








key: cord-198395-v15queyh
authors: Storch, David-Maximilian; Timme, Marc; Schroder, Malte
title: Incentive-driven discontinuous transition to high ride-sharing adoption
date: 2020-08-25
journal: nan
DOI: nan
sha: 
doc_id: 198395
cord_uid: v15queyh

Ride-sharing - the combination of multiple trips into one - may substantially contribute towards sustainable urban mobility. It is most efficient at high demand locations with many similar trip requests. However, here we reveal that people's willingness to share rides does not follow this trend. Modeling the fundamental incentives underlying individual ride-sharing decisions, we find two opposing adoption regimes, one with constant and one with decreasing adoption as demand increases. 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 requests in New York City and Chicago illustrates that both regimes coexist across the cities, consistent with our model predictions. These results suggest that current incentives for ride-sharing may be near the boundary to the high-sharing regime such that even a moderate increase in the financial incentives may significantly increase ride-sharing adoption.

Sustainable mobility [1] [2] [3] [4] [5] [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 urban mobility [9] . Recent analyses suggest that large-scale ride-sharing is specifically suited for densely populated urban areas [9] [10] [11] [12] [13] . By combining two or more individual trips into a shared ride served by a single vehicle, ride-sharing increases the average utilization per vehicle, reduces the total number of vehicles required [11] and thereby mitigates congestion and environmental impacts of urban mobility [14] . Hence, embedding ride-sharing for trips that would otherwise 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 efficiency gains derived from aggregating rides [9, 12, 13] . Generally, matching individual rides into shared ones without large detours becomes easier with more users, increasing both the economic and environmental efficiency as well as the service 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] [18] [19] [20] [21] [22] . In particular, it is unclear how to encourage an ever growing number of ride-hailing users to choose shared rides over their current individual mobility options [23] [24] [25] .

In this article, we disentangle the complex incentive structure that governs ride-hailing users' decisions to share their rides -or not. In a game theoretic model of a one-to-many demand constellation we illustrate how the interactions 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 the population shares their rides independent of demand. Analyzing ride-sharing decisions from approximately 250 million ride-requests in New York City and an additional 110 million in Chicago suggests that both adoption regimes coexist in these cities, consistent with our theoretical predictions. Our findings indicate that current financial incentives are nearly, but not fully, sufficient to stimulate a transition towards the high-sharing regime. 1 . Contrasting ride-sharing adoption despite high request rate in New York City. Fraction of shared ride requests from different origins (red) served by the four major for-hire vehicle transportation service providers in New York City by destination zone (January -December 2019) [27] . Gray areas were excluded from the analysis due to insufficient data (see Methods). The fraction of shared ride requests differs significantly by origin and destination, even though the average overall request rate is similar for all four origin locations. a,b Some areas, such as East Village and Crown Heights North, show a high adoption of ride-sharing services. c,d Despite a similarly high request rate, other locations, such as JFK and LaGuardia airports, 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 urban environment, demand structure and socio-economic factors that govern the adoption of ride-sharing. To disentangle these complex interactions, we introduce and analyze a game theoretic model capturing essential features of ridesharing incentives, disincentives as well as topological demand structure. Trade-offs between incentives determine the decision to share a ride, or not. a Shared rides offer advantages and disadvantages compared to single rides. On the one hand, they offer financial discounts typically proportional to the distance of a direct single ride (blue, dotted). On the other hand, rides shared with strangers may be inconvenient due to other passengers in the car (e.g. loss of privacy or less space, green) and may include detours compared to a direct trip to pickup or deliver these other passengers (orange, solid compared to dotted). b The decision to book a shared ride depends on the balance of all three factors. If the expected utility difference E[∆u] = E[u share ] − E[u single ] between a shared and a single ride is positive, the financial discounts overcompensate detour and inconvenience effects; users share. If E[∆u] is negative (as illustrated), users prefer to book single rides.

The decision of ride-hailing users to request a single or a shared ride reflects the balance of three fundamental incentives ( Fig. 2) [18, 22] :

Discounts. Ride-sharing is incentivized by financial discounts granted on the single ride trip fare, partially passing on savings of the service cost to the user. Often, these discounts are offered as percentage discounts on the total fare such that the financial incentives u share fin > 0 are proportional to the distance or duration d single of the requested ride, u share fin = d single , where denotes the per-distance financial incentives. In many cases, these discounts are also granted if 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 share det < 0 increases with the detour d det . Inconvenience. Sharing a ride with another user may be inconvenient due to spending time in a crowded vehicle or due to loss of privacy [18, 20, 21] . This disincentive u share inc < 0 scales with the distance or duration d inc users ride together.

In the following we take u share det ∼ d det and u share inc ∼ d inc , describing the first order approximation of these disincentives and 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 mode of transport. The overall utility of a shared ride is then given by

where the utility u single for a single ride describes the benefit of being transported, as well as the cost and time spent on the ride. The factors , ξ and ζ denote the user's preferences. By rescaling the utilities (measuring in monetary units), directly denotes the relative price difference between single and shared rides whereas ζ and ξ quantify the importance of 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 a given discount factor . In contrast, detour and inconvenience contributions depend on the destinations and sharing decisions of other users. Their magnitude depends on where these users are going and on the route the vehicle is taking for a shared ride (see Methods). The decision to share a ride is determined by the expected utility difference (see Fig. 2 )

where E[·] signifies the expectation value over realizations of other users' destinations and sharing decisions conditional on one's own sharing decision.

To understand how these incentives determine the adoption of ride-sharing, we study sharing decisions in a stylized city network [30] with a common origin o (e.g., from a central downtown location) in the center and multiple destinations d (illustrated in Fig. 3 ). Two rings define urban peripheries equidistant from the city center. Branches represent 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 to single rides (see Methods). Pairing at most two users who request a shared ride, the problem of matching shared ride requests reduces to a minimum-weight-matching with an efficient solution, eliminating the influence of heuristic matching 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 requests within some small time window τ . Therefore, we consider a game with S = s τ users travelling to a uniformly chosen destination location, where s denotes the average request rate. These users have the option to book a single ride or a 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 the equilibrium sharing probabilities π * (d) with which users from destination d adopt ride-sharing to maximize their expected utility (see Methods for details).

At fixed discount and preferences ζ and ξ ride-hailing users may decrease their overall adoption of ride-sharing π * as the total number S of users increases (see Fig. 3a , blue), even though ride-sharing becomes more efficient with higher user numbers. Here · denotes the average over all destinations d. While for small request rates everybody is requesting shared rides (Fig. 3b) , a distinctive sharing/non-sharing pattern emerges along the branches of the city network upon higher demand (Fig. 3c,d) , before the adoption of ride-sharing eventually fades out for high request rates, S 1 (Fig. 3e) . This observation offers a novel perspective on the prevalent conclusion that increased demand improves the shareability of rides [9, 13] . While more rides are potentially shareable, less people may be willing to share them.

The underlying incentives explain this phenomenon: Ideally, a user wants to book a shared ride (financial incentive) but without actually sharing the ride (inconvenience and detour). The expected detour and inconvenience mediate an interaction between ride-hailing users, turning ride-sharing decisions into a complex anti-coordination game. For small request rates, i.e. small numbers of concurrent users S, the probability p match (d) for a user with destination d to be matched with other users is low (see Fig. 3a , gray). Consequently, the expected detour

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 requesting shared 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, resulting in more requests that are actually matched with another user (see Fig. 3a ). Consequently, the expected detour and inconvenience also increase. However, instead of reducing the average adoption of ride-sharing homogeneously across all destinations, neighboring destinations adopt opposing sharing strategies (see Fig. 3b ). In this sharing pattern, only destinations in identical cardinal direction can and will be matched into a shared ride, minimizing the detours for shared requests and simultaneously disincentivizing other users to start sharing due to high expected detours ( Fig. 3c-e bottom) . As the number of users S increases further, the probability p match (d) would also increase at given sharing adoption π(d). This leads to an adoption of mixed sharing strategies where the financial discounts In this configuration, users requesting a shared ride never suffer any detour while users that do not share are disincentivized from doing so due to their high expected detour (compare bottom part of panel c). For high numbers of users (S = 12 and 30, panels d and e), the probability to be matched with another user when requesting a shared ride increases and the financial incentives cannot fully compensate the expected inconvenience. The adoption of ride-sharing decreases until the financial incentives exactly balance the expected inconvenience (panels d and e, bottom). Illustrated here for financial discount = 0.2 and inconvenience and detour preferences ζ = 0.3 and ξ = 0.3. and the expected inconvenience are exactly in balance ( Fig. 3d and e) . Further numerical simulations demonstrate that this transition robustly exists also for heterogeneous demand distribution across the destinations and different origin locations within the network (see Supplementary Information) . Naturally, if the discount is sufficiently large such that the financial incentives completely compensate the expected inconvenience, > ζ, all users share also in the high request rate limit, S → ∞. In this limit, d single = d inc as detours disappear, 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 financial discount and number of users S, illustrating under which conditions the users adopt ride-sharing (high-sharing regime) and under which conditions the users only share partially or not at all (low-sharing regime).

For fixed values of financial discounts , different behavior emerges for different inconvenience preferences ζ. If ζ is sufficiently 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 4c illustrates the scaling of S share in both states as S increases. In the partial sharing state, S share becomes constant for large S, such that S share /S → 0 as S → ∞ (compare Fig. 3a) , implying a discontinuous phase transition between low-sharing and high-sharing regimes for large S when the financial incentives exactly balance the inconvenience, c /ζ c = 1 (see Supplementary Information for details). Two qualitatively different regimes of ride-sharing adoption. a,b Phase diagram of fraction of shared rides S share /S for different inconvenience preferences ζ. Ride-sharing is adopted dominantly if the financial discount fully compensates the expected inconvenience (high-sharing, dark blue). Otherwise, the total number of shared ride requests saturates and the overall adoption of ride-sharing decreases with increasing number of users S (low-sharing, compare Fig. 3a ). In the limit of an infinite number of requests S → ∞ the transition becomes discontinuous (see Supplementary Information) . c With identical financial discounts = 0.2, different sharing behavior emerges for different inconvenience preferences ζ. When ζ < all users request shared rides (S share = S, dark blue triangles, red line in panel a). When ζ > the system is in a low-sharing regime where users request shared rides at low numbers of users S but the number of shared ride requests saturates and becomes constant at high S (S share < S, light green triangles, red line in panel b). In the low-sharing regime, spatially heterogeneous patterns of ride-sharing adoption emerge (compare Fig. 3b -e).

In a real city with heterogeneous preferences across different locations and constant financial discounts , the sharing decisions may, on an aggregate level, appear to be in a hybrid state between the high-and low-sharing phases predicted by our model. Indeed, the ride-sharing adoption across different origin locations in New York City and 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 linearly with more requests, s share ∼ s. At high request rates, sharing decisions differ by origin zone (compare Fig. 1 ): for Crown Heights North and East Village the linear scaling prevails, indicating is sufficiently large to compensate the expected inconvenience and detour effects completely. In fact, the spatial pattern of fraction of rides shared appears to be largely homogeneous across destinations as expected in this state (Fig. 5a) . Other origins with a similarly high request rate, such as JFK and LaGuardia airports, accumulate on a horizontal line with a constant number of shared ride requests per time. For these zones s share has saturated for the given financial incentives and will not increase with 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, these observations suggest that financial incentives in New York City are at the phase boundary between the high-and low-sharing regime and slightly higher discounts may significantly increase sharing in some areas. Ride-sharing adoption in New York City and Chicago is consistent with the predicted high-and lowsharing regimes. a,b Sharing decisions for New York City and Chicago (blue dots) accumulate on two branches corresponding to the predicted high-and low-sharing regime as a function of request rate (compare Fig. 4) . At low request rates, the number of 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 inconvenience preferences ζ are naturally heterogeneous in the cities, adoption is in a hybrid low/high-sharing state. c,d For origins in the high-sharing state a spatially homogeneous pattern of ride-sharing adoption emerges across destinations. e,f For origins in the low-sharing state a spatially heterogeneous pattern forms. The agglomeration of most data points on the high-sharing branch for New York City suggests that the financial discounts are close to the boundary of the high-sharing phase. However, the slope of the high-sharing branch indicates that only about 20% of ride-hailing users consider ride-sharing as an option. While about 40% of requests are shared in the high-sharing regime in Chicago, this potential is largely not realized. Most data points at locations with high request rates accumulate on the horizontal line representing the low-sharing regime. Seven large and busy zones in Chicago with up to 50 requests per minute (not shown) fall in between the high-and low-sharing state (see Supplementary Information for details).

An analysis for the ride-sharing adoption across more than 110 million trips in Chicago (see Methods and Supplemental 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 (corresponding to the slope of the diagonal branch in Fig. 5a,b) is below 100%. In terms of our ride-sharing game, the remaining fraction of requests for single rides corresponds to a user group that does not consider ride-sharing as a potential option at all and, hence, is not captured by our model.

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 include financial savings potentials, detour or delay preferences, various types of inconveniences, as well as sustainability, security and uncertainty [18, [20] [21] [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 million ride-sharing decisions from New York City and Chicago. A basic model includes three core incentive types: financial benefits, potential detours (and thus effectively slower service) and other inconveniences such as reduced privacy resulting from sharing a vehicle. This setting may already reflect many additional factors influencing ride-sharing adoption on an aggregate level. For example, sustainability or uncertainty preferences to first approximation scale with the additional distance driven and may thus be incorporated into the detour preferences. Similarly, alternative public transport options may be captured by modifying the effective financial discount and relative inconvenience preferences for individual destinations. As such, we expect the qualitative dynamics to be robust even in more detailed settings taking 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 conditions and 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 strong financial incentives, the number of shared ride requests increases linearly with increasing demand. However, if the financial incentives are weak compared to inconvenience disincentives, the number of requests for shared rides saturates with increasing demand (regime of low ride-sharing adoption). This observation is independent of the choice of origin locations or the specific demand distribution (see Supplementary Information) and stands in stark contrast to the increasing shareability of rides with high demand [9, 12, 13] . In the limit of large demand, the transition between the two regimes becomes discontinuous, switching abruptly from the low adoption to the high adoption regime with a small change of the incentives.

Ride-sharing adoption observed across New York City and Chicago is consistent with these predictions and demonstrates that both regimes exist across the cities. The data suggest that even a moderate increase of financial incentives may strongly improve ride-sharing adoption in some areas currently in the low-sharing regime. Still, the overall low fraction of shared ride requests, even in the high-sharing regime, suggests that an additional societal change towards acceptance of shared mobility is required [31] to make the full theoretical potential of ride-sharing accessible [9, 12] . A carefully designed incentive structure for ride-sharing users adapted to local user preferences is essential to drive this change and to avoid curbing user adoption or stimulating unintended collective states [32, 33] . This is particularly relevant 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 by regulating and adapting incentives to promote ride-sharing in place of motorized individual transport.

New York City ride-sharing data. We analyzed trip data of more than 250 million transportation service requests delivered through high-volume For-Hire Vehicle (HVFHV) service providers in New York City in 2019. The data is provided by New York City's Taxi & Limousine Commission (TLC) [27] and consists of origin and destination zone per request, pickup and dropoff times, as well as a shared request tag, denoting a request for a single or shared ride. We compute the average request rate across all data throughout 2019 taking 16 hours of demand per day as an approximate average.

For fixed origin-destination pairs we determine the sharing fraction as the ratio of the total number of shared ride requests and the total number of requests. Departure and destination zones represent the geospatial taxi zones defined by TLC [27] . However, we exclude zones without geographic decoding, nor name tag defined by TLC. For each 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 and destination zone, we exclude that destination zone from the analysis to avoid excessive stochastic fluctuations (see Supplementary Information for details) .

Chicago ride-sharing data. We additionally analyzed more than 110 million trips delivered by three service providers 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 a shared 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 average request rate across all data for 2019 taking 16 hours of demand per day as an approximate average reference time and repeat the analysis explained for New York City.

City topology. For our ride-sharing model we construct a stylized city topology that combines star and ring topology [30] . Starting from a central origin node, rides can be requested to 12 destinations distributed equally across two rings of radius 1 (inner ring) and 2 (outer ring), as depicted in Figure 3 . The distances between neighboring nodes on the same branch are set to unity. Correspondingly, the distances between neighboring nodes are π/3 on the inner, and 2π/3 on the outer ring.

Ride-sharing adoption. We compute the equilibrium state of ride-sharing adoption by evolving the adoption probabilities π(d, t) following discrete-time replicator dynamics [35, 36] π(d, t + 1) = r(d, t) π(d, t),

where the reproduction rate r(d, t) at destination d and time t is

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 a constant utility of a single ride u single (d) = 10 to ensure positivity of Eqn. (5) . To evolve Eqn. (4), we numerically compute E[u share (d, t)] = E[u(d, t)|share] at each replicator time step t: We generate n = 100 samples of ride requests of size S of which at least one goes to destination d and requests a shared ride. The other S − 1 requests are drawn from a uniform destination distribution. Each of them realizes a sharing decision in line with the current probability distribution π(d , t) at their respective destination d 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 this procedure 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 replicator time steps, corresponding to 150000 game realizations per destination. We then measure the average adoption for 1000 replicator time steps, representing a proxy for the stationary solution π * (d) of Eqn. (4) and plotted as the sharing fraction 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 optimal pairwise matching of requests as follows: For shared requests we construct a graph whose nodes correspond to requests and edges encode the distance savings potential of matching the two requests. To determine the distance savings potential we assume that, independent of single or shared ride, the provider has to return to the origin of the trip.

After constructing the shared request graph we employ the 'Blossom V' implementation of Edmond's Blossom algorithm to determine the maximum weight matching of highest distance savings potential [37] . The matching determines the routing and the realization of inconvenience and detour (see Supplementary Information for more details).

In the Main Manuscript of this article we disentangle the incentive structure of urban ride-sharing and demonstrate how it leads to emergence of two qualitatively different regimes of sharing adoption. A game theoretic model reproduces key features of the ride-sharing activity in New York City and Chicago, including spatially heterogeneous patterns of ride-sharing adoption, saturation in the number of shared rides upon increasing demand and provides insights on the underlying mechanisms. This Supplementary Information provides additional details on the model, methods and results presented, and is structured as follows: In 2019, four high-volume for-hire vehicle (HVFHV) companies (Uber, Lyft, Via, Juno) served more than 250 million transportation service requests in New York City, corresponding to approximately 700000 trips per day conducted by a population of 8.4 million people [38, 41] . In this Supplementary Note, we unveil the spatiotemporal demand patterns underlying this macroscopic number of transportation requests.

The flux matrix W (∆t) formalizes the spatiotemporal demand for transportation services between different locations of an urban environment. Its entries W o,d denote the number of transportation requests originating at location o and going to location d within a specific time window ∆t.

W (∆t) decomposes into

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

of shared rides to absolute number of rides. Note that Eqn. (7) is only defined if the total flux between origin and destination exceeds a threshold w min to reduce bias from fluctuations in statistical analyses of P o,d (∆t).

We determine W single , W shared and P for the 265 taxi zones in New York City from the New York City Taxi Fig. 6b ) and evening (6 pm -3 am) encompassing leisure activity hours ( Supplementary Fig. 6c ). Independent of daytime, all four origins exhibit complex 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 rides shared to all destinations throughout the day. For Crown Heights North and East Village only few rides are undertaken to far distance destinations in the morning and midday time window (gray areas representing W o,d < w min ). In the evening, more rides are requested overall, also to far distance destinations. Overall, the qualitative patterns of ridesharing adoption do not vary significantly with the time of day (compare Fig. 1 

Across the full set of origin zones in New York City, Supplementary Figure 8 suggests an overall trend to higher absolute demand for transportation services in the evening. The fraction of shared rides, however, is not affected by this trend. It is approximately constant throughout the day as illustrated in Supplementary Figure 7 . The average standard 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, an aggregate analysis is representative. a linear scaling between S share and S indicates sufficient financial incentives to compensate the expected negative effects of ride-sharing with increasing demand. A decrease in slope and eventual saturation corresponds to a situation where financial incentives, expected detour and inconvenience are in balance. S share will not increase upon higher demand for given incentives (compare Fig. 1 in Main Manuscript as well as large S regime in Supplementary Fig. 8 ). Consider for example Times Square/Theatre District and Alphabet City (top left and bottom right): While for the first only approximately one in nine ride requests is shared, it is one in three for the latter. The spatial pattern of fraction of rides shared follows this trend. It is similar for regions with saturated shared ride request rate and starts to deviate the more the origin zone resembles a high-sharing regime (compare Fig. 1 Fig. 4 in the Main Manuscript). Ride-sharing is only fully adopted if the financial discount compensates the expected inconvenience fully (full-sharing, dark blue). Otherwise, the adoption of ridesharing decreases with increasing number of users S (partial sharing). c Sharing decisions in Chicago exhibit a hybrid state between low-and partial-sharing states (panel c, blue dots). At low request rates, the number of requests for shared rides increases linearly with the total number of requests. At higher request rates, the sharing decisions saturate (horizontal orange curve), indicating a partial sharing regime. Few communities cross the horizontal branch, hinting at hardly any zones in a full-sharing regime, but generally low adoption of ride-sharing for the given financial incentives. The inset in panel c includes communities North East Side, Loop, Near West Side, Lake View, West Town, Lincoln Park, and trips originating outside of the boundaries of the City of Chicago, whose request rates significantly exceed those of the other communities by up to one order of magnitude (not shown in the main panel, green border).

In 2019, three transportation service providers (Uber, Lyft, Via) served more than 110 million transportation service requests in the City of Chicago, corresponding to approximately 300000 trips per day [42] . In this Supplementary Note we demonstrate that the ride-sharing adoption in the city reproduces the hybrid sharing states observed for New York 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 a function of the total request rate for rides. As illustrated for New York City in the Main Manuscript, Chicago's different communities exhibit spatially heterogeneous ride-sharing adoption. While there exists a subset of communities for which 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 ride requests 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 upper branch of ride-sharing adoption (compare Supplementary Fig. 10 inset) . This means the different communities are in a partial-or non-sharing regime. In other words, financial discounts seem to be insufficient to compensate the inconvenience preferences in Chicago, explaining that the majority of communities is not in a full-sharing state. The low number of locations on the upper branch suggests that a larger increase of the financial incentives is required to trigger the transition to the high-sharing regime to overcome the inconvenience preference ζ.

In this Supplementary Note we formally define the ride-sharing anti-coordination game introduced in the Main Manuscript. We introduce a replicator dynamics governing the evolution of the population's willingness to share their rides. The resulting network dynamics unveils spatially heterogeneous sharing patterns, emerging from dynamic symmetry breaking.

Denote by G = (V, E) a mathematical graph of an urban street network composed of a node set V and an edge set E. Nodes can be identified with individual intersection, census tracts or qualitatively similar zones embedded in space. Edges correspond to streets connecting the different zones and are weighted by the geographical distance between them. The distance matrix D bundles the pairwise (shortest path) distances. In the following we consider a one-to-many setting where S people request transportation from a single origin o ∈ V to a destination d ∈ V \{o} on G.

Per destination node d ∈ V \{o} the probability π(d, t) ∈ [0, 1] defines the local population's ride-sharing adoption when embarking from origin o at time t. π(d, t) is an aggregate measure for people's ride-sharing willingness, describing the average ride-sharing behavior of people with the same origin-destination combination. The ride-sharing adoption evolves under discrete-time replicator dynamics

with reproduction factor 

where d single (d) = D o,d is the shortest path distance between origin o and destination d, d det (d) is the detour from sharing for destination d at time t and d inc (d, t) is the distance spent together on a shared ride. While the first distance is deterministic, the latter two are stochastic and depend on the overall demand for shared rides on the network. Hence, they mediate a coupling between destinations on the network. A rescaling of Eqn. (11)

shows that the dimensionless parameters ξ/ and ζ/ as well as 11 . Pairing rides is a maximum weight matching problem. The provider's matching algorithm solves the maximum weight matching problem. a A shared ride request graph defines potentially shareable rides σ share with edge weights defining the saved distance of a combined ride. b The provider pairs requests to maximize his saved distance. c Per matching the provider defines the shared route to minimize the distance driven and customer 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 choices across 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 simultaneous transportation requests from o. σ is a random variable governed by the origin-destination distribution W o . It impacts 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 is equivalent to a maximum weight matching problem on a mathematical graph: Shared ride requests define the nodes 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 defines the edge weight. Here, we assume that both for single and shared ride requests the provider needs to return to the trip origin, consistent with the one-to-many setting. The provider's matching algorithm determines the matching 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 indifferent whom to drop first, he will deliver the passenger with the shorter distance first to minimize customer inconveniences (see Supplementary Fig. 11c ). If, again, he is indifferent he tosses a fair coin to determine the order of the shared ride.

While the adoption of ride-sharing in general depends on the underlying street network and the destination distribution, the problem simplifies in the limit of many concurrent users, S → ∞. In particular, a necessary and sufficient condition for full adoption of ride-sharing to be a stable equilibrium in this limit is that the financial incentives compensate the inconvenience (see Supplementary Fig. 12 ). In this limit and with full sharing, detours disappear as users 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 incentive must be ζ/ < 1.

A dominant equilibrium strategy in sharing, π(d) * = 1, implies positive expected utility difference E[∆u(d) * ] > 0. The limit of infinite request number yields At ζc/ c = 1, a spatially heterogeneous pattern of ride-sharing adoption forms for finite S and detour preferences ξ > 0 where adjacent branches alternate between the low-and high-sharing regimes. For ζ/ > 1 ride-sharing adoption fades out along the branches that are still in the high-sharing regime. In the limit S → ∞ (or for ξ = 0) the transition becomes discontinuous at ζc/ c = 1. Simulation parameters: ξ = 0.3, = 0.2.

where we used that π(d) * = 1 for which S → ∞ corresponds to zero-detour matching to destination d. Consequently,

which implies ζ < 1.

The case S = 3

The case S = 3 produces equilibrium adoption of ride-sharing qualitatively different than adjacent values of S, as discussed in Figure 3a in the the Main Manuscript. For sufficiently high the population has a dominant sharing strategy in this configuration which is induced by the fact that the service provider can at most pair two of the three ride requests into a shared one. The left-over request will enjoy the benefits of a single ride at discounted trip fare, inducing an incentive to become this request which fuels both the ride-sharing adoption as well as the matching probability.

As S increases beyond S = 3 the incentive of gambling on being a left-over request reduces drastically as far less corresponding constellations exist. Thus, S = 3 produces a behavior in Figure 3a of the Main Manuscript that looks qualitatively different than for other values of S. Robustness of ride-sharing adoption for radially asymmetric origin-destination demand. Radial asymmetry of the destination demand distribution does not qualitatively affect the equilibrium ride-sharing adoption. a Dense core setting, inner ring destinations (gray shading) are visited twice as often as outer ring destinations. Increased request rate reduces the destination's ride-sharing adoption from inside to outside. b Urban sprawl setting, outer ring destinations (gray shading) are visited twice as often as inner ring destinations. As S increases the ride-sharing adoption ceases from outside to inside. In both cases branches of high ride-sharing adoption emerge in a random direction (compare Fig. 3 In all settings, the results are qualitatively the same as for homogeneous origin-destination demand (compare Fig. 3 in the Main Manuscript). Naturally, sufficiently high financial incentives overcome this partial-sharing phase and result in full sharing, reproducing the two phases of ride-sharing adoption (see Supplementary Fig. 16 Robustness of ride-sharing adoption for azimuthally asymmetric origin-destination demand. An azimuthally 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 as likely (gray shading) as the other branches. The high-demand branches are sharing while the low demand branches quit sharing due to their high expected detour. Increased request rate reduces the destination's ride-sharing adoption from inside to outside. b Heterogeneous settlement setting, inner and outer ring destination nodes on the same branch alternate between being requested twice as often (gray shading) as the other one. Also in this setting branches of high ride-sharing adoption emerge, driven by the outermost destinations. As S increases the ride-sharing adoption ceases from outside to inside (compare Supplementary Fig. 13b) . Parameters: = 0.2, ζ = 0.3, ξ = 0.3. 

In the one-to-many ride-sharing game, the relative position of the origin defines the scale of average distances to different destinations and the possible combinations in which requests for shared rides are matched. Hence, it impacts expected detours and inconvenience. Here, we consider the stylized city topology introduced in the Main Manuscript with a decentral origin at the periphery. Supplementary Figure 16 illustrates a one-to-many situation of homogeneous transportation demand from the northernmost node in the stylized city topology. Again, a distinct spatial pattern of ride-sharing adoption emerges in the partial sharing regime (Supplementary Fig. 16a ). When the financial incentives are sufficiently large, we recover the full sharing phase (Supplementary Fig. 16b ). In this setting, the sharing pattern is symmetric about the north-tosouth axis, where nodes in the direction of the city center share dominantly. They have no expected detours since in all constellations where they are matched, they will be dropped first. This is not the case anymore for destinations on the opposite side of the city center. Hence, these destinations do not share. For the remaining destinations the decision to 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 and not sharing (compare Fig. 3 in Main Manuscript).

Again, ride-sharing adoption behaves qualitatively similar compared to the constellation for central origins.

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We thank the Network Science Group from the University of Cologne and Nora Molkenthin for helpful discussions and Christian Dethlefs for help with simulations. D.S. acknowledges support from the Studienstiftung des Deutschen Volkes. M.T. acknowledges support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) through the Center for Advancing Electronics Dresden (cfaed).

In the Main Manuscript we demonstrated that the ride-sharing anti-coordination game reproduces opposing regimes of ride-sharing adoption in a simple setting. In this section we demonstrate the robustness of these results under different 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 and azimuthally asymmetric destination demand on the ride-sharing adoption from a joint origin. We distinguish between four 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 in radial 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.

Urban sprawl : In situations where distant destinations from the city center make up the majority of ride requests the 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 from an 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 significantly by cardinal direction.

Heterogeneous settlement: Urban constellations where both radial as well as azimuthal destination demand gradients exist might describe heterogeneously grown environments, e.g. because of natural obstacles or staged development.Figs. 13 and 14 correspond to the four scenarios. For given financial discount an increase in request rate S gives rise to a spatially heterogeneous sharing/non-sharing pattern and decreasing overall adoption of ride-sharing in all scenarios, independent of the destination demand distributions (compare Fig. 3 in the Main Manuscript). As second order effects, the origin-destination distribution determines (i) whether the cardinal direction of the sharing pattern is random ( Supplementary Fig. 13 for radially asymmetric destination demand), or aligned with the highest destination demand ( Supplementary Fig. 14 for azimuthally asymmetric destination demand), and (ii) whether close-by or distant destinations 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/nonsharing pattern is solely driven by random fluctuations breaking the azimuthal symmetry. The destination demand 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, but this time increasing the request rate reduces the willingness to share from the outside (i.e. high destination demand).

In the presence of azimuthal destination demand gradients the sharing pattern forms along the branches of high demand (see Supplementary Fig. 14a ). The dominance of those destinations in the replicator dynamics 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 from in-to outside as in the uniform case analyzed in the Main Manuscript.

In this section of the Supplementary Information we provide detailed insight into the data used, cleansing procedures applied and simulation methods implemented.

Data sources. The New York City Taxi & Limousine Commission (TLC) publishes trip records for high-volume for-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 10000 trips 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 due to regulatory changes effective in 2019 [25] , potentially impacting ride-hailing behavior, and data from 2020 due to changed 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 whether a shared ride has been authorized by the requester. Our analysis encompasses the time-span between January and December 2019, as chosen for New York City, and includes more than 110 million trip requests served by three transportation service providers (Uber, Lyft, Via).In our geospatial analysis we restrict ourselves to Chicago's 77 community areas, as well as trips leaving or entering the official city borders.Data preparation. We use TLC's data as-is. Our data cleansing procedure removes trip records for which trip information 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 we normalize the total annual flux between origin and destination zones o and d by 16 hour days to obtain a per-minuterequest 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) .

Equilibration. In this article, we focus on the equilibrium properties of the replicator dynamics underlying the ridesharing game on networks. To equilibrate the system we evolve Eqn. (8) . We discard a transient of 1500 replicator time steps before starting measurements of equilibrium values of observables.Per replicator time step and per destination node we repeat the ride-sharing game for 100 times for the current configuration of π(d, t) to generate a reliable numerical estimate for the expected utility increment of sharing E[∆u(d, t)] that is being used to update π(d, t + 1).Matching. After generating the shared ride request graph (see Supplementary Note 3) we implement Edmond's Blossom algorithm to determine a maximum weight matching [37] . Since the algorithm used implements a minimum cost perfect matching, we reduce our non-perfect matching problem to a perfect one as described in [40, Ch. 1.5.1] .

