- Email: [email protected]
A process model of fare decision making for integrated transit systems
00384121/85 $3.00 + .oO 0 1985 Pergamon Press Ltd.
Socio-Econ. Plan. Sci. Vol. 19, No. 2, pp. 109-116, 1985 Printed in the U.S.A.
A PROCESS MODEL OF FARE DECISION MAKING FOR INTEGRATED TRANSIT SYSTEMS
DAN B. RINKS Quantitative Business Analysis, Louisiana State University, Baton Rouge, LA 70803, U.S.A. and FREDRICK C. SCHERR College of Business and Economics, West Virginia University, Morgantown, WV 26505, U.S.A. (Received 5 July 1984) Abstract-Based on interviews with transit officials in several American cities, a process model for evaluating proposals to integrate local transit properties into an area-wide transportation system is developed. Relevant factors for a data base are discussed and the information processing flows are described. In this model, negotiation between the participants plays an important role in constructing and assessing revenue allocation plans.
1. INTRODUCTION Many metropolitan areas are served by several different transit properties. Some of these properties are privately owned, while others are governmental or quasi-governmental institutions. In some cases, the transit properties have exclusive rights to certain territories, while other properties serve overlapping territories. These complicated transit structures are often the result of arbitrary political boundaries or historical accident. The transit patron has little concern for the political, physical, and legal constraints under which the various properties in his/her area operate. His or her goal is to move from one point to another with a minimum of time, cost and inconvenience. The end result of this is, often, the choice of private transportation as the transport mode most consistent with these goals. One way to reduce inconvenience, and thus increase ridership, is to integrate the transit systems of a metropolitan area. Routes can be co-planned for easier rider transfers. Cost savings may result from the elimination of redundant routes. Integrated fare collection may be easier for riders and save the systems collection expense as well. Several issues are involved in fare setting and collection for integrated systems. 1. The methodology for pricing transit trips for an integrated system 2. The fare collection methodology for an integrated system 3. The revenue allocation among participants in an integrated system. For properties considering integrating their systems, the sequence of the decision analysis might logically follow the order listed above. First, the decisions on a joint pricing structure are made. Next, the methods for collection the fare are determined. Finally, the participants decide on how to allocate the joint revenues. However, it should be noted that these three important decisions are likely to be coincident because the decisions may interact with one another. For instance, fare collection technology may prohibit
certain pricing structures, such as an exact distancebased fare for a bus system. Furthermore, due to its importance to the well-being of the participants, an agreement in principle on revenue sharing is probably necessary before final determinations are made on fare structure and fare collection. Indeed, we view the revenue distribution plan as a primary factor in whether or not autonomous units will agree to be participants in an integrated system. As pointed out by Cowing and Holtmann [I], the combination of governmental service units can benefit one part of the combination while being a disadvantage to others, even if the combination as a whole benefits the public. For example, a city and a county bus system might combine and form a new routing plan which better serves the public, but this plan’s cost might bear more heavily on one unit without a totally compensating revenue increase. Another possibility is that this new routing system might greatly aid one unit’s consuming public while causing minor losses to the other unit’s public. In cases such as this, there is an incentive on an overall basis to combine, even though the combination is disadvantageous to one of the combining units. Cowing and Holtmann suggest that “enticement strategies”, which involve subsidization of those units made worse off by those units made better off, can be used to affect the combination. We view the division of revenues and costs among units as a way to implement such an enticement strategy. In other words, the plan for the distribution of revenues and costs for the combined system, made before the combination, should be such that there is an incentive for the combination to take place, for the coalition of units to form an integrated system. The remainder of this paper describes a process model for integrated transit decision making that was constructed based on interviews with transit officials of the major transit property in Baltimore, Md., Washington, D.C., Pittsburg, Pa., and Atlanta, Ga., and from officials of six properties in the San Francisco, Cal., Bay Area. 109
110
D.B. RINKS and F.C. SCHERR 2. PROCESS MODELS FOR FARE INTEGRATION DECISIONS
The model presented in this paper is an effort to describe the information processing that takes place coincident with the setting of fares and division of revenues for an integrated transit system. Because it attempts to simulate the structural form and behavior of the decision process, the model has been labeled a process model in the sense of Larkey [2]. Similar models, referred to as positive models, have been developed by Crecine [3, 4, 51. Thompson and Willilams [6] state that the typical justification for this type of model is “that the public expenditure process deviates too much from the idealized conditions assumed by the utility-maximizing model and that the complexity of the process renders the utilitymaximizing model useless to explain public expenditure and tax decisions”. The pros and cons of using utility-maximizing models versus process models for public decision making is discussed in [6]; Downs and Larkey [7], commenting on [6], also discuss the issue. Since our primary purpose is to describe the process and not prescribe solutions, this approach has been deemed appropriate. An important benefit of constructing a descriptive model is that seemingly “obvious” relationships are “discovered”. The relationships are obvious in retrospect, but rarely are so a priori. What happens, of course, is that by asking such questions as “What data are required?’ and “How is it used?’ the information processing patterns emerge. In the subsection “Critical Dimensions . . .” the data base for integrated transit system decision making is described. After this, the subsection entitled “The Process Model” outlines the information-processing patterns that emerged.
been a major problem in developing fare integration among the San Francisco systems [S].
Frequency of trip length. This refers to the diversity of length of trips taken. If trips are principally of one length (as where a transit system serves mostly the central business district from a series of similardistance suburbs), a flat-fare system, perhaps with tokens, should be strongly considered, regardless of the philosophy of the systems involved regarding distance-based fares. If the trip distribution is broad, the systems’ philosophies with regard to fares come into play, as do simplicity and in-place collection technology effects. Operating philosophy and practices of properties. While there are some very strong commonalities in current philosophy and practice, it is equally obvious that a great deal of diversity also exists. This presents significant problems in negotiating price structures for integrated routes, in deciding on appropriate collection technologies and in dividing revenues. For example, if one property uses and believes in flat fares and tokens, while another uses and believes in distance-based fares and strip cards, there is a basic philosophical problem which will cause significant disagreement.
2.1 Critical dimensions in developing an integrated fare structure
Proportion of integrated trips taken. This dimension refers to the proportion of the properties’ routes to be affected. Obviously, if the potential market for integrated trips is a very small proportion of total ridership, the degree of change in fare policy and collection technology that can be cost justified is proportionately small. In such cases, integrated fares and collection systems must be compatible with existing fare structures and collection systems; a simple transfer system suggests itself. On the other hand, if the potential or actual market for integrated trips is large, then a revamping of significant parts of fare collection structures may be justified, and compatibility deemphasized.
A preliminary review of the critical factors we believe to be relevant in developing an integrated system is appropriate at this point. These important parameters of the system to be integrated are the primary influences on the types of fare structures, collection methodologies and allocation methods that can be feasibly used for integrated routes. Modes involved. The modes involved (bus, trolley, limited-access light rail, etc.) affect primarily the collection methodology of an integrated fare system, rather than fare structure or revenue allocation. If the integrated system involves a limited-access light rail system in conjunction with bus feeders, collection technology must be compatible with practical collection methods on these two modes, but the modality has little effect on revenue allocation and only a moderate effect on fare structure policy. Existing/new systems hardware. Some integrations will involve the use of existing hardware only, while in other cases some new hardware will be purchased. Where existing hardware is predominant, the fare structure and collection methods must be compatible with it. Distance-based fares, for example, are easier to administer if one has a strip-reader system. Revenue division is little affected by this dimension except as the dimension affects fare structure and revenue produced. The prevalence of existing hardware has
This dimension refers to the public or private ownership of the transit properties and the structure of any subsidies involved. With regard to fare structure, privately held units will be more likely than public units to be much more concerned with revenue effects than with other aspects of transit price structure (simplicity, equity, etc.). The major effect here, however, is on revenue division. Privately held units will likely veto any revenue division formula that does not hold constant or increase their profitability, regardless of ridership increases, on integrated routes. Public units are likely to be more flexible in this regard, being willing to bear some revenue reductions and/or cost increase if ridership is significantly enhanced. Also, the effects of integration and ridership on subsidies occur on this dimension. If subsidies are based partially on ridership, or if there is agency funding available for integration projects (as through the MTA in San Francisco), these costs and revenue effects will certainly influence fare structure, the amount that can be spent on new collection technology, and the bargaining involved in the revenue allocation plan. Size of the system. This refers to population served and potential ridership; the primary effect is on
Organizational structures of integrated properties.
Fare decision making for integrated transit systems
collection technology. There are returns to scale in the development of many collection technologies, and they can be practically used only in conjunction with larger systems. The population factor, along with the proportion of integrated trips taken, determines ridership for integrated trips. 2.2 The process modes A process model for fare integration is presented in Fig. 1. The model is composed of two parts. Within the dashed portion is the part of the model referring to the development of the revenue allocation formula. This is appropriate only if there is a revenue allocation problem, i.e. if the integration is going to be by formal agreement and not a combination by fiat. In the discussion that follows, we shall assume that a formal agreement (federation) is to be employed, as this utilizes the model in full. The model starts when properties decide to evaluate the possibilities of route integration. The “TRIAL” boxes refer to the generation of one or more plans and forecasts on a particular topic. Discussion proceeds by box number (see Fig 1). q Data collection. At this point data is collected on potential integration routes and schemes. To a great extent, this data is information on the critical dimensions discussed earlier. Data on these dimensions serve to parameterize the problem, eliminate obviously infeasible solutions and suggest the tradeoffs involved in various potential integration schemes. Data which are obviously necessary are: (a) Possible route rationalizations and integrations (b) Current ridership along routes that are candidates for integration
111
(c) Elasticity data for area and routes, by modes, groups, time of day, etc. (d) Cost data for potential rationalizations/integrations (e) Collection hardware in place. On-board studies, studies of past price changes, cost analysis, etc., are useful here. q Trial: route ratio~a~i~at~~~/i~tegration and rescheduling. Here, a series of plans for route rationalizations are developed. We do not review the mathematical methodology here, which is a part of Transportation Economics. Plans will involve various route changes and costs. Thus, this plan or plans has a major impact on the properties’ cost picture (Box q) via the dollars saved or spent in the integration process. The major flow of the process at this point, however, is to Box q, estimates of the effect on ridership. q Ridership estimates. Here the ridership effects of the rationalization/integration plans are evaluated. The effects are, of course, heavily influenced by the price structure on the integrated routes (Box a). •I Trial: pricing structure. Here several price structures are developed for the various route rationalization/integration plans and their ridership effects evaluated via the elasticities. Revenue, equity and simplicity factors are included in the fo~ulation and evaluation of the various plans. •I Trial: fare collection methodology. At this point, several fare collection methodologies are discussed for the various transit plans. There is a great deal of co-determination involved here with the fare structure plan; different fare structures require different collection systems.
Fig. 1. A process model of fare integration.
112
D.B. RINKS andF.C. SCHERR
q Are the various price and coifection structures feasible? At this point, a great number of price/ collection alternatives will be discarded based on the relevant dimensions: the expected ridership is too small to justify their cost; they entail too much modification of the present system for the ridership involved, etc. Discarded plans are sent back to price and collection planners for review/revision. Remaining feasible rou~efp~cefcoll~on plans go to the “Revenue Allocation Fo~ulation” section of the model. In this section of the model, various system philosophies and revenue-cost-profit data are used to develop one or more revenue allocation plan(s). q Trial: joint revenues. Here the joint revenues from the integrated/rationalized routes are calculated for the various route/price/collection plans. q Consideration of potential variables in revenue division formula. These are philosophical considerations: What “should” be included in such a formula? Some formulas are quite complex, others simple. The attitudes of properties come into play here (see Rinks et al [9]). q Relevant ridership data. This is one of the pragmatic considerations in a revenue allocation plan. Whose ridership will change? How? What will the effect on subsidies be? The major input here is the ridership estimates from Box a. @ Relevant costs. Changes in costs for the various systems will depend on collection methodologies employed and route shifts/eliminations/len~henin~. If changes in total revenue are small (which would be the case if relatively few new rides were generated and the fare structure remained relatively unchanged), cost changes may be the major effect of any integration. q Relevant profitability data. A separate box has been included to account for the explicit profitability concern of private systems. Revenue and cost data also reach the allocation box via Boxes q and 8. Any changes in ridership-based subsidies are also included in Box q I. @i Trial: revenue a~l~at~on plan. Here an allocation plan is formulated via the considerations encompassed in Boxes q through a. Methods for the formulation for such a plan are found in [9]. There may be several such plans if several route/price/collection alternatives have survived to this point. q Plan for the division of joint fares. The major output of a revenue allocation plan is the division of fafes on the integrated/rationalized routes. We assume here that the properties continue to run their various modes. Any joint int~ation costs (for example, a stalf to administer the division of revenues according to the allocation formula) are likely to be very small in comparison to the operating cost. q Plan for division of integration subsidies. This deals with where these occur. @I Evaluation of revenue allocation plan(s). Here a formal evaluation of the various route/price/collection/allocation plans is made by the staffs and managers of the properties. Plans may be rejected and sent back to various planning stages for reconsideration. If a plan or plans are acceptable, the political process starts (submission to transit boards, public hearings, etc.). While this is not included in the
model, it may also involve feedback to various planning stages. •I Evaluation criteria. These are the factors by which route/price/collection/allocation plans are judged by the properties. Several are suggested in the box; they may vary from property to property. 3. THE ROLE OF NEGOTIATION IN THE PROCESS
Although this is not usually stated explicitly: negotiation between the participants in the proposed integrated system plays an important role in the process of constructing and assessing revenue allocation plans. First, the variables that directly and indirectly have an effect on revenue sharing and cost allocation for participants in an integrated transportation are identified and described. This step clarifies those issues which are important and must receive major attention in the construction of a formula. Next, as trial formulas are put forward, each party must try to assess how the allocation formula will affect its operation. Finally, through negotiations, an agreement that is acceptable to all the participants is reached, or failing this, the area-wide transportation system will remain unintegrated with respect to joint fares. Figure 2 depicts the flow of activities in the model just described. Of vital concern to each carrier is how the integration is going to affect its revenue and operating costs. In our approach to revenue allocation, control of its costs is the ~s~nsibility of each carrier (see Rinks et al. 193). Operating costs may, however, be an appropriate issue for discussion in the negotiation process. Consider a very simple integration situation. The only significant change caused by the integration is that the originating carrier collects the entire joint fare (fA), remits the appropriate amount (fAB),to the second carrier, and retains the portion of the joint fare that is due it (fAA). As Fig. 3 and Table I illustrate, there were no route changes from the int~tion that would have caused a change in the operating costs. Also, it is assumed that the fare collection and transfer costs are not materially affected by the integration in this example. A straightforward joint-fare policy, in this case, would be to leave the price charged to the rider unchanged. Thus, a rider boarding at point a would pay $1.00 for the trip to point b, which would involve a transfer at point x. System A would remit $0.50 to System B for their share of the revenue generated by the system trip. fA = f*B + fM $1.00 = $0.50 + $0.50 A somewhat more complicated situation occurs when the integration causes significant changes in operating costs and/or ridership patterns. Integration of transit systems offers the potential for operating improvements such as rerouting and route rationalizationf~onsohdation. Figure 4 depicts a circ~stan~e where both System A and System B have made changes to their route structures to accommodate the integration; Table 2 shows how these changes alIect the operating costs. Although the changes result in
Fare decision making for inte~at~
113
transit systems
STEP 1 IDENTIFICATlON OF CRITICALVARIABLES COSTS RIDERSHIP PROFITS (LOSSES) EASE OF ~MINIsTRATIoN POLITICAL CONSIDERATIONS
STEP 2
FORMULATION OF TRIAL REVENUEALLOCATION PLANS
4
,
NO
OF TRIALFORMA:
REVENUE ALLOCATION PLAN ACCEPTED IN PRINCIPLE
INTEGRATION OF AREA WIDE TRANSPORTATION NOT POSSIBLE WITH RESPECT TO JOINTFARES
Fig. 2. The role of negotiation in revenue allocation.
overall efficiency (total post-integration operating costs per trip are $1.90, versus $2.00 for pre-integration),
the net change is quite different for the two systems. System A’s operating costs have increased by $0.25 per trip, whereas the operating costs of System B have decreased by $0.35 per trip. What is the proper price for the system trip, and how much should each system receive as its share? For the sake of simplicity, assume that the system trip is priced at $1.00. Now, the state of affairs is one of distributive bargaining. That is, the two participants are trying to decide how to divide the fixed joint fare;
SYSTEM A
TRIP
what one gains by negotiation, the other one loses. One must define the reservation prices as the least favorable amounts the Systems are willing to accept for their share of the joint fare, or else they will not participate in the integrated transit system. For instance, System A and System B might establish their reservation prices so that their revenue-t~o~rating cost ratio after the integration is no worse than it was prior to the integration. This guideline implies that System A should receive at least $0.625 from the joint fare and System B should receive at least $0.325. Since the reservation prices for both particiSYSTEM B
TRANSFER
ORIGIN
TRIP DESTINATION
-----
POST-INTEGRATION PRE-INTEGRATION
Fig. 3. Typical route in an integration where route changes are unnecessaq.
114
D. B. RINKS and F. C. SCHERR
Table 1. Comparison of operating cost/trip and fares for the integration depicted in Fig. 3 Prior to the Integration
After the Integration
System A Operating Cost/Trip Fare/Trip
$1.00 $0.50
$1.00 $0.50
System B Operating Cost/Trip Fare/Trip
$1.00 $0.50
$1.00 $0.50
pants can be satisfied, there is a zone of agreement (Fig. 5). Other reservation prices are, of course, possible. For example, System B might believe that the projected cost savings to its system will not materialize and, correspondingly, set its reservation price at $0.50 to maintain its revenue-to-cost ratio at 50%. If System A’s reservation price is $0.625, then an agreement is not possible, as there is no zone of agreement (Fig. 6). Naturally, each participant desires to do better than its reservation price. For situations where there is a zone of agreement, the final negotiated settlement will depend on many factors, not the least of which are the respective bargaining skills. The sequence of offers by one side and counter-offers by the other is sometimes referred to as the “negotiation dance”. For the previous situation, where there was a zone of agreement, the negotiation dance might proceed as depicted in Fig. I. System A proclaims that its operating costs are projected to increase by $0.25, so it believes that it should receive $0.70 of the joint fare. Initial negotiating positions often contain a certain amount of puffery. System A probably does not expect to get this amount, and System B likely guesses as much.
Table 2. Comparison of operating cost/trip and fares for the integration depicted in Fig. 4 Prior to the Integration
After the Integration
System A Operating Cost/Trip Fare/Trip
$1.00 $0.50
$1.25 ?
System B Operating Cost/Trip Fare/Trip
$1.00 $0.50
$0.65 ?
System B might respond in kind with an initial negotiating offer of $0.40 for its share of the joint fare. Note that each of the initial offers leaves the other System with less than its reservation price; hence, there will have to be concessions from these initial positions for an agreement to be reached. System A might counter with an offer of $0.685 as its share, and System B might respond with $0.38 as its second position. System A’s third offer of $0.675 as its share would leave System B with an amount equal to its reservation price. Thus, System B could accept this offer and conclude the negotiations. On the other hand, System B might believe it possible to gain further concessions from System A and respond with an offer of $0.36. Note that this offer satisfies the reservation prices of both parties. Eventually, the two systems might negotiate an agreement, whereby each would improve its revenue-to-operating cost ratio to 52.6% ($1 .OO/$ 1.90), with System A receiving $0.658 of the joint fare and System B receiving $0.342. Numerous other negotiated outcomes are, of course, possible. Raiffa [lo] discusses in detail compromise agreements reached through negotiation. Simply because there is a zone of agreement does not necessarily mean that an agreement will actually
TRANSFER
TRANSFER
_----
POST- INTEGRATION PRE-INTEGRATION
Fig. 4. Affected route in an integration where route changes are necessary.
Fare
115
decision making for integrated transit systems ZONE OF AGREEMENT I Uk I II
80,625 I SYSTEM A's RESERVATION PRICE
$0.325
I
I 1 SYSTEM B's RESERVATION PRICE
$l,OO--JOINT FARE THAT Is To BE ALLOCATED
Fig. 5. Distributive bargaining
with a
zone of agreement.
$0,50
I
k SYSTEM B's RESERVATION PRICE
$0,625
cl
I SYSTEM A's RESERVATION PRICE
$l,OO--JOINT FARE THAT Is To BE ALLOCATED
Fig. 6. Distributive bargaining with no zone of agreement.
,40
t 0
UII
>, /I
Bl
8,625
$,325
SYSTEMA's
SYSTEMB's
RESERVATICNPRICE RESERVATKM PRICE I I I I I ZONEOFASREWENT I I I I ,38 1 ,36
3; I
I
% I
I
I
I I I
$460
30 I
I
n-4
u.00
A3 4 Al ,675 ,685 ,70
I--c Ai DENOTESSYSTw A's iTH OFFER
AsTHEIR SHARE OF ME
JOINT FARE
Fig.
7. The
B, DENOTESSYSTW B's ITH OFFER
4
ASlHEIRStlWEoF ME JOINT FARE
negotiation dance.
be struck. For instance, poor negotiating skills, incorrect assessment of the opponent’s reservation price, etc., may cause the negotiations to be broken off without an agreement. On the other hand, a distributive bargaining situation without a zone of agreement can sometimes end in agreement if the domain of negotiation can be enlarged. As pointed out previously, revenue distribution for integrated transit systems involves many diverse issues. Hence, there are typically several issues subject to negotiation. One strategy is to divide the issues and negotiate them separately. While this allows the parties to focus on one issue at a time, if there is no zone of agreement on one of the issues, negotiations can stall and/or break off. Including another issue or issues in a distributive bargaining situation can sometimes resolve the no-zone-of-agreement dilemma. A negotiation involving multiple parties and multiple issues is referred to as integrative bargaining. An important facet of integrative bargaining is that it offers the potential for each party to obtain more than it would if the issues were negotiated separately. Thus, unlike distributive bargaining, where the parties are strict competitors for a fixed sum, the enlargement of the domain of negotiation may make it possible to enlarge what is going to be divided. Central to
integrative bargaining is that the parties value the issues differently; this allows the parties to make tradeoffs and concessions across issues. A full description of integrative bargaining as it relates to integrated transit negotiations is the subject of a future paper. Interested readers are referred to Raiffa [lo] for a lucid discussion on the general nature of integrative bargaining. 4. CONCLUSIONS This paper addresses a significant problem faced by many urban transit officials-namely, how to go about evaluating proposals to integrate the transit systems of a metropolitan area. American cities are notorious for their fragmented service; consequently, there are no good existing models on how this should be accomplished. Nevertheless, cities continue to struggle with this problem; Atlanta and Washington, D.C., have recently completed mass transit projects, and numerous other cities have either projects or studies underway. The process model we have presented is an effort to describe the information processing that takes place coincident with the setting of fares and division of revenues for an integrated transit system. The model was conceived based on interviews with transit
116
D. B. RINKS and F. C.
officials of major properties in several American cities. In this model, negotiation between the participants plays an important role in the process of constructing and assessing revenue allocation plans. In attempting to integrate local transit units into an area-wide transportation system, the perceived equity in the plan for disbursal of costs and revenues is anticipated to be a major factor in the issue of whether the integration takes place. The question one would expect each unit to ask itself is: “Given the expected level of service to my constituency, is this a fair division of costs and revenues?’ The revenueand cost-sharing plan should be such, for each unit, that the answer is yes. This research provides a framework within which transit officials can explore these issues. Acknowledgments-Support
for this research was provided by the Urban Mass Transportation Administration of the U.S. Department of Transportation under grant number UMTA-WV-l l-0003. C. M. Logar and G. Pitman participated in this study, and the contribution of G. Pitman in the interviewing of transit officials is acknowledged. REFERENCES 1. T. G. Cowing and A. G. Holtmann, The Economics of Local Public Service Consolidation. Lexington Books, Lexington, Massachusetts (1976).
SCHERR
P. D. Larkey, Process models of governmental resource
5. 6.
7.
8.
allocation and program evaluation. Policy Sci. 8, 269301 (1977). J. P. Crecine, A computer simulation model of municipal budgeting. Management Sci. 13, 786-8 15 (1967). J. P. Crecine, A simulation of municipal budgeting: the impact of problem environment. In Simulation in the Studv of Politics (Edited bv W. D. Coulin). Markham Publishing Co., Chicago (1968). _ J. P. Crecine, Governmental Problem Solving. Rand McNally, Chicago (1969). F. Thompson and R. Williams, A horse race around a Mobius strip: a review and a test of utility-maximizing and organizational process models of public expenditure decisions. Policy Sci. 11, 119-142 (1979). G. Downs and P. Larkey, Theorizing about public expenditure decision making: (as) if wishes were horses. . . . Policy Sci. 11, 143-156 (1979). W. G. Homburger and J. A. Desveaux, Conceptual Plan for Multi-Operator Transit Fares in the San Francisco Bay Area, Vol. III. Urban Mass Transportation Admin-
istration, Washington, D.C. (1982). 9. D. B. Rinks, F. C. Scherr, C. M. Logar and G. Pitman,
Revenue distribution methods for integrated transit systems: Part II-A general revenue sharing model based on ridership. Louisiana State University, Quantitative Business Analysis Working Paper 830027 (1983). 10. H. Raiffa, The Art and Science of Negotiation. Harvard University Press, Cambridge, Massachusetts (1982).
Socio-Econ. Plan. Sci. Vol. 19, No. 2, pp. 109-116, 1985 Printed in the U.S.A.
A PROCESS MODEL OF FARE DECISION MAKING FOR INTEGRATED TRANSIT SYSTEMS
DAN B. RINKS Quantitative Business Analysis, Louisiana State University, Baton Rouge, LA 70803, U.S.A. and FREDRICK C. SCHERR College of Business and Economics, West Virginia University, Morgantown, WV 26505, U.S.A. (Received 5 July 1984) Abstract-Based on interviews with transit officials in several American cities, a process model for evaluating proposals to integrate local transit properties into an area-wide transportation system is developed. Relevant factors for a data base are discussed and the information processing flows are described. In this model, negotiation between the participants plays an important role in constructing and assessing revenue allocation plans.
1. INTRODUCTION Many metropolitan areas are served by several different transit properties. Some of these properties are privately owned, while others are governmental or quasi-governmental institutions. In some cases, the transit properties have exclusive rights to certain territories, while other properties serve overlapping territories. These complicated transit structures are often the result of arbitrary political boundaries or historical accident. The transit patron has little concern for the political, physical, and legal constraints under which the various properties in his/her area operate. His or her goal is to move from one point to another with a minimum of time, cost and inconvenience. The end result of this is, often, the choice of private transportation as the transport mode most consistent with these goals. One way to reduce inconvenience, and thus increase ridership, is to integrate the transit systems of a metropolitan area. Routes can be co-planned for easier rider transfers. Cost savings may result from the elimination of redundant routes. Integrated fare collection may be easier for riders and save the systems collection expense as well. Several issues are involved in fare setting and collection for integrated systems. 1. The methodology for pricing transit trips for an integrated system 2. The fare collection methodology for an integrated system 3. The revenue allocation among participants in an integrated system. For properties considering integrating their systems, the sequence of the decision analysis might logically follow the order listed above. First, the decisions on a joint pricing structure are made. Next, the methods for collection the fare are determined. Finally, the participants decide on how to allocate the joint revenues. However, it should be noted that these three important decisions are likely to be coincident because the decisions may interact with one another. For instance, fare collection technology may prohibit
certain pricing structures, such as an exact distancebased fare for a bus system. Furthermore, due to its importance to the well-being of the participants, an agreement in principle on revenue sharing is probably necessary before final determinations are made on fare structure and fare collection. Indeed, we view the revenue distribution plan as a primary factor in whether or not autonomous units will agree to be participants in an integrated system. As pointed out by Cowing and Holtmann [I], the combination of governmental service units can benefit one part of the combination while being a disadvantage to others, even if the combination as a whole benefits the public. For example, a city and a county bus system might combine and form a new routing plan which better serves the public, but this plan’s cost might bear more heavily on one unit without a totally compensating revenue increase. Another possibility is that this new routing system might greatly aid one unit’s consuming public while causing minor losses to the other unit’s public. In cases such as this, there is an incentive on an overall basis to combine, even though the combination is disadvantageous to one of the combining units. Cowing and Holtmann suggest that “enticement strategies”, which involve subsidization of those units made worse off by those units made better off, can be used to affect the combination. We view the division of revenues and costs among units as a way to implement such an enticement strategy. In other words, the plan for the distribution of revenues and costs for the combined system, made before the combination, should be such that there is an incentive for the combination to take place, for the coalition of units to form an integrated system. The remainder of this paper describes a process model for integrated transit decision making that was constructed based on interviews with transit officials of the major transit property in Baltimore, Md., Washington, D.C., Pittsburg, Pa., and Atlanta, Ga., and from officials of six properties in the San Francisco, Cal., Bay Area. 109
110
D.B. RINKS and F.C. SCHERR 2. PROCESS MODELS FOR FARE INTEGRATION DECISIONS
The model presented in this paper is an effort to describe the information processing that takes place coincident with the setting of fares and division of revenues for an integrated transit system. Because it attempts to simulate the structural form and behavior of the decision process, the model has been labeled a process model in the sense of Larkey [2]. Similar models, referred to as positive models, have been developed by Crecine [3, 4, 51. Thompson and Willilams [6] state that the typical justification for this type of model is “that the public expenditure process deviates too much from the idealized conditions assumed by the utility-maximizing model and that the complexity of the process renders the utilitymaximizing model useless to explain public expenditure and tax decisions”. The pros and cons of using utility-maximizing models versus process models for public decision making is discussed in [6]; Downs and Larkey [7], commenting on [6], also discuss the issue. Since our primary purpose is to describe the process and not prescribe solutions, this approach has been deemed appropriate. An important benefit of constructing a descriptive model is that seemingly “obvious” relationships are “discovered”. The relationships are obvious in retrospect, but rarely are so a priori. What happens, of course, is that by asking such questions as “What data are required?’ and “How is it used?’ the information processing patterns emerge. In the subsection “Critical Dimensions . . .” the data base for integrated transit system decision making is described. After this, the subsection entitled “The Process Model” outlines the information-processing patterns that emerged.
been a major problem in developing fare integration among the San Francisco systems [S].
Frequency of trip length. This refers to the diversity of length of trips taken. If trips are principally of one length (as where a transit system serves mostly the central business district from a series of similardistance suburbs), a flat-fare system, perhaps with tokens, should be strongly considered, regardless of the philosophy of the systems involved regarding distance-based fares. If the trip distribution is broad, the systems’ philosophies with regard to fares come into play, as do simplicity and in-place collection technology effects. Operating philosophy and practices of properties. While there are some very strong commonalities in current philosophy and practice, it is equally obvious that a great deal of diversity also exists. This presents significant problems in negotiating price structures for integrated routes, in deciding on appropriate collection technologies and in dividing revenues. For example, if one property uses and believes in flat fares and tokens, while another uses and believes in distance-based fares and strip cards, there is a basic philosophical problem which will cause significant disagreement.
2.1 Critical dimensions in developing an integrated fare structure
Proportion of integrated trips taken. This dimension refers to the proportion of the properties’ routes to be affected. Obviously, if the potential market for integrated trips is a very small proportion of total ridership, the degree of change in fare policy and collection technology that can be cost justified is proportionately small. In such cases, integrated fares and collection systems must be compatible with existing fare structures and collection systems; a simple transfer system suggests itself. On the other hand, if the potential or actual market for integrated trips is large, then a revamping of significant parts of fare collection structures may be justified, and compatibility deemphasized.
A preliminary review of the critical factors we believe to be relevant in developing an integrated system is appropriate at this point. These important parameters of the system to be integrated are the primary influences on the types of fare structures, collection methodologies and allocation methods that can be feasibly used for integrated routes. Modes involved. The modes involved (bus, trolley, limited-access light rail, etc.) affect primarily the collection methodology of an integrated fare system, rather than fare structure or revenue allocation. If the integrated system involves a limited-access light rail system in conjunction with bus feeders, collection technology must be compatible with practical collection methods on these two modes, but the modality has little effect on revenue allocation and only a moderate effect on fare structure policy. Existing/new systems hardware. Some integrations will involve the use of existing hardware only, while in other cases some new hardware will be purchased. Where existing hardware is predominant, the fare structure and collection methods must be compatible with it. Distance-based fares, for example, are easier to administer if one has a strip-reader system. Revenue division is little affected by this dimension except as the dimension affects fare structure and revenue produced. The prevalence of existing hardware has
This dimension refers to the public or private ownership of the transit properties and the structure of any subsidies involved. With regard to fare structure, privately held units will be more likely than public units to be much more concerned with revenue effects than with other aspects of transit price structure (simplicity, equity, etc.). The major effect here, however, is on revenue division. Privately held units will likely veto any revenue division formula that does not hold constant or increase their profitability, regardless of ridership increases, on integrated routes. Public units are likely to be more flexible in this regard, being willing to bear some revenue reductions and/or cost increase if ridership is significantly enhanced. Also, the effects of integration and ridership on subsidies occur on this dimension. If subsidies are based partially on ridership, or if there is agency funding available for integration projects (as through the MTA in San Francisco), these costs and revenue effects will certainly influence fare structure, the amount that can be spent on new collection technology, and the bargaining involved in the revenue allocation plan. Size of the system. This refers to population served and potential ridership; the primary effect is on
Organizational structures of integrated properties.
Fare decision making for integrated transit systems
collection technology. There are returns to scale in the development of many collection technologies, and they can be practically used only in conjunction with larger systems. The population factor, along with the proportion of integrated trips taken, determines ridership for integrated trips. 2.2 The process modes A process model for fare integration is presented in Fig. 1. The model is composed of two parts. Within the dashed portion is the part of the model referring to the development of the revenue allocation formula. This is appropriate only if there is a revenue allocation problem, i.e. if the integration is going to be by formal agreement and not a combination by fiat. In the discussion that follows, we shall assume that a formal agreement (federation) is to be employed, as this utilizes the model in full. The model starts when properties decide to evaluate the possibilities of route integration. The “TRIAL” boxes refer to the generation of one or more plans and forecasts on a particular topic. Discussion proceeds by box number (see Fig 1). q Data collection. At this point data is collected on potential integration routes and schemes. To a great extent, this data is information on the critical dimensions discussed earlier. Data on these dimensions serve to parameterize the problem, eliminate obviously infeasible solutions and suggest the tradeoffs involved in various potential integration schemes. Data which are obviously necessary are: (a) Possible route rationalizations and integrations (b) Current ridership along routes that are candidates for integration
111
(c) Elasticity data for area and routes, by modes, groups, time of day, etc. (d) Cost data for potential rationalizations/integrations (e) Collection hardware in place. On-board studies, studies of past price changes, cost analysis, etc., are useful here. q Trial: route ratio~a~i~at~~~/i~tegration and rescheduling. Here, a series of plans for route rationalizations are developed. We do not review the mathematical methodology here, which is a part of Transportation Economics. Plans will involve various route changes and costs. Thus, this plan or plans has a major impact on the properties’ cost picture (Box q) via the dollars saved or spent in the integration process. The major flow of the process at this point, however, is to Box q, estimates of the effect on ridership. q Ridership estimates. Here the ridership effects of the rationalization/integration plans are evaluated. The effects are, of course, heavily influenced by the price structure on the integrated routes (Box a). •I Trial: pricing structure. Here several price structures are developed for the various route rationalization/integration plans and their ridership effects evaluated via the elasticities. Revenue, equity and simplicity factors are included in the fo~ulation and evaluation of the various plans. •I Trial: fare collection methodology. At this point, several fare collection methodologies are discussed for the various transit plans. There is a great deal of co-determination involved here with the fare structure plan; different fare structures require different collection systems.
Fig. 1. A process model of fare integration.
112
D.B. RINKS andF.C. SCHERR
q Are the various price and coifection structures feasible? At this point, a great number of price/ collection alternatives will be discarded based on the relevant dimensions: the expected ridership is too small to justify their cost; they entail too much modification of the present system for the ridership involved, etc. Discarded plans are sent back to price and collection planners for review/revision. Remaining feasible rou~efp~cefcoll~on plans go to the “Revenue Allocation Fo~ulation” section of the model. In this section of the model, various system philosophies and revenue-cost-profit data are used to develop one or more revenue allocation plan(s). q Trial: joint revenues. Here the joint revenues from the integrated/rationalized routes are calculated for the various route/price/collection plans. q Consideration of potential variables in revenue division formula. These are philosophical considerations: What “should” be included in such a formula? Some formulas are quite complex, others simple. The attitudes of properties come into play here (see Rinks et al [9]). q Relevant ridership data. This is one of the pragmatic considerations in a revenue allocation plan. Whose ridership will change? How? What will the effect on subsidies be? The major input here is the ridership estimates from Box a. @ Relevant costs. Changes in costs for the various systems will depend on collection methodologies employed and route shifts/eliminations/len~henin~. If changes in total revenue are small (which would be the case if relatively few new rides were generated and the fare structure remained relatively unchanged), cost changes may be the major effect of any integration. q Relevant profitability data. A separate box has been included to account for the explicit profitability concern of private systems. Revenue and cost data also reach the allocation box via Boxes q and 8. Any changes in ridership-based subsidies are also included in Box q I. @i Trial: revenue a~l~at~on plan. Here an allocation plan is formulated via the considerations encompassed in Boxes q through a. Methods for the formulation for such a plan are found in [9]. There may be several such plans if several route/price/collection alternatives have survived to this point. q Plan for the division of joint fares. The major output of a revenue allocation plan is the division of fafes on the integrated/rationalized routes. We assume here that the properties continue to run their various modes. Any joint int~ation costs (for example, a stalf to administer the division of revenues according to the allocation formula) are likely to be very small in comparison to the operating cost. q Plan for division of integration subsidies. This deals with where these occur. @I Evaluation of revenue allocation plan(s). Here a formal evaluation of the various route/price/collection/allocation plans is made by the staffs and managers of the properties. Plans may be rejected and sent back to various planning stages for reconsideration. If a plan or plans are acceptable, the political process starts (submission to transit boards, public hearings, etc.). While this is not included in the
model, it may also involve feedback to various planning stages. •I Evaluation criteria. These are the factors by which route/price/collection/allocation plans are judged by the properties. Several are suggested in the box; they may vary from property to property. 3. THE ROLE OF NEGOTIATION IN THE PROCESS
Although this is not usually stated explicitly: negotiation between the participants in the proposed integrated system plays an important role in the process of constructing and assessing revenue allocation plans. First, the variables that directly and indirectly have an effect on revenue sharing and cost allocation for participants in an integrated transportation are identified and described. This step clarifies those issues which are important and must receive major attention in the construction of a formula. Next, as trial formulas are put forward, each party must try to assess how the allocation formula will affect its operation. Finally, through negotiations, an agreement that is acceptable to all the participants is reached, or failing this, the area-wide transportation system will remain unintegrated with respect to joint fares. Figure 2 depicts the flow of activities in the model just described. Of vital concern to each carrier is how the integration is going to affect its revenue and operating costs. In our approach to revenue allocation, control of its costs is the ~s~nsibility of each carrier (see Rinks et al. 193). Operating costs may, however, be an appropriate issue for discussion in the negotiation process. Consider a very simple integration situation. The only significant change caused by the integration is that the originating carrier collects the entire joint fare (fA), remits the appropriate amount (fAB),to the second carrier, and retains the portion of the joint fare that is due it (fAA). As Fig. 3 and Table I illustrate, there were no route changes from the int~tion that would have caused a change in the operating costs. Also, it is assumed that the fare collection and transfer costs are not materially affected by the integration in this example. A straightforward joint-fare policy, in this case, would be to leave the price charged to the rider unchanged. Thus, a rider boarding at point a would pay $1.00 for the trip to point b, which would involve a transfer at point x. System A would remit $0.50 to System B for their share of the revenue generated by the system trip. fA = f*B + fM $1.00 = $0.50 + $0.50 A somewhat more complicated situation occurs when the integration causes significant changes in operating costs and/or ridership patterns. Integration of transit systems offers the potential for operating improvements such as rerouting and route rationalizationf~onsohdation. Figure 4 depicts a circ~stan~e where both System A and System B have made changes to their route structures to accommodate the integration; Table 2 shows how these changes alIect the operating costs. Although the changes result in
Fare decision making for inte~at~
113
transit systems
STEP 1 IDENTIFICATlON OF CRITICALVARIABLES COSTS RIDERSHIP PROFITS (LOSSES) EASE OF ~MINIsTRATIoN POLITICAL CONSIDERATIONS
STEP 2
FORMULATION OF TRIAL REVENUEALLOCATION PLANS
4
,
NO
OF TRIALFORMA:
REVENUE ALLOCATION PLAN ACCEPTED IN PRINCIPLE
INTEGRATION OF AREA WIDE TRANSPORTATION NOT POSSIBLE WITH RESPECT TO JOINTFARES
Fig. 2. The role of negotiation in revenue allocation.
overall efficiency (total post-integration operating costs per trip are $1.90, versus $2.00 for pre-integration),
the net change is quite different for the two systems. System A’s operating costs have increased by $0.25 per trip, whereas the operating costs of System B have decreased by $0.35 per trip. What is the proper price for the system trip, and how much should each system receive as its share? For the sake of simplicity, assume that the system trip is priced at $1.00. Now, the state of affairs is one of distributive bargaining. That is, the two participants are trying to decide how to divide the fixed joint fare;
SYSTEM A
TRIP
what one gains by negotiation, the other one loses. One must define the reservation prices as the least favorable amounts the Systems are willing to accept for their share of the joint fare, or else they will not participate in the integrated transit system. For instance, System A and System B might establish their reservation prices so that their revenue-t~o~rating cost ratio after the integration is no worse than it was prior to the integration. This guideline implies that System A should receive at least $0.625 from the joint fare and System B should receive at least $0.325. Since the reservation prices for both particiSYSTEM B
TRANSFER
ORIGIN
TRIP DESTINATION
-----
POST-INTEGRATION PRE-INTEGRATION
Fig. 3. Typical route in an integration where route changes are unnecessaq.
114
D. B. RINKS and F. C. SCHERR
Table 1. Comparison of operating cost/trip and fares for the integration depicted in Fig. 3 Prior to the Integration
After the Integration
System A Operating Cost/Trip Fare/Trip
$1.00 $0.50
$1.00 $0.50
System B Operating Cost/Trip Fare/Trip
$1.00 $0.50
$1.00 $0.50
pants can be satisfied, there is a zone of agreement (Fig. 5). Other reservation prices are, of course, possible. For example, System B might believe that the projected cost savings to its system will not materialize and, correspondingly, set its reservation price at $0.50 to maintain its revenue-to-cost ratio at 50%. If System A’s reservation price is $0.625, then an agreement is not possible, as there is no zone of agreement (Fig. 6). Naturally, each participant desires to do better than its reservation price. For situations where there is a zone of agreement, the final negotiated settlement will depend on many factors, not the least of which are the respective bargaining skills. The sequence of offers by one side and counter-offers by the other is sometimes referred to as the “negotiation dance”. For the previous situation, where there was a zone of agreement, the negotiation dance might proceed as depicted in Fig. I. System A proclaims that its operating costs are projected to increase by $0.25, so it believes that it should receive $0.70 of the joint fare. Initial negotiating positions often contain a certain amount of puffery. System A probably does not expect to get this amount, and System B likely guesses as much.
Table 2. Comparison of operating cost/trip and fares for the integration depicted in Fig. 4 Prior to the Integration
After the Integration
System A Operating Cost/Trip Fare/Trip
$1.00 $0.50
$1.25 ?
System B Operating Cost/Trip Fare/Trip
$1.00 $0.50
$0.65 ?
System B might respond in kind with an initial negotiating offer of $0.40 for its share of the joint fare. Note that each of the initial offers leaves the other System with less than its reservation price; hence, there will have to be concessions from these initial positions for an agreement to be reached. System A might counter with an offer of $0.685 as its share, and System B might respond with $0.38 as its second position. System A’s third offer of $0.675 as its share would leave System B with an amount equal to its reservation price. Thus, System B could accept this offer and conclude the negotiations. On the other hand, System B might believe it possible to gain further concessions from System A and respond with an offer of $0.36. Note that this offer satisfies the reservation prices of both parties. Eventually, the two systems might negotiate an agreement, whereby each would improve its revenue-to-operating cost ratio to 52.6% ($1 .OO/$ 1.90), with System A receiving $0.658 of the joint fare and System B receiving $0.342. Numerous other negotiated outcomes are, of course, possible. Raiffa [lo] discusses in detail compromise agreements reached through negotiation. Simply because there is a zone of agreement does not necessarily mean that an agreement will actually
TRANSFER
TRANSFER
_----
POST- INTEGRATION PRE-INTEGRATION
Fig. 4. Affected route in an integration where route changes are necessary.
Fare
115
decision making for integrated transit systems ZONE OF AGREEMENT I Uk I II
80,625 I SYSTEM A's RESERVATION PRICE
$0.325
I
I 1 SYSTEM B's RESERVATION PRICE
$l,OO--JOINT FARE THAT Is To BE ALLOCATED
Fig. 5. Distributive bargaining
with a
zone of agreement.
$0,50
I
k SYSTEM B's RESERVATION PRICE
$0,625
cl
I SYSTEM A's RESERVATION PRICE
$l,OO--JOINT FARE THAT Is To BE ALLOCATED
Fig. 6. Distributive bargaining with no zone of agreement.
,40
t 0
UII
>, /I
Bl
8,625
$,325
SYSTEMA's
SYSTEMB's
RESERVATICNPRICE RESERVATKM PRICE I I I I I ZONEOFASREWENT I I I I ,38 1 ,36
3; I
I
% I
I
I
I I I
$460
30 I
I
n-4
u.00
A3 4 Al ,675 ,685 ,70
I--c Ai DENOTESSYSTw A's iTH OFFER
AsTHEIR SHARE OF ME
JOINT FARE
Fig.
7. The
B, DENOTESSYSTW B's ITH OFFER
4
ASlHEIRStlWEoF ME JOINT FARE
negotiation dance.
be struck. For instance, poor negotiating skills, incorrect assessment of the opponent’s reservation price, etc., may cause the negotiations to be broken off without an agreement. On the other hand, a distributive bargaining situation without a zone of agreement can sometimes end in agreement if the domain of negotiation can be enlarged. As pointed out previously, revenue distribution for integrated transit systems involves many diverse issues. Hence, there are typically several issues subject to negotiation. One strategy is to divide the issues and negotiate them separately. While this allows the parties to focus on one issue at a time, if there is no zone of agreement on one of the issues, negotiations can stall and/or break off. Including another issue or issues in a distributive bargaining situation can sometimes resolve the no-zone-of-agreement dilemma. A negotiation involving multiple parties and multiple issues is referred to as integrative bargaining. An important facet of integrative bargaining is that it offers the potential for each party to obtain more than it would if the issues were negotiated separately. Thus, unlike distributive bargaining, where the parties are strict competitors for a fixed sum, the enlargement of the domain of negotiation may make it possible to enlarge what is going to be divided. Central to
integrative bargaining is that the parties value the issues differently; this allows the parties to make tradeoffs and concessions across issues. A full description of integrative bargaining as it relates to integrated transit negotiations is the subject of a future paper. Interested readers are referred to Raiffa [lo] for a lucid discussion on the general nature of integrative bargaining. 4. CONCLUSIONS This paper addresses a significant problem faced by many urban transit officials-namely, how to go about evaluating proposals to integrate the transit systems of a metropolitan area. American cities are notorious for their fragmented service; consequently, there are no good existing models on how this should be accomplished. Nevertheless, cities continue to struggle with this problem; Atlanta and Washington, D.C., have recently completed mass transit projects, and numerous other cities have either projects or studies underway. The process model we have presented is an effort to describe the information processing that takes place coincident with the setting of fares and division of revenues for an integrated transit system. The model was conceived based on interviews with transit
116
D. B. RINKS and F. C.
officials of major properties in several American cities. In this model, negotiation between the participants plays an important role in the process of constructing and assessing revenue allocation plans. In attempting to integrate local transit units into an area-wide transportation system, the perceived equity in the plan for disbursal of costs and revenues is anticipated to be a major factor in the issue of whether the integration takes place. The question one would expect each unit to ask itself is: “Given the expected level of service to my constituency, is this a fair division of costs and revenues?’ The revenueand cost-sharing plan should be such, for each unit, that the answer is yes. This research provides a framework within which transit officials can explore these issues. Acknowledgments-Support
for this research was provided by the Urban Mass Transportation Administration of the U.S. Department of Transportation under grant number UMTA-WV-l l-0003. C. M. Logar and G. Pitman participated in this study, and the contribution of G. Pitman in the interviewing of transit officials is acknowledged. REFERENCES 1. T. G. Cowing and A. G. Holtmann, The Economics of Local Public Service Consolidation. Lexington Books, Lexington, Massachusetts (1976).
SCHERR
P. D. Larkey, Process models of governmental resource
5. 6.
7.
8.
allocation and program evaluation. Policy Sci. 8, 269301 (1977). J. P. Crecine, A computer simulation model of municipal budgeting. Management Sci. 13, 786-8 15 (1967). J. P. Crecine, A simulation of municipal budgeting: the impact of problem environment. In Simulation in the Studv of Politics (Edited bv W. D. Coulin). Markham Publishing Co., Chicago (1968). _ J. P. Crecine, Governmental Problem Solving. Rand McNally, Chicago (1969). F. Thompson and R. Williams, A horse race around a Mobius strip: a review and a test of utility-maximizing and organizational process models of public expenditure decisions. Policy Sci. 11, 119-142 (1979). G. Downs and P. Larkey, Theorizing about public expenditure decision making: (as) if wishes were horses. . . . Policy Sci. 11, 143-156 (1979). W. G. Homburger and J. A. Desveaux, Conceptual Plan for Multi-Operator Transit Fares in the San Francisco Bay Area, Vol. III. Urban Mass Transportation Admin-
istration, Washington, D.C. (1982). 9. D. B. Rinks, F. C. Scherr, C. M. Logar and G. Pitman,
Revenue distribution methods for integrated transit systems: Part II-A general revenue sharing model based on ridership. Louisiana State University, Quantitative Business Analysis Working Paper 830027 (1983). 10. H. Raiffa, The Art and Science of Negotiation. Harvard University Press, Cambridge, Massachusetts (1982).