Incentive-driven transition to high ride-sharing adoption
IIncentive-driven discontinuous transition to high ride-sharing adoption
David-Maximilian Storch, Marc Timme,
1, 2 and Malte Schr¨oder ∗ Chair for Network Dynamics, Institute for Theoretical Physics and Center for Advancing Electronics Dresden (cfaed),Technical University of Dresden, 01062 Dresden, Germany Lakeside Labs, Lakeside B04b, 9020 Klagenfurt, Austria (Dated: August 26, 2020)Ride-sharing – the combination of multiple trips into one – may substantially contribute towardssustainable urban mobility. It is most efficient at high demand locations with many similar triprequests. However, here we reveal that people’s willingness to share rides does not follow thistrend. Modeling the fundamental incentives underlying individual ride-sharing decisions, we findtwo opposing adoption regimes, one with constant and one with decreasing adoption as demandincreases. In the high demand limit, the transition between these regimes becomes discontinuous,switching abruptly from low to high ride-sharing adoption. Analyzing over 360 million ride requestsin New York City and Chicago illustrates that both regimes coexist across the cities, consistentwith our model predictions. These results suggest that current incentives for ride-sharing may benear the boundary to the high-sharing regime such that even a moderate increase in the financialincentives may significantly increase ride-sharing adoption.
INTRODUCTION
Sustainable mobility [1–6] is essential for ensuring socially, economically and environmentally viable urban life[7, 8]. Ride-sharing constitutes a promising alternative to individual motorized transport currently dominating urbanmobility [9]. Recent analyses suggest that large-scale ride-sharing is specifically suited for densely populated urbanareas [9–13]. By combining two or more individual trips into a shared ride served by a single vehicle, ride-sharingincreases the average utilization per vehicle, reduces the total number of vehicles required [11] and thereby mitigatescongestion and environmental impacts of urban mobility [14]. Hence, embedding ride-sharing for trips that wouldotherwise be conducted in a single-occupancy motorized vehicle, is preferable from a systemic perspective.Previous research focused on algorithms to implement large-scale ride-sharing [15] as well as the potential efficiencygains derived from aggregating rides [9, 12, 13]. Generally, matching individual rides into shared ones without largedetours becomes easier with more users, increasing both the economic and environmental efficiency as well as theservice quality of the ride-sharing service [12, 13, 16]. Yet, if and under which conditions people are actually willing to adopt ride-sharing remains elusive [17–22]. In particular, it is unclear how to encourage an ever growing numberof ride-hailing users to choose shared rides over their current individual mobility options [23–25].In this article, we disentangle the complex incentive structure that governs ride-hailing users’ decisions to sharetheir rides – or not. In a game theoretic model of a one-to-many demand constellation we illustrate how the inter-actions between individual ride-hailing users give rise to two qualitatively different regimes of ride-sharing adoption:one low-sharing regime where the adoption decreases with increasing demand and one high-sharing regime where thepopulation shares their rides independent of demand. Analyzing ride-sharing decisions from approximately 250 millionride-requests in New York City and an additional 110 million in Chicago suggests that both adoption regimes coexistin these cities, consistent with our theoretical predictions. Our findings indicate that current financial incentives arenearly, but not fully, sufficient to stimulate a transition towards the high-sharing regime.
RESULTSContrasting ride-sharing adoption
Currently, only a small fraction of people adopts ride-sharing even in high-demand situations, despite all its positiveaspects [26]. For example, among more than 250 million ride-hailing requests served in New York City in 2019 lessthan 18% were requests for shared transportation [27]. Moreover, the city’s ride-sharing activity varies strongly acrossdifferent parts of the city, in particular at locations with a high number of ride-hailing requests (see Fig. 1): Forinstance, in the East Village and Crown Heights North the fraction of shared ride requests is relatively high, whileit is low at both John F. Kennedy and LaGuardia airports, locations that would intuitively be especially efficientfor sharing rides. Several other location throughout New York City as well as Chicago exhibit similarly contrasting a r X i v : . [ phy s i c s . s o c - ph ] A ug total requestsshared requests 11.3 / min1.1 / min JFK Airport total requestsshared requests 12.1 / min2.7 / min
East Village total requestsshared requests 10.4 / min2.5 / min
Crown Heights North total requestsshared requests 12.1 / min1.1 / min
LaGuardia Airportfraction of shared ride requests a bc d
FIG. 1.
Contrasting ride-sharing adoption despite high request rate in New York City . Fraction of shared riderequests from different origins (red) served by the four major for-hire vehicle transportation service providers in New York Cityby destination zone (January - December 2019) [27]. Gray areas were excluded from the analysis due to insufficient data (seeMethods). The fraction of shared ride requests differs significantly by origin and destination, even though the average overallrequest rate is similar for all four origin locations. a,b
Some areas, such as East Village and Crown Heights North, show ahigh adoption of ride-sharing services. c,d
Despite a similarly high request rate, other locations, such as JFK and LaGuardiaairports, show a significantly lower adoption of ride-sharing services with a complex spatial pattern across destinations. ride-sharing adoption (see Supplementary Information for details). These findings hint at a complex interplay of urbanenvironment, demand structure and socio-economic factors that govern the adoption of ride-sharing. To disentanglethese complex interactions, we introduce and analyze a game theoretic model capturing essential features of ride-sharing incentives, disincentives as well as topological demand structure. ab $ s h a r e s i n g l e D e t o u r I nconven i ence + sharesingle OR ? E [ u single ] E [ u share ] Utility u D i s c o un t FIG. 2.
Trade-offs between incentives determine the decision to share a ride, or not. a
Shared rides offeradvantages and disadvantages compared to single rides. On the one hand, they offer financial discounts typically proportionalto the distance of a direct single ride (blue, dotted). On the other hand, rides shared with strangers may be inconvenient dueto other passengers in the car (e.g. loss of privacy or less space, green) and may include detours compared to a direct trip topickup or deliver these other passengers (orange, solid compared to dotted). b The decision to book a shared ride depends onthe balance of all three factors. If the expected utility difference E [∆ u ] = E [ u share ] − E [ u single ] between a shared and a singleride is positive, the financial discounts overcompensate detour and inconvenience effects; users share. If E [∆ u ] is negative (asillustrated), users prefer to book single rides. Ride-sharing incentives
The decision of ride-hailing users to request a single or a shared ride reflects the balance of three fundamentalincentives (Fig. 2) [18, 22]:
Discounts.
Ride-sharing is incentivized by financial discounts granted on the single ride trip fare, partially passingon savings of the service cost to the user. Often, these discounts are offered as percentage discounts on the total faresuch that the financial incentives u sharefin > d single of the requested ride, u sharefin = (cid:15) d single , where (cid:15) denotes the per-distance financial incentives. In many cases, these discounts are also grantedif the user cannot actually be matched with another customer into a shared ride [28, 29]. Detours.
Potential detours d det to pickup or to deliver other users on the same shared ride discourage sharing.The magnitude of this disincentive u sharedet < d det . Inconvenience.
Sharing a ride with another user may be inconvenient due to spending time in a crowded vehicleor due to loss of privacy [18, 20, 21]. This disincentive u shareinc < d inc users ridetogether.In the following we take u sharedet ∼ d det and u shareinc ∼ d inc , describing the first order approximation of these disincentivesand matching the linear scaling of the financial incentives with the relevant distance or time.These incentives for a shared ride describe the difference ∆ u in utility compared to a single ride or another modeof transport. The overall utility of a shared ride is then given by u share = u single + ∆ u (1)= u single + u sharefin + u sharedet + u shareinc = u single + (cid:15) d single − ξ d det − ζ d inc (2)where the utility u single for a single ride describes the benefit of being transported, as well as the cost and time spent onthe ride. The factors (cid:15) , ξ and ζ denote the user’s preferences. By rescaling the utilities (measuring in monetary units), (cid:15) directly denotes the relative price difference between single and shared rides whereas ζ and ξ quantify the importanceof inconvenience and detours relative to the financial incentives (see Supplementary Information for details).For a given origin-destination pair with fixed single ride distance d single , financial incentives are constant for agiven discount factor (cid:15) . In contrast, detour and inconvenience contributions depend on the destinations and sharingdecisions of other users. Their magnitude depends on where these users are going and on the route the vehicle istaking for a shared ride (see Methods). The decision to share a ride is determined by the expected utility difference(see Fig. 2) E [∆ u ] = E [ u share ] − E [ u single ] (3)where E [ · ] signifies the expectation value over realizations of other users’ destinations and sharing decisions conditionalon one’s own sharing decision. Ride-sharing coordination game on networks
To understand how these incentives determine the adoption of ride-sharing, we study sharing decisions in a styl-ized city network [30] with a common origin o (e.g., from a central downtown location) in the center and multipledestinations d (illustrated in Fig. 3). Two rings define urban peripheries equidistant from the city center. Branchesrepresent cardinal directions of destinations. Requests for shared rides will only be matched along adjacent branches,if the shared ride reduces the total distance driven to deliver the users and to return to the origin compared tosingle rides (see Methods). Pairing at most two users who request a shared ride, the problem of matching sharedride requests reduces to a minimum-weight-matching with an efficient solution, eliminating the influence of heuristicmatching algorithms [13, 15] (see Methods for details).In this one-to-many setting, users requesting a shared ride would only share a ride if they make their requestswithin some small time window τ . Therefore, we consider a game with S = s τ users travelling to a uniformly chosendestination location, where s denotes the average request rate. These users have the option to book a single ride ora shared ride at discounted trip fare. Their decision to share depends on their expected utility difference E [∆ u ( d )][Eq. (3)], now depending on their respective destination d . From the utility differences E [∆ u ( d )], we compute theequilibrium sharing probabilities π ∗ ( d ) with which users from destination d adopt ride-sharing to maximize theirexpected utility (see Methods for details).At fixed discount (cid:15) and preferences ζ and ξ ride-hailing users may decrease their overall adoption of ride-sharing (cid:104) π ∗ (cid:105) as the total number S of users increases (see Fig. 3a, blue), even though ride-sharing becomes more efficient withhigher user numbers. Here (cid:104)·(cid:105) denotes the average over all destinations d . While for small request rates everybodyis requesting shared rides (Fig. 3b), a distinctive sharing/non-sharing pattern emerges along the branches of the citynetwork upon higher demand (Fig. 3c,d), before the adoption of ride-sharing eventually fades out for high requestrates, S (cid:29) S , the probability p match ( d ) for a userwith destination d to be matched with other users is low (see Fig. 3a, gray). Consequently, the expected detour E [ d det ( d )] = p match ( d ) E [ d det ( d ) | match] is also small (analogously for the inconvenience). As illustrated in Fig. 3b,bottom, financial incentives outweigh the expected disadvantages of ride-sharing such that everybody is requestingshared rides, π ∗ ( d ) = 1 for all destinations d , but is only rarely matched with another user. As the number of users S increases, the provider can pair ride requests more efficiently given constant sharing decisions, ∂p match ( d ) /∂S >
0, re-sulting in more requests that are actually matched with another user (see Fig. 3a). Consequently, the expected detourand inconvenience also increase. However, instead of reducing the average adoption of ride-sharing homogeneouslyacross all destinations, neighboring destinations adopt opposing sharing strategies (see Fig. 3b). In this sharing pat-tern, only destinations in identical cardinal direction can and will be matched into a shared ride, minimizing thedetours for shared requests and simultaneously disincentivizing other users to start sharing due to high expecteddetours (Fig. 3c-e bottom). As the number of users S increases further, the probability p match ( d ) would also increaseat given sharing adoption π ( d ). This leads to an adoption of mixed sharing strategies where the financial discounts S = 2 S = 30 I n c e n t i v e s Inner:share Inner:share Outer:share Outer:share
Inner:single Inner:share Outer:single Outer:share Inner:single Inner:mixed Outer:single Outer:mixed Inner:single Inner:mixed Outer:single Outer:mixed Sharingfraction * Willingness to share Matched riderequests a S = 12 S = 4 b c d e Total requests S Financial Detour Inconvenience
FIG. 3.
Adoption of ride-sharing decreases with request rate.
In a stylized city topology (panels b - e ) users requesttransportation from a single origin (gray) to destinations in the city periphery homogeneously (results are robust for alternativesettings, see Supplementary Information). a The global equilibrium adoption of ride-sharing decreases as the number of usersincreases (blue) while the number of actually shared rides becomes constant (gray). The kink for S = 3 is an artefact relatedto the small and odd number of requests and matching of exactly two requests per vehicle such that one request can neverbe paired (see Supplementary Information for details). b-e As the number of users increases a sharing/non-sharing patternemerges around the origin (top), resulting from the equilibrium incentive balance (bottom) and possible matching constellations:Requests for shared rides are only matched when travelling to the same or to neighboring branches when the combined tripand return is shorter than the sum of individual trips. With few requests ( S = 2, panel b ), all users request a shared ride.The expected detour and inconvenience is small since it is unlikely to be matched with another user. As the number of usersincreases ( S = 4, panel c ), half of the destinations stop sharing in an alternating sharing/non-sharing pattern around the origin.In this configuration, users requesting a shared ride never suffer any detour while users that do not share are disincentivizedfrom doing so due to their high expected detour (compare bottom part of panel c ). For high numbers of users ( S = 12 and30, panels d and e ), the probability to be matched with another user when requesting a shared ride increases and the financialincentives cannot fully compensate the expected inconvenience. The adoption of ride-sharing decreases until the financialincentives exactly balance the expected inconvenience (panels d and e , bottom). Illustrated here for financial discount (cid:15) = 0 . ζ = 0 . ξ = 0 . and the expected inconvenience are exactly in balance (Fig. 3d and e). Further numerical simulations demonstratethat this transition robustly exists also for heterogeneous demand distribution across the destinations and differentorigin locations within the network (see Supplementary Information).Naturally, if the discount (cid:15) is sufficiently large such that the financial incentives completely compensate the expectedinconvenience, (cid:15) > ζ , all users share also in the high request rate limit, S → ∞ . In this limit, d single = d inc as detoursdisappear, E [ d det ] →
0, due to an abundance of similar requests.Figure 4a-b summarizes these results in a phase diagram for the ride-sharing decisions as a function of financialdiscount (cid:15) and number of users S , illustrating under which conditions the users adopt ride-sharing (high-sharingregime) and under which conditions the users only share partially or not at all (low-sharing regime).For fixed values of financial discounts (cid:15) , different behavior emerges for different inconvenience preferences ζ . If ζ issufficiently small (Fig. 4a), the system is in the high-sharing phase and the number of users requesting a shared ride is S share = S . Otherwise, the system switches from the high- to the low-sharing state (Fig. 4b, compare Fig 3). Figure 4cillustrates the scaling of S share in both states as S increases. In the partial sharing state, S share becomes constantfor large S , such that S share /S → S → ∞ (compare Fig. 3a), implying a discontinuous phase transition betweenlow-sharing and high-sharing regimes for large S when the financial incentives exactly balance the inconvenience, (cid:15) c /ζ c = 1 (see Supplementary Information for details). S S = 0.3, = 0.215 ca b Sharingfraction0.00.20.40.60.81.0
High-sharing Low-sharing S T o t a l s h a r e d r i d e s S S h a r e = 0.155= 0.215 High-sharingLow-sharing FIG. 4.
Two qualitatively different regimes of ride-sharing adoption. a,b
Phase diagram of fraction of sharedrides S share /S for different inconvenience preferences ζ . Ride-sharing is adopted dominantly if the financial discount (cid:15) fullycompensates the expected inconvenience (high-sharing, dark blue). Otherwise, the total number of shared ride requests saturatesand the overall adoption of ride-sharing decreases with increasing number of users S (low-sharing, compare Fig. 3a). In thelimit of an infinite number of requests S → ∞ the transition becomes discontinuous (see Supplementary Information). c Withidentical financial discounts (cid:15) = 0 .
2, different sharing behavior emerges for different inconvenience preferences ζ . When ζ < (cid:15) all users request shared rides ( S share = S , dark blue triangles, red line in panel a ). When ζ > (cid:15) the system is in a low-sharingregime where users request shared rides at low numbers of users S but the number of shared ride requests saturates and becomesconstant at high S ( S share < S , light green triangles, red line in panel b ). In the low-sharing regime, spatially heterogeneouspatterns of ride-sharing adoption emerge (compare Fig. 3b-e). Ride-sharing activity in New York City and Chicago
In a real city with heterogeneous preferences across different locations and constant financial discounts (cid:15) , thesharing decisions may, on an aggregate level, appear to be in a hybrid state between the high- and low-sharingphases predicted by our model. Indeed, the ride-sharing adoption across different origin locations in New York Cityand Chicago, illustrated in Fig. 5, matches the qualitative sharing behaviors at different preferences in our model(compare Fig. 4c). At locations with a low request rate s , the fraction of shared ride requests increases linearlywith more requests, s share ∼ s . At high request rates, sharing decisions differ by origin zone (compare Fig. 1): forCrown Heights North and East Village the linear scaling prevails, indicating (cid:15) is sufficiently large to compensate theexpected inconvenience and detour effects completely. In fact, the spatial pattern of fraction of rides shared appearsto be largely homogeneous across destinations as expected in this state (Fig. 5a). Other origins with a similarly highrequest rate, such as JFK and LaGuardia airports, accumulate on a horizontal line with a constant number of sharedride requests per time. For these zones s share has saturated for the given financial incentives and will not increasewith higher request rate. The sharing decisions in these locations are spatially heterogeneous across the city (Fig. 5c),consistent with the low-sharing state observed in our model (compare Fig. 3). Together with Fig. 4a and b, theseobservations suggest that financial incentives in New York City are at the phase boundary between the high- andlow-sharing regime and slightly higher discounts may significantly increase sharing in some areas. EastVillageCrownHeightsNorth UnionSq
LGAJFKTimesSqWestVillageMidtownNorthAlphabetCity PennStationLenox HillEast ClintonEast a New York City s S h a r e [ r i d e s / m i n ] O'HareLogan SquareNear South SideUptownGar fi eld RidgeHyde ParkLower West SideAustin d Chicago s S h a r e [ r i d e s / m i n ] s [rides/min] s [rides/min] LGAEast Village fraction of shared ride requests
O'HareAustin fraction of shared ride requests b c ef
FIG. 5.
Ride-sharing adoption in New York City and Chicago is consistent with the predicted high- and low-sharing regimes. a,b
Sharing decisions for New York City and Chicago (blue dots) accumulate on two branches correspondingto the predicted high- and low-sharing regime as a function of request rate (compare Fig. 4). At low request rates, the numberof requests for shared rides increases linearly with the total number of requests (compare red diagonal). At high request rates,the sharing decisions differ between locations (compare Fig. 1 and 4, see also Supplementary Information). As inconveniencepreferences ζ are naturally heterogeneous in the cities, adoption is in a hybrid low/high-sharing state. c,d For origins in thehigh-sharing state a spatially homogeneous pattern of ride-sharing adoption emerges across destinations. e,f
For origins in thelow-sharing state a spatially heterogeneous pattern forms. The agglomeration of most data points on the high-sharing branchfor New York City suggests that the financial discounts (cid:15) are close to the boundary of the high-sharing phase. However, theslope of the high-sharing branch indicates that only about 20% of ride-hailing users consider ride-sharing as an option. Whileabout 40% of requests are shared in the high-sharing regime in Chicago, this potential is largely not realized. Most data pointsat locations with high request rates accumulate on the horizontal line representing the low-sharing regime. Seven large andbusy zones in Chicago with up to 50 requests per minute (not shown) fall in between the high- and low-sharing state (seeSupplementary Information for details).
An analysis for the ride-sharing adoption across more than 110 million trips in Chicago (see Methods and Sup-plemental Material for details) shows similar results (Fig. 5d), highlighting the existence of the low-sharing regime(horizontal branch s share = const . ).Even in the high-sharing regime, s share ∼ s , the ride-sharing adoption in New York City and Chicago (correspondingto the slope of the diagonal branch in Fig. 5a,b) is below 100%. In terms of our ride-sharing game, the remainingfraction of requests for single rides corresponds to a user group that does not consider ride-sharing as a potentialoption at all and, hence, is not captured by our model. DISCUSSION
The adoption of ride-sharing is governed by the complex interplay between demand patterns, matching algorithms,available transportation services, urban environments and the relevant incentive structure. Incentives may includefinancial savings potentials, detour or delay preferences, various types of inconveniences, as well as sustainability,security and uncertainty [18, 20–22]. We have introduced a model capturing essential incentives for and against ride-sharing, predicting two qualitatively different regimes of ride-sharing adoption consistent with an analysis of 360 millionride-sharing decisions from New York City and Chicago. A basic model includes three core incentive types: financialbenefits, potential detours (and thus effectively slower service) and other inconveniences such as reduced privacyresulting from sharing a vehicle. This setting may already reflect many additional factors influencing ride-sharingadoption on an aggregate level. For example, sustainability or uncertainty preferences to first approximation scale withthe additional distance driven and may thus be incorporated into the detour preferences. Similarly, alternative publictransport options may be captured by modifying the effective financial discount and relative inconvenience preferencesfor individual destinations. As such, we expect the qualitative dynamics to be robust even in more detailed settingstaking into account additional conditions (compare Supplementary Information for different demand distributions).Specific, district-level policy recommendations naturally require a more detailed description of the traffic conditionsand alternative transport options, capturing all the above-mentioned dependencies.In particular, we predict the existence of two distinct regimes of ride-sharing adoption. For sufficiently strongfinancial incentives, the number of shared ride requests increases linearly with increasing demand. However, if thefinancial incentives are weak compared to inconvenience disincentives, the number of requests for shared rides saturateswith increasing demand (regime of low ride-sharing adoption). This observation is independent of the choice of originlocations or the specific demand distribution (see Supplementary Information) and stands in stark contrast to theincreasing shareability of rides with high demand [9, 12, 13]. In the limit of large demand, the transition betweenthe two regimes becomes discontinuous, switching abruptly from the low adoption to the high adoption regime witha small change of the incentives.Ride-sharing adoption observed across New York City and Chicago is consistent with these predictions and demon-strates that both regimes exist across the cities. The data suggest that even a moderate increase of financial incentivesmay strongly improve ride-sharing adoption in some areas currently in the low-sharing regime. Still, the overall lowfraction of shared ride requests, even in the high-sharing regime, suggests that an additional societal change towardsacceptance of shared mobility is required [31] to make the full theoretical potential of ride-sharing accessible [9, 12]. Acarefully designed incentive structure for ride-sharing users adapted to local user preferences is essential to drive thischange and to avoid curbing user adoption or stimulating unintended collective states [32, 33]. This is particularlyrelevant in the light of increasing demand as urbanization progresses [1].Overall, the approach introduced above can serve as a framework to work towards sustainable urban mobility byregulating and adapting incentives to promote ride-sharing in place of motorized individual transport.
METHODS
New York City ride-sharing data . We analyzed trip data of more than 250 million transportation servicerequests delivered through high-volume For-Hire Vehicle (HVFHV) service providers in New York City in 2019. Thedata is provided by New York City’s Taxi & Limousine Commission (TLC) [27] and consists of origin and destinationzone per request, pickup and dropoff times, as well as a shared request tag, denoting a request for a single or sharedride. We compute the average request rate across all data throughout 2019 taking 16 hours of demand per day as anapproximate average.For fixed origin-destination pairs we determine the sharing fraction as the ratio of the total number of sharedride requests and the total number of requests. Departure and destination zones represent the geospatial taxi zonesdefined by TLC [27]. However, we exclude zones without geographic decoding, nor name tag defined by TLC. Foreach individual analysis, we fix the destination zone and compute the fraction of shared rides to destination zones.For a given departure zone, if the total number of requests is less than 100 trips in the considered time interval anddestination zone, we exclude that destination zone from the analysis to avoid excessive stochastic fluctuations (seeSupplementary Information for details).
Chicago ride-sharing data . We additionally analyzed more than 110 million trips delivered by three serviceproviders in Chicago in 2019. The data is provided through the City of Chicago’s Open Data Portal and contains,amongst others, information of trip origin, destination, pickup and dropoff times as well as information whether ashared ride has been authorized [34]. We restrict ourselves to geospatial decoding of the city’s 77 community areas,as well as trips leaving or entering the official city borders. In analogy to New York City, we compute the averagerequest rate across all data for 2019 taking 16 hours of demand per day as an approximate average reference time andrepeat the analysis explained for New York City.
City topology . For our ride-sharing model we construct a stylized city topology that combines star and ringtopology [30]. Starting from a central origin node, rides can be requested to 12 destinations distributed equally acrosstwo rings of radius 1 (inner ring) and 2 (outer ring), as depicted in Figure 3. The distances between neighboringnodes on the same branch are set to unity. Correspondingly, the distances between neighboring nodes are π/ π/ Ride-sharing adoption . We compute the equilibrium state of ride-sharing adoption by evolving the adoptionprobabilities π ( d, t ) following discrete-time replicator dynamics [35, 36] π ( d, t + 1) = r ( d, t ) π ( d, t ) , (4)where the reproduction rate r ( d, t ) at destination d and time t is r ( d, t ) = E [ u share ( d, t )] E [ u ( d, t )] = u single ( d ) + E [∆ u ( d, t )] u single ( d ) + π ( d, t ) E [∆ u ( d, t )] (5)and E [ X ] represents the expectation value of random variable X .We prepare the system in an initial state π ( d,
0) = 0 .
01 of ride-sharing adoption for all destinations d and set aconstant utility of a single ride u single ( d ) = 10 to ensure positivity of Eqn. (5). To evolve Eqn. (4), we numericallycompute E [ u share ( d, t )] = E [ u ( d, t ) | share] at each replicator time step t : We generate n = 100 samples of ride requestsof size S of which at least one goes to destination d and requests a shared ride. The other S − π ( d (cid:48) , t ) at their respective destination d (cid:48) at time t . Shared ride requests are matched pairwise (see below).From these n = 100 game realizations, we compute the conditional expected utility of sharing. We repeat thisprocedure for all destinations d and then update all probabilities π ( d, t ) according to Eqn. (4).Before performing measurements on the system’s equilibrium observables, we discard a transient of 1500 replicatortime steps, corresponding to 150000 game realizations per destination. We then measure the average adoption for1000 replicator time steps, representing a proxy for the stationary solution π ∗ ( d ) of Eqn. (4) and plotted as the sharingfraction in Figs. 3 and 4 (see Supplementary Information for details). Matching . Each request set of size S decomposes into single and shared ride requests. We realize the optimalpairwise matching of requests as follows: For shared requests we construct a graph whose nodes correspond torequests and edges encode the distance savings potential of matching the two requests. To determine the distancesavings potential we assume that, independent of single or shared ride, the provider has to return to the origin of thetrip.After constructing the shared request graph we employ the ’Blossom V’ implementation of Edmond’s Blossomalgorithm to determine the maximum weight matching of highest distance savings potential [37]. The matching0determines the routing and the realization of inconvenience and detour (see Supplementary Information for moredetails). ACKNOWLEDGEMENTS
We thank the Network Science Group from the University of Cologne and Nora Molkenthin for helpful discussionsand Christian Dethlefs for help with simulations. D.S. acknowledges support from the Studienstiftung des DeutschenVolkes. M.T. acknowledges support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG)through the Center for Advancing Electronics Dresden (cfaed).
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In the Main Manuscript of this article we disentangle the incentive structure of urban ride-sharing and demonstratehow it leads to emergence of two qualitatively different regimes of sharing adoption. A game theoretic model reproduceskey features of the ride-sharing activity in New York City and Chicago, including spatially heterogeneous patternsof ride-sharing adoption, saturation in the number of shared rides upon increasing demand and provides insights onthe underlying mechanisms. This Supplementary Information provides additional details on the model, methods andresults presented, and is structured as follows:
Supplementary Note 1. Ride-Sharing adoption in New York City–
Origin-destination demand – Spatiotemporal ride-sharing activity in New York City – Scaling properties of shared ride requests – Spatial patterns of ride-sharing adoption in high and saturated sharing phases
Supplementary Note 2. Ride-Sharing adoption in ChicagoSupplementary Note 3. Ride-sharing anti-coordination game on networks–
Urban environment – Replicator dynamics – Expected detour and inconvenience – The case S = 3 Supplementary Note 4. Robustness of phases of ride-sharing adoption–
Ride-sharing adoption for non-homogeneous origin-destination demand – Ride-sharing adoption for decentral origin
Supplementary Methods–
Data acquisition, structure and treatment – Numerical simulations of ride-sharing anti-coordination games3
Supplementary Note 1. Ride-sharing adoption in New York City
In 2019, four high-volume for-hire vehicle (HVFHV) companies (Uber, Lyft, Via, Juno) served more than 250 milliontransportation service requests in New York City, corresponding to approximately 700000 trips per day conductedby a population of 8.4 million people [38, 41]. In this Supplementary Note, we unveil the spatiotemporal demandpatterns underlying this macroscopic number of transportation requests.
Origin-destination demand
The flux matrix W (∆ t ) formalizes the spatiotemporal demand for transportation services between different locationsof an urban environment. Its entries W o,d denote the number of transportation requests originating at location o andgoing to location d within a specific time window ∆ t . W (∆ t ) decomposes into W (∆ t ) = W single (∆ t ) + W shared (∆ t ) (6)where W single (∆ t ) and W shared (∆ t ) are the flux matrices describing trip requests tagged as single or shared rides,respectively. We define the fraction of rides shared as the relative ratio P o,d (∆ t ) = W shared o,d (∆ t ) W single o,d (∆ t ) + W shared o,d (∆ t ) = W shared o,d (∆ t ) W o,d (∆ t ) , if W o,d (∆ t ) > w min (7)of shared rides to absolute number of rides. Note that Eqn. (7) is only defined if the total flux between origin anddestination exceeds a threshold w min to reduce bias from fluctuations in statistical analyses of P o,d (∆ t ). Spatiotemporal ride-sharing activity in New York City
We determine W single , W shared and P for the 265 taxi zones in New York City from the New York City Taxi &Limousine Commission’s (TLC) HVFHV aggregate trip records between January and December 2019 (compare Fig. 1in the Main Manuscript, see Data section in Supplementary Methods for more details on data acquisition, structureand treatment). If not stated otherwise, we choose a value w min = 100.Supplementary Figure 6 illustrates P o,d for the four origin zones o with highest average demand for transportationservices, (cid:80) d =1 d (cid:54) = o W o,d , for three different time windows ∆ t : morning (6-10 am) including commuting hours (Supple-mentary Fig. 6a), midday (12-4 pm) reflecting off-peak afternoon hours (Supplementary Fig. 6b) and evening (6 pm -3 am) encompassing leisure activity hours (Supplementary Fig. 6c). Independent of daytime, all four origins exhibitcomplex spatial patterns of ride-sharing adoption across destinations.For JFK and LaGuardia airport these patterns are robust for all time windows, indicating stable fraction of ridesshared to all destinations throughout the day. For Crown Heights North and East Village only few rides are undertakento far distance destinations in the morning and midday time window (gray areas representing W o,d < w min ). In theevening, more rides are requested overall, also to far distance destinations. Overall, the qualitative patterns of ride-sharing adoption do not vary significantly with the time of day (compare Fig. 1 in the Main Manuscript).Across the full set of origin zones in New York City, Supplementary Figure 8 suggests an overall trend to higherabsolute demand for transportation services in the evening. The fraction of shared rides, however, is not affected bythis trend. It is approximately constant throughout the day as illustrated in Supplementary Figure 7. The averagestandard deviation of fraction of rides shared across all taxi zones is less than 1.9% between the three time windows,suggesting an equilibrated system.An aggregate analysis will naturally be dominated by the high overall demand in the evening and night time.Still, the data suggests that the average ride-sharing adoption in New York City is stable across the day. Hence, anaggregate analysis is representative. Scaling properties of shared ride requests
Supplementary Figure 8 illustrates a dominant linear scaling in the total number of ride requests S and the numberof shared requests S share across origin zones in New York City for all time windows (morning, midday, evening). Such4 fraction of shared ride requests a b total requestsshared requests 6.7 / min1.4 / min total requestsshared requests 6.0 / min0.7 / min JFK Airport total requestsshared requests 4.4 / min0.4 / min
LaGuardia Airport total requestsshared requests 5.8 / min1.5 / min
East Village total requestsshared requests 6.1 / min1.2 / min total requestsshared requests 6.9 / min1.7 / min total requestsshared requests 8.0 / min0.7 / min total requestsshared requests 10.0 / min0.9 / min c total requestsshared requests 9.5 / min2.1 / min total requestsshared requests 10.3 / min2.3 / min total requestsshared requests 10.3 / min0.9 / min total requestsshared requests 11.4 / min1.0 / min Crown Heights North
FIG. 6.
Intraday variation of ride-sharing behavior in New York City a-c , Morning (6-10 am), midday (12-4 pm) andevening (6 pm - 3 am) average adoption of ride-sharing for the four most requested origin zones of ride-hailing activity in NewYork City (red). Gray zones indicate insufficient total number of requests and are excluded from the analysis. a Intraday ride-sharing adoption b Intraday variation in ride-sharing adoption F r a c t i o n o f r i d e s s h a r e d MorningMiddayEvening S t a n d a r d d e v i a t i o n o ff r a c t i o n o f r i d e s s h a r e d FIG. 7.
Stable ride-sharing adoption intraday in New York City . (left) During morning, midday and evening timewindows the fraction of shared rides is approximately the same for the different origin zones. As before, origin-destination pairswith less than w min total requests have been excluded from the analysis. (right) The standard deviation of the fraction of ridesshared across the three time intervals hints at average intraday fluctuations of less than 2% in the fraction of rides shared (reddashed line). a linear scaling between S share and S indicates sufficient financial incentives to compensate the expected negativeeffects of ride-sharing with increasing demand. A decrease in slope and eventual saturation corresponds to a situationwhere financial incentives, expected detour and inconvenience are in balance. S share will not increase upon higherdemand for given incentives (compare Fig. 1 in Main Manuscript as well as large S regime in Supplementary Fig. 8). a Morning (6-10 am) b Midday (12-4 pm) S [rides/min]0.00.51.01.5 S s h a r e [ r i d e s / m i n ] EastVillage LGAJFKLenox HillEast CrownHeightsNorth S [rides/min]0.00.51.01.5 S ss h a r e [ r i d e s / m i n ] S [rides/min]012 S s h a r e [ r i d e s / m i n ] Evening (6 pm- 3 am) c FIG. 8.
Intraday sharing decisions in New York City reflect ride-sharing adoption close to full-sharing regime .During morning, midday and evening time windows the number of shared rides scales approximately linear in the total numberof ride requests, indicating sufficient financial incentives to compensate expected detour and inconvenience effects. For locationsin the high request rate limit (compare Fig. 4d in the Main Manuscript) a saturation of S share indicates insufficient financialincentives for given negative aspects of sharing. total requestsshared requests 8.6 / min1.0 / min Times Square/Theatre District total requestsshared requests 6.8 / min1.1 / min
West Village total requestsshared requests 5.9 / min0.8 / min
Midtown North total requestsshared requests 6.0 / min1.0 / min
Penn Station/Madison Sq Garden total requestsshared requests 8.6 / min1.7 / min
Clinton East total requestsshared requests 9.6 / min1.5 / min
Union Square total requestsshared requests 5.1 / min1.5 / min
Lenox Hill East total requestsshared requests 3.1 / min0.9 / min
Alphabet Cityfraction of shared ride requests
FIG. 9.
Ride-sharing adoption is heterogeneous across New York City . Origin zones with saturated ride-sharingadoption (top row, compare horizontal line in Fig. 4d in Main Manuscript) yield qualitatively similar spatial patterns offraction of rides shared across New York City, despite differences in absolute request rate. Origin zones with yet unsaturatedride-sharing adoption (bottom row, compare Fig. 4d in Main Manuscript) yield different patterns and much higher fraction ofrides shared.
Spatial patterns of ride-sharing adoption in high and saturated sharing phases
Supplementary Figure 9 shows daily averages for W o,d for the origins in New York City highlighted in Figure 4d ofthe Main Manuscript, assuming 16 hours of daily activity as a normalization for the time window ∆ t . SupplementaryFigure 9 (top row) represents origins for which the number of shared rides has saturated (compare horizontal line inFig. 4d in Main Manuscript), while Supplementary Figure 9 (bottom row) illustrates origins on the linear ascendingbranch of the ride-sharing adoption curve.Consider for example Times Square/Theatre District and Alphabet City (top left and bottom right): While for thefirst only approximately one in nine ride requests is shared, it is one in three for the latter. The spatial pattern offraction of rides shared follows this trend. It is similar for regions with saturated shared ride request rate and startsto deviate the more the origin zone resembles a high-sharing regime (compare Fig. 1 in the Main Manuscript).7 ab c SharingfractionTotal requests S S s [rides/min]0.00.51.01.52.02.5 s S h a r e [ r i d e s / m i n ] Chicago s [rides/min]0246 s S h a r e [ r i d e s / m i n ] O'HareNear South SideUptown Logan SquareGar fi eld RidgeAustinLower West Side Hyde Park AustinO'Hare fraction of shared ride requests
FIG. 10.
Chicago exhibits hybrid sharing adoption . a,b Phase diagram of the number of shared ride requests S share for different inconvenience preferences ζ (compare Fig. 4 in the Main Manuscript). Ride-sharing is only fully adopted if thefinancial discount compensates the expected inconvenience fully (full-sharing, dark blue). Otherwise, the adoption of ride-sharing decreases with increasing number of users S (partial sharing). c Sharing decisions in Chicago exhibit a hybrid statebetween low- and partial-sharing states (panel c, blue dots). At low request rates, the number of requests for shared ridesincreases linearly with the total number of requests. At higher request rates, the sharing decisions saturate (horizontal orangecurve), indicating a partial sharing regime. Few communities cross the horizontal branch, hinting at hardly any zones in afull-sharing regime, but generally low adoption of ride-sharing for the given financial incentives. The inset in panel c includescommunities
North East Side, Loop, Near West Side, Lake View, West Town, Lincoln Park , and trips originating outside ofthe boundaries of the City of Chicago, whose request rates significantly exceed those of the other communities by up to oneorder of magnitude (not shown in the main panel, green border).
Supplementary Note 2. Ride-Sharing adoption in Chicago
In 2019, three transportation service providers (Uber, Lyft, Via) served more than 110 million transportation servicerequests in the City of Chicago, corresponding to approximately 300000 trips per day [42]. In this SupplementaryNote we demonstrate that the ride-sharing adoption in the city reproduces the hybrid sharing states observed for NewYork City and exhibits spatially heterogeneous patterns in ride-sharing adoption.Chicago consists of 77 community areas [42]. Supplementary Fig. 10c illustrates request rate for shared rides as afunction of the total request rate for rides. As illustrated for New York City in the Main Manuscript, Chicago’s differentcommunities exhibit spatially heterogeneous ride-sharing adoption. While there exists a subset of communities forwhich the number of shared ride requests scales linearly in the total number of requests, other origin communities(e.g. Lower West Side, Hyde Park, Uptown, Near South Side, O’Hare) form a branch where the number of shared riderequests has saturated and does not increase with the overall number of ride requests. Similarly to New York City,we observe partial and full-sharing regimes that give rise to spatially heterogeneous patterns of ride-sharing adoption(compare Supplementary Fig. 10c right).Other than in New York City, there are hardly any locations in the full-sharing regime indicated by the upperbranch of ride-sharing adoption (compare Supplementary Fig. 10 inset). This means the different communities arein a partial- or non-sharing regime. In other words, financial discounts seem to be insufficient to compensate theinconvenience preferences in Chicago, explaining that the majority of communities is not in a full-sharing state. Thelow number of locations on the upper branch suggests that a larger increase of the financial incentives is required totrigger the transition to the high-sharing regime to overcome the inconvenience preference ζ .8 Supplementary Note 3. Ride-sharing anti-coordination game on networks
In this Supplementary Note we formally define the ride-sharing anti-coordination game introduced in the MainManuscript. We introduce a replicator dynamics governing the evolution of the population’s willingness to sharetheir rides. The resulting network dynamics unveils spatially heterogeneous sharing patterns, emerging from dynamicsymmetry breaking.
Urban environment
Denote by G = ( V, E ) a mathematical graph of an urban street network composed of a node set V and an edgeset E . Nodes can be identified with individual intersection, census tracts or qualitatively similar zones embeddedin space. Edges correspond to streets connecting the different zones and are weighted by the geographical distancebetween them. The distance matrix D bundles the pairwise (shortest path) distances. In the following we consider aone-to-many setting where S people request transportation from a single origin o ∈ V to a destination d ∈ V \{ o } on G . Replicator dynamics
Per destination node d ∈ V \{ o } the probability π ( d, t ) ∈ [0 ,
1] defines the local population’s ride-sharing adoptionwhen embarking from origin o at time t . π ( d, t ) is an aggregate measure for people’s ride-sharing willingness, describingthe average ride-sharing behavior of people with the same origin-destination combination. The ride-sharing adoptionevolves under discrete-time replicator dynamics π ( d, t + 1) = r ( d, t ) π ( d, t ) (8)with reproduction factor r ( d, t ) = E [ u share ( d, t )] E [ u ( d, t )] (9)with the expected utility of sharing E [ u share ( d, t )] and the population average utility E [ u ( d, t )] from both shared andsingle rides. For simplicity, we assume the utility derived from single rides to be constant. Hence, E [ u single ( d, t )] = u single ( d ) >
0, allowing to express E [ u share ( d, t )] = u single ( d ) + E [∆ u ( d, t )]. With this shorthand notation the repro-duction factor becomes r ( d, t ) = u single ( d ) + E [∆ u ( d, t ) u single ( d ) + π ( d, t ) E [∆ u ( d, t )] . (10)In the limit u single ( d ) → ∞ , Eqn. (8) becomes equivalent to the continuous-time version of the replicator equation[36].Depending on the equilibrium value of E [∆ u ( d ) ∗ ] the dynamics converges to a pure strategy equilibrium π ( d ) ∗ =0 , if E [∆ u ( d ) ∗ ] < π ( d ) ∗ = 1 , if E [∆ u ( d ) ∗ ] >
0, or a mixed strategy equilibrium π ( d ) ∗ ∈ (0 ,
1) if E [∆ u ( d ) ∗ ] = 0.The incremental utility of sharing decomposes into E [∆ u ( d, t )] = (cid:15)d single ( d ) − ξE [ d det ( d, t )] − ζE [ d inc ( d, t )] (11)where d single ( d ) = D o,d is the shortest path distance between origin o and destination d , d det ( d ) is the detour fromsharing for destination d at time t and d inc ( d, t ) is the distance spent together on a shared ride. While the firstdistance is deterministic, the latter two are stochastic and depend on the overall demand for shared rides on thenetwork. Hence, they mediate a coupling between destinations on the network.A rescaling of Eqn. (11) E [∆ u ( d, t )] = (cid:15)d single ( d ) (cid:18) − ξ(cid:15) E [ d det ( d, t )] d single ( d ) − ζ(cid:15) E [ d inc ( d, t )] d single ( d ) (cid:19) (12)shows that the dimensionless parameters ξ/(cid:15) and ζ/(cid:15) as well as the relative detour E [ d det ( d, t )] /d single ( d ) and inconve-nience E [ d inc ( d, t )] /d single ( d ) at time t govern whether E [∆ u ( d, t )] is positive or negative. The quantities measure thetrade-off between inconvenience or detour to financial discount of a shared ride, respectively. If the financial discountis sufficiently high to compensate for the detours and inconvenience, the replicator dynamics in Eqn. (8) will amplifythe local population’s adoption of ride-sharing.9 a Sharing request graph b Maximum weight matching c Routing
FIG. 11.
Pairing rides is a maximum weight matching problem . The provider’s matching algorithm solves the maximumweight matching problem. a A shared ride request graph defines potentially shareable rides σ share with edge weights definingthe saved distance of a combined ride. b The provider pairs requests to maximize his saved distance. c Per matching theprovider defines the shared route to minimize the distance driven and customer inconvenience.
Expected detour and inconvenience
The expected detour and inconvenience of shared rides originating from origin o ∈ V , going to destination d ∈ V ,depend on (i) the configuration of destinations in the request set S at time t , (ii) the realization of sharing choicesacross all users, and (iii) the service provider’s matching and routing algorithm.( i ) Origin-destination distribution . Denote by σ ∈ V S the destination request configuration of the S simultaneoustransportation requests from o . σ is a random variable governed by the origin-destination distribution W o . Itimpacts where users travel and which users may potentially be matched when sharing a ride.( ii ) Adoption of ride-sharing . Depending on the user’s individual decisions to share their rides, σ decomposes into σ share and σ single . The realization of destinations in σ share determines the potentially shareable rides.( iii ) Matching and routing algorithm . Providers match ride requests based on distance savings potentials, which isequivalent to a maximum weight matching problem on a mathematical graph: Shared ride requests define thenodes of this graph. If two rides offer a distance savings potential to the provider compared to two single rides,the ride requests are connected by an edge (see Supplementary Fig. 11a). The distance savings potential definesthe edge weight. Here, we assume that both for single and shared ride requests the provider needs to returnto the trip origin, consistent with the one-to-many setting. The provider’s matching algorithm determines thematching of shared ride requests that maximizes the saved distance (see Supplementary Fig. 11b).Per matched request pair, the provider defines the trip route to minimize the distance driven. If he is indifferentwhom to drop first, he will deliver the passenger with the shorter distance first to minimize customer inconve-niences (see Supplementary Fig. 11c). If, again, he is indifferent he tosses a fair coin to determine the order ofthe shared ride.While the adoption of ride-sharing in general depends on the underlying street network and the destination distri-bution, the problem simplifies in the limit of many concurrent users, S → ∞ . In particular, a necessary and sufficientcondition for full adoption of ride-sharing to be a stable equilibrium in this limit is that the financial incentivescompensate the inconvenience (see Supplementary Fig. 12). In this limit and with full sharing, detours disappear asusers will always be matched with other users with the same destination. Formally:
Theorem 1 (Full sharing in high-demand limit) . If lim S →∞ π ( d ) ∗ = 1 the ratio of inconvenience to financial incentivemust be ζ/(cid:15) < .Proof. A dominant equilibrium strategy in sharing, π ( d ) ∗ = 1, implies positive expected utility difference E [∆ u ( d ) ∗ ] >
0. The limit of infinite request number yieldslim S →∞ E [∆ u ( d ) ∗ ] = d single ( d ) (cid:15) (cid:18) − lim S →∞ ξ(cid:15) E [ d det ( d )] d single ( d ) − lim S →∞ ζ(cid:15) E [ d inc ( d )] d single ( d ) (cid:19) = d single ( d ) (cid:15) (cid:18) − ζ(cid:15) (cid:19) (13)0 S S h a r e / S S = 20 S = 30 S = 40 S = 50 FIG. 12.
Discontinuous phase transition in the ride-sharing adoption in the high-demand limit . The controlparameter ζ/(cid:15) governs the ride-sharing adoption. For ζ/(cid:15) < S Share /S = 1).At ζ c /(cid:15) c = 1, a spatially heterogeneous pattern of ride-sharing adoption forms for finite S and detour preferences ξ > ζ/(cid:15) > S → ∞ (or for ξ = 0) the transition becomes discontinuous at ζ c /(cid:15) c = 1. Simulation parameters: ξ = 0 . , (cid:15) = 0 . where we used that π ( d ) ∗ = 1 for which S → ∞ corresponds to zero-detour matching to destination d . Consequently, E [ d inc ( d )] = d single ( d ) and 0 < E [∆ u ( d ) ∗ ] = d single ( d ) (cid:15) (cid:18) − ζ(cid:15) (cid:19) (14)which implies ζ(cid:15) < The case S = 3 The case S = 3 produces equilibrium adoption of ride-sharing qualitatively different than adjacent values of S , asdiscussed in Figure 3a in the the Main Manuscript. For sufficiently high (cid:15) the population has a dominant sharingstrategy in this configuration which is induced by the fact that the service provider can at most pair two of thethree ride requests into a shared one. The left-over request will enjoy the benefits of a single ride at discounted tripfare, inducing an incentive to become this request which fuels both the ride-sharing adoption as well as the matchingprobability.As S increases beyond S = 3 the incentive of gambling on being a left-over request reduces drastically as far lesscorresponding constellations exist. Thus, S = 3 produces a behavior in Figure 3a of the Main Manuscript that looksqualitatively different than for other values of S .1 Supplementary Note 4. Robustness of phases of ride-sharing adoption
In the Main Manuscript we demonstrated that the ride-sharing anti-coordination game reproduces opposing regimesof ride-sharing adoption in a simple setting. In this section we demonstrate the robustness of these results under dif-ferent conditions, including non-homogeneous demand constellations and for different origin locations in the network,illustrating that the underlying mechanisms balancing incentives remain identical.
Ride-sharing adoption for non-homogeneous origin-destination demand
Using the stylized city topology introduced in the Main Manuscript, we investigate the impact of radially andazimuthally asymmetric destination demand on the ride-sharing adoption from a joint origin. We distinguish betweenfour scenarios representative for different types of urban settlements:1.
Dense core : Starting from a joint origin in the city center, a gradient of decreasing destination demand inradial direction mimics urban environments with densely populated city core. Further distance destinations(e.g. suburbs) are less often requested, e.g. because of sparser population density.2.
Urban sprawl : In situations where distant destinations from the city center make up the majority of ride requeststhe radial destination demand gradient is reversed. Theses scenarios represent constellations of urban sprawl,or situations where the city core is only sparely populated, e.g. because of high real-estate prices.3.
Sparse settlement : Urban environments may exhibit azimuthal gradients in destination demand starting froman origin in the city center, e.g. stretched out residential settlements that have formed next to existing road,river banks etc. In that case destination demands in radial direction might be similar, but differ significantlyby cardinal direction.4.
Heterogeneous settlement : Urban constellations where both radial as well as azimuthal destination demandgradients exist might describe heterogeneously grown environments, e.g. because of natural obstacles or stageddevelopment.Figs. 13 and 14 correspond to the four scenarios. For given financial discount (cid:15) an increase in request rate S givesrise to a spatially heterogeneous sharing/non-sharing pattern and decreasing overall adoption of ride-sharing in allscenarios, independent of the destination demand distributions (compare Fig. 3 in the Main Manuscript). As secondorder effects, the origin-destination distribution determines (i) whether the cardinal direction of the sharing pattern israndom (Supplementary Fig. 13 for radially asymmetric destination demand), or aligned with the highest destinationdemand (Supplementary Fig. 14 for azimuthally asymmetric destination demand), and (ii) whether close-by or distantdestinations reduce their willingness to share first upon increased request rate.1. Dense core : For dense core settings (see Supplementary Fig. 13a) the cardinal direction of the sharing/non-sharing pattern is solely driven by random fluctuations breaking the azimuthal symmetry. The destinationdemand gradient leads to a reduction of willingness to share from inside to outside as S increases.2. Urban sprawl : Phenomenologically, urban sprawl (see Supplementary Fig. 13b) corresponds to dense core , butthis time increasing the request rate reduces the willingness to share from the outside (i.e. high destinationdemand).3.
Sparse settlement : In the presence of azimuthal destination demand gradients the sharing pattern forms alongthe branches of high demand (see Supplementary Fig. 14a). The dominance of those destinations in the replicatordynamics guides the symmetry breaking into letting low demand destinations reduce their willingness to share,which reduces the expected detour for sharing branches. As S increases the willingness to share reduces fromin- to outside as in the uniform case analyzed in the Main Manuscript.4. Heterogeneous settlement : For heterogeneous settlements the combination of radial and azimuthal gradients inthe destination demand orients the cardinal direction of the sharing pattern in line with high demand outsidedestinations (see Supplementary Fig. 14b). Outside destinations have a higher utility gain of sharing (incentivesproportional to distance), resulting in faster adjustments in the replicator dynamics and an effective first-moveradvantage such that these destinations determine the spatial pattern in the partial sharing state.2
Inner:share Inner:share Outer:share Outer:share Inner:single Inner:mixed Outer:single Outer:mixed a I n c e n t i v e s Inner:single Inner:mixed Outer:single Outer:mixedInner:share Outer:share Outer:shareInner:share Inner:share Inner:single Outer:share Outer:singleInner:mixed Outer:share Outer:shareInner:single Inner:mixed Inner:single Outer:mixed Outer:single I n c e n t i v e s Dense core b Urban sprawl
Sharingfraction
Inner:mixed Outer:share Outer:singleInner:single
FIG. 13.
Robustness of ride-sharing adoption for radially asymmetric origin-destination demand . Radialasymmetry of the destination demand distribution does not qualitatively affect the equilibrium ride-sharing adoption. a Densecore setting, inner ring destinations (gray shading) are visited twice as often as outer ring destinations. Increased request ratereduces the destination’s ride-sharing adoption from inside to outside. b Urban sprawl setting, outer ring destinations (grayshading) are visited twice as often as inner ring destinations. As S increases the ride-sharing adoption ceases from outsideto inside. In both cases branches of high ride-sharing adoption emerge in a random direction (compare Fig. 3 in the MainManuscript). Parameters: (cid:15) = 0 . , ζ = 0 . , ξ = 0 . In all settings, the results are qualitatively the same as for homogeneous origin-destination demand (compare Fig. 3 inthe Main Manuscript). Naturally, sufficiently high financial incentives overcome this partial-sharing phase and resultin full sharing, reproducing the two phases of ride-sharing adoption (see Supplementary Fig. 16, compare Fig. 4 inthe Main Manuscript).3 a I n c e n t i v e s I n c e n t i v e s Inner:share Inner:share Outer:share Outer:share Inner:single Inner:share Outer:single Outer:shareInner:share Inner:share Outer:share Outer:share Inner:share Inner:mixed Outer:share Outer:single Inner:share Inner:single Outer:mixed Outer:single Inner:mixed Inner:single Outer:mixed Outer:singleInner:single Inner:mixed Outer:single Outer:mixed Inner:single Inner:mixed Outer:single Outer:mixed
Sparse settlement b Heterogeneous settlement
Sharingfraction
FIG. 14.
Robustness of ride-sharing adoption for azimuthally asymmetric origin-destination demand . Anazimuthally asymmetric destination demand distribution predetermines the emergence of sharing/non-sharing branches. a Sparse settlement setting, neighboring branches of destinations in radial direction alternate between being visited twice aslikely (gray shading) as the other branches. The high-demand branches are sharing while the low demand branches quitsharing due to their high expected detour. Increased request rate reduces the destination’s ride-sharing adoption from insideto outside. b Heterogeneous settlement setting, inner and outer ring destination nodes on the same branch alternate betweenbeing requested twice as often (gray shading) as the other one. Also in this setting branches of high ride-sharing adoptionemerge, driven by the outermost destinations. As S increases the ride-sharing adoption ceases from outside to inside (compareSupplementary Fig. 13b). Parameters: (cid:15) = 0 . , ζ = 0 . , ξ = 0 . S T o t a l s h a r e d r e q u e s t s S s h a r e =0.15=0.3 S =0.15=0.3 S =0.15=0.3 S =0.15=0.3 a Dense core b Urban sprawl Sparse settlement c Heterogeneous d settlement FIG. 15.
Phases of ride-sharing adoption . All settings reproduce the partial sharing phase ( S share = const [green]) andfull sharing phase ( S share = S [blue]) illustrated in the Main Manuscript for financial incentives that (do not) compensate theinconvenience (compare Fig. 4c in the Main Manuscript). Ride-sharing adoption for decentral origin
In the one-to-many ride-sharing game, the relative position of the origin defines the scale of average distances todifferent destinations and the possible combinations in which requests for shared rides are matched. Hence, it impactsexpected detours and inconvenience. Here, we consider the stylized city topology introduced in the Main Manuscriptwith a decentral origin at the periphery. I n c e n t i v e s S = 4 S = 6 S = 18 FinancialDetourInconvenience
Sharing fraction
Ride-sharing adoption for decentral origin Scaling of shared ride requests a b S T o t a l s h a r e d r i d e s S S h a r e =0.115=0.135=0.155 FIG. 16.
Robustness of ride-sharing adoption for decentral origin . A decentral origin node (gray) defines a new scale ofaverage distances to the destinations in the stylized city topology, but the ride-sharing adoption remains qualitatively the same. a In the partial sharing regime a spatially heterogeneous pattern emerges in the ride-sharing adoption across origins, and fadesout as the number of rides S increases (Parameters: (cid:15) = 0 . , ζ = 0 . , ξ = 0 . b If financial incentives (cid:15) overcompensateinconvenience effects ζ we recover a full-sharing phase (Parameters: (cid:15) = 0 . , ξ = 0 . Supplementary Figure 16 illustrates a one-to-many situation of homogeneous transportation demand from thenorthernmost node in the stylized city topology. Again, a distinct spatial pattern of ride-sharing adoption emerges inthe partial sharing regime (Supplementary Fig. 16a). When the financial incentives are sufficiently large, we recoverthe full sharing phase (Supplementary Fig. 16b). In this setting, the sharing pattern is symmetric about the north-to-south axis, where nodes in the direction of the city center share dominantly. They have no expected detours since in allconstellations where they are matched, they will be dropped first. This is not the case anymore for destinations on theopposite side of the city center. Hence, these destinations do not share. For the remaining destinations the decisionto share, or not, results in a zero-sum game very soon as S increases (compare Supplementary Fig. 16a, center panel)and eventually reproduces a ride-sharing adoption pattern where neighboring branches alternate between sharing andnot sharing (compare Fig. 3 in Main Manuscript).Again, ride-sharing adoption behaves qualitatively similar compared to the constellation for central origins.6 Supplementary Methods
In this section of the Supplementary Information we provide detailed insight into the data used, cleansing proceduresapplied and simulation methods implemented.
Data acquisition, structure, and treatment
Data sources.
The New York City Taxi & Limousine Commission (TLC) publishes trip records for high-volumefor-hire vehicles (HVFHV) on a monthly basis. The data includes trip information on pickup time, origin zone,drop-off time, destination zone as well as a shared ride request label for providers completing more than 10000trips per day [41]. Our analysis is based on the aggregate HVFHV activity between January and December 2019,independent of service provider, including more than 250 million transportation services. We exclude older data dueto regulatory changes effective in 2019 [25], potentially impacting ride-hailing behavior, and data from 2020 due tochanged transportation service activity in the course of the COVID-19 pandemic [39].TLC partitions New York City into 265 taxi zones and provides geospatial information about zone boundaries,names and jurisdictions [41]. We adopt the definition of these zones in all of our analyses.Additionally, the City of Chicago publishes ride-hailing trip records on its Open Data Portal [42]. The data contains,amongst others, information about trip origin, destination, pickup and dropoff times as well as information whethera shared ride has been authorized by the requester. Our analysis encompasses the time-span between January andDecember 2019, as chosen for New York City, and includes more than 110 million trip requests served by threetransportation service providers (Uber, Lyft, Via).In our geospatial analysis we restrict ourselves to Chicago’s 77 community areas, as well as trips leaving or enteringthe official city borders.
Data preparation.
We use TLC’s data as-is. Our data cleansing procedure removes trip records for which tripinformation is decoded as not available. Furthermore, we omit trip records for zones 264 and 265 in our analysis.While the dataset contains trip requests labeled by these zones, there is no geographic decoding specified by TLC,nor do the zones have names.Similarly, we use the Chicago trip records as-is.For our analyses, we determine the total flux matrix specified in Eqn. (6) per city. When showing daily averages wenormalize the total annual flux between origin and destination zones o and d by 16 hour days to obtain a per-minute-request rate, assuming hardly any request activity for 8 hours per day. In case of specifically defined time windows(see Supplementary Note 1 and 2), we normalize the total flux by the window size.We compute the fraction of shared as specified in Eqn. (7). Numerical simulations of ride-sharing anti-coordination games