DDrivers learn city-scale dynamic equilibrium
Ruda Zhang ∗ and Roger Ghanem Department of Civil and Environmental Engineering,University of Southern California, Los Angeles, CA, USA. (Dated: August 26, 2020)Understanding collective human behavior and dynamics at urban-scale has drawn broad interestin physics, engineering, and social sciences. Social physics often adopts a statistical perspective andtreats individuals as interactive elementary units, while the economics perspective sees individuals asstrategic decision makers. Here we provide a microscopic mechanism of city-scale dynamics, interpretthe collective outcome in a thermodynamic framework, and verify its various implications empirically.We capture the decisions of taxi drivers in a game-theoretic model, prove the existence, uniqueness,and global asymptotic stability of Nash equilibrium. We offer a macroscopic view of this equilibriumwith laws of thermodynamics. With 870 million trips of over 50k drivers in New York City, we verifythis equilibrium in space and time, estimate an empirical constitutive relation, and examine thelearning process at individual and collective levels. Connecting two perspectives, our work shows apromising approach to understand collective behavior of subpopulations.
Keywords: urban-scale dynamics, collective behavior of individuals, human behavior change, social physics
I. INTRODUCTION
Quantitative understanding of collective human behav-ior and its dynamics has been the focus of a long-standing,interdisciplinary collection of research. Such efforts spanacross physics [1–15], engineering [16–20], economics [21–27], and planning [28–30]. Cities, as agglomerations ofhuman beings, are the natural setting for such studies.Thanks to ubiquitous data sensing techniques, a host ofurban phenomena, ranging from traffic [5] to buildingenergy use [30] and transaction patterns [31], can now beanalyzed quantitatively. Research in the physical sciencesoften value simplistic models that fit data well. Social sci-ences, on the other hand, emphasize individual incentiveand realistic theory. Models that connect these perspec-tives and are backed by empirical data would thereforebe a valuable contribution to this literature.Social physics [1] or social dynamics [7, 9, 10] are ap-proaches that use physical models to describe collectivehuman behavior. A useful framework for this is statisticalphysics, where individuals are seen as elementary unitsand collective phenomena emerge from their interactions[10]. Random walk models are used to study human mobil-ity at individual level [4]. Non-equilibrium phase transitionis observed in the transition of urban traffic from free flowto congestion [6]. Scaling laws of socioeconomic and infras-tructure indicators on population have been discoveredand generative mechanisms proposed [2, 11]. Based onnon-equilibrium statistical physics, sociodynamics modelsthe evolution of city configuration with a master equation[7], and finds application in studying the impact of trans-portation system on regional development [8]. Entropymodels have been used to study spatial structure andinteraction [29]. Other tools in statistical physics such ∗ [email protected]; also at the Statistical and Applied Mathemat-ical Sciences Institute, Durham, NC, USA. as spin-like models also find use in the science of cities[14]. Other theories in physics have been adopted as well.Social force and social field models quantify the most prob-able behavioural change of individuals [9]. Force fieldsresulting from infrastructure and socioeconomic layoutsaffect urban morphology [12]. Gravity and radiation haveinspired models for human mobility at population level[13]. In particular, a potential field can be well-definedfrom commute flows in a city [15].Game theory and economics, on the other hand, con-sider collective phenomena as results of strategic decisionmaking of individuals, and it is therefore crucial to cap-ture individual incentives realistically. Discussion on traf-fic equilibrium predates modern game theory [21, 22], andhas been formalized since [16, 23, 24]. Spatial equilibriumgeneralizes this idea to explain many problems in urbaneconomics [25]. Game theory has also been used to studysocial network formation [3]. Equilibrium outcomes areoften different from social optimal, and there are manyefforts to quantify such inefficiency [18, 19] and find waysto minimize it [5].These perspectives from physical and social sciencesrarely come together, partly because it is difficult to ac-count for the different incentive structures within a popu-lation and provide a simplistic, universal understanding.Consider urban transportation. People move for variousreasons and often do not minimize their travel time asassumed in traffic equilibrium. But a model that accountsfor heterogeneous incentives often evades an equilibriumsolution. Here we bridge the two perspectives by study-ing the behavior and dynamics of a subpopulation—taxidrivers. As ride service providers and independent con-tractors, these drivers move around a city to maximizeincome. Using GPS data, recent work has shown that theirrevenue are affected by their strategies in search, delivery,and service-region preference [20]. Gender earnings gapamong drivers of a ride-hailing company is explained bytheir driving speed, experience, and service location andtime [26]. Moreover, drivers with higher incomes benefit a r X i v : . [ phy s i c s . s o c - ph ] A ug significantly from their ability to learn from local andglobal demand information [27].In this paper, we regard the regularity in urban trans-portation as the equilibrium outcome of individual deci-sion making in response to transportation demand and ser-vices [32]. In particular, we provide a microscopic, game-theoretic model for taxi transportation and interpret itsequilibrium in a macroscopic framework of thermodynam-ics. Given exogenous traffic speeds and passenger demand,the income maximization of drivers leads to an economicequilibrium. We formalize drivers’ decision making as anon-cooperative game, prove the existence and unique-ness of its Nash equilibrium, and show its stability undersimple learning dynamics such as adaptive learning andsocial learning. We also provide an interpretation of thisequilibrium as a thermodynamic equilibrium, and describethe laws of thermodynamics, constitutive relation, andfundamental thermodynamic relation. With five yearsof New York City (NYC) taxi trip records, we validatethe equilibrium in space and over time, and estimatean empirical constitutive relation. We also examine thelearning process of individual drivers, as well as theiradjustment as a group to a new system. We discuss theeconomic efficiency of taxi transportation, as well as analternative formulation of strategy. This paper thereforeoffers a microscopic mechanism of city-scale dynamics,and empirically verify its various implications. Connectingsocial and physical science perspectives, our work showsa promising approach to understand collective humanbehavior and dynamics. II. RESULTSA. Game model
We assume that each driver chooses their driving strat-egy to maximize income, which we show to be strategicallyequivalent to revenue maximization. Taxi activity variesin space (see fig. 1a-b), and we define the strategy of adriver in service to be how they allocate their service timeacross the city. Let s ix be the proportion of service timedriver i allocates on street segment x and E be the setof street segments in the road network, then the driver’sstrategy is s i = ( s ix ) x ∈ E , whose components sum to 1.Figure 1c illustrates driver strategy. We can formalize adriver’s decision as an optimization problem:maximize π i ( s i ; s − i , E )subject to s i ≥ s i · = 1 (1)Here, with hour as the unit of time, π i is the expectedhourly revenue of the driver, s − i = (cid:80) j (cid:54) = i s j is the aggre-gate strategy of other drivers, and E is environment con-dition which includes traffic speed and passenger demand.For details of driver decision making, see SupplementaryNote A 1. Competition among drivers could lead to specificchoices of strategies, called equilibrium. If we see everystreet segment as a distinct market and every driver inservice as a multi-market firm, we can abstract eq. (1) asa game of multi-market competition among firms of equalcapacity. This game has a unique Nash equilibrium (NE),where all drivers use the same strategy and marginaldriver revenue are uniform across all searched segments.Moreover, the equilibrium is globally asymptotically sta-ble under adaptive learning [33] and/or imitative learning[1, 34, 35]. Denote this equilibrium as S ∗ = ( s ∗ i ) i ∈ N , where N is the set of drivers in service. Because all driversuse the same strategy, let s = (cid:80) i ∈ N s i , we can write S ∗ = n − s ∗ T n , where n = | N | is the number of drivers inservice. The equilibrium can be determined such that s ∗ is the unique point that maximizes a potential function:Φ( s ) = (cid:88) x ∈ E (cid:90) s x φ x ( t ) d t (2)where φ x ( s ∗ x ) = ( ∂π i /∂s ix )( S ∗ ) is marginal driver revenueon a segment at equilibrium, see fig. 1d. For details of thegame model, see Supplementary Note A 2. B. Thermodynamic interpretation
We can interpret the Nash equilibrium of drivers asa thermodynamic equilibrium. This establishes a macro-scopic equilibrium where aggregate behavior is perceivedas a transport phenomenon built up from individualchoices. This macroscopic view ignores the decision mak-ing and competition of drivers, but helps understand theoutcome of a social system from the perspective of aphysical system.We regard drivers as interchangeable particles withidentical behavior at equilibrium. Regard total servicetime s , which equals the number of drivers in service, astotal energy of the taxi transportation system. Regardpotential function Φ of the game as entropy of the system.And regard the reciprocal of equilibrium marginal driverrevenue, ψ = 1 /φ , as temperature. Then s , ψ , and Φare all state variables of the system at equilibrium givenenvironment condition E .Being a state variable and intensive property, temper-ature ψ is the driving force of the transport of servicetime s over the street segments. As we mentioned ear-lier, the learning process of the game always increasesthe potential function Φ( s ), which is maximized at equi-librium. When two systems at equilibrium are put intocontact with an interface permeable to the transfer ofservice time, s will flow from the system with higher ψ to the one with lower ψ . At equilibrium, ψ is uniformacross all searched segments. In summary, we can makethe following statements of thermodynamics. Zeroth law:two taxi systems in contact have the same equilibriummarginal driver revenue. ψ = ψ (3) FIG. 1. Driver strategy. a-b , Manhattan street network used in this paper, showing characteristics of taxi activity: a , quantileof drop-off per segment length; b , log2 pickup-dropoff ratio. Black bold line marks the north border of core Manhattan. c ,drivers allocate their service time across the segments, which can differ. d , model of equilibrium. With s drivers, marginal driverrevenue ν on a segment at equilibrium is determined by (cid:80) x ∈ E s x ( ν ) = s . Equilibrium allocation on each segment can then bedetermined by s ∗ x = s x ( ν ). First law: taxi transportation is the transfer process oftotal service time s , which is a conserved quantity.d s = (cid:88) x ∈ E d s x (4)Second law: under fixed demand and traffic state, a closedtaxi system maximizes its potential function.dΦ ≥ δsψ (5)Zeroth law defines equivalent classes of equilibrium,which are strictly totally ordered by state variable ψ . Themanifold of equilibrium is thus one-dimensional, parame-terized by ψ , and any other state variable must dependon it. This means that state space (Φ , s, ψ ) | E has onlyone degree of freedom, and this dependency is the consti-tutive relation of the system given environment condition E , which can be written explicitly as(Φ , ψ )( s ) | E (6)Rearranging the exact differential of s (Φ) | E gives thefundamental thermodynamic relation of the equilibrium:d s = ψ dΦ (7)We test various implications of this theory of thermo-dynamics in our empirical results. C. Verification of spatial equilibrium
To verify that drivers actually follow the theoreticalequilibrium, we proceed in two parts. First, all drivers use the same strategy. Second, given that drivers use thesame strategy, marginal driver revenue is uniform acrossall searched segments.Although driver strategy—the spatial distribution ofservice time—is not directly observed, it is proportionalto driver pickup probability on each segment. If all driversuse the same strategy, each driver’s pickup probabilitydistribution across segments shall be the same as that ofthe overall distribution. Then each driver’s actual pickupsshall be a sample of the corresponding categorical randomvariable. Since there are 6,001 street segments, pickuprecords of each driver is not enough to test the probabilitymodel. We partition the segments into 10 equi-probablegroups, so pickup counts in these groups shall be a multi-nomial random variable with the same probability foreach group. Drivers’ pickup counts in these groups can betested by a corrected log likelihood ratio of multinomialdistributions [36]. For each driver, the pickup counts arenormalized into a probability vector x = ( x , . . . , x ),which is then summarized by the 1-norm (cid:107) x (cid:48) (cid:107) , where x (cid:48) = x − /
10. We consider a strategy to be a largedeviation if (cid:107) x (cid:48) (cid:107) exceeds 0.3. Note that (cid:107) x (cid:48) (cid:107) = 0 . (cid:107) x (cid:48) (cid:107) .Only 3.66% of drivers have statistically significant largedeviations. Although the threshold for large deviation isarbitrary, the result shows that most drivers use similarstrategies. Therefore we can regard drivers to be particleswith identical behavior.Now we verify that all segments have the same marginaldriver revenue, or equivalently, the same temperature. We FIG. 2. Verification of dynamic equilibrium. Using trip records in Spring 2011 and Spring 2012. a , probability distribution of1-norm of driver deviation from average strategy, in Tue-Thu PM peaks, 6pm–10pm, grouped by p -values. 3.66% of drivers havestatistically significant ( p > .
05) large deviations ( (cid:107) x (cid:48) (cid:107) > . b , log of search time–revenue ratio on street segments, Mon-Fri6pm–7pm, shifted to a reference value. Local regression (red) and prediction intervals (shade). Distribution of log revenue onsegments (margin). c-d , average number of drivers and driver revenue, Wed 5am–Thu 5am: c , time series, rectangles mark AMshift (8:30am–4pm) and PM shift (6pm–4am); d , trajectory, colored by the hour, red line shows a linear regression for 5pm. note that when n (cid:29) φ x ≈ π x /s x , where π x is the rev-enue originated on a segment and s x is the total servicetime attributable to the segment. Because at any momentthe number of drivers in service in Manhattan is in thethousands, this approximation is suitable. So it sufficesto show that π x is proportional to s x , which is the sum ofsearch time t sx and trip time t px per unit time. Becausethe majority of trips are metered, which is calculated fromtrip distance and time in slow traffic, driver revenue fromeach trip is highly correlated to trip duration regardlessof driver strategy, especially when traffic speed is holdstationary. To avoid the influence of this fact, considertrip time as a linear function of trip revenue, then π x ∝ s x is equivalent to π x ∝ t sx , and we try to show the latter.Because search routes are not recorded in the trip records,we take trip records between 6pm and 7pm on weekdays inspring, and estimate search routes between trips by short-est distance routing. We consider this approach acceptablebecause during the selected hours, traffic is roughly ata uniform congested speed while average search time isthe shortest, so route deviation from the shortest pathis unlikely. Figure 2b shows log(˜ t sx /π x ) versus log( π x ),where ˜ t sx is the estimated search time. The majority ofstreet segments have similar search-revenue ratios, while segments with low revenue appear to be over-suppliedand those with very high revenue under-supplied. For seg-ments with low revenue, marginal driver revenue mightnot be equilibrated since they contribute little to driverrevenue. Our estimation assigns search time equally toeach segment on route, which may underestimate theactual search time near the pickup location, and there-fore underestimate search time on high revenue segments.Moreover, shortest path routing provides a single routefor trips with the same origin and destination, so the esti-mated search time may be concentrated on a few streetsegments, which contributes to estimation error. D. Dynamic equilibrium
As environment condition E varies over times of aday, the equilibrium will also vary. If drivers are freeto choose when to work and are indifferent about workingat different times of a day, by zeroth law eq. (3), driversupply s will adjust so that temperature ψ is station-ary throughout a day. Equivalently, φ stays the samethroughout a day. Note that marginal driver revenue ona segment and average driver revenue are approximately FIG. 3. Learning and adjustment of equilibrium. a-c , percentile of strategy deviation of new drivers joined in a b c d , pickup probability in the region bordering core Manhattan, in the second halves of2012 (black) and 2013 (red), 7-day rolling value with 90% bootstrap confidence band. Significant events and period marked bydashes and shade. the same at equilibrium: because φ x ≈ π x /s x , therefore φ ≈ (cid:80) x π x / (cid:80) x s x = π/s . This means that, given theassumptions, average driver revenue is the same through-out a day. To verify this, we examine the trajectory ofaverage driver revenue and number of drivers throughouta typical weekday, shown in Fig. 2c. Average driver rev-enues during 8:30am–4pm and 6pm–4am center around$29/hour and $33.5/hour respectively, and are constantin the sense that its overall variation is about the same asits short-term variation. The difference between averagedriver revenue for these two periods can be explained bytwo factors. First, the total number of taxis is limited andnot all is available for the night shift, so not all driverswho would like to work at night can get a taxi. Second,the lease rate for day shifts is less than those of nightshifts, so the difference in average driver income betweenthe two periods is less than that of average driver rev-enue. During 4pm–6pm most double-shifted taxis changedrivers, which means supply decisions during this periodis not up to the drivers, so the average driver revenue isnot constant. During 3am–6am very few drivers are atwork, and the high average driver revenue justifies thecost of working when most people prefer to be sleeping.During 6am–8:30am most day shift drivers start working,and although the average driver revenue is not constant,it stabilizes as more drivers become active.In contrast to the equilibration of average driver rev- enue over time, by constitutive relation eq. (6), marginaldriver revenue on a segment at equilibrium is a decreas-ing function of the number of drivers given environmentcondition: φ ( s ) | E . This constitutive relation is hard tomeasure without controlled experiment, but can be mea-sured from observational data if the number of drivers isforced to change much faster than the environment does,such as during shift transition. In fig. 2d, the downwardtrend in 5pm–6pm reflects φ ( s ) for that time of day, whenpeople leave work and taxis return for the night shift. E. Individual learning
It is natural to ask if drivers learn to use the samestrategy that results in a spatially uniform marginal rev-enue. We use drivers’ first appearance in trip records toinfer if they are new or experience drivers. The rate ofnew drivers stabilizes around September 2009, with about10.23 new drivers each day since. For new drivers joinedeach spring from 2010 to 2012, we compute the 1-norm oftheir strategy deviation, (cid:107) x (cid:48) (cid:107) , and compare it with thegroup of experienced driver who worked through 2010-2013. In particular, we group each year’s new drivers bytheir eventual consecutive years of driving up to 2013,and track their percentile of (cid:107) x (cid:48) (cid:107) against the experienceddrivers. Figure 3a-c provide box plots for the groups. Notethat the experienced drivers, if plotted, would always havethe median and the first and third quartiles at 50, 25,and 75, respectively. For all groups of new drivers whostayed for at least a year, their strategy deviation decreasesignificantly in the second year, with the median reducingbetween 10 to 20 percentile. For new drivers who stayedthrough 2013 and for at least two years, their strategydeviation stabilize in the later years and are smaller or thesame as the experienced drivers. Moreover, new driverswho stay longer always have smaller strategy deviationthan their cohorts. We see that new drivers learn theequilibrium strategy within one year of driving. F. Group adjustment
Changes in taxi regulation affect the equilibrium, whichprovide unique opportunities to test the implications ofthe theory. On 2013-08-08, NYC TLC launched StreetHail Livery, also known as green cabs. The new systemis allowed to pick up street-hail passengers outside coreManhattan, defined as south of West 110th Street andEast 96th Street, see fig. 1a-b. This change graduallyincreased the supply of street-hail service outside coreManhattan, and by constitutive relation eq. (6) this shoulddecrease the marginal driver revenue on segments therein.By zeroth law eq. (3), segments within core Manhattanshould also have marginal driver revenue decreased tothe same level, which implies more supply of yellow cabsin core Manhattan where they have exclusive rights toservice. By first law eq. (4), the proportion of service timeyellow cab drivers spent outside core Manhattan shoulddecrease. Figure 3d compares the time-series of percentageof pickups in the region bordering core Manhattan in 2012and 2013. This percentage slightly reduced after the 2012fare raise, greatly increased during Hurricane Sandy, andmoderately increased during Thanksgiving and Christmas.Excluding irregularities due to Hurricane Sandy and theholidays, the percentage is stable in the last two monthsof both years, with a robust decline in 2013.
III. DISCUSSION
The equilibrium is not socially optimal in general. Asocially optimal outcome would maximize total revenue,whereas the equilibrium maximizes the potential function.In fact, if total revenue is maximized, marginal segmentrevenue ∂π x /∂s x should be the same for all searchedsegments. At Nash equilibrium, a weighted average ofmarginal and average segment revenue is made uniforminstead, with more weight on the latter as the numberof drivers in service increases. This difference implies aninefficiency of the equilibrium, except for special choices of π x ( s x ). For details of the inefficiency, see SupplementaryNote A 3. This phenomenon of difference between coopera-tive and competitive decisions has been studied for a longtime under different names. Economic inefficiency [37, 38] refers to a situation where total income, or social wealth,is not maximized. The problem of social cost [21, 22] isthe divergence between private and social costs or value.Later developments include external effect [39], rent dissi-pation [40], market failure [41], and transaction cost [42].Algorithmic game theory uses price of anarchy [18, 19] andprice of stability for this inefficiency of equilibria. Despitethe various terminology, the essence of the problem is thesame: when individuals do not have incentive to maximizethe total revenue, equilibrium naturally will differ fromthe optimum set, which by definition results in less totalrevenue. If people put a moderate weight towards thetotal outcome, much of the inefficiency can be avoided[5]. Here we propose the main takeaway for the case ofmulti-market oligopoly: if a property is heterogeneous inproductivity, the owner cannot obtain the optimal rent byleasing to multiple tenants without contracting on theirallocation of effort.Now we point out how a driver would implement asearch strategy. Picture a driver i who is familiar withcity traffic and hailer and driver distributions throughouta day. To earn more money, the driver has a plan onhow much time to spend searching different places forhailers; the plan may vary for different time of day. At thebeginning of i ’s shift, the driver heads to the region wherethe plan allocates the most search time. After deliveringthe first pickup, the driver is likely to be in a regionwith less planned search time. To avoid over-searching thecurrent region, i drives back to the preferred region. If i goes through the preferred region without a pickup, thedriver would circle around and continue the search, as longas the total search time within the region is not too longcompared with the plan. The driver does not always searchor immediately go back to the region with the highestplanned search time, but would balance the allocation ofrealized search time to approximate the plan. But when i drops off at a location with very little planned search time,the driver would directly head to a place nearby wherethe plan gives more search time, since a single pass wouldtypically suffice for the drop-off location. Because thetotal search time is limited for any given shift, the driverwould not be able to perfectly implement the strategyin one shift. But aggregated over time, the distributionof realized search time could reasonably approximate anintended strategy.Driver’s search strategy can be alternatively formalizedas a Markov chain. That is, depending on the currentlocation, the driver chooses probabilistically a neighboringlocation to search. If drivers are non-strategic, a nullhypothesis for the Markov strategy would be random walk.However, fig. 1b suggests that drivers tend to move backto the area with more pickups. We test out this hypothesisin fig. 4. We see that, regardless of trip origin, locationswith more pickups tend to be popular destinations as well.Overall the drop-off distribution is more spread out thanthe pickup distribution. On the other hand, the searchmatrix is diagonal dominated and skews towards morepopular locations. In particular, group 1 accounts for 42% FIG. 4. Markov strategy. Using trip records in Spring 2011 and Spring 2012, Mon-Fri 6pm–7pm. a , trip origin–destinationmatrix among 10 equal-sized groups of street segments in decreasing order of pickups. Rows normalized by 1-norm to showtransition probability. Margins show pickup and drop-off counts in each group. b , search start–end matrix among the groups.Rows normalized by max-norm. of pickups and 32% of drop-offs, and comparatively veryfew drivers find their next pickup in other groups. As thedrop-off location gets less popular, the skew away from lesspopular groups and towards more popular ones becomemore prominent. Because search time is typically shortin the PM peak, most drivers do not need to drive far tofind passengers. Although a Markovian search strategymay be simple to describe and implement, it is difficultto estimate and does not allow a simple thermodynamicinterpretation. Moreover, it is equivalent to our definitionof search strategy. For details of the Markov strategy, seeSupplementary Note A 4. IV. METHODSA. Taxi trip records
The New York City (NYC) Taxi and Limousine Com-mission (TLC) started its Taxicab Passenger Enhance-ment Program (TPEP) in late 2008, which collects elec-tronic trip record of its Medallion taxis (aka yellow cabs).TLC releases TPEP records to the public per the Freedomof Information Law of New York State. We have gatheredthe records from 2009 to 2013, the first five calendar yearssince TPEP devices were installed in all 13,237 Medalliontaxis. The data set contains over 870 million trips and50,297 frequent drivers. Each trip record contains medal-lion ID (for vehicles), hack license (for drivers), latitude,longitude and time stamp of pickup and drop-off, tripdistance, fare amounts, and other attributes. We use theID fields to link a taxi between consecutive trips, andderive new attributes for use in our study. The originaland processed data are available for reuse at [43].
B. Road network and map matching
We use OpenStreetMap (OSM) data for the publicnon-freeway vehicular road network in NYC. Specifically,we include OSM ways whose highway tag take one ofthe following values: trunk , primary , secondary , tertiary , unclassified , residential . To make the road network stronglyconnected, we removed tunnels, bridges, and link roads.The filtered OSM map has 8,928 locations and 11,458edges. We use Open Source Routing Machine (OSRM)to create a compressed graph of 6,001 edges. We exploitanother module in OSRM to match GPS locations tothe nearest segment, where longitudes and latitudes aretransformed in Mercator projection for isotropic localscales of distance. The modified code is available at https://github.com/rudazhan/osrm-backend . Appendix A: Supplementary notes1. Taxi driver decision making
The transportation decision of a taxi driver can besimply expressed as: taxi drivers maximize their incomeby choosing their driving strategy. We ignore the exitdecision of taxi drivers, and assume that individuals whodrive a taxi can earn at least as much income as theircost, i.e. their alternative income. When this conditiondoes not hold, rational individuals would not be drivinga taxi. We show in the following that driver’s objectivefunction is strategically equivalent to trip revenue, andformalize driver’s decision as an optimization problem.Income structure of a taxi driver differs by the propertyrights of the taxi in use. Owner-drivers are Medallionowners who also drive their taxis, so they have no leaseto pay. Drivers of driver-owned vehicle (DOV) leasea Medallion from fleets, agents, or Medallion owners,and either own or finance the purchase of the vehicle,at different lease costs. Other drivers lease both theMedallion and the vehicle. In any kind of such leases,the driver pays a fixed amount of money either per shiftwhich lasts 12 hours, or per week in longer-term leases.(See TLC Rules § ) Taxi lease type can be inferredfrom driver names on the taxi’s rate card: if a taxi hasnamed drivers, its owner typically uses long-term lease; ifit has unspecified driver, its owner typically uses shiftlease. Table I shows the number of NYC taxis in 2005 bytheir manager and driver types, derived from [44]. TABLE I. NYC taxis by manager-driver type, 2005Owner-driver Named driver UnspecifiedOwner 3730 1210 -Fleet - 1481 635Agent - 1435 4305
Taxi drivers also pay for fuel usage, which depends onvehicle model, vehicle speed and acceleration, air tem-perature, and air conditioning. As of vehicle model, afterthe 2008-05-02 TLC auction, 275 of the 13237 Medal-lions are restricted to alternative fuel vehicles, but manyunrestricted Medallion owners voluntarily converted toclean-fuel vehicles (see Table II, data from TLC 2008-2013 Annual Reports ). For gasoline/hybrid lightpassenger vehicles operating at urban traffic speed (16-40 km/h, or 10-25 mph), fuel consumption per hour isalmost constant, see [45]. This means that fuel cost perservice time can be seen as a constant for each taxicabregardless of speed — we do not consider taxis parked bythe curb with engine off actively in service. Even withoutthis observation, fuel cost per service time would still beapproximately constant for a driver in one shift, as long asthe driver has consistent driving speeds and accelerationpatterns.
TABLE II. NYC taxis by vehicle fuel typeEnd of year Gasoline Hybrid-electric Diesel CNG * * CNG: compressed natural gas
A taxi driver earns the remaining fare and tips afterpaying for lease, fuel, or both. Formally, the hourly income u i of driver i derives from hourly trip revenue π i , minushourly fuel cost f i , minus amortized hourly lease payment r i : u i = π i − f i − r i (A1)The amortized hourly lease payment by the driver is r i = R i /T i , where T i denotes driver total service timeduring the lease term, and R i denotes lease payment, i.e.rent of the Medallion taxicab. Depending on the lease, f i or r i may be zero. Since f i and r i are constant fordriver i in any given shift, they do not affect the driver’sdriving strategy. Thus, driver’s objective is strategicallyequivalent to trip revenue π i . We note that althoughvehicle maintenance is another cost to drivers who ownthe vehicle, it is not relevant to the driver strategy of ourinterest.To define taxi drivers’ driving strategy, we first analyzetaxi transportation. Taxis in service are either vacantor occupied: when vacant, drivers search the streets forhailers; when occupied, drivers take the passengers to theirdestination. Taxi drivers can freely choose how they spendtheir search time over the street network. Once they findhailers, drivers will stop searching and pick them up. (Inreal life, not all taxi drivers pick up every hailer they meet.They may discriminate hailers based on the destination,race, or other factors, due to profitability, security, or end-of-shift concerns. See NYC 311 records for complaintsabout taxis service denial.) Taxi fare rate is set by thecity government, which may be metered or has a flat rate,depending on the destination. Under flat rate, drivers arebest off taking the fastest path. Metered rates charge bydistance or duration, based on a speed threshold, whichare typically set such that drivers have no incentive todrive slow. Although drivers do have an incentive to takeroutes longer than the fastest path, passengers typicallyare motivated to supervise trip duration. In case of driverfraud, detouring is not a common strategy [46]. Thus,we assume that taxi driver’s delivery strategy is to takepassengers to their destination via the fastest path, so tripduration between two specific locations only depend ontraffic speed. We can see that the only strategic elementfor taxi drivers is how they allocate their search time.Now we formalize drivers’ driving strategy. Let N bethe set of taxi drivers currently in service. Let G = ( V, E )be the road network within the urban area being studied,where V is the set of intersections and dead ends, and E is the set of street segments. Street segment x ∈ E haslength l x , with traffic speed v x and taxi search speed ˜ v x .Define demand rate µ dxy as the frequency of hailers starthailing on segment x who are going to segment y ; sucha group of hailers have impatience µ txy = 1 / E T xy , thereciprocal of hailer mean patience. Within a short timeinterval, environment condition E = ( v , µ d , µ t ) can beconsidered as constant, where v is the vector of trafficspeeds, and µ d and µ t are matrices of hailer demandand impatience. Strategy for driver i can be defined asthe spatial distribution of supply rates µ si , where µ six =( µ si ) x is the frequency at which driver i enters segment x as a vacant taxi. Equivalently, driver strategy can bedefined as the distribution of driver’s search time per unittime: t six t = l x ˜ v x µ six (A2)This shows that on each segment, driver search time islinearly related to driver supply rate. Define pickup rate µ pixy as the frequency at which driver i picks up passen-gers on x going to y . These attributes naturally aggregateson each segment: µ px = (cid:80) i (cid:80) y µ pixy , µ sx = (cid:80) i µ six , µ dx = (cid:80) y µ dxy , and µ tx = 1 / E T x . Pickup rate can thusbe expressed as a function of supply rate, demand rateand hailer impatience: µ px ( µ sx , µ dx , µ tx ). [32] proposed aclass of pickup models and proved that the pickup ratefunctions are increasing, strictly concave, and arbitrarilydifferentiable, with respect to supply rate; for three rep-resentative models, analytical forms of the pickup ratefunctions are also provided.We now relate driver strategy with driver revenue. LetΠ xy be the revenue of a single trip from x to y , which onlydepends on traffic speeds v . We can write hourly revenueoriginated on x as π x = (cid:80) y Π xy µ pxy and average revenueof a trip originated on x as Π x = π x /µ px . Assume patienceand destination are approximately uncorrelated for hailerswith the same origin, which means ∀ x, y ∈ E, µ tx ≈ µ txy .Then hailers on the same segment have an equal chanceof being picked up regardless of their destination: ∀ x ∈ E, µ pxy ∝ µ dxy , ∀ y ∈ E Thus, the average revenue for a trip originated on x onlydepends on traffic speeds and demand rates: Π x ( v , µ dx ) = (cid:80) y Π xy µ dxy /µ dx . Since drivers are assumed not to dis-criminate hailers: ∀ i ∈ N, ∀ x ∈ E, µ pixy ∝ µ pxy , ∀ y ∈ E Driver revenue originated on a segment π ix = (cid:80) y Π xy µ pixy can thus be written as π ix = (cid:80) y Π xy µ pxy µ pix /µ px = Π x µ pix . Since eachpass of a vacant taxi has an equal chance of picking up ahailer regardless of the driver: ∀ x ∈ E, µ pix ∝ µ six , ∀ i ∈ N We have π ix = Π x µ pix = Π x µ px µ six /µ sx . Driver hourlytrip revenue can thus be expressed with explicit functiondependency as: π i = (cid:88) x ∈ E π ix = (cid:88) x ∈ E Π x ( v , µ dx ) µ px ( µ sx , µ dx , µ tx ) µ six µ sx (A3)A more analytically convenient definition of driverstrategy is driver’s allocation of service time. Servicetime t ix = t six + t pix is the total time driver i spendssearching and delivering trips originated on x duringa period of time t . The rationale of using service time distribution as driver strategy instead of supply rate orsearch time is that: service time is a conserved quan-tity and identical for all drivers; meanwhile, service timeis monotonic in supply rate and preserves propertiesof the pickup rate function. Let t xy be the trip dura-tion from x to y , which only depends on traffic speeds v . The average duration of a trip originated on x is t x ( v , µ dx ) = (cid:80) y t xy µ dxy /µ dx = (cid:80) y t xy µ pxy /µ px , withreasoning similar to average trip revenue Π x . The propor-tion of time driver i spends delivering trips originated on x is thus t pix /t = (cid:80) y t xy µ pixy = (cid:80) y t xy µ pxy µ pix /µ px = t x µ pix = t x µ px µ six /µ sx , with reasoning similar to π ix .Together with Equation A2, the proportion of servicetime driver i allocates on x can thus be written as: s ix = t six + t pix t = (cid:18) l x ˜ v x + t x µ px µ sx (cid:19) µ six (A4)This shows that on each segment, driver service time isalso linearly related to driver supply rate: ∀ x ∈ E, s ix ∝ µ six , ∀ i ∈ N . From Equation A4, service time on a seg-ment s x = µ sx l x / ˜ v x + µ px t x . With pickup rate function µ px ( µ sx , µ dx , µ tx ) and constant environment condition E , pickup rate is implicitly a function of service time: µ px ( s x , E ). Each taxi driver must allocate all the servicetime among the street segments: (cid:80) x t ix = t , or equiva-lently (cid:80) x s ix = 1. The driving strategy of taxi driver i isthus s i ∈ S i , where the strategy space S i = ∆ | E |− , a sim-plex of dimension one less than the number of segments.Now we can formally write the optimization problem of ataxi driver:maximize (cid:88) x ∈ E Π x ( v , µ dx ) µ px ( s x , E ) s ix s x subject to s i ≥ s i · = 1 (A5)Now we prove that pickup rate µ px ( s x , E ) is also increas-ing, strictly concave, and arbitrarily differentiable withrespect to s x . With constant environment condition E ,the implicit function can be abstracted to z = ax + by ,where z = s x , x = µ sx , y = µ px , a = l x / ˜ v x , and b = t x ; y ( x ) is increasing, strictly concave, and arbitrarily differ-entiable, while a, b > y ( z ) is also increasing, strictly concave,and arbitrarily differentiable. Differentiability is simplypreserved by the linear relation. Since z ( x ) = ax + by ( x )is increasing, its inverse x ( z ) is thus also increasing; bycomposition, y ( z ) = y ( x ( z )) is also increasing. By im-plicit differentiation, d y/ d z = y (cid:48) ( x ) / ( a + by (cid:48) ( x )), andthus d y/ d z = ay (cid:48)(cid:48) ( x ) / ( a + by (cid:48) ( x )) . Since y (cid:48) ( x ) > y (cid:48)(cid:48) ( x ) < y (cid:48)(cid:48) ( z ) <
0, which means y ( z ) is also strictlyconcave.
2. Multi-market oligopoly
In this section we formalize the game of multi-marketcompetition among firms of equal capacity, and prove that0the game has Nash equilibrium (NE), which is symmetricand essentially unique in that marginal player payoffs areuniform across all invested markets.We use subscript x to denote a market, or product; sub-script i for a firm, or player; subscript − i for opponents offirm i . Boldface denotes a vector; single subscript indicatessummation. Conditions in parentheses are optional.Game setup of multi-market oligopoly. For firms i ∈ N , | N | = n , each distributing a unit of resources over markets x ∈ E , | E | = m :1. Total payoff in a market u x ( s x ), s x ≥ u x (0) = 0,is (increasing) non-decreasing, differentiable, and(strictly) concave;2. Payoff per investment in a market p x ( s x ) = u x /s x , s x >
0, is (decreasing) non-increasing; not necessar-ily convex;3. Player payoff in a market u ix ( s ix ; s − ix ) = p x ( s x ) s ix , s ix ∈ [0 , s − ix = (cid:80) j (cid:54) = i s jx ∈ [0 , n − u i ( s i ; s − i ) = (cid:80) x u ix ( s ix ; s − ix ), s i ∈ S i = ∆ m − , s − i = (cid:80) j (cid:54) = i s j ∈ S − i = ( n − m − ;Here ∆ m − = { v ∈ R m | v ≥ , v · = 1 } is the( m − φ x ( s x ) = p x ( s x ) + p (cid:48) x ( s x ) s x /n , or equivalently φ x ( s x ) = u (cid:48) x ( s x ) /n + (1 − /n ) u x ( s x ) /s x , is (posi-tive) non-negative, (decreasing) non-increasing;6. Potential function Φ( s ) = (cid:80) x (cid:82) s x φ x ( t ) d t , s ∈ n ∆ m − , thus Φ( s ) = (cid:80) x (cid:2) u x ( s x ) /n + (1 − /n ) (cid:82) s x u x ( t ) /t d t (cid:3) ;Multi-market oligopoly is similar to Cournot oligopoly[47], but differs in significant ways. In Cournot oligopoly,each player chooses a production level of the same prod-uct, whose marginal return decreases with total pro-duction; while the multi-market oligopoly can be seenas a multi-product Cournot game, where all playershave the same total productivity. Formally, Cournotoligopoly can be written as: G c = { N, Q, u } , where playerstrategy q i ∈ Q i = R ≥ , and player payoff function u i ( q i , q − i ) = p ( q ) q i − cq i ; price p ( q ) is a decreasing func-tion on total productivity q = (cid:80) i q i , and marginal cost c is assumed to be constant. The multi-market oligopolyinstead has m products, and each player distributes oneunit of resource s among the products, earning payofffrom all products invested.To prove that multi-market firms of the same capac-ity have a unique and symmetric NE, we follow a list ofpropositions shown below. Before getting into the details,we point out the keys to the proof: convex game guaran-tees NE exists; equal capacity leads to symmetry; andmonotonic marginal payoffs provide a unique solution. Proposition 1. Φ( s ) is (strictly) concave. Proof. Let P x ( s x ) = (cid:82) s x p x ( t ) d t . Since P x ( s x ) is a differ-entiable real function with a convex domain, it is (strictly)concave if and only if it is globally (strictly) dominatedby its linear expansions: ∀ s > , ∀ s x ≥ , s x (cid:54) = s , P x ( s x ) − [ P x ( s ) + p x ( s )( s x − s )]= (cid:90) s x s p x ( t ) d t − p x ( s )( s x − s )= (cid:90) s x s p x ( t ) − p x ( s ) d t ≤ p x ( s x ) is (decreasing) non-increasing.Because Φ( s ) is a positive linear transformation of P x ( s x )and u x ( s x ) which is also (strictly) concave, it impliesthat Φ( s ) is (strictly) concave on the non-negative cone R m ≥ . Because simplex n ∆ m − is a convex subset of thenon-negative cone R m ≥ , it implies that Φ( s ) is (strictly)concave on the simplex n ∆ m − . This proves Proposition 1. Proposition 2. u i ( s i ; s − i ) is (strictly) concave, ∀ i, ∀ s − i ∈ S − i .Proof. Because simplex S i is a convex subset of the non-negative cone R m ≥ , if u i ( s i ; s − i ) is (strictly) concave on R m ≥ , ∀ i, ∀ s − i ∈ S − i , then u i ( s i ; s − i ) is also (strictly) con-cave on S i , ∀ i, ∀ s − i ∈ S − i . It suffices to prove the formerstatement without constraints on opponent strategies: u i ( s i ; s − i ) is (strictly) concave on R m ≥ , ∀ i, ∀ s − i ∈ R m ≥ .Because u i ( s i ; s − i ) is a positive linear transformationof u ix ( s ix ; s − ix ), x ∈ E , it suffices if u ix ( s ix ; s − ix ) is(strictly) concave on R ≥ , ∀ x, ∀ i, ∀ s − ix ≥
0. To simplifynotations, this is equivalent to u ix ( s ; c ) = p x ( s + c ) s (strictly) concave on R ≥ , ∀ x, ∀ c ≥
0. This can be provedby definition, and we do not include the proof here becauseit is straightforward but tedious. The key to this proof isthat u x ( s x ) is (strictly) concave and p x ( s x ) is (decreas-ing) non-increasing; either of the optional conditions canguarantee strict concavity. This proves Proposition 2. Proposition 3.
Multi-market oligopoly is a convex game.Proof.
A convex game is a game where each player hasa convex strategy space and a concave payoff function u i ( s i ; s − i ) for all opponent strategies. In this game, playerstrategy space is the same simplex S i = ∆ m − for allplayers, which is convex. Together with Proposition 2,this is proves Proposition 3. Proposition 4.
Multi-market oligopoly has NE, (allstrict).Proof.
A convex game has NE if it has a compact strategyspace and continuous payoff functions, see [48]. Becausethe product space of simplices is compact, this game has acompact strategy space S ≡ (cid:81) i S i = (cid:81) i ∆ m − . Because u x ( s x ) is continuous ∀ x , player payoff u i ( s ) is thus contin-uous ∀ i . This game thus has NE. If u i ( s i ; s − i ) is strictlyconcave, all NEs are strict. This proves Proposition 4.1 Proposition 5.
Multi-market oligopoly can only havesymmetric NE.Proof.
Given a NE s ∗ , for all player i , equilibrium strategy s ∗ i solves the convex optimization problem:maximize u i ( s i ; s ∗− i )subject to s i ≥ s i · = 1 (A6)Since this convex optimization problem is strictly feasi-ble, by Slater’s theorem, it has strong duality. Since theobjective function u i ( s i ; s ∗− i ) is differentiable, the Karush-Kuhn-Tucker (KKT) theorem states that optimal pointsof the optimization problem is the same with the solutionsof the KKT conditions: ∇ u i + λ i − ν i = 0 (saddle point) s i ≥ λ i ≥ s i ◦ λ i = 0 (complementary slackness) s i · = 1 (primal constraint 2) (A7)Here operator ◦ denotes the Hadamard product: ( x ◦ y ) i = x i y i . Given the saddle point conditions ∂u i /∂s ix + λ ix − ν i = 0 , ∀ x , the dual constraint implies that themarginal payoff for player i in market x is bounded above: ∂u i /∂s ix ≤ ν i , ∀ x . If player i invests in market x , s ix > i are uniformin all markets i invests. Together, marginal player payoffsat equilibrium have relation: ∂u i ∂s iy ≤ ∂u i ∂s ix = ν i , ∀ i, ∀ x, y, s ∗ ix > s ∗ iy ≥ ∂u i /∂s ix = p x ( s x ) + p (cid:48) x ( s x ) s ix , and p (cid:48) x <
0, this isequivalent to p x ( s ∗ x ) ≤ ν i + | p (cid:48) x ( s ∗ x ) | s ∗ ix , ∀ x , with equalityif s ∗ ix >
0. If player i invests more in market x than player j does, s ∗ ix > s ∗ jx ≥
0, this implies ν i + | p (cid:48) x ( s ∗ x ) | s ∗ ix ≤ ν j + | p (cid:48) x ( s ∗ x ) | s ∗ jx . But because the players have the samecapacity, player i must have invested less in some market y than player j does: s ∗ jy > s ∗ iy ≥
0, which implies ν j + | p (cid:48) y ( s ∗ y ) | s ∗ jy ≤ ν i + | p (cid:48) y ( s ∗ y ) | s ∗ iy . Together, these inequalitiesimply | p (cid:48) x ( s ∗ x ) | ( s ∗ ix − s ∗ jx ) + | p (cid:48) y ( s ∗ y ) | ( s ∗ jy − s ∗ iy ) ≤
0. Thiscontradicts our assumption on player resource allocation,thus all players must have the same strategy in equilibrium.This proves Proposition 5.
Proposition 6.
Multi-market oligopoly have a (unique)essentially unique NE, in that marginal player payoffs areuniform across all invested markets.Proof.
From Proposition 4 and 5, multi-market firms haveNE, which are symmetric. For a symmetric NE s ∗ , playerstrategy s ∗ i = s ∗ /n , and marginal player payoffs in in-vested markets are the same for all players: ν i = ν, ∀ i .Now the relation among equilibrium marginal player pay-offs can be rewritten as: p x ( s ∗ x ) + p (cid:48) x ( s ∗ x ) s ∗ x /n ≤ ν, ∀ x , with equality in invested markets, s ∗ x >
0. Because φ x ( s x ) = p x ( s x ) + p (cid:48) x ( s x ) s x /n , this is equivalent to φ x ( s ∗ x ) ≤ ν, ∀ x (A9)with equality in invested markets. Since u x ( s x ) is a uni-variate differentiable (strictly) concave function, u (cid:48) x ( s x )is (decreasing) non-increasing. Because p x ( s x ) is also(decreasing) non-increasing, φ x ( s x ) = u (cid:48) x ( s x ) /n + (1 − /n ) p x ( s x ) is (decreasing) non-increasing. Define inversefunction φ − x : R ≥ → R ≥ , so that φ − x ( ν ) = 0 for ν > φ − x (0). The function is non-increasing (decreasingfor ν ≤ φ − x (0)) and the equilibrium satisfies: s ∗ x = φ − x ( ν ) , ∀ x (A10)Since total investment equals the number of players, (cid:80) x s x = n , marginal player payoff in invested markets ν is determined by: (cid:88) x φ − x ( ν ) = n (A11)Because the left-hand side of Equation A11 is decreasingfor ν ≤ max x φ − x (0) where the left-hand side is positive,the equation gives a unique solution ν . Thus Equation A10gives a (unique) essentially unique s ∗ . Figure 1d showsthis process graphically. This proves Proposition 6. Proposition 7.
The equilibrium of multi-marketoligopoly is globally asymptotically stable under gradientadjustment process.Proof.
The gradient adjustment process [33] is a heuristiclearning rule where players adjust their strategies accord-ing to the local gradient of their payoff functions, projectedonto the tangent cone of player strategy space. Formally,gradient adjustment process is a dynamical system:d s i d t = P T ( s i ) ∇ i u i ( s ) , ∀ i (A12)Here ∇ i denotes the gradient with respect to player strat-egy s i , T ( s i ) is the tangent cone of player strategy space S i at point s i , and P is the projection operator. For allinterior points of the player strategy space, P T ( s i ) is sim-ply the centering matrix, M = I − T /m . To prove thatthe dynamical system is globally asymptotically stable,we show that V ( s ) = Φ( s ∗ ) − Φ( s ) is a global Lyapunovfunction: a function that is positive-definite, continuouslydifferentiable, and has negative-definite time derivative.From Proposition 1 and KKT theorem, the maximalpoints of the potential function Φ( s ) is determined thesame way as Equation A9. That is, the maximal set of thepotential function is identical to the NE of multi-marketoligopoly. This means V ( s ) > V ( s ) is positive-definite. V ( s ) is clearly continuously2differentiable, and its time derivatived V ( s )d t = −∇ Φ( s ) · d s d t = − m (cid:88) x =1 n (cid:88) i =1 (cid:32) ∂u ix ∂s ix − m m (cid:88) y =1 ∂u iy ∂s iy (cid:33) φ x = − n m (cid:88) x =1 (cid:32) φ x − m m (cid:88) y =1 φ y (cid:33) φ x = − mn (cid:16) φ − φ (cid:17) Here φ = (cid:80) mx =1 φ x /m and φ = (cid:80) mx =1 φ x /m . Thus, φ − φ ≥
0, with equality if and only if φ x are all equal. FromEquation A9, we can see that d V / d t ≤
0, with equalityonly at equilibrium points s ∗ . We have thus shown that V ( s ) is a global Lyapunov function of the dynamicalsystem, which immediately implies Proposition 7.The stability result in Proposition 7 is only intendedto show that under a simple and plausible learning rule,global asymptotic stability of Nash equilibrium is pos-sible in multi-market oligopoly so that the equilibriumcan be empirically observed. The gradient adjustmentprocess adopted in this paper is not meant to be the exactlearning rule used in real life, which is hard to determine.But compared with Bayesian or best-response learningrules, it is less demanding on the players as it does notrequire complete information of the game or long-termmemory of the players. And even if some players adoptalternative, non-economic learning rules, the stability ofthe equilibrium may well be preserved. For example, newdrivers may simply choose imitative learning [34, 35], orin other words “follow the older drivers”. In this case theNash equilibrium is still the stable focus as all playersadopt the same strategy and the rational payoff-improvingplayers adjust to the equilibrium. By imitative learning,new drivers save the possibly long process of strategyadjustment and quickly converge to the equilibrium strat-egy. This allows the equilibrium remain stable under anevolving set of drivers.We note that the payoff in multi-market oligopoly isnot strictly diagonally concave, so the uniqueness andstability results cannot follow [49]. But the eigenvaluesof the Jacobian ∇ (d s / d t ) are always negative, so localasymptotic stability at the equilibrium is guaranteed un-der gradient dynamics with individual-specific adjustmentspeeds. We also note that unlike the Cournot game, multi-market oligopoly is not an aggregate game defined in[50] or later generalizations, because player strategies aremulti-dimensional. Thus it does not inherit the stabilityunder discrete-time best-response dynamics. Multi-marketoligopoly is also not a potential game and thus does notinherit the general dynamic stability properties in [51]. In-stead, we provided a “potential function” that is a globalLyapunov function for the gradient dynamics.
3. Inefficiency of the equilibrium
We notice that in multi-market oligopoly the totalpayoff u ( s ) = (cid:80) i ∈ N u i (cid:54) = Φ( s ), which means total payoffis generally not maximized in Nash equilibrium, thus notsocially optimal. In fact, if total payoff is maximized, then ∂u x /∂s x ≥ ∂u y /∂s y , ∀ x, y ∈ E, s x >
0, which meansmarginal payoff are the same for all invested markets.Compare with Equation A9 and the definition of φ x ( s x ),a weighted average of marginal and average payoff isbalanced instead. With n (cid:29) φ ≈ u x /s x . So at equilibrium the averagesegment revenue per service time are effectively the samefor all searched segments.This is similar to the Cournot game. The total payoffin the Cournot game is u ( q ) = (cid:80) i ∈ N u i = p ( q ) q − cq .Assuming p ( q ) differentiable, the social optimum is u ∗ =( p ( q ∗ ) − c ) q ∗ , where q ∗ satisfies p (cid:48) ( q ∗ ) q ∗ + p ( q ∗ ) = c . TheNash equilibrium is q i = q † /n, ∀ i ∈ N , where q † satisfies p (cid:48) ( q † ) q † /n + p ( q † ) = c . This makes q † > q ∗ and u ( q † ) , p ∈ (0 , u = (cid:80) x ∈ E a x s px andplayer payoff u i = (cid:80) x ∈ E a x s p − x s ix . At social equilib-rium, ∂u x /∂s x = a x ps p − x is a constant for all markets x ∈ E . Since (cid:80) x ∈ E s x = (cid:80) i ∈ N (cid:80) x ∈ E s ix = n , social op-timal strategy is s ∗ x = na / (1 − p ) x / (cid:80) y ∈ E a / (1 − p ) y , ∀ x ∈ E .At Nash equilibrium, for all players i ∈ N , let u ix = u x s ix /s x , then ∂u ix /∂s ix = a x (cid:0) ( p − s p − x s ix + s p − x (cid:1) is a constant for all markets x ∈ E . This means (cid:80) i ∈ E ∂u ix /∂s ix = ( p − n ) a x s p − x is a constant forall markets x ∈ E , which gives the same aggregate strat-egy s † x = na / (1 − p ) x / (cid:80) y ∈ E a / (1 − p ) y , ∀ x ∈ E , so the Nashequilibrium is social optimal. Use the condition again, wefind Nash equilibrium s † ix = s † x /n, ∀ i ∈ N, x ∈ E .
4. Markov strategy
Here we show the equivalence of Markov strategy andsearch time allocation vector. A Markov strategy can berepresented by a search transition matrix Q xy , a rightstochastic matrix that gives the transition probabilitiesfrom every segment x to every neighboring segment y while the driver is searching for passengers. Let P xy bethe empirical pickup transition matrix, i.e. row-normalizedtrip origin-destination matrix. Let p x be the probabilityof pickup per search on a segment x , and P x = diag { p x } .We note that P xy can be easily computed, see e.g. fig. 4a,while p x and Q xy can be computed if high resolutiontrajectory data is available. The equilibrium search timeallocation vector s ∗ and the Markov strategy Q xy satisfythe equation: s ∗ = s ∗ [ P x P xy + ( I − P x ) Q xy ] (A13)3To see this, think of an ensemble of vacant vehicles, eachsearching for pickup. After one time step, most continueto search on neighboring streets; some become occupiedand exit the ensemble, and join the ensemble again afterdrop-off. Note that the drivers are non-interacting. Since we are only counting the allocation of search time, in long-time limit each driver would have the same allocation ofsearch time, which equals the ensemble average at anytime, and satisfies the equation as written. Because p x and P xy are determined by the environment condition, Q xy uniquely determines s ∗ . [1] A. Pentland, Social physics: how good ideas spread — thelessons from a new science (Scribe Publications, 2014).[2] W. Pan, G. Ghoshal, C. Krumme, M. Cebrian, andA. Pentland, Urban characteristics attributable to density-driven tie formation, Nature Communications , 1961(2013).[3] Y. Yuan, A. Alabdulkareem, and A. S. Pentland, An in-terpretable approach for social network formation amongheterogeneous agents, Nature Communications , 4704(2018).[4] M. C. Gonzalez, C. A. Hidalgo, and A.-L. Barabasi, Un-derstanding individual human mobility patterns, Nature , 779 (2008).[5] S. Colak, A. Lima, and M. C. Gonzalez, Understandingcongested travel in urban areas, Nature Communications , 10.1038/ncomms10793 (2016).[6] L. E. Olmos, S. Colak, S. Shafiei, M. Saberi, and M. C.Gonzalez, Macroscopic dynamics and the collapse of urbantraffic, Proceedings of the National Academy of Sciences , 12654 (2018).[7] W. Weidlich, Sociodynamics: a systematic approach tomathematical modelling in the social sciences , 1st ed. (Har-wood Academic, Amsterdam, 2000).[8] W. Weidlich and G. Haag,
An integrated model oftransport and urban evolution: with an application to ametropole of an emerging nation , 1st ed. (Springer-VerlagBerlin Heidelberg, 1999).[9] D. Helbing,
Quantitative sociodynamics: stochastic meth-ods and models of social interaction processes , 2nd ed.(Springer, 2010).[10] C. Castellano, S. Fortunato, and V. Loreto, Statisticalphysics of social dynamics, Reviews of Modern Physics , 591 (2009).[11] L. M. A. Bettencourt, The origins of scaling in cities,Science , 1438 (2013).[12] M. Lee, H. Barbosa, H. Youn, P. Holme, and G. Ghoshal,Morphology of travel routes and the organization of cities,Nature Communications , 2229 (2017).[13] H. Barbosa, M. Barthelemy, G. Ghoshal, C. R. James,M. Lenormand, T. Louail, R. Menezes, J. J. Ramasco,F. Simini, and M. Tomasini, Human mobility: models andapplications, Physics Reports , 1 (2018).[14] M. Barthelemy, The statistical physics of cities, NatureReviews Physics , 406 (2019).[15] M. Mazzoli, A. Molas, A. Bassolas, M. Lenormand, P. Co-let, and J. J. Ramasco, Field theory for recurrent mobility,Nature Communications , 3895 (2019).[16] J. G. Wardrop, Some theoretical aspects of road trafficresearch, Proceedings of the Institution of Civil Engineers , 325 (1952).[17] J. W. Forrester, Urban dynamics (M.I.T. Press, Cam-bridge, Mass., 1969). [18] E. Koutsoupias and C. Papadimitriou, Worst-case equi-libria, in
STACS 99 (Springer Berlin Heidelberg, 1999)pp. 404–413.[19] T. Roughgarden,
Selfish routing , Ph.D. thesis, Departmentof Computer Science, Cornell University, Ithaca, New York(2002).[20] D. Zhang, L. Sun, B. Li, C. Chen, G. Pan, S. Li, andZ. Wu, Understanding taxi service strategies from taxiGPS traces, IEEE Transactions on Intelligent Transporta-tion Systems , 123 (2015).[21] A. C. Pigou, The economics of welfare , 1st ed. (Macmillanand Co., London, 1920).[22] F. H. Knight, Some fallacies in the interpretation of socialcost, Quarterly Journal of Economics , 582 (1924).[23] M. J. Beckmann, C. B. McGuire, and C. B. Winsten, Stud-ies in the economics of transportation (Yale UniversityPress, 1956).[24] D. E. Boyce, H. S. Mahmassani, and A. Nagurney, A ret-rospective on Beckmann, McGuire and Winsten’s Studiesin the Economics of Transportation, Papers in RegionalScience , 85 (2005).[25] E. L. Glaeser, Cities, agglomeration, and spatial equilib-rium (Oxford University Prress, 2008).[26] C. Cook, R. Diamond, J. Hall, J. A. List, and P. Oyer,
Thegender earnings gap in the gig economy: evidence fromover a million rideshare drivers , Working Paper 24732(National Bureau of Economic Research, 2018).[27] Y. Zhang, B. Li, and K. Ramayya, Learning individualbehavior using sensor data: the case of GPS traces andtaxi drivers (2020).[28] R. L. Meier, The metropolis as a transaction-maximizingsystem, Daedalus , 1292 (1968).[29] A. Wilson, Entropy in urban and regional modelling:retrospect and prospect, Geographical Analysis , 364(2010).[30] E. Barbour, C. C. Davila, S. Gupta, C. Reinhart, J. Kaur,and M. C. Gonzalez, Planning for sustainable cities by es-timating building occupancy with mobile phones, NatureCommunications , 3736 (2019).[31] R. Di Clemente, M. Luengo-Oroz, M. Travizano, S. Xu,B. Vaitla, and M. C. Gonzalez, Sequences of purchasesin credit card data reveal lifestyles in urban populations,Nature Communications , 3330 (2018).[32] R. Zhang and R. Ghanem, Demand, supply, and perfor-mance of street-hail taxi, IEEE Transactions on IntelligentTransportation Systems , 1 (2019).[33] K. J. Arrow and L. Hurwicz, Stability of the gradientprocess in n-person games, Journal of the Society forIndustrial and Applied Mathematics , 280 (1960).[34] A. E. Roth and I. Erev, Learning in extensive-form games:experimental data and simple dynamic models in theintermediate term, Games and Economic Behavior , 164(1995). [35] D. Fudenberg and D. K. Levine, Learning and equilibrium,Annual Review of Economics , 385 (2009).[36] P. J. Smith, D. S. Rae, R. W. Manderscheid, and S. Sil-bergeld, Approximating the moments and distribution ofthe likelihood ratio statistic for multinomial goodness offit, Journal of the American Statistical Association ,737 (1981).[37] J. S. Mill, On liberty (J. W. Parker and Son, 1859).[38] H. Sidgwick,
Principles of political economy (Macmillanand co, 1883).[39] J. E. Meade, External economies and diseconomies in acompetitive situation, Economic Journal , 54 (1952).[40] H. S. Gordon, The economic theory of a common-propertyresource: the fishery, Journal of Political Economy ,124 (1954).[41] F. M. Bator, The anatomy of market failure, QuarterlyJournal of Economics , 351 (1958).[42] R. H. Coase, The problem of social cost, Journal of Lawand Economics , 1 (1960).[43] R. Zhang, New York City taxi trip records, 2009-2013(2019).[44] B. Schaller, The New York City taxicab fact book , Tech.Rep. (Schaller Consulting, 2006).[45] S. C. Davis, S. W. Diegel, and R. G. Boundy,
Transporta-tion energy data book (Oak Ridge National Laboratory,2017). [46] L. Balafoutas, R. Kerschbamer, and M. Sutter, Second-degree moral hazard in a real-world credence goods mar-ket, Economic Journal , 1 (2015).[47] A. A. Cournot,