Entropy production of selfish drivers: implications for efficiency and predictability of movements in a city

Here we describe the process of moving from the origins (Os) to destinations (Ds) in a single day:

• The starting times of the OD trips are distributed uniformly in the origin time interval ΔT_O. We assume that ΔT_O is the same for all origins. In each time step, the time increases by Δt = 1. Driver i starts its trip and becomes active at time t_O(i) ∈ ΔT_O. The trip will become inactive when the driver reaches its destination at time t_D(i) ∈ ΔT_D. The destination time interval ΔT_D is determined by the system structure and dynamics (see figure 1). The arrival times of the drivers at destination site a determine the destination time interval of that site ΔT_D(a). The travel time from origin to destination for driver i is denoted by Δt_OD(i) = t_D(i) − t_O(i).
• An active driver i at site a chooses the next site as follows: with probability α the next site is chosen randomly and uniformly from the set of neighbouring sites. With probability 1 − α the neighbour that minimizes the expected travel time to the destination D(i) is selected. Here, the probability α controls the degree of selfishness of individuals. The expected travel time on each directed edge (ab) in day d is denoted by ${\tilde{t}}_{ab}(d)$. The expected travel times in day d are estimated by using the actual travel times t_ab (d − 1) from the previous day:
  $$\begin{equation}{\tilde{t}}_{ab}(d)=\lambda {t}_{ab}(d-1)+(1-\lambda ){\tilde{t}}_{ab}(d-1).\end{equation} \tag{ 2 }$$
  For the initial day ${\tilde{t}}_{ab}(0)={t}_{ab}(0)$, where the t_ab (0) are the travel times for free lines (with no flow). We shall assume that all edges in the two-dimensional lattice are the same and set t_ab (0) = 1. In practice t_ab (0) is given by the length of the line and the velocity limit. Note that when λ = 0 the expected travel times do not change with day and the drivers always choose the shortest path according to the t_ab (0).
• Let flow F_ab (t) be the number of drivers that enter edge (ab) at time step t. Given the flows, the actual travel times are obtained from
  $$\begin{equation}{t}_{ab}({F}_{ab})={t}_{ab}(0)\left(1+g{\left(\frac{{F}_{ab}}{{F}_{ab}^{{\ast}}}\right)}^{\mu }\right),\end{equation} \tag{ 3 }$$
  to model the influence of flows on the travel times [27–29]. There are other models which try to describe the relation between the observed travel times and the flows [30]. The above relation allows us to control the strength of interactions by the parameter g and the degree of nonlinearity in the interactions by the μ. The actual travel time t_ab is the time that a person spends on edge (ab). Here ${F}_{ab}^{{\ast}}$ is a measure of the line capacity. For simplicity, we assume that t_ba (0) = t_ab (0) and ${F}_{ba}^{{\ast}}={F}_{ab}^{{\ast}}$. In the following, we take μ = 3 and ${F}_{ab}^{{\ast}}=M/\vert E\vert$ in all the numerical simulations. Note that the empirical values of μ are usually greater than 2 [26, 30]. The selected value of μ is to capture the nonlinearity of the relation and it is expected to represent the main qualitative behaviour of the system. The idea behind the above capacities is that the links should nearly behave like free lines when all the movements (here M) are uniformly distributed among all the directed edges (here |E|). The capacity of a line is of course given by the number of its lanes, but this should somehow depend on the population density (or local density). As mentioned before, in the following we work with a fixed population density M/N = 10³. This means that the line capacities are fixed and do not change.

Predictability of the movements can be studied by characterizing the entropy or mutual information of the relevant quantities [10, 32–34]. For instance, it is interesting to know how much the travel times depend on the geometrical distances and the starting times of the trips. We also study a time series of the travel times in different days to check a measure of predictability in the stationary state of the process.

For a single stochastic variable one may use the (estimated) entropy to quantify the variable uncertainty [35]. A measure of predictability for two stochastic variables x, x' is provided by the mutual information of the two variables,

$$\begin{equation}\mathrm{M}\mathrm{I}(x,{x}^{\prime })=\sum\limits _{x,{x}^{\prime }}P(x,{x}^{\prime })\mathrm{log}\frac{P(x,{x}^{\prime })}{P(x)P({x}^{\prime })},\end{equation} \tag{ 4 }$$

where P(x, x'), P(x) and P(x') are the joint and marginalized probability distributions of the variables. Note that large values of mutual information do not necessary mean that accurate prediction is an easy task [36].

Let us define the mutual information between the destination and origin times in one day (i.e. along the vertical direction in figure 1),

$$\begin{equation}{{\Pi}}_{t,t}=\mathrm{M}\mathrm{I}({t}_{\text{O}},{t}_{\text{D}}),\end{equation} \tag{ 5 }$$

and the mutual information between the travel time and the geometrical distance,

$$\begin{equation}{{\Pi}}_{x,t}=\mathrm{M}\mathrm{I}({\Delta}{x}_{\text{OD}},{\Delta}{t}_{\text{OD}}).\end{equation} \tag{ 6 }$$

Figure 3 shows the results of numerical simulations for these quantities in a square lattice of linear size L = 40. The above measures of predictability diminish with increasing g and α, or by using the travel information from the previous days (λ = 0.5) in the absence of randomness (α = 0). The difference between the cases λ = 0 and λ = 0.5 gets smaller for larger α, where Π_t,t is even a bit larger in the latter case. Note that when λ = 0 all edges have the same expected travel times ${\tilde{t}}_{ab}(d)={t}_{ab}(0)=1$, where the shortest path is strongly correlated with the geometrical distance Δx_OD. When λ = 0.5, there are some edges with small travel times in the previous day and all drivers are aware of this information which is used to determine the shortest path to their destinations. Therefore, one expects to observe less correlations here between the geometrical distance and the actual travel times. However, the same global information could make the arrival times at the destinations more dependent on the departure times and results to larger mutual information between the two time variables.

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Next, we study the changes in the travel times along the links for different days (i.e. along the horizontal direction in figure 1). Recall that the trips in each day rely on the expected travel times ${\tilde{t}}_{ab}(d)$ from the previous days. The actual travel times t_ab (d) in that day could however be different from the estimations we started with. The relative difference of the two quantities thus provides a measure of unpredictability in the movement process,

$$\begin{equation}{D}_{d}(t,\tilde{t})=\frac{1}{\vert E\vert }\sum\limits _{(ab)}\frac{\vert {t}_{ab}(d)-{\tilde{t}}_{ab}(d)\vert }{{\tilde{t}}_{ab}(d)}.\end{equation} \tag{ 7 }$$

Consider a time series of the above quantity in terms of d. The power spectrum S(ω) of such a time series as a function of frequency ω is reported in figure 4. Here $S(\omega )\propto \vert \hat{D}(\omega ){\vert }^{2}$ where $\hat{D}(\omega )$ is the discrete Fourier transform of the above time series. The appearance of a dominant peak at high frequencies for larger interactions g signals a high level of variation at the shortest time scales (two consecutive days). Figure 5 shows the stationary values of ${D}_{d}(t,\tilde{t})$ versus the model parameters g and λ. Interestingly, the above quantity exhibits a minimum for very small but nonzero values of λ with a discontinuity at λ = 0. This means that some level of information sharing could be helpful even in the absence of any coordination. Moreover, the stronger interactions (larger g) result in smaller values and ranges of beneficial λ, which can be used to reduce the above measure of unpredictability. On the other hand, for a fixed λ, the deviations from the expected travel times grow with the strength of interactions as expected.

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Perhaps the most important quantities of the process regarding its efficiency are the travel times and costs. The average travel time in the process is given by

$$\begin{equation}{\tau }_{\text{OD}}=\frac{1}{M}\sum\limits _{i}{\Delta}{t}_{\text{OD}}(i),\end{equation} \tag{ 8 }$$

where the sum is over all individuals i = 1, ..., M. The average number of travels per person in a day is obtained from the sum of all the input flows F_ab (t) for different edges and times,

$$\begin{equation}{\sigma }_{\text{OD}}=\frac{1}{M}\sum\limits _{t}\sum\limits _{(ab)}{F}_{ab}(t).\end{equation} \tag{ 9 }$$

This is expected to be proportional to the total energy (e.g. the total amount of fuel) that is consumed in the process. A measure of efficiency then is defined by ratio of the two quantities,

$$\begin{equation}{\eta }_{\text{OD}}=\frac{1/{\tau }_{\text{OD}}}{{\sigma }_{\text{OD}}}.\end{equation} \tag{ 10 }$$

This efficiency takes its maximum value for g = 0, α = 0, when both the travel times and the number of edge travels are minimum. For each person i, we can also define the velocity v_OD(i) = Δx_OD(i)/Δt_OD(i), given the geometrical origin to destination distance Δx_OD(i). Then, the average velocity is

$$\begin{equation}{v}_{\text{OD}}=\frac{1}{M}\sum\limits _{i}{v}_{\text{OD}}(i),\end{equation} \tag{ 11 }$$

which can be regarded as another measure of the process efficiency [37].

In the following, we are also interested in the amount of disorder that is generated by the movements. A measure of increase in the system disorder or uncertainty (entropy production) is provided by the relative entropy of the origin and destination time intervals,

$$\begin{equation}{\Delta}{S}_{\text{OD}}=\langle \mathrm{log}\enspace {\Delta}{T}_{\text{D}}\rangle -\langle \mathrm{log}\enspace {\Delta}{T}_{\text{O}}\rangle =\frac{1}{N}\sum\limits _{a}\mathrm{log}\enspace {\Delta}{T}_{\text{D}}(a)-\mathrm{log}\enspace {\Delta}{T}_{\text{O}}.\end{equation} \tag{ 12 }$$

As mentioned before, ΔT_O is the same for all sites a. Other measures of entropy production (or irreversibility) of the process can be defined for example by the relative entropy of the forward (origin to destination) and backward (destination to origin) flows [21]. In this study, we focus on the above measure ΔS_OD, which only needs the destination time intervals at the end of the forward process, to compare with the origin time intervals.

Figure 6 shows how the above quantities change with the origin time interval ΔT_O when λ = 0. We see that by increasing ΔT_O (slower process) the efficiency increases because the interaction parameter g becomes effectively irrelevant for very large time intervals. In this limit, the relative entropy ΔS_OD is nearly zero but it grows continuously as ΔT_O decreases, separating two phases of high and low efficiency (or velocity).

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In figure 7, we report the lattice size effects when the parameters g and α are varied for λ = 0. We keep the population density M/N = 1000 and the links' capacity ${F}_{ab}^{{\ast}}=M/\vert E\vert$ for different linear sizes L = 20, 30, 40. It is observed that the efficiency and velocity go to zero very rapidly with L even when we increase ΔT_O to take the ratio L/ΔT_O fixed. As figure 8 shows, the efficiency scales roughly as L^−1.8 for g values around 1. This means that, under the above conditions, the city size is limited by the efficiency or velocity of the movement process. A similar phenomenon is observed in [38], where infrastructure costs put an upper bound on the size of a city (in terms of social interactions). The key point is that here the population is increasing with the size and we have a strongly heterogeneous population distribution concentrated around the center of the lattice. More importantly, here the link capacities ${F}_{ab}^{{\ast}}$ are the same in whole the lattice and do not change with the size of the system. These assumptions explain why the efficiency approaches zero by increasing L and why in practice we have to adjust the capacities according to the population size and distribution. The question then is how we should set the local capacities such that the efficiency approaches a desirable limit as the population grows.

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Another interesting observation in figure 7 is that ΔS_OD initially decreases with the randomness parameter α and displays a minimum that is more pronounced for larger lattice sizes. This behaviour is better exhibited in figure 9, where we report the results for a fixed L = 40 with different values of g and λ. In fact, the relative entropy increases with α for small values of the interaction parameter. But, for g larger than a critical value, which depends on L and the other model parameters (ΔT_O and λ), a bit of randomness in the movements can reduce the relative entropy ΔS_OD. That is, random deviations from the shortest paths in this regime allow to have more order in the destination time intervals. When λ = 0.5 this behaviour is observed for a smaller g, as using the travel information from the previous days results in a larger effective interaction between the drivers. Besides the minimum displayed by ΔS_OD, in figure 9 we also observe a maximum in the efficiency and velocity for λ = 0.5. It means that in this case the randomness is also reducing the total travel times by avoiding the congested links of the network. Note that the minimum of entropy production and the maximum of efficiency occur for different values of α. So, definition of an optimal α depends on the quantity of interest. Recall that the parameter α represents the randomness in the local movements of the agents. In practice, such a randomness can be effectively modelled by random local reroutings or by some level of noise in the shared travel information.

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In general, the efficiency is positively correlated with the probability of making a right decision which is expected to increase by reducing the travel uncertainties. This relation should be exhibited by reasonable measures of the two quantities (efficiency and uncertainty), see figure 10. As the figure shows, there is however a region where both the efficiency and entropy production decrease with increasing α. For λ = 0, this happens when randomness in the movements reduces the relative entropy ΔS_OD. For λ = 0.5, this region shrinks to a smaller interval, that is from the maximum of efficiency to the minimum of ΔS_OD. In other words, a negative correlation between the efficiency and entropy production is better displayed in the more realistic case when the shortest paths are chosen according to the available travel information.

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