Commuting is, in general, not an enjoyable activity [58] and each journey has some degree of “undesirability” [6, 46]. This undesirability, or cost, varies from person to person and across transport methods [59]. For example, waiting time is particularly important in public transport since users view it as more burdensome than the same amount of time spent travelling [60]. Here, we assume that any undesirability of a journey can be measured, compared across users and expressed as the minutes that the journey takes.

Every person in a city elects a mode of transport from a pool of κ transport modes. Consider a city with N (fixed) individuals who, at time t, choose to commute using a particular mode of transportation. Let x _i (t) be the number of users of mode i at time t, with i = 1, 2, … , κ. Since everyone picks a mode of transportation, then we only consider points inside the simplex X _N = {x | ∑ _i x _i = N}. We assume that the level-of-service of a mode of transportation can be measured in travel time and, thus, all costs can be expressed in commuting minutes of an average journey for each transport mode [9]. The commuting time of each transport mode is divided into three components: 1) a baseline commuting time, which represents the free-flow travel time, i.e., the commuting time as if the city were “empty,” and two variable costs, representing the additional time due to congestion, that depend on 2) the number of users in the same mode of transportation and 3) the number of users in the other modes. If the modal share, i.e., the distribution of users over different modes at time t, is given by the vector x(t) = (x ₁(t), x ₂(t), …, x _κ (t)), then the cost of using mode i can be expressed in minutes as C i x = B i + D i x + I i x , (1) where B _i is a constant baseline commuting time for mode i, the function D _i is the direct cost that users of i impose to the users on the same mode i, and I _i are the indirect costs imposed by the users of other transport modes. In general, D _i ≥ 0 and, if the rate of change of D _i with respect to x is denoted by D i ′ , then we assume that D i ′ ≥ 0 so that costs are non-negative and do not decrease with more users. Similarly with I _i , so more users of any other mode impose higher indirect costs.

The induced and direct costs are not linear and a variety of functions have been proposed to link travel times and traffic flow [61–63]. For example, the Bureau of Public Roads function was developed by fitting speeds on freeways and it captures a steep increase in travel times with high levels of traffic [63–65]. At a city level, measuring travel time is more complicated. Speed rises with distance from the city centre [6] but also, it has been suggested that the average modal journey time per day and per person has remained constant for decades [59], perhaps due to a polycentric transition of cities [66]. However, although we observe a process that is non-linear at a microscopic level, we assume linear costs at a macroscopic level to simplify our model. Assuming that the induced costs functions are linear, we can express the commuting time for all transport modes as C x = B + M x , (2) where C(x) is the commuting time vector (a function), B are the fixed baseline commuting times and M is a matrix with the arranged marginal costs (with the direct costs on the diagonal).

In general, more car users imply a higher commuting time for driving, even if that means fewer cyclists or walkers, and similarly for other modes of transport. Thus, we assume that each entry of M are non-negative, and in general, the direct costs are higher than the induced costs of other modes, so the elements in the diagonal will be the largest values of each row.

We assume that there is some modal share for all transport modes for which the commuting time is lower than other transport modes. That is, if there are enough walkers, cyclists and drivers, then using public transport becomes faster. With enough drivers and public transport users, then walking is faster and so on. Under such conditions, no commuting mode is dominant, i.e., no transportation mode is faster than the others for all modal shares x. Rather, for some modal shares, different transport modes compete with others. If everyone uses a car, it will be the worst (slowest) commuting method. The same applies to public transport, walking and cycling. If everyone uses one transport method, it will be the slowest option due to the imposed costs. Also, there exists an internal point, where the number of users of all transport modes is greater than zero, x°, such that C _i (x°) = C _j (x°), for all pairs i ≠ j, with i, j = 1, 2, … , κ, meaning that there is some internal point for which all commuting methods take the same travel time.

For modal share x, the average commuting time of a trip is given by μ(x) = (1/N)x ^⊤ C(x), i.e., the commuting time of each mode weighted by its share of users. We can write the average commuting time function as μ x = 1 N x ⊤ B + x ⊤ M x . (3)

The function μ(x) is a paraboloid. It has higher values (commuting times) in μ ( N , 0 , … , 0 ) = B 1 + M 11 N , which is a local maximum (and similarly for μ(0, N, … , 0), and for other modes of transport μ(0, 0, … , N, 0, … , 0)). Thus, there is some modal share, x ^*, for which the average commuting time is minimum. That is, μ(x) has some global minimum, μ ^* = μ(x ^*) for modal share x ^*.

Shorter travel times is one of the key factors that make a transport mode more attractive [10, 11, 15, 16, 67]. People often try different commuting options, particularly if others manage to commute faster using a different mode [9]. If cycling, for example, is much faster than using a car, then some drivers might try to commute by bicycle and then make it a habit. Similarly, if the public transport is too slow, people might try other methods and choose a faster option. Users switching becomes a dynamic process at a collective level. More people attracted by the benefits of cycling means higher direct costs, less cycling space, and so slower times for cyclists and fewer costs for drivers or public transport users. More car drivers mean more congestion, and more public transport users lead to more delays, queues, etc. Choosing a transport mode can be thought of as an iterative, multiplayer, evolutionary and non-cooperative game [68, 69]. The strategies are the mode of transportation that each person chooses each day, and their outcome is the commuting time. At each iteration, here taken to be a day, some players switch between the distinct transport alternatives to achieve a faster commute. Users switching between methods alter the modal distribution and the commuting time for all users as a result. A common way in which such types of evolutionary games are modelled is through the so-called replicator equation. The core aspect of the model is that users copy better strategies [70].

Each traveller (or player) chooses their transport method (or strategy) and experiences some commuting time (obtains a payoff, or some “gain” according to the strategy). This results in a dynamical process, where the rate of change of a modal share is given by x i ̇ = ρ x i μ x − C i , (4) where ρ > 0 is a speed parameter, so people switch commuting methods on some time scale. The overdot represents differentiation with respect to time, t. Here, μ(x) − C _i is the difference between the average commuting time and the commuting time for mode i, so that the commuting method i becomes more popular ( x i ̇ > 0 ) when it is faster than the average.

Equation 4 defines a dynamical behaviour that is frequently used to study the adaptation and co-evolution of biological populations [71–73] possessing some important properties. Firstly, the set X _N = {x | ∑ _i x _i = N} is invariant under the dynamics, meaning that a trajectory which begins in X _N , never leaves it [70]. The corners of the set X _N , which are x _i = N and x _j = N for j ≠ i, are fixed points of Equation 4 which might also be termed as an equilibrium. There might also be a fixed point on the edges, where C _i = C _j = μ for any two modes, i and j and x _k = 0 for the others, and there might be a fully internal fixed point x°, where C _i = μ for all modes. Such an internal fixed point, if it exists, forms a Nash equilibrium in the dynamics [74] so that no user wants to change their commuting mode, and if they do, their commuting time is longer (so they or someone else takes the city back to its equilibrium). The corners are not necessarily a Nash equilibrium. Although the corners (or some points along the edge) might form an equilibrium, it is generally unstable to small perturbations. Considering discrete units of time (days, for example) gives a first-order difference equation [75].

A unique internal equilibrium exists (with x _i > 0 for all i = 1, 2, … , κ) if there is a point in X _N such that C _i (x) = C _j (x) for every pair i ≠ j. It is possible to see that, from an arbitrary start, the dynamics will converge to the fixed point x°, hence it becomes asymptotically stable.

For some baseline commuting time B, a cost matrix M and some initial conditions, x ₀, the number of users of each transport mode evolves whilst some are attracted by faster commuting methods. Figure 1 shows plots based on three modes of transport. In the long run, the commuting time of all transport modes converges to the same value if and when an equilibrium is reached.

Eventually, users converge to some equilibrium x°, which has the same commuting time for all transport modes (Figure 2). In general, regardless of the initial distribution, all trajectories converge to some equilibrium distribution, although it could be a cyclic behaviour depending on the parameters. With a small value of ρ > 0, the process takes all internal trajectories to some consensual distribution [9]. The rate of convergence depends on the values of ρ, where smaller values take a longer time.

If there is a modal share for which each method has the lowest commuting time, that is, if all commuting methods are desirable for some modal share and have the worst commuting time if everyone uses them, then an internal equilibrium exists. See the Methods 4.2 for the details.

The replicator equation is based on the principle that faster commuting methods become more popular since individuals switch to reduce their commuting time. For some initial conditions, x ₀, the system produces a trajectory in the space of transport modes X _N . As people switch commuting methods, the collective modal share changes accordingly. Therefore, a trajectory is defined for x(t) as t increases. Also, as time passes, and users switch between commuting methods, the corresponding commuting time for each transport mode changes and the mean commuting time also changes. Combining Equations 4, 3 gives the evolution of the mean commuting time with respect to time d μ d t = ρ N B ⊤ + x ⊤ M + M ⊤ x ̇ . (5)

For different initial conditions x ₀, the observed trajectories x ( t ) x 0 and the mean commuting time curve μ x 0 ( t ) are different, although for any internal initial conditions, trajectories will converge to the fixed point x°. Therefore, all mean commuting time curves μ x 0 ( t ) eventually have the same final value μ _x°.

The mean commuting time μ _x varies according to the distribution of users of different transport modes (Figure 3). Equation 3 is a paraboloid in X _N , a function with a single minimum point that increases in all directions. With different values of B and M , the shape of the second-degree polynomial changes and the location of the minimum average commuting time x ^* also changes. It is possible to find a point in X _N where the average commuting time is minimum. See the Methods 4.2 for the details.

The mean commuting time curve μ x 0 ( t ) is a continuous function for t ≥ 0 since it is a trajectory over the domain of a second-degree polynomial. It begins at x ₀ and, as time evolves, the trajectory converges to x°.

Each time a person switches from one commuting method to another, they reduce their travelling time. However, as a result of that switching, the average commuting time changes, and it might increase. This scenario happens when higher induced costs imposed on other users compensate for the personal gain from the switching. For example, if a person decides to drive instead of walking, they might save some time by doing this, but the payoff is only for the individual and creates even more traffic for those who already drive. An individual gain becomes a collective loss.

The city’s transport evolves as visualised by trajectories x(t) formed by people trying to decrease their commuting time, although the collective result does not always reduce the mean commuting time. To account for this, here the region where trajectories increase the average commuting times is called tragic. When a city is in a tragic area, the individual action of switching between methods for a faster commute results in a slower collective system, with higher commuting times on average. In the case of only two transport modes, it was observed that the equilibrium, where everyone has minimised their own commuting time, might be reached at a point with the highest mean commuting time for society [9], forming a transport paradox. Here, with the replicator dynamics, it is observed that some trajectories are such that μ ̇ > 0 , meaning that the average commuting time increases. For any modal share x, we say that it is tragic if d μ d t | x > 0 , (6) and we call x to be rewarding if the derivative is not positive. A trajectory might switch between being tragic and rewarding as it moves and evolves in X _N .

In general x° ≠ x ^*, so the system will converge to a non-optimum distribution. There is only one scenario where the equilibrium is optimum. If the baseline commuting time is the same for all methods, the direct costs are equal, and the induced costs are the same for all cross mode users, then the stable point is also the minimum mean commuting time. This scenario means that only if the commuting times can be simplified as C _i (x _i ) = η + ω x _i , with the same η and ω for all transport methods, then the stable point is optimum. Furthermore, the optimum distribution is x° = x ^* = (N/κ, N/κ, … , N/κ) so all methods have the same number of users. See the Methods 4.3 for the details.

Unless x° = x ^* (so the equilibrium is the optimum modal share distribution), the modal share of a city converges to a sub-optimum point, where the commuting time for all users is the same but the mean commuting time is higher than the optimum. We can consider the trajectory that begins at the optimum point x ^* but eventually converges to x°. It forms a continuous function μ x ⋆ ( t ) that begins with a lower mean commuting time, but increases for some interval with t > 0. Therefore, except for a unique set of parameters (where x° = x ^*), there is always at least one trajectory that begins in the optimum and increases its commuting time as the steps evolve. The tragedy is almost inevitable.

Depending on the initial conditions and the baseline and costs of the transport modes, the system might have many trajectories where average commuting times increase during some periods. Also, some trajectories might switch between being tragic and not being tragic as time evolves (Figure 4).

The optimum point x ^* might be relatively close to the equilibrium point x°, in which case the price of anarchy is limited. However, the distance between both points might be large, meaning that a city might converge to a modal share that is far from the optimum (Figure 5).

With different baseline commuting time B and induced and direct costs M , the trajectories are distinct and form different regions with tragedy (Figure 6).

The emergent tragedy of the commons can be thought of as the result of a multi-person prisoner’s dilemma [76] where each player picks their best strategy, reducing the collective gain. Playing the iterative game and allowing some players to switch their strategy between one round and the next, our results show that the distribution of users per strategy will converge to some equilibrium, but also, that the collective dilemma imposes a high burden on society. An individual gain results in a collective loss.

Introducing a new technology to a society, even if it offers some gain at first, rather oddly might result in higher commuting times. This result appears to imply that a community would be better if that new technology never existed. For example, contrary to some expectations, it was observed that the introduction of ride-hailing services to compete with more socially desirable modes of transportation, saw trips carried out that otherwise would have been made by public transport or walking [77–80], as well as extra vehicle miles, travelled before each pick-up [81], contributing to growing road congestion and pollution. If the new technology is attractive since it has a small baseline commuting time, then as this becomes more popular, it will impose a high burden on others, and the novel method might be tragic. Eventually, given the right conditions, most people will be attracted by the novel technology, but the collective result is that everyone has to pay the subsequent high induced and direct costs. Thus, a novel commuting transport might be attractive, but it could have a tragic outcome.

The borders of the simplex X _N enable us to compare a system with and without a particular transport mode. Within the replicator dynamics, the borders are invariant, meaning that a trajectory which begins on the border (with some transport mode x _j (0) = 0) never leaves it (so x _j (t) = 0 for all t ≥ 0). We compare the modal share and the average commuting time between two systems with equal commuting times, one with an initial condition of x _j (0) = 0, and one with x _j (0) = θ, with θ > 0 small (Figure 7). After a few days, the mean commuting time might be much lower when the novel method does not exist x _j (0) = 0, than with the new technology. This scenario happens when the new transport mode j is attractive (with a small baseline commuting time), becomes popular (so x _j (τ) becomes large for some τ > 0) and, more importantly, has high induced costs imposed on other transport modes and on the same mode. People are attracted to use that novel mode j due to its lower commuting times, but each user switching imposes a cost on other modes and other users.

The scenario where most users are tempted by a “fast” commuting mode that imposes high direct and indirect costs on others (left panel of Figure 7) might create trajectories that are mostly tragic (bottom right panel of Figure 6). The imposed costs compensate whatever gain that switching individuals might have, so the mean commuting time increases after switching. Under these conditions, the city will converge to most users in mode 3, experiencing high commuting times (on average, 100 min) against the scenario where the commuting method remains unknown (right panel of Figure 7), where the average commuting times remain close to 20 min.

Novel modal shares might alter the equilibrium, although it takes time for the replicator dynamics to fully adhere them into the system. For example, a city with N = 1, 000, 000 inhabitants, with x ₁(0) = 1, with x ₂(0) = 400, 000 and x ₃(0) = 599, 999, the equilibrium between only two modes is reached quickly and for many steps, the number of x ₁ remains almost negligible (Figure 7). After a significant time, x ₁ is “a hidden gem” since it has a lower commuting time (since almost nobody knows about it) and remains fast but unpopular. Eventually, x ₁ becomes a popular strategy and then even a dominant modal share, reducing the size of x ₂ and x ₃ . With any x ₁(0) > 0 small, the modal share eventually attracts most people. However, if x ₁(0) = 0, the equilibrium would remain on the boundary, with x ₁ (t) = 0 for all t ≥ 0.

Changing commuting behaviour is not easy [37]. Two elements are needed to detect the tragedy of transport in a city: first, we need to understand the reasons why people drive and second, also understanding the attitudes of those who do not drive. Using origin-destination surveys that capture details about the journeys and attitudes, it is possible to uncover current and future trends for mobility in a city. Different socioeconomic groups make travel mode decisions based on various factors [89]. For example, a study showed that nearly one in five people in Mexico City would use a car if they could pay for it [88, 90]. Although the Metro System in Mexico City has nearly five million passengers per day, a considerable part of the public transport users in the city is made up of captive riders who do not have an alternative mode of travel [20]. However, it is likely that many people will shift to driving as soon as they can afford it.

All transport modes have some degree of undesirability [46] and although we have used time units to quantify this factor, understanding the main reasons for choosing a commuting method is critical for reducing the size of the tragedy. Many people might avoid public transport if it is perceived as unsafe, unreliable, uncomfortable or inaccessible [91], including pedestrian safety [92], neighbourhood amenities [93] and the urban form [94].

Without loss of generality, considering only κ = 3 commuting options, for method i, the commuting time is higher when all users choose that mode. To see this, let x = (x, y, z). We can write the commuting time as C i x , y , z = B i + α x + β y + γ z , with i = 1,2 and  3 , (7) where x + y + z = N, and where α, β and γ are parameters. For x = N we get that C _i (N, 0, 0) = B _i + αN, but, if x = N − r, it means r users with a different method, so C _i (N − r, s, r − s) = B _i + α(N − r) + βs + γ(r − s), thus, the commuting time is reduced in r units of α and increases in either βs and γ(r − s), for some s ∈ [0, r]. Let δ = max(β, γ) < α be the largest induced cost. Then, C i N , 0,0 = B i + α N (8) > B i + α N − r + δ s + δ r − s (9) ≥ B i + α N − r + β s + γ r − s = C i N − r , s , r − s . (10)

Therefore, when everyone uses method i, its commuting time C _i (N, 0, 0) is higher than the commuting time of other modes.

Assuming that for all transport modes, there is some modal share for which the commuting time is lower than other transport modes, means that all transport modes are the slowest and the fastest method for different shares. Consider only two transport modes. Their commuting time surface C ₁ and C ₂ (a plane from the simplex X ₃ to R ⁺) intersects on a non-empty line, that is, a set U _1,2 for which the commuting time of both transport modes are equal. The set U _1,2 divides the simplex into two parts, one for which mode 1 has a higher commuting time (which contains the corner of S ₃ where all journeys are via mode 1) and one where mode 2 has a higher commuting time (which contains the corner of S ₃ where all journeys are on mode 2). Since U _1,2 divides the simplex into two parts, it intersects the edge where no one uses mode 3 into two parts, so that the corresponding corners of S ₃ are slower for each transport mode. For modes 2 and 3, there is also a set U _2,3 which also divides X ₃ into two parts as well that intersects the edge where nobody uses transport 1. We will show that U _1,2 and U _2,3 intersect at some point by contradiction. Suppose they do not intersect. This means that U _2,3 is fully contained either in the set where mode 1 is slower or fully contained in the set where mode 2 is slower. Suppose it is fully contained in the set where mode 2 is slower. This means that the simplex can be divided into three regions: one in which mode 2 is slower than both methods (the region which contains the corner of X ₃ where all journeys are on transport 2), one in which mode 1 is faster than the other two methods (and contains the corner where all journeys are on mode 3) and one in which mode 3 is faster than the other two methods (and contains the corner where all journeys are on mode 1). This means that X ₃ is divided into three parts, where mode 2 is slower than one or both of the distinct transport modes, which is a contradiction since it is assumed that mode 2 is faster than the other modes for some modal share. Similarly, if we suppose that U _2,3 is fully contained where mode 1 is slower. Then we divide the space X ₃ into three regions where mode 1 is always slower than one or both of the other methods, which is also a contradiction. Therefore, there exists an internal equilibrium.

Thus, what we have shown is that, with linear costs and a fixed baseline, all modal shares are desirable when they have zero users and the worst option when everyone uses them., and also, that a unique internal equilibrium exists. Therefore, we can also ensure that all trajectories with an internal initial condition, will converge to that equilibrium.

Let f(r, s) = rC ₁(r, s, N − r − s) + sC ₂(r, s, N − r − s) + (N − r − s)C ₃(r, s, N − r − s) be the total commuting time of the modal share (r, s, N − r − s) for some 0 ≤ r + s ≤ N. The function f can then be expressed as f r , s = D r 2 + E r s + F s 2 + G r + H s + I , (11) where D = M 11 − M 13 − M 31 + M 33 (12) E = M 12 − M 13 + M 21 − M 23 − M 31 − M 32 + 2 M 33 (13) F = M 22 − M 23 − M 32 + M 33 (14) G = B 1 − B 3 + N M 13 + N M 31 − 2 N M 33 (15) H = B 2 − B 3 + N M 23 + N M 32 − 2 N M 33 , and (16) I = N B 3 + N 2 M 33 . (17)

Evaluating f(N, 0) = N ² M ₁₁ + NB ₁ and similarly, for f(0, N) = N ² M ₂₂ + NB ₂, gives the total commuting time for N users of the first and second commuting methods. The expression f(0, 0) = N ² M ₃₃ + NB ₃ gives the total commuting time when the third method has N users. The function f is a quadratic polynomial that has three local maximum values in the corners of its domain 0 ≤ r + s ≤ N and therefore, there exists a point (r ^*, s ^*) that minimises the values of f.

The gradient of f gives ∇ f r , s = ∂ f ∂ r , ∂ f ∂ s = 2 D r + E s + G , E r + 2 F s + H , (18) and by setting the values equal to (0, 0) we get that r * = 2 F G − E H E 2 − 4 D F , and  s * = 2 D H − E G E 2 − 4 D F . Therefore, we can compute N − r ^* − s ^* and obtain the optimum modal share, (r ^*, s ^*, N − r ^* − s ^*).

Using the same function f(r, s) as introduced before, we know that the system is in equilibrium if C ₁(r, s, N − r − s) = C ₂(r, s, N − r − s) = C ₃(r, s, N − r − s) from which we get that r° = (WY − VZ)/(UY − VX) and s° = (WX − UZ)/(VX − UY), where the coefficients are U = M 11 − M 13 − M 21 + M 23 (19) V = M 12 − M 13 − M 22 + M 23 (20) W = B 2 − B 1 + N M 23 − N M 13 (21) X = M 11 − M 13 − M 31 + M 33 (22) Y = M 12 − M 13 − M 32 + M 33 , and (23) Z = B 3 − B 1 + N M 33 − N M 13 . (24)

From these two equations for (r°, s°, N − r° − s°) the conditions for an equilibrium point in the modal share can now be found. In general, r° ≠ r ^* and s° ≠ s ^* meaning that the modal share with minimum mean commuting time is not the modal share observed at equilibrium. Under certain conditions, however, it is possible that r° = r ^* and s° = s ^*. In terms of the induced costs, there is one case, where M ₁₂ = M ₁₃ = M ₂₁ = M ₂₃ = M ₃₁ = M ₃₂, in terms of the direct costs, M ₁₁ = M ₂₂ = M ₃₃ and the baseline commuting time B ₁ = B ₂ = B ₃. Such a scenario happens when all transport modes have the same baseline commuting time and the same induced and direct costs with respect to the other modes. Since P ₁ + P ₂ + P ₃ = N, then the commuting time for some transport mode can be expressed as C i ( P i ) = B i 0 + m P i , where the commuting time only depends on the number of users of the chosen method of transport, B i 0 considers all induced costs and the gradient m is the difference between the direct and induced costs. In such a case, the matrix M in Eq. 2 is diagonal. The equilibrium from the system x° = (r°, s°, N − r° − s°) and the optimum modal share, x ^* = (r ^*, s ^*, N − r ^* − s ^*) are the same. Moreover, the optimum and stable equilibrium is reached with a modal share corresponding to (N/3, N/3, N/3), meaning that all transport modes attract the same number of users. Except for this scenario, we observe that (r°, s°, N − r° − s°) ≠ (r ^*, s ^*, N − r ^* − s ^*), meaning that the stable point in the replicator dynamics is not the system optimum.