the industry structure in Chile, where the average ¯eet size went from 1.33 in 1985 to 1.67 in 1990 (DarbeÂra, 1993). However, all bus operators, with the exception of one or two, belong to a route association. These are non-pro®t organizations whose main role is to dispatch buses and provide services like insurance, legal protection, etc. Fierce competition exists between associations in terms of frequencies of services because their respective routes largely overlap. The diversity of deregulation schemes has motivated the international research community to develop analytic tools capable of providing a better insight into, and understanding of the mechanisms arising in such diverse forms of competition (Evans, 1987). The purpose of this paper is to present a network equilibrium model and a solution procedure for a deregulated transit system inspired from the experience in Santiago. The model is characterized by a full representation of the transit network with a small number of private transit ®rms providing the totality of the urban mass transportation services. Competition among the ®rms is based solely on the frequency of services as the model considers a ®xed origin±destination matrix of demand and fares are assumed constant. The solution procedure seeks an overall Nash-equilibrium state *Fax: 001 819 822 9720; e-mail: [email protected] y This work was partially funded by the National Science and Engineering Research Council of Canada (NSERC).
413
414
L. Zubieta
where passengers minimize their expected total travel time and transit ®rms frequencies maximize their pro®ts. In the transit deregulation literature, much work is devoted to providing descriptive studies of some particular city or country, and to issues such as whether or not private participation ought to be allowed. Also we ®nd some assessments of the bene®ts and/or drawbacks observed or expected in private vs public urban mass transit systems (Beesley, 1990; Mackie et al., 1995; White, 1995). The prediction of market prices, market structure and service levels in a deregulated scenario has also received some attention. Viton (1982) developed equilibrium models for the analysis of a duopoly which consisted of two ®rms competing on a simpli®ed corridor between a residential area and a central business district connected by a limited-access highway. Kanemoto (1984) provided another theoretical model for railway ®rms competing on a radial network. An important extension of Viton's model was proposed by Harker (1988). His representation of corridors is more general in that it allows passengers to embark/disembark at points other than the extremities of the corridor. Each corridor represents a dierent transit service (automobile, subway, private mini-bus, public bus, etc.) with no transfers between services; competition is present among the dierent types of services oered. FernaÁndez and Marcotte (1992) proposed the ®rst full network equilibrium model within a deregulated context. The two main advantages of this model are the full representation of the city network and the modeling of a free market competition between operators within and among transit services. When an operator chooses a line to run competition exists with other bus operators running on the same line but it also exists with other operators running on lines `attractive' to passengers. In this model, car drivers and passengers are assumed to behave according to Wardrop's First Principle (user equilibrium) while bus operators seek to maximize their individual pro®t under conditions of perfect competition. Marcotte et al. (1990) proposed an iterative scheme to solve this model, which was successfully applied to the transit network of the city of Santiago de Chile. In this paper, we modify the perfect competition assumption of the above model to re¯ect a speci®c case of market imperfection: the situation where a few companies have control of the transit market. A private company may own a ¯eet of city buses, or various members may form a cooperative in which they share garages and mechanics to decrease their operating costs. In both cases, we assume that each company has exclusive rights to run a particular bus line. In contrast to the perfect competition case, the decisions of one company aect the market as a whole, as the frequencies of service will determine, for each origin±destination (O±D) pair, the set of `attractive lines' for passengers. 2. MODEL FORMULATION
2.1. Notation and assumptions We consider a transit network where nodes represent bus stops and links represent the distance between two consecutive bus stops. Each line is operated by one private company whose only decision variable is the number of buses to run per unit of time (frequency) in order to maximize its pro®t. There is no restriction on the frequency levels, so neither a minimum nor a maximum number of buses is required. We assume that demand is exogenous, represented by a ®xed O±D matrix, and that bus fares for each company are constant. For every O±D pair, competition among bus lines is generated by the frequency of services, as passengers have several possibilities (strategies) for reaching their destination. As the bus itineraries are not disjoint, transfers between lines are allowed at bus stops. However, every time a passenger boards a vehicle the fare must be paid. Congestion is implicitly taken into account as the variable operating costs for the companies and the round trip travel times are assumed to depend on vehicle ¯ows (cars and buses). In the sections below we use the following notation: R N ; A L 1; 2; . . . ; L S
Transportation network with node set N and arc set A Set of transit lines (sequence of adjacent arcs) Strategy vector (passengers' decision vector)
Network equilibrium in city bus services
``"L V a S; a"A fir C ; V S; ` ; V S; `"L ` fij `
415
Bus frequency vector Passenger volume per time unit Passenger ¯ow boarding at node i and with ®nal destination r Passengers total transit time (travel+waiting time), (a continuous function of S and ) Operators' pro®t function (vector) Bus fare on line ` Number of passengers at node i with destination j Total operating costs per time unit for line `
2.2. The passengers sub model As stated before, passengers strive to minimize their total transit time or generalized transit cost, which includes waiting time and in-vehicle time. Given a vector of frequencies and the O±D matrix, the passenger assignment to the bus lines can be easily calculated by the state-of-the-art transit assignment models (Spiess and Florian, 1989; Nguyen and Pallottino, 1988). A passenger's strategy is a set of lines and transfer points he has in mind when going from a point of origin to a destination. At each bus stop, a strategy is characterized by a subset of available lines called the set of attractive lines. At each bus stop, a passenger boards the ®rst incoming bus belonging to this set. If bus arrivals follow a Poisson distribution (exponential headways), a passenger will choose the set of attractive lines by minimizing his expected transit time to his destination, taking into account transfers, waiting and in-vehicle travel times. For further details, see Spiess (1981) or Nguyen and Pallottino (1988). An equilibrium state is attained when passengers travel on shortest routes, according to a frequency vector , and thus the optimal strategies can be written as the solution of the following linear program parameterized in (Marcotte and Blain, 1990): min FC ; V S; s"
X a"A
ca a S;
X
fir S; ir S;
1
i"N ;r"N
where denotes the set of admissible strategies, ca is the travel time function on arc a, and ir is the waiting time for passengers boarding at node i having ®nal destination r. This is a well known transit assignment problem (Smith, 1979; Dafermos, 1980) that may be formulated as a variational inequality in terms of strategies rather than ¯ows: C ; V S; V S ; ÿ V S; 40
8S"
2
2.3. Formulation of the bus operators sub model A solution to the passengers sub model (1) can provide two useful sets of information for each node i: the number of boardings at node i with destination j; fij , and the set of `attractive lines', Uij , which compete through their frequencies to transport these passengers. If we consider fij and Uij ®xed, then the total number of boarding passengers for a given line ` can be written as a function of the frequency vector by assigning the demand at each node to the attractive lines, in proportion to their frequencies: XX ` fij `ij 3 f` k k"U ij i"N j"N where `ij
1; if line ` stops at node i 0; otherwise
We notice here that if a line is not attractive for any destination then the value of f` will be zero. The pro®t functions are then de®ned as the dierence between revenues and costs:
416
L. Zubieta
` ` f` ÿ `
4
An equilibrium state is obtained when no operator can increase his pro®t by modifying his frequency ÿ of service ` , given the present frequencies of the rest of the operators. Let ` 1 ; 2 ; . . . ; ` ; `1 ; . . . ; L 50 for each ` 1; . . . ; L. Then ` is an optimal solution of the following problem: ÿ 5 max ` ` ` 50
It can be shown that if ` "C1 ; ` 1; . . . ; L then each pro®t function ` is concave in ` and the ®rst order optimality conditions for the set of problems de®ned by (5) are: r` ` ` 0;
` 1; . . . ; L
` ` 0 ` ; ` 5 0:
6
Conditions (6) constitute a nonlinear complementarity problem, which is equivalent (Karamardian, 1971) to the following variational inequality problem: ; V S ; ÿ 40;
850
7
Given the passengers behaviour observed after a transit assignment, the bus operators can be seen as players of an L-person, noncooperative game for which we seek a Nash equilibrium. The connection between Nash equilibrium and variational inequalities was shown by Gabay and Moulin (1980) and Harker (1984). 2.4. The global model An equilibrium state for both passengers and bus, companies consists of a vector of strategies S and a vector of frequencies satisfying (2) and (7), which results in the following system of variational inequalities: C ; V S ; V S ; ÿ V S; 40;
; V S ; ÿ 40; 8 S; "ÿ
8
where ÿ denotes the set of feasible S and . The economic interpretation of such an equilibrium state follows the classical micro economic model of an oligopoly (Moulin, 1986; Friedman, 1977). For every bus operator, marginal pro®t equal marginal cost [see eqn (6)], and no ®rm can increase its pro®t by providing a dierent output if the other ®rms' output remains constant. This model does not take into account any kind of collusion among bus companies and so, the equilibrium in (8) need not be Pareto optimal. For the passengers, the equilibrium state means that no strategy other than the optimal one can reduce the expected transit time for every O±D pair. 2.5. Existence and uniqueness of solutions Given a frequency vector , the existence of at least one solution for the passenger sub model follows from the fact that C is a linear function with a bounded feasible set. Although the set of feasible ¯ows is not technically bounded in the usual formulation of the problem, it is easy to show that an optimal solution cannot occur at in®nity, given the assumption of ®xed (exogenous) demand. For the bus operators sub model, standard arguments would ensure the existence of a solution provided that the feasible set is compact and the functions are continuous. The set of feasible frequencies (50) can be limited to a compact set since the ®xed demand and the concavity of the pro®t functions imply that large frequencies would yield negative pro®ts, clearly a nonequilibrium solution. In fact, the total number of passengers boarding a bus line is obviously limited by the total number of passengers, so the revenue is bounded by a constant depending only on the problem data. As a consequence of the above, the Brouwer±Kakutani ®xed point theorem (Ortega and Rheinbolt, 1970) ensures the existence of at least one solution to the global model (8).
Network equilibrium in city bus services
417
Uniqueness of a solution can be proven for the bus operator sub model if the pro®t functions are strictly concave. For the global model, however, it would be quite surprising to ®nd reasonable conditions to the uniqueness of a solution, given that the pro®t functions are not monotone and that the passengers sub model is linear. Reasonable conditions should not restrict the topology of the bus lines to situations such as disjoint paths or non-intersecting lines. These cases are not observed in city networks. To ®nd an equilibrium (S ; ) for the global model (8) is not straightforward but we can exploit two characteristics. First, for given frequency vector , state-of-the-art transit assignment models can provide the reaction of passengers: the set of `attractive lines' Uij and the number of passengers boarding at each node of the network fij . Now, for a given Uij and fij , there is a wide body of algorithms for variational inequality problems that can be used to obtain a new vector of frequencies that maximize operators pro®t (e.g. Glowinski et al., 1976). The new frequency vector may produce a change in the passengers behaviour by revealing a dierent set of attractive lines and thus a dierent number of passengers boarding at each node of the network. A solution procedure based on these characteristics is presented next. 3. HEURISTIC PROCEDURE
In order to solve the global model (8), the following iterative scheme is proposed: start with an initial frequency vector; solve the passengers sub model to obtain the passengers reaction. Solve the operators sub model to obtain a new frequency vector. If no major dierences exist between the new and the current frequency vector, stop. Otherwise solve again the passengers sub model. Step 0. Step 1. Step 2. Step 3.
ÿ Let 0 50; 0 6 0 ; " > 0; 0 < 41; k : 0. Solve the passengers sub model (1) using k Solve the bus operators sub model (6) given the optimal strategies and passenger ¯ows obtained fromÿ Step 1. Let 0 be the solution. Let k1 k 0 ÿ k . If jk1 ÿ k j < " then STOP Otherwise k : k 1 and return to Step 1.
The above procedure is a Gauss±Seidel approach easy to implement since both Step 1 and Step 2 can be eciently solved by well-known algorithms. Solution to Step 1 is easily obtained by the state-of-the-art transit assignment software EMME/2 (1994) and Step 2 can be obtained by an iterative scheme such as Newton±Jacobi. The introduction of the relaxation parameter a is required in order to avoid abrupt changes in the optimal strategies which may prevent the procedure from converging. 4. NUMERICAL EXAMPLE
To illustrate the heuristic method proposed, consider the transit system shown in Fig. 1. It consists of 29 nodes, 86 arcs. (transit segments) and 24 O±D pairs distributed over the network. Eight bus lines serve this city network and we assume the bus fares to be $1 for all lines. The assignment results were obtained from the EMME/2 transit assignment model. The operating cost functions and the travel time functions considered are adapted from FernaÂndez and Marcotte (1992) by multiplying their cost functions per vehicle by the number of vehicles running on each line. Let FC` fixed operating cost per unit of time on line `; G` variable operating cost on line `; per unit of time : X a a a Ba 2 `a ; ` 1; . . . ; L a"A
T` round trip travel time on line ` : X a a fa Ba 2 `a ; ` 1; . . . ; L a"A
418
L. Zubieta
Fig. 1. Small network.
with `a
1; 0;
if arc a is on line ` itinerary otherwise
is the bus±car equivalence, fa is the car ¯ow on arc a; Ba ` B` `a is the total number of buses and are given in the Appendix ( 3). The operrunning on arc a, and the parameters ; ; , ating cost function is de®ned as ` ` G` T` FC` ;
` 1; . . . ; L
A particular. line may experience very little or no competition at all if it is the only `attractive line' for a subset of O±D pairs. In such cases, the revenue may become independent of the frequency vector and the latter may be driven to zero by the heuristic. In order to avoid this situation, we added one dummy `walk line' per regular line in the passengers sub model, with a ®xed frequency of 0.001 and running at 3 Km/h. These walk lines introduce a type of modal split as they represent the possibility of passengers walking instead of taking a bus if the frequency of service becomes too low. In Fig. 2 the initial value for the bus frequencies was set to 0.1 buses per min or equivalently, headways are set to one bus every 10 min. Each iteration represents the solution to the bus operators sub model (k1 ), given the transit assignment results obtained with the previous frequency vector (k ). After a few iterations it is clear that the four bottom lines rapidly converge to an equilibrium point. A second group of bus lines, with headways ranging between 6 and 9 mins, do not converge to a value but stay around a range of values. The upper bus line is a clear case of a monopoly for which there is a built-in barrier to the highest headway value passengers accept before turning away public transportation: the walk lines. In this case the revenue function is independent of frequencies and therefore pro®t maximization pulls costs to zero by increasing headway to a point where only a fraction of the clientele will continue to ride. This in turn forces the line to reduce the headway in order to increase pro®t and eventually the cycle restarts. Similar results were obtained in Fig. 3, where the initial solution was set to 1.0 min instead of 10.0 min. Again, the heuristic
Network equilibrium in city bus services
419
Fig. 2. Evolution of headways (initial value=10.0 min).
Fig. 3. Evolution of headways (initial value=1.0 min).
found the same pattern of behaviour although we have no reason to believe that this `pseudo equilibrium' is unique. The three groups of lines shown in Figs 2 and 3 can be related to the amount of competition. Table 1 shows headways, market shares and the interaction among bus lines, at the last iteration of the heuristic. We can observe that lines 102, 104, 107 and 108, having the lowest headway values, face the strongest competition: directly, by sharing passengers with another line, or indirectly, by inheriting clientele coming from other bus companies. The second group of lines (101, 103, and 105) are more aected by the transfers they receive from other bus lines than by direct competition with other lines, although line 101 shares a few passengers with line 108. Finally the monopoly situation of line 106 is clear given that this line neither shares or
420
L. Zubieta Table 1. Competition and market shares
Line
Boardings (%)
Headway (min)
Compete with
Transfers from
102 104 107 108 101 103 105 106
7.55 4.35 21.96 8.01 14.45 15.21 13.50 15.00
1.97 2.75 2.91 2.48 7.15 4.89 7.69 17.55
104,108 102,107 104 101,102 108 ± ± ±
106 101 102,103 102 103,104,106 104,107 101,107 ±
inherits any passengers. We must point out that the solution procedure proposed in this paper is memoryless in terms of all the previous headway values experienced, only the present value is considered to get the next one. Hence, the evolution patterns observed should not be interpreted as dynamic behaviour of the public transportation market. This paper presents a model and a solution procedure of a deregulated transit system where bus lines are assumed to compete for passengers in a city network. The results obtained with a small example provided a better insight into the mechanisms of such forms of competition faced by the bus companies and to explicitly relate those levels to frequencies of service. A natural extension of the model would be to allow for variable demand in the passengers sub model, an analysis which will be presented in a forthcoming paper. AcknowledgementsÐThe author wishes to thank Patrice Marcotte and an anonimous referee for their comments and suggestions. REFERENCES Beesley, M. E. (1990) Collision, prediction, and merger in the UK bus industry. Journal of Transport Economics and Policy 24, 295±310. Dafermos, S. (1980) Trac equilibria and variational inequalities. Transportation Science 14, 42±54. DarbeÂra, R. (1993) Deregulation of urban transport in Chile: what have we learned in the decade 1979±1989? Transport Reviews 13, 45±59. INRO Consultants (1995) EMME/2, User Reference Manual. Evans, A. (1987) A theoretical comparison of competition with other economic regimes for bus services. Journal of Transport Economics and Policy 21, 7±36. Evans, A. (1990) Competition and the structure of local bus markets. Journal of Transport Economics and Policy 24, 255± 281. FernaÂndez, J. E. and Marcotte, P. (1992) Operators-users equilibrium model in a partially regulated transit system. Transportation Science 26, 95±105. Gabay, D. and Moulin, H. (1980) On the uniqueness and stability of Nash-equilibria in noncooperative games. In Applied Stochastic Control in Econometrics and Management Science, eds A. Bensoussan, P. Kleindorfer and C. S. Tapiero. North Holland, Amsterdam. Glowinski, R., Lions, J. L. and TremolieÁrs, R. (1976) Analyse NumeÂrique des IneÂquations Variastionnelles, Dunod, Paris. Harker, P. T. (1984) A variational inequality approach for the determination of oligopolistic market equilibrium. Mathematical Programming 30, 105±111. Harker, P. T. (1988) Private market participation in urban mass transportation: application of computable equilibrium models of network competition. Transportation Science 22, 96±111. Kanemoto, Y. (1984) Pricing and investment policies in a system of competitive commuter railways. Review of Economic Studies 51, 665±681. Kang, Y. (1995) The development of urban bus deregulatory policy in Korea revisited. Transport Reviews 15, 357±370. Karamardian, S. (1971) Generalized complementarity problem. Journal of Optimization Theory and Applications 8, 161±168. Mackie, P., Preston, J. and Nash, C. (1995) Bus deregulation: ten years on. Transport Reviews 15(3), 229±251. Marcotte, P. and Blain, M. (1990) A Stackelberg±Nash model for the design of deregulated transit system. Fourth International Symposium of Dierential Games and Applications, eds Raimo P. HaÈmaÈlaÈinen and Harri Ehfamo, Pre prints Vol. 1. Helsinki University of Technology, Systems Analysis Laboratory, June. Marcotte, P., Zubieta, L. and Drissi-Kaitouni, O. (1990) Iterative methods for solving and equilibrium problem arising in transit deregulation. Transportation Research-B 24, 45±55. Moulin, H. (1986) Game Theory for the Social Sciences, 2nd edn. New York University Press, New York. Nguyen, S. and Pallottino, S. (1988) Equilibrium trac assignment for large scale transit networks. EJOR 37, 176±186. Ortega, J. and Rheinboldt, W. (1970) Iterative Solutions of Nonlinear Equations in Several Variables. Academic Press, New York. Smith, M. J. (1979) Existence, uniqueness and stability of trac equilibria. Transportation Research-B 13, 295±304. Spiess, H. (1981) Contributions aÁ la theÂorie et aux outils de plani®cation des reÂseaux de transport urban, Ph. D. thesis, Universite de MontreÂal, Canada.
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Spiess, H. and Florian, M. (1989) Optimal strategies: a new assignment model for transit networks. Transportation Research-B 23, 83±102. Viton, P. A. (1982) Privately-provided urban transport services. Journal of Transportation Economics and Policy 16, 85±94. White, P. (1995) Deregulation of local bus services in Great Britain: an introductory review. Transport Reviews 15(2), 185± 209.
APPENDIX
Table A1. Arc parameters ARC 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
ORIG
DEST
Car flow
1001 1002 1003 1004 1005 1006 1005 1004 1003 1002 1007 1024 1008 1025 1009 1026 1010 1011 1010 1026 1009 1025 1008 1024 1019 1002 1008 1012 1013 1012 1008 1002 1002 1017 1018 1005 1010 1015 1016 1015 1010 1005 1018 1017 1014 1013 1009 1004 1018 1020 1018
1002 1003 1004 1005 1006 1005 1004 1003 1002 1001 1024 1008 1025 1009 1026 1010 1011 1010 1026 1009 1025 1008 1024 1007 1002 1008 1012 1013 1012 1008 1002 1019 1017 1018 1005 1010 1015 1016 1015 1010 1005 1018 1017 1002 1013 1009 1004 1018 1020 1018 1004
15.0 15.0 10.0 9.8 9.4 24.4 10.0 10.0 10.0 10.0 11.0 9.8 10.0 20.0 9.4 9.4 8.0 10.0 20.0 10.2 22.0 10.8 8.0 9.4 5.0 9.8 9.4 14.0 9.8 5.0 8.0 11.0 14.0 23.4 10.6 7.0 17.0 9.4 23.4 20.0 9.4 9.0 6.0 10.2 20.0 10.2 8.0 6.8 9.8 11.0 15.6
7.5 7.5 5.0 4.9 4.7 12.2 5.0 5.0 5.0 5.0 5.5 4.9 5.0 10.0 4.7 4.7 4.0 5.0 10.0 5.1 11.0 5.4 4.0 4.7 2.5 4.9 4.7 7.0 4.9 2.5 4.0 5.5 7.0 11.7 7.8 3.5 8.5 4.7 11.7 10.0 4.7 4.5 3.0 5.1 1.0 5.1 4.0 3.4 4.9 5.5 7.8
0.390E-06 0.390E-06 0.270E-06 0.260E-06 0.260E-06 0.660E-06 0.2700E-06 0.270E-06 0.270E-06 0.270E-06 0.300E-06 0.260E-06 0.270E-06 0.540E-06 0.260E-06 0.260E-06 0.220E-06 0.270E-06 0.540E-06 0.280E-06 0.600E-06 0.290E-06 0.220E-06 0.260E-06 0.140E-06 0.260E-06 0.260E-06 0.360E-06 0.260E-06 0.140E-06 0.220E-06 0.300E-06 0.360E-06 0.630E-06 0.420E-06 0.189E-06 0.450E-06 0.260E-06 0.630E-06 0.540E-06 0.260E-06 0.240E-06 0.160E-06 0.276E-06 0.540E-06 0.276E-06 0.216E-06 0.183E-06 0.264E-06 0.297E-06 0.420E-06
0.130E-06 0.130E-06 0.900E-07 0.880E-07 0.850E-07 0.220-06 0.900E-07 0.900E-07 0.900E-07 0.900E-07 0.130E-07 0.130E-07 0.900E-07 0.180E-06 0.850E-07 0.850E-07 0.720E-07 0.900E-07 0.180E-06 0.920E-07 0.200E-06 0.970E-07 0.720E-07 0.850E-07 0.450E-07 0.880E-07 0.850E-07 0.120E-06 0.880E-07 0.450E-07 0.720E-07 0.990E-07 0.120E-06 0.210E-06 0.140E-06 0.630E-07 0.150E-06 0.850E-07 0.210E-06 0.180E-06 0.850E-07 0.800E-07 0.540E-07 0.920E-07 0.180E-06 0.920E-07 0.720E-07 0.613E-07 0.880E-07 0.990E-07 0.140E-06
40 40 50 40 40 40 70 40 40 90 40 40 40 40 50 40 40 30 40 50 40 40 40 86 40 40 40 40 49 40 40 48 40 40 57 40 40 40 40 35 40 40 40 59 40 40 40 40 40 40 40 (continued)
422
L. Zubieta Table A1Ðcontd
ARC 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87
ORIG
DEST
Car flow
1004 1009 1013 1016 1028 1011 1029 1006 1020 1021 1006 1029 1011 1028 1007 1008 1009 1010 1027 1015 1027 1010 1009 1008 1016 1011 1006 1023 1005 1022 1018 1022 1005 1023 1006 1011
1009 1013 1014 1028 1011 1029 1006 1020 1021 1006 1029 1011 1028 1016 1008 1009 1010 1027 1015 1027 1010 1009 1008 1007 1011 1006 1023 1005 1022 1018 1022 1005 1023 1006 1011 1016
11.0 11.0 11.0 9.4 13.4 13.2 10.2 13.4 14.8 9.4 6.6 9.3 9.4 6.6 17.8 6.8 24.4 17.8 11.4 14.8 8.0 6.8 1.0 10.4 5.0 5.0 8.0 7.0 16.0 9.0 6.0 6.6 8.0 6.8 8.0 6.8
5.5 5.5 15.5 4.7 6.7 6.6 5.1 6.7 7.4 4.7 3.3 4.9 4.7 3.3 8.9 3.4 12.2 8.9 5.7 7.4 4.0 3.4 5.0 5.4 2.5 2.5 4.0 3.5 8.0 4.5 3.0 3.3 4.0 3.4 4.0 3.4
0.300E-06 0.300E-06 0.300E-06 0.260E-06 0.360E-06 0.330E-06 0.280E-06 0.360E-06 0.390E-06 0.260E-06 0.180E-06 0.260E-06 0.260E-06 0.180E-06 0.480E-06 0.180E-06 0.660E-06 0.480E-06 0.300E-06 0.390E-06 0.220E-06 0.180E-06 0.270E-06 0.280E-06 0.140E-06 0.140E-06 0.220E-06 0.190E-06 0.420E-06 0.240E-06 0.160E-06 0.180E-06 0.220E-06 0.180E-06 0.220E-06 0.180E-06
0.990E-07 0.990E-07 0.990E-07 0.850E-07 0.120E-06 0.110E-06 0.920E-07 0.120E-06 0.130E-06 0.850E-07 0.600E-07 0.880E-07 0.850E-07 0.600E-07 0.160E-06 0.610E-07 0.220E-06 0.160E-06 0.100E-06 0.130E-06 0.720E-07 0.610E-07 0.900E-07 0.940E-07 0.450E-07 0.450E-07 0.720E-07 0.680E-07 0.140E-06 0.810E-07 0.540E-07 0.600E-07 0.720E-07 0.610E-07 0.720E-07 0.610E-07
40 40 40 40 46 40 40 40 57 40 40 40 49 40 40 40 40 5 40 40 40 40 40 40 46 40 40 40 40 43 40 40 40 49 40 40