Bus operators in competition: a directed location approach
BBUS OPERATORS IN COMPETITION: A DIRECTED LOCATIONAPPROACH
FERNANDA HERRERA † AND SERGIO I. L ´OPEZ*
Abstract.
We present a directed variant of Salop (1979) model to analyze bus transport dynamics.The players are operators competing in cooperative and non-cooperative games. Utility, like inmost bus concession schemes in emerging countries, is proportional to the total fare collection.Competition for picking up passengers leads to well documented and dangerous driving practicesthat cause road accidents, traffic congestion and pollution. We obtain theoretical results thatsupport the existence and implementation of such practices, and give a qualitative description ofhow they come to occur. In addition, our results allow to compare the current or base transportsystem with a more cooperative one. Introduction
In this work, we model the competition of bus operators for passengers in a public transportconcession scheme. The models -which are directed variants of the Salop model [18], in turn a circuitadaptation of the classic Hotelling model [13]- are a characterization of Mexico City’s transportsystem, where 74.1% of the trips made by public transport are carried out on buses with concessioncontracts [15].Much like in other Latin American cities, the contracts laying the responsibilities, penalties andservice areas are rarely enforced by the corresponding authorities, and in these instances, the maindriver determining the planning and operations tend to be the operator’s profit margins [12] p. 9.Leaving such task to the companies or in some cases individuals, has lead to what [10] refer toas curious old practices : driving habits adopted by bus operators, whose salary is proportional tothe fare collection, to maximize the number of users boarding the unit. While these practices wereobserved and recorded in the United Kingdom in the 1920s, they are very much present in currenttimes, particularly in cities with emerging economies and suboptimal concession plans. We mentionthe practices enlisted by [10] pertaining to driving:(1)
Hanging back or Crowling . Buses go slowly to pick up as much people possible that wouldotherwise take the bus behind. A variant is to stop altogether until the bus is fully loaded,or the bus behind appears in the horizon.(2)
Racing . A driver decides there are too few passengers in a station to make a stop, renderingit more profitable to continue and collect passengers ahead.(3)
Overtaking, Tailing or Chasing . Attempting to pass the bus ahead, trying to cut in andpick up the users ahead.(4)
Turning . When a bus is empty or nearly empty it turns around before the end of the route,and drives back in the opposite direction.
Date : January 5, 2021.2020
Mathematics Subject Classification.
C710, C720, C730, R410.
Key words and phrases.
Transport, Bus, Location games, Nash equilibrium, Mixed Strategies.Sergio I. L´opez was partially funded by Conacyt-SNI 215989 grant. a r X i v : . [ ec on . T H ] J a n FERNANDA HERRERA AND SERGIO I. L ´OPEZ
Many of these practices have negative consequences on the perception of public transport, and inparticular, that with concession contracts. The 2019 Survey on victimization in public transport [5]carried out in Mexico City and its metropolitan area, showed that 50% of the interviewees perceivedthe quality of concession transport to be Bad, and 15% to be Very Bad, while 27% labeled travelingin concessioned transport Somewhat Dangerous, and 60% labeled it Very Dangerous. In bothdimensions, public transport concession schemes did worse than any other public or private form oftransport. The matter is pressing enough that the current administration of Mexico City stressed inits Strategic Mobility Plan of 2019 [19] p. 9:
The business model that governs this (transport) sector,in which profits are individual and exclusively per transported passenger, produces competition inthe streets for users, which results in the pick up and drop off of passengers in unauthorized places,increased congestion and a large number of traffic incidents each year.
To reduce public expenditure on transport, Margaret Thatcher introduced the Transport Act1985 [1], a privatization scheme that demanded companies to keep its vehicles in good condition,avoid dangerous driving, establish service routes, and publish timetables, in addition to forcingdrivers to adhere to routes and timetables through regulatory mechanisms established by the com-panies. This type of regulation and its implementation appear distant for Mexico, so with this workwe aim to shed new light on the implications of a transport system where operators compete forpassengers without much regulation. We obtain a simple interpretation of the results, consistentwith the driving practices mentioned above.To the best of our knowledge, our approach where bus operators are independent players compet-ing to maximize their utility, which is proportional to the potential advantage of driving at differentspeeds is novel, and it allows us to model a variety of scenarios and obtain explicit descriptionsof their equilibria. Furthermore, we are able to explore the time-evolution of the adopted strate-gies. All the results are expressed in terms of the behavior of the operators, and not in terms ofglobal economic variables. Given the tractability of our models, some natural theoretical questionsemerge.Relevant literature on transport problems includes [17] modeling of the optimal headway busservice from the point of view of a central dispatcher. In the historical context of Transport Act1985 [1], several scientific articles analyzed the effect of the privatization. Under the assumptionof the existence of an economic equilibrium in the competition system, [10] classify the drivingpractices into two categories: those consistent with the equilibrium, and those who are not. Theyanalyzed the expected timetables in the deregulated scenario. In [9] is presented a comparativeanalysis of fare and timetable allocation in competition, monopoly and net benefit maximization(both restricted and unrestricted to a zero profit) models. Building on from this, [16] introduces theconsumer’s perspective and obtains the equilibria prices and number of services offered by transportcompanies. The possibility of predatory behavior between two enterprises competing through faresand service level, is analyzed by [7], using the data from the city of Inverness. In [8] the authorsstudy the optimal policies of competing enterprises in terms of fares, and the bus service headway,in a unique bus stop and destination scenario. They also introduce the concept of demand coor-dination which can be implemented through timetables. Assuming a spatial directed model witha single enterprise, [6] finds the timetable that minimizes the costs associated to service delays.The work of [4] analyzes flight time data and finds empirical evidence to support Hotelling models.From a non-economic perspective, [3] models competing buses in a circuit behaving like randomparticles with repulsion between them (meaning they could not pass each other). A contemporaryreview on transport market models using game theory is given by [2], and a general review ofcontrol problems which arise in buses transport systems is presented in [14].
US OPERATORS IN COMPETITION: A DIRECTED LOCATION APPROACH 3 The model
The assumptions of the game are the following. There are n ≤ n driversalong a route. There is only one type of bus and one type of driver, meaning that the buseshave identical features, and that the drivers are homogeneous in terms of skill and other relevantcharacteristics.The speed of a bus v , is bounded throughout every time and place of the road by:(2.1) 0 < v min ≤ v ≤ v max , where the constants v min and v max are fixed, and determined by exogenous factors like the conditionof the bus, Federal and State laws and regulations, the infrastructure of the road, etc.Drivers can pick up passengers along any point on the route at any given time. In other words,there are no designated bus stations, nor interval-based time schedules in place. This scenario isan approximation to a route with a large number of homogeneously distributed bus stops.We allow for infinite bus capacity, so drivers can pick up any number of passengers they comeacross. Alternatively, one can assume that passengers alight from the bus almost right after board-ing it, so the bus is virtually empty and ready to pick up users at any given time. The importantpoint to note is that passengers that have boarded a bus will not hop on the next, either becausethey never descended it in the first place, or because they already reached their final destination ifthey did.Bus users reach their pick up point at random times, so demand for transport is proportional tothe time elapsed between bus arrivals. Let λ > p ≥ c ≥ { v ≥ v min ≤ v ≤ v max } , where v min and v max are given in (2.1). We define a mixed strategy to be a random variable takingvalues in the space Γ.In what follows we define the expected utility of drivers given a set of assumptions on the numberof players and their starting positions, the fixed variables of the models, and route characteristics.Relevant notation and concepts are introduced when deemed necessary.2.1. Single player games.
We first consider a game with only one driver picking up passengersalong the road. Importantly, the fact that only one bus is covering the route implies that commutershave no option but to wait for its arrival, the player is aware of this. • Fixed-distance game
A single bus departs the origin of a route of length D . We adopt the convention thatthe initial time is whenever the bus departs the origin. We define the expected utility ofdriving at a given speed v to be(2.3) u ( v ) := ( pλ ) T − cT, where T = Dv is the time needed to travel the distance D at speed v . FERNANDA HERRERA AND SERGIO I. L ´OPEZ
Note that since there is no other bus picking up passengers, the expected number ofpeople waiting for the bus in a fixed interval of the road increases proportionally with time.From this, one infers that the expected total number of passengers taking the bus is pro-portional to the time it takes the bus to reach its final destination. The conclusion and itsimplication can be expressed rigorously using a space-time Poisson process, see for example[11] pp. 283-296. • Fixed-time game
Suppose now that the driver chooses a constant speed v satisfying (2.1) in order to drivefor T units of time. The bus then travels the distance D = T v , which clearly depends on v . We define the expected utility of driving at a given speed v to be(2.4) u ( v ) := ( pλ ) D − cT. The underlying assumption is that for sufficiently small T , there are virtually no new arrivalsof commuters to the route, so effectively, the number of people queuing for the bus remainsthe same as that of the previous instant. The requirement is that T is small compared tothe expected interarrival times of commuters.It follows that the total amount of money collected by the driver is proportional to thetotal distance traveled by the bus.2.2. Two player games.
There are two buses picking up passengers along a route, which weassume is a one-way traffic circuit. An advantageous feature of circuits is that buses that returnfrom any point on the route to the initial stop may remain in service; this is generally not thecase in other types of routes. In particular, we assume that the circuit is a one-dimensional torusof length D . For illustration purposes and without loss of generality, from now on we require thedirection of traffic to be clockwise.We define the D -module of any real number r as( r ) mod D := rD − (cid:106) rD (cid:107) , where (cid:98) z (cid:99) is the greatest integer less than or equal to z .The interpretation of ( r ) mod D is the following: if starting from the origin, a bus travels the totaldistance r , then ( r ) mod D denotes its relative position on the torus. Indeed, r may be such thatthe bus loops around the circuit many times, nonetheless ( r ) mod D is in [0 , D ) for all r . We referto r as the absolute position of the bus, and to ( r ) mod D as the relative position (with respect tothe torus). Note that the origin and the end of the route share the same relative position, since(0) mod D = 0 = ( D ) mod D .Let x and y denote the two players of the game, and let x, y be their respective relative positions.The directed distance function d x is given by d x ( x, y ) := (cid:26) y − x if x ≤ y,D + y − x if x > y. (2.5)Equation (2.5) has a key geometrical interpretation: it gives the distance from x to y consideringthat traffic is one-way. The interest of this is that the potential amount of commuters x picks upis proportional to the distance between x and y , namely d x ( x, y ). See Figure 1.A straightforward observation is that for any real number r , we have(2.6) d x (( x + r ) mod D , ( y + r ) mod D ) = d x ( x, y ) . This justifies the first summand in (2.3).
US OPERATORS IN COMPETITION: A DIRECTED LOCATION APPROACH 5
This asserts that if we shift the relative position of the two players by r units (either clockwise orcounterclockwise, depending on the sign of r ), then the directed distance d x is unchanged.One can define the directed distance d y analogously, d y ( x, y ) := (cid:26) x − y if y ≤ x,D + x − y if y > x. By definition, there is an intrinsic symmetry between d x and d y : we have d x ( x, y ) = d y ( y, x ) and d y ( x, y ) = d x ( y, x ). Roughly speaking, this means that if one were to swap all the labels, namely x to y , x to y , and vice versa, then it suffices to plug the new labels into the previous definitionsto obtain the directed distances.Another immediate observation is that for any pair of different positions ( x, y ), the sum of thetwo directed distances gives the total length of the circuit,(2.7) d x ( x, y ) + d y ( x, y ) = D. This is portrayed in Figure 1.
Figure 1.
Directed distancesLet us assume that players x and y have starting positions x and y in [0 , D ). The initialminimal distance is defined to be(2.8) d := min { d x ( x , y ) , d y ( x , y ) } . Now suppose that starting from x and y , operators drive at the respective speeds v x and v y ,with v x , v y in Γ, for T units of time. Their final relative positions are then x T = ( x + T v x ) mod D and y T = ( y + T v y ) mod D . Let us orient the maximum displacement of buses by requiring
T v max , with v max given in (2.1),to be small compared to D . The reason for this is to be consistent with our assumption of constantspeed strategies, since they are short-term. More precisely, we require(2.9) T v max < D . Lastly, we define the escape distance by(2.10) d := T ( v max − v min ) . Importantly, this switches the relative positions of the players.
FERNANDA HERRERA AND SERGIO I. L ´OPEZ
This gives a threshold such that if the distance between the players is shorter than d , then thebuses can catch up to the other, given an appropriate pair of strategies. If the distance is greaterthan d , this cannot occur.We now proceed to define the expected utility of players given the type of game being played,namely, whether it is cooperative or non-cooperative. • Non-cooperative game
We define the utility of x given the initial positions of players x and y , and the strategies v x and v y , to be(2.11) u x ( x , v x , y , v y ) := (cid:26) pλ d x ( x T , y T ) − cT if x T (cid:54) = y T ,pλ D − cT if x T = y T . The definition above includes two summands: the first one gives the (gross) expectedincome of x , since the factor pλ is the expected income per unit of distance. The secondfactor gives the total driving cost.It is worth pointing out that for simplicity, we have assumed that the expected incomedepends only on the relative final positions x T and y T . A more precise account wouldconsider the entire trajectory of the buses. Nevertheless, even if this could be describedwith mathematical precision, the model would grow greatly in complexity without addingto the economic interpretation.Similarly, we define(2.12) u y ( x , v x , y , v y ) := (cid:26) pλ d Y ( x T , y T ) − cT if x T (cid:54) = y T ,pλ D − cT if x T = y T . By equation (2.7) and the definition of the utility functions (2.11), (2.12), we have thatthe sum u x + u y is a constant that does not depend on the driving speeds nor on the initialpositions. For this reason, we analyze the game as a zero-sum game. • Cooperative game
Players aim to maximize the collective payoff, and this amounts to solving the globaloptimization of the sum U x + U y , which includes the utility functions in the non-cooperativegame (2.11) and (2.12). Since the non-cooperative game is a zero-sum game, we introduce anextra term in the utility, which gives the discomfort players derive from payoff inequality.This assumption can be imagined in a situation where equity in payments is desirable,specially since players have complete information.We define the utility function to be(2.13) u ( x , v x , y , v y ) := u x ( x , v x , y , v y ) + u y ( x , v x , y , v y ) − k | u x ( x , v x , y , v y ) − u y ( x , v x , y , v y ) | , where k is a non-negative constant, and all the other elements being as in the non-cooperativegame.2.2.1. Mixed strategies and ε -equilibria. For the solution of two player games it is convenient to define the expected utility of randomizingover the set of strategies. We also introduce the definition of ε -equilibrium.Suppose that players x and y use the mixed strategies X and Y . We define the utility of player x to be U x ( x , X, y , Y ) := E [ u x ( x , X, y , Y )] . Recall that a mixed strategy is a random variable taking values in the set Γ = { v ≥ v min ≤ v ≤ v max } . US OPERATORS IN COMPETITION: A DIRECTED LOCATION APPROACH 7
An analogous definition can be derived for player y .Let ε >
0. We say that a pair of pure strategies ( v ∗ x , v ∗ y ) is an ε -equilibrium if for every v x and v y we have u x ( x , v x , y , v ∗ y ) ≤ u x ( x , v ∗ x , y , v ∗ y ) + ε, and u y ( x , v ∗ x , y , v y ) ≤ u y ( x , v ∗ x , y , v ∗ y ) + ε. This means that any unilateral deviation from the equilibrium strategy leads to a gain of no morethan ε ; this is why an ε -equilibrium is also called near-Nash equilibrium. Note that in particular,an ε -equilibrium with ε = 0 gives the standard definition of Nash equilibrium.A mixed strategies ε -equilibrium ( X, Y ) is similarly defined by replacing the utility functionswith the expected utility functions in the last definition.3.
Results
In what follows, we analyze the speeds that drivers choose, both in the short-run and in longertime periods. Results on the short-term are crucial for the latter analysis, which involves theimplementation of the obtained equilibria.3.1.
Single player games.
The single player games have pure strategy Nash equilibria. We statethe driver’s optimal strategy in each game.
Theorem 1.
Let v ∗ in Γ be the driving speed that maximizes the utility of the driver. We providean explicit description of v ∗ .a) Fixed-distance game. Given the utility function defined in (2.3) , we have v ∗ = v min if pλ > c,v min ≤ v ≤ v max if pλ = c,v max if pλ < c. b) Fixed-time game. Given the utility function defined in (2.4) , we have v ∗ = v max .Proof. The proofs are straightforward, nevertheless we outline the main ideas. Note that in thefixed-distance game, pλ − c gives the driver’s expected net income per unit of time. If this amountis positive, then the player maximizes her utility by driving for the longest time, or equivalently, bydriving at the lowest possible speed. Conversely, a negative expected net income leads to drivingat the highest speed. Lastly, a null expected income makes the driver indifferent between any givenspeed in the range.In the fixed-time game, the total revenue is proportional to the traveled distance, so the drivermaximizes her utility by driving at the highest speed. (cid:3) Two-player games.
The strategies adopted by the players strongly depend on the initialminimal distance defined in (2.8). We cover all the cases.
Theorem 2.
Non-cooperative game.
Assume, without lose of generality, that d = d x ( x , y ) .We then have:a) If d = 0 , that is, if the initial positions of the players are the same, then the pair ofstrategies ( v max , v max ) is the only Nash equilibrium. FERNANDA HERRERA AND SERGIO I. L ´OPEZ b) If < d < d < d y ( x , y ) , with d the escape distance in 2.10, then for sufficiently small ε ,the mixed strategies ε -equilibria ( X, Y ) are of the form P ( X = V ) = (cid:40) − d − d D if V = v mind − d D if V = U X and P ( Y = W ) = q if W = v min q if W = U Y − dD if W = v max − d T + εT , where U X is a uniform random variable on (cid:16) v min + d T , v max (cid:17) , q and q are non-negativenumbers such that q + q = dD and q ≤ d − d D , and U Y is a uniform random variable on (cid:16) v max − d T − q DT , v max − d T (cid:17) .c) If < d = d < d y ( x , y ) , then for sufficiently small ε , the mixed strategies ε -equilibrium ( X, Y ) are of the form P ( X = V ) = (cid:40) − εD if V = v min εD if V = v max and P ( Y = W ) = (cid:40) dD if W = v min − dD if W = v min + εT . d) If d < d , then the pair of strategies ( v min , v min ) is the unique Nash equilibrium.Proof. The proof is in Appendix A. (cid:3)
By assumption (2.9), this result covers all the possible initial positions ( x , y ), so we have acomplete and explicit characterization of the equilibria. Simply put, the theorem asserts that if theplayers have the same starting point, they drive at the maximum speed. If their positions differ byat most the escape distance, then they play mixed strategies. Lastly, if the distance between themis greater than the escape one, they drive at the minimum speed. See Figure 2 for an illustrationof the result and its cases. Theorem 3.
Cooperative game.
Without loss of generality we assume that d = d x ( x , y ) .a) If d = 0 , then the optimal driving speeds are ( v min , v max ) , and ( v max , v min ) .b) If < d and d + d < D , then the only optimal strategies are ( v min , v max ) .c) If d + d > D , then any pair ( v x , v y ) such that T ( v y − v x ) = D is an optimal strategy.Proof. The proof is direct. Since the sum u x ( x , v x , y , v y ) + u y ( x , v x , y , v y ) is equal to a constantfor any pair ( v x , v y ), the only quantity left to optimize is − k | u x ( x , v x , y , v y ) − u y ( x , v x , y , v y ) | .Minimization occurs when the distance between the final positions x F and y F is the greatestpossible. It is easy to check that the driving speeds listed above do just this. (cid:3) An important observation is that in the case where d = D , accounted for in c), all the optimalstrategies are of the form ( v, v ) for feasible speed v . Intuitively, this means that if the players havediametrically opposite initial positions, then any speed is optimal, as long as both adopt it.3.3. Time evolution.
Let us recall that the previous results are obtained for small enough T , theformal requirement being stated in (2.9). It is of interest to know what happens in longer periods,and in particular, in the long-run. To this end, we repeat the base game, but update the locationof the players accordingly. It is convenient to define a recursive process, and to introduce a fewvariables.Consider the initial positions of x and y , ( x , y ) with d defined in (2.8). Let { ( x n , y n ) } n ≥ be astochastic process with the following property: the pair ( x k +1 , y k +1 ) gives the final locations of the US OPERATORS IN COMPETITION: A DIRECTED LOCATION APPROACH 9
Figure 2.
Time-space depictions of Theorem 2. On the right ( y -axis at time T ),are the final positions of players: blue for x and red for y , when they implement theequilibrium strategies. Points represent probability mass atoms, while continuousintervals give the range of the uniform random variables.players after they have played their optimal strategies, taking ( x k , y k ) as their starting positions.Given that equilibria in Theorem 2 involve mixed strategies, randomness is present in the process.We may define the distance between the buses at any (non-negative integer) time as:(3.1) d n := min { d x ( x n , y n ) , d y ( x n , y n ) } ∀ n ≥ . We also consider the first time in which d n exceeds the escape distance d (defined in (2.10)), N defined as follows N = min { n ≥ d n > d } . Theorem 4.
Non-cooperative game. If d (cid:54) = 0 , d , we have P ( N > k ) ≤ (cid:16) dD (cid:17) k for all k ≥ . If d = d , then there exists a geometrically distributed random time M with parameter − (1 − εD )( dD ) , taking values in the natural numbers, with ε satisfying the ε -equilibrium conditions inTheorem 2, with the property that d k = d for all k < M , and d M = with probability εdD (cid:16) − (cid:16) − εD (cid:17)(cid:16) dD (cid:17)(cid:17) ,> d with complementary probability.Proof. For the proof we refer the reader to Appendix A. (cid:3)
Explicitly, this means that for most starting points, playing the game repeatedly leads to a busgap greater than the escape distance. From Theorem 2, we conclude that in this case, drivers endup driving at the minimum speed. There are two exceptions to this: if the drivers have the samestarting position, or if the initial distance between them is exactly that of escape. In the formercase, the drivers choose to go at the maximum speed forever, and in the latter, they maintaintheir distance for some random time, and from then on reach the escape distance, and drive atthe minimum speed. It is with very little probability (proportional to ε ) that this scenario doesnot occur. Figure 3 shows the evolution of the distance process { d n : n ≥ } given a few initialdistances d . Figure 3.
Evolution of the process { d n : n ≥ } for different initial positions. Theorem 5.
Cooperative game.
We have N ≤ (cid:100) D d (cid:101) , where N = min (cid:110) n ≥ d n = D (cid:111) , and (cid:100) z (cid:101) is the least integer greater than or equal to the real number z .Proof. First note that N gives the time in which the buses reach diametrically opposite positionsin the circuit. It is also worth noting that playing the optimal strategies in Theorem 3, increasesthe distance between the buses by d . Hence, repeating the game eventually leads to reaching thediametric distance. This means that N is at most the number of steps of size d necessary to goover D . Once that some diametrical positions are reached, that distance is preserved. (cid:3) US OPERATORS IN COMPETITION: A DIRECTED LOCATION APPROACH 11
Extension.
It is possible to account for perturbations like traffic lights, congestion, or acci-dents in the model, by introducing a random noise to the displacement of buses. One could do thisdefining(3.2) x T = ( x + T v x + σZ x ) mod D and y T = ( y + T v y + σZ y ) mod D , where Z x and Z y are independent standard normal random variables and σ ≥ • Non-cooperative game.
Given that the expected value of the final positions is unchanged,Theorem 2 remains valid. However, the repetition of this new game leads to an interestingresult. Since the probability of maintaining a null or escape distance d at any positive timeis zero, the long-run analysis is reduced to two distinct cases: 0 < d < d and d < d .Arguments similar to that in the proof of Theorem 4 show that if 0 < d < d , we have d N ≥ d in an exponentially fast time. If d < d , then the process { d n } n ≥ is above d for a random time M , but falls below it eventually. The expected time above is inverselyproportional to σ . • Cooperative game.
The analysis collapses to the cases b) and c) of Theorem 3. So, theplayers try to reach the diametrically opposite positions, though with probability one thisdoes not occur. 4.
Concluding remarks
Our theoretical results are consistent with the driving practices mentioned in the Introduction. Inparticular, Theorem 2. a corresponds to (2) Racing, Theorem 2. b , c to (2) Racing and (3) Overtaking,Tailing or Chasing, and Theorem 2. d to (1) Hanging back or Crowling. It is worth noting that allof the aforementioned are short-term strategies. As far as time-evolution goes, Theorem 4 assertsthat in the long run and with high probability, both operators end up hanging back. Theorems 3and 5 are intended to contrast the drivers’ optimal strategies and ultimately the equilibria whencooperation is desired.In subsection 3.4, we extended the model to allow for randomness in displacement. In thisscenario no equilibrium is lasting, so the operators alternate between racing, hanging back andchasing from time to time. We believe this is precisely what happens in Mexico City, althoughproving this would require a data driven approach analysis.There are a few open problems worth exploring. First, one could increase the number of players,and investigate whether equilibria still exists, and if so, try to characterize it. Second, one mayvary the distribution of passengers along the route, dispensing with the homogeneous assumption.Along these lines, one may introduce traffic congestion by making the utility function depend onspace in a non-homogeneous manner. This would potentially require strategies to depend on theplayer’s position. Lastly, one could introduce decision variables like tariffs and timetables; doing sowould allow to compare the results with some that have already been addressed in the literature. Declaration of interest.
None.
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Appendix A. Computations
To prove Theorem 2, it is convenient to introduce the following Lemma.
Lemma 1.
Let X be a mixed strategy of x and Y be a mixed strategy of y . We define Z tobe a mixed random variable in the Probability theory sense: it has both discrete and continuouscomponents. In particular, Z is of the form Z = (cid:26) z i with probability p i , for i ∈ I,W with probability − (cid:80) i ∈ I p i , where I is a finite or numerable set, and W is a continuous random variable with density f W ( t ) onits support, denoted by supp ( f W ) . Then, U x ( x , X, y , Y ) = (cid:88) i ∈ I E ( u x ( x , X, y , Y ) | Z = z i ) p i + (cid:16) − (cid:88) i ∈ I p i (cid:17) (cid:90) supp ( f W ) E ( u x ( x , X, y , Y ) | Z = w ) f W ( w ) dw. (A.1) If Z = X and ( X, Y ) is a mixed strategy Nash equilibrium, then (A.2) E ( u x ( x , X, y , Y ) | X = z i ) = (cid:90) supp ( f W ) E ( u x ( x , X, y , Y ) | X = w ) f W ( w ) dw ∀ i ∈ I, and (A.3) E ( u x ( x , X, y , Y ) | X = w ) = E ( u x ( x , X, y , Y ) | X = w ) ∀ w , w ∈ supp ( f W ) . US OPERATORS IN COMPETITION: A DIRECTED LOCATION APPROACH 13
Proof.
Equation (A.1) is straightforwardly obtained by computing the conditional expectancy ofthe random variable u x ( x , X, y , Y ) given the values of Z .Note that if (A.2) does not occur, then there exist two different values z i and z j , such that E ( u x ( x , X, y , Y ) | X = z i ) (cid:54) = E ( u x ( x , X, y , Y ) | X = z j ). This means that U x can be increasedby placing all the probability on the value that gives the highest expectation. This leads to acontradiction with the form of the mixed strategy X . Similar arguments apply to the case where(A.2) is violated through the continuous component.Likewise, if condition (A.3) is not fulfilled, then there are two values w and w such that E ( u x ( x , X, y , Y ) | X = w i ) are different. Then, U x can be increased by restricting the support of f W to the points where the maximum of the function g ( w ) = E ( u x ( x , X, y , Y ) | X = w ) is reached.Here, the form of the mixed strategy X is violated. (cid:3) Proof of Theorem 2 :First, note that for optimizing the utility function (2.11), (2.12) the terms pλ and c are irrelevant,since the arg min of any function is invariant under linear transformations. Thus, there is no lossof generality in assuming that pλ = 1 and c = 0.By equation (2.6), we may actually assume that 0 = x ≤ y < D . We then have d = d x ( x , y ) = y and d y ( x , y ) = D − y . Under the above assumption and using (2.1), (2.9) in cases a), b), c) and d), it happens that0 < x T , y T < D , so we can get rid of all the D -modules in the computations.For computing the ε -equilibrium, we will consider the ε -best reply, defined as follows. Let ε bea positive number. We say that a strategy v ∗ x is x ’s ε -best reply to y ’s strategy v y , if u x ( x , v x , y , v y ) ≤ u x ( x , v ∗ x , y , v y ) + ε, for all strategies v x .To simplify notation, we write u x ( v x , v y ) and u x ( X, Y ) in the case of mixed strategies, insteadof u x ( x , v x , y , v y ) and u x ( x , X, y , Y ) if the computations do not depend on the fixed initial po-sitions. • Case a)
We assume that x = y = 0. Let player y pick the strategy v y = v max . Then, u x ( x , v x , y , v min ) = d x ( T v x , T v max ) = T v max − T v x ≤ T v max . Using (2.9), we obtain the bound u x ( x , v x , y , v min ) ≤ D u x ( x , v max , y , v max ) . Explicitly, this means that the strategy v x = v max is the best reply to v y = v max . Bysymmetry, we conclude that ( v max , v max ) is a Nash equilibrium.To check the uniqueness of the equilibrium, we note that y ’s ε -best reply to a given speed v x < v max chosen by x , is v y = v x + (cid:15) for sufficiently small ε . On the other hand, x ’s ε -bestreply to v y = v x + (cid:15) is v x = v y + ε . Therefore the only equilibrium is ( v max , v max ). • Case b)
Let us denote by B x ( v ) x ’s best reply when y plays v . It is straightforward to show that B x ( v ) = v + d T + εT if v min ≤ v < v max − d T ,v max if v = v max − d T ,v min if v max − d T < v, and B y ( v ) = (cid:26) v min if v < v min + d T ,v − d T + εT if v min + d T ≤ v ≤ v max , under hypothesis b ).If ( X, Y ) is a mixed strategy Nash equilibrium, then the support of the random variable X should be contained in the set of x ’s best replies, the corresponding is true for variable Y . In this particular case, X has support on { v min } ∪ ( v min + d T , v max ) ∪ { v max } , while Y has support on { v min } ∪ ( v min , v max − d T ) ∪ { v max − d T + εT } .Hence, a mixed strategy X with the support obtained is of the form X = v min with probability p ,U with probability p ,v max with probability 1 − p − p , where p , p ∈ [0 , U is a continuous random variable with density f U ( u ) and supportcontained in ( v min + d T , v max ). Similarly, a mixed strategy Y with the desired support is Y = v min with probability q ,V with probability q ,v max − dT + εT with probability 1 − q − q , where q , q ∈ [0 , V is a continuous random variable with density f V ( v ) with supportcontained in ( v min , v max − d T ).To compute the density of U , we apply (A.3) to Y . Let us compute E ( u y ( X, Y ) | Y = v )when v ∈ ( v min − d T , v max − d T ): E ( u y ( X, Y ) | Y = v ) = E ( u y ( X, v ))= p u y ( v min , v ) + p E ( u y ( U, v )) + (1 − p − p ) u y ( v max , v )= p ( D + T v min − T v − d )+ p (cid:104) (cid:90) v + d T v min ( D + T u − T v − d ) f U ( u ) du + (cid:90) v max v + d T ( T u − T v − d ) f U ( u ) du (cid:105) + (1 − p − p )( T v max − T v − d )= p D + p DF U (cid:16) v + d T (cid:17) + p T v min + (1 − p − p ) T v max + p T E ( U ) − T v − d , where F U ( u ) is the cumulative probability distribution function of the random variable U .By (A.3), we have(A.4) p D + p DF U (cid:16) v + d T (cid:17) + p T v min + (1 − p − p ) T v max + p T E ( U ) − T v − d = k, for some constant k .Since F U ( v max ) = 1, when we plug v = v max − d T , we obtain its value US OPERATORS IN COMPETITION: A DIRECTED LOCATION APPROACH 15 (A.5) k = ( p + p ) D + p T v min − ( p + p ) T v max + p T E ( U ) . On substituting k into (A.4) we obtain F U (cid:16) v + d T (cid:17) = 1 − T ( v max − v ) − d p D .
Let u = v + d T . Then, u ∈ ( v min + d T , v max ) and F U ( u ) = 1 − T ( v max − u ) p D . From this we have u ∗ = v max − p DT is the value such that F U ( u ∗ ) = 0.The conclusion is that U is uniformly distributed on the interval ( v max − p DT , v max ), thus(A.6) E ( U ) = v max − p D T .
In the same manner we can see that V has uniform distribution on the interval ( v max − d T − q T , v max − d T ), with expectancy given by(A.7) E ( V ) = v max − d T − q D T .
To compute the values of p and p necessary for the ε -equilibrium, we use (A.2). Wefirst compute the conditional expectancy of u y ( X, Y ) given Y , E ( u y ( X, Y ) | Y = v min ) = p u y ( v min , v min ) + p E ( u y ( U, v min )) + (1 − p − p ) u y ( v max , v min )= p ( D − d ) + p (cid:104) (cid:90) v max v max − p DT ( T u − T v min − d ) f U ( u ) du (cid:105) + (1 − p − p )( T v max − T v min − d )= p ( D − d ) + p ( T E ( U ) − T v min − d ) + (1 − p − p )( T v max − T v min − d ) . By (A.6), we have(A.8) E ( u y ( X, Y ) | Y = v min ) = p D − d + (1 − p )( T v max − T v min ) − p D . Computing E ( u y ( X, Y ) | Y = V ) yields E ( u y ( X, Y ) | Y = V ) = (cid:90) v max − d T v max − d T − q DT E ( u y ( X, Y ) | Y = v ) f V ( v ) dv. Since we know that the integrand is constant and its value is given by equations (A.5) and(A.6), we directly obtain(A.9) E ( u y ( X, Y ) | Y = V ) = ( p + p ) D − p ( T v max − T v min ) − p D . We are left with the task of determining the expected value of u y ( X, Y ) conditioned on thevalue Y = v max − d T + εT , E (cid:16) u y ( X, Y ) | Y = v max − d T + εT (cid:17) = p u y (cid:16) v min , v max − d T + εT (cid:17) + p E (cid:16) u y (cid:16) U, v max − d T + εT (cid:17)(cid:17) + (1 − p − p ) u y (cid:16) v max , v max − d T + εT (cid:17) = p ( D − T ( v max − v min ) − ε )+ p (cid:90) v max v max − p DT ( D − T ( v max − u ) − ε ) f U ( u ) du + (1 − p − p )( D − ε )= p ( D − T ( v max − v min ) − ε )+ p ( D − T v max + T E ( U ) − ε ) + (1 − p − p )( D − ε )= D − ε − p T ( v max − v min ) − p D , (A.10)where we used (A.6) in the last equality.Lemma (A.2) implies that in order to have an ε -equilibrium, the expressions (A.8), (A.9)and (A.10) must be equal. This system of equations has the unique solution p = 1 − T ( v max − v min ) − d D , p = T ( v max − v min ) − d D , − p − p = 0 . We now apply this argument again, to obtain the expectancy of the random variable u x ( X, Y ) conditioned on the values of X , as well as the values q , q necessary to havean ε -equilibrium. In this case, there are many solutions. Indeed, any combination q , q satisfying0 ≤ q , q , q + q = T ( v max − v min ) D , − q − q = 1 − T ( v max − v min ) D , fulfills equation (A.2).Given that the support of V is (cid:16) v max − d T − q DT , v max − d T (cid:17) ⊆ (cid:16) v min , v max − d T (cid:17) , it isnecessary to impose the condition q ≤ d − d D . • Case c)
From the conditions stated in c ), it follows that B x ( v ) = (cid:26) v max if v = v min ,v min if v > v min . Intuitively, under hypothesis c ), it always happens that x T ≤ y T for every pair of strate-gies v x , v y . Equality holds only when v x = v max and v y = v min .Similarly, one can check that B y ( v ) = (cid:26) v min if v < v max ,v max + εT if v = v max , where last case is an ε -best reply.To find the ε -equilibria, we define X to be a random variable such that P ( X = v min ) = p, P ( X = v max ) = 1 − p, for some probability p ∈ [0 , US OPERATORS IN COMPETITION: A DIRECTED LOCATION APPROACH 17
Similarly, we define a random variable Y such that P ( Y = v min ) = q, P (cid:16) Y = v min + εT (cid:17) = 1 − q, for q ∈ [0 , ε -equilibrium requires E ( u x ( v min , Y )) = E ( u x ( v max , Y )), which is exactly the condi-tion (A.2) when there is no continuous part for X .Since E ( u x ( v min , Y )) = qu x ( v min , v min ) + (1 − q ) u x (cid:16) v min , v min + εT (cid:17) = qd + (1 − q )( d + ε ) , and E ( u x ( v max , Y )) = qu x ( v max , v min ) + (1 − q ) u x (cid:16) v max , v min + εT (cid:17) = q (cid:16) D (cid:17) + (1 − q )( ε ) , we can equalize the two equations and solve to obtain q = dD . Note that (2.1) implies that0 < q < E ( u y ( X, v min )) = E (cid:16) u y (cid:16) X, v min + εT (cid:17)(cid:17) . The explicit formulasbeing E ( u y ( X, v min )) = pu y ( v min , v min ) + (1 − p ) u y ( v max , v min ) = p ( D − d ) + (1 − p ) D , and E (cid:16) u y (cid:16) X, v min + εT (cid:17)(cid:17) = p u y (cid:16) v min , v min + εT (cid:17) +(1 − p ) u y (cid:16) v max , v min + εT (cid:17) = p ( D − d − ε )+(1 − p )( D − ε ) . Matching and solving the two yields 0 < − p = εD < • Case d)
Assume that player y chooses strategy v y satisfying (2.1). Then(A.11) u x ( x , v x , y , v y ) = d x ( T v x , y + T v y ) . By assumption d ), we have T ( v x − v y ) ≤ T ( v max − v min ) < d x ( x , y ) = y , so y + T v y − T v x > v x , v y . Then, (A.11) is equal to u x ( x , v x , y , v y ) = y + T ( v y − v x ) , which is bounded by u x ( x , v x , y , v y ) ≤ y + T ( v y − v min ) = u x ( x , v min , y , v y ) . We conclude that v x = v min is x ’s best reply to any strategy v y played by y .Similarly, if x chooses strategy v x , then(A.12) u y ( x , v x , y , v y ) = d y ( T v x , y + T v y ) . We have already proven that y + T v y − T v x > v x , v y , so (A.12) is equal to u y ( x , v x , y , v y ) = D + T v x − y − T v y . We can bound the last expression by u y ( x , v x , y , v y ) = D − y + T ( v x − v y ) ≤ D − y + T ( v x − v min ) = u y ( x , v x , y , v min ) . This implies v y = v min is y ’s best reply to any strategy v x played by x . The conclusionis that ( v min , v min ) is the unique Nash equilibrium. (cid:3) Proof of Theorem 4 :First, note that d > d implies N ≡
0, and the result holds trivially.Assume that 0 < d < d , and suppose that 0 < d k < d for some k ≥
0. Then, the strategies(
U, v min ) , ( U, V ) , ( U, v max − d k T + εT ) , ( v min , v min ) , ( v min , V ) lead to 0 < d k +1 < x and y are instead ( v min , v max − d k T − εT ), then d k +1 = d + ε . We can uniformlybound from below the probability that the players adopt these strategies by P (cid:16) ( X, Y ) = (cid:16) v min , v max − d k T − εT (cid:17)(cid:17) = (cid:16) − d − d k D (cid:17)(cid:16) − d k D (cid:17) ≥ (cid:16) − dD (cid:17) , ∀ < d k < d, where the inequality can be obtained by calculus (or by noting that this probability is an invertedparabola, as a function of d k ). Therefore, P ( N > k ) ≤ P ( G > k ) , where G is a geometric random variable with parameter 1 − dD , and the result follows.Finally, assume that d = d . If players x and y choose ( v min , v min ), then d = d . Any otherstrategy choice yields d (cid:54) = d .Define M = min { n ≥ d n (cid:54) = d } . By the above remark, M has geometric distribution on thenatural numbers with parameter 1 − (cid:16) − εD (cid:17)(cid:16) dD (cid:17) . After M trials, we are on the conditional spacewhere x and y do not play ( v min , v min ), instead they choose( v max , v min ) with probability (cid:16) εD (cid:17)(cid:16) dD (cid:17) − (cid:16) − εD (cid:17)(cid:16) dD (cid:17) , ( v min , v min + εT ) with probability (cid:16) − εD (cid:17)(cid:16) − dD (cid:17) − (cid:16) − εD (cid:17)(cid:16) dD (cid:17) , ( v max , v min + εT ) with probability (cid:16) εD (cid:17)(cid:16) − dD (cid:17) − (cid:16) − εD (cid:17)(cid:16) dD (cid:17) . The first election leads to d M +1 = 0, while the other two give d M +1 > d . This concludes the proof. (cid:3) ( † ) Centro de Estudios Econ´omicos, El Colegio de M´exico, C.P. 14110, Ciudad de M´exico, M´exico (*)
Departamento de Matem´aticas, Facultad de Ciencias, UNAM, C.P. 04510, Ciudad de M´exico,M´exico
Email address , † : [email protected] Email address , *:, *: