# Evaluating incentives for public transit in large gatherings

Omar Meebed, Hassen Fourati, Alain Kibangou

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Omar Meebed, Hassen Fourati, Alain Kibangou. Evaluating incentives for public transit in large gatherings. ECC 2025 - 23rd European Control Conference, Jun 2025, Thessaloniki, Greece. pp.2495-2500, 10.23919/ECC65951.2025.11186915. hal-05028699

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Submitted on 10 Apr 2025

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# Evaluating incentives for public transit in large gatherings

Omar Meebed<fnref n="1" />, Hassen Fourati<fnref n="1" /> and Alain Kibangou<fnref n="1" />,²

*Abstract*— Large crowds at events disrupt city transportation networks, and incentivizing public transit to mitigate these impacts remains challenging. This paper presents a method to design and assess public transit incentives for large events. This is done by first developing a traveller mobility model that describes how information from routing apps data, public transit fare and frequency guides the choice of travellers between two modes and updates the traffic state based on their mode choice. Traveller's wait time, departure time, public transit capacity, road capacity, road congestion, and parking capacity of private vehicles at destination are all considered when updating the number of users who completed their trip. The influence of public transit fare and frequency on the population's overall travel time, modal split and parking overcapacity is studied, along with the influence of the ratio of travellers informed with routing apps. Finally, incentives are selected by including operational cost and user satisfaction in an optimization approach.

## I. INTRODUCTION

For decades, engineers, mathematicians, urban planners, and economists have studied urban mobility to enhance traveller experience while meeting profitability and sustainability goals. Modern cities now offer diverse transportation options, prompting the development of models to understand travellers' mode choices. In particular, discrete choice models [1] estimate the travel behaviour of individuals by measuring the benefits and costs of each alternative. This has been applied in various works aiming at identifying key factors influencing mode choice [2], evaluating demand under different pricing scenarios [3], and exploring several nudging strategies to promote public transportation [4].

Information Communication and Technology (ICT) can assist travellers deciding between different modes and routes. In particular, navigation apps nowadays play a critical role in influencing traveller's choices, as shown in [5]. They not only recommend specific routes, but specific modes that usually promote selfish goals such as minimizing the commuter's travel time. In [6], the authors demonstrate the effect of navigation apps on road congestion and show that for a given rate of penetration and obedience to such apps, inefficiencies such as partial demand transfer can appear. This can be worsened during large gatherings. Indeed, it is well known that mobility infrastructure is significantly strained during planned special events that attract large crowds such as football games, concerts, trade fairs and other major gatherings that contribute to the cultural vibrancy of cities [7].

Several strategies have been outlined to alleviate the pressures of peak mobility demand during large gatherings [8], [9], [10]. Measures like redirecting traffic, providing real-time travel information, and enhancing public transit can ease congestion around events but may impact nearby residents. Redirected traffic may increase noise, pollution, and safety concerns in adjacent areas, while crowded public transit can disrupt locals' commutes. The additional transit options adds significant costs and may not bring noticeable results if many travellers still prefer using private cars.

This paper focuses on analysing user behaviour in response to mode-choice incentives, leading to an optimal strategy for expanding public transit during high demand. A methodology is proposed to assess transit incentives with informed users, aimed at mobility managers tasked with transporting large crowds to events. With limited parking spots for private vehicles available at the venue and costly bus alternatives, mobility managers must find a way to limit disturbances at the venue created by private vehicles parking on the road while reducing costs of transporting people with buses. The problem is addressed as follows:

* Developing an activity-based mobility model similar to [11], with one origin and one destination. Compared to [11], the model is extended by adding mode choice and accounting for flow limits, wait times and road congestion in addition to the existing departure flow and destination capacity limits.

* Assessing the impact of informed and undecided travellers on travel time, mode split, and parking capacity, with recommendations to optimize mobility costs and traveller satisfaction.

This paper is organized as follows: Section II introduces the mobility model for travellers in a large gathering. Section III presents a behavioural analysis of the model. Section IV develops the optimization problem used to select incentives and illustrates how optimal parameters mitigates negative effects of selfish informed travellers. Section V summarizes the contributions and suggests future investigations.

## II. MODEL DESCRIPTION

Given an initial number of travellers $N_0$ at an origin $O$, the model presented in this section describes their transfer towards the destination $D$, the gathering location, using two different modes of transportation, namely public transit $P$ and private vehicle $C$. Travellers at $O$ are divided in three categories: travellers $P_p(t)$ who are potential users of mode $P$, travellers $C_p(t)$ who are potential users of mode $C$ and undecided travellers $U(t)$ who decide whether to use mode $P$ or $C$ based on the attractiveness of each alternative at

This work was supported by the French National Research Agency through grant COCOON ANR-22-CE48-0011.

<footnote n="1">¹Univ. Grenoble Alpes, CNRS, Inria, Grenoble INP, GIPSA-lab, Grenoble, France. ²Faculty of Science, University of Johannesburg (Auckland Park Campus), Johannesburg 2006, South Africa.</footnote>
their departure time $t$. Let $h(\mu)$ be the probability of users choosing mode $P$ at $t$, with $\mu$ a set of parameters influencing this decision. At $t = 0$, $P_p(0)$ and $C_p(0)$ represent captive (or stubborn) users of modes $P$ and $C$ respectively, while $U(0)$ represent the total number of undecided travellers. All travellers eventually become active users $P_a(t)$ and $C_a(t)$ to arrive at the destination $\mathcal{D}$. Travellers in mode $P$ are always satisfied users, while travellers in mode $C$ are satisfied users $C_s$ if the occupied parking capacity $\eta C_a(t)$ remains less than the maximum capacity $C_{\max}$, with $\eta$ denoting the vehicle to passenger ratio, implying that a parking spot is available. Otherwise, they are dissatisfied users $C_d(t)$ who are forced to park elsewhere. A schematic representation of the traveller mobility model, showing population groups and flows between them, is given in Fig. 1.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Schematic representation of the traveller mobility model dynamics.</figcaption></figure>

The following paragraphs will detail the expressions of outflows $\dot{U}(t)$, $\dot{P}_a(t)$ and $\dot{C}_a(t)$ from $U(t)$, $P_p(t)$ and $C_p(t)$ respectively, as well as detail each group's expressions. It is assumed that transportation demand is weighted by a shaping function $s(t)$ distributing demand in time.

### A. Travellers using public transit

Public transit flow is expressed through boarding time $t_b$, in which travellers leave the origin and bus period $T$ defining the time between two consecutive busses. These times are encoded in $\Pi(t)$. Additionally, public transit flow is bounded by a maximum flow $F_p$. The outflow of active users $\dot{P}_a$ of mode $P$ from potential users $P_p(t)$ is therefore expressed as:

$$ \dot{P}_a(t) = \min(F_p\Pi(t), s(t)P_p(t)). \quad (1) $$

It is assumed here that road congestion does not impede the flow of mode $P$, a plausible assumption if a path between $\mathcal{O}$ and $\mathcal{D}$ is dedicated for this public transit line.

### B. Travellers using private vehicles

Let $F_c$ denote the maximum flow when using private vehicles. Contrarily to mode $P$, the flow in mode $C$ is further limited by road congestion. This congestion is represented not only in $F_c$ but in an additional expression $r(t) \in [0, 1)$ portraying a flow reduction fraction resulting from congestion from other travellers in the city who are using the road to reach different destinations. Precisely, a low congestion is given by a value close to 1 and a high congestion is given a

value close to 0. Similarly to (1), the outflow $\dot{C}_a(t)$ of mode $C$ users from potential users $C_p(t)$ is written as:

$$ \dot{C}_a(t) = \min(F_c r(t), s(t)C_p(t)). \quad (2) $$

Accounting for limited parking capacity and the number of travellers leaving $\mathcal{O}$ using mode $C$, the number of satisfied active users $C_s(t)$ of mode $C$ is expressed as:

$$ C_s(t) = \frac{1}{\eta} \min(C_{\max}, \eta C_a(t)) \quad (3) $$

with $\eta$ the vehicle to passenger ratio. Therefore, the expression for dissatisfied travellers $C_d(t)$ is:

$$ C_d(t) = C_a(t) - C_s(t). \quad (4) $$

### C. Travellers deciding between two modes

A simplifying assumption is made on the rate in which travellers choose between two modes, namely the outflow $\dot{U}(t)$ from $U(t)$ depends on a constant rate $\lambda$:

$$ \dot{U}(t) = -\lambda U(t) \quad (5) $$

Travellers in $U(t)$ are separated into two categories: informed ($I$) and noninformed ($N$). Each category relies on a different set of parameters $\mu$ to decide their preferred mode. Indeed, informed travellers rely on the demand on each mode $s(t)P_p(t)$ and $s(t)C_p(t)$, road congestion state $r(t)$, available parking spaces $C_{\max} - \eta C_s(t)$, public transit fare $p$ and period $T$. Noninformed travellers only rely on static parameters, namely $p$ and $T$. Let $\gamma$ be the fraction of informed travellers and $1 - \gamma$ the fraction of noninformed ones.

To compute the probability $h(\mu)$, disutility functions describing the undesirability of each alternative are defined. For informed users the disutility functions are expressed by:

$$ V_{P,I}(t) = P_{p,norm}(t) + p_{norm} + T_{norm} \quad (6) $$

$$ V_{C,I}(t) = C_{p,norm}(t) + (1 - r(t)) + \frac{C_{crit}}{C_{\max} - \eta C_s(t)} \quad (7) $$

with

$$ P_{p,norm}(t) = \frac{s(t)P_p(t)}{N_0} \quad (8) $$

$$ C_{p,norm}(t) = \frac{s(t)C_p(t)}{N_0} \quad (9) $$

$$ p_{norm} = \frac{p}{p_{max}} \quad (10) $$

$$ T_{norm} = \frac{T}{T_{max}} \quad (11) $$

Equations (6) and (7) portray that disutility for public transportation is high when the demand $s(t)P_p(t)$ is high, when the price $p$ is high or when bus frequency $1/T$ is low. For private vehicles, the disutility function is driven, among others, by the undesirability of the external congestion expressed by $1 - r(t)$ and also by the available parking places. Indeed, the term $\frac{C_{crit}}{C_{\max} - \eta C_s}$ expresses an impossibility for travellers to use mode $C$ if all parking spots are occupied
since if $\eta C_s = C_{\max}$, $V_{C,I} \to \infty$. The divergence rate is controlled by a critical number of free parking spots $C_{\text{crit}}$.

The disutility function for non informed user is simply:

$$V_{P,N} = p_{\text{norm}} + T_{\text{norm}} \quad (12)$$

$$V_{C,N} = 0 \quad (13)$$

These expressions of disutility in (6), (7), (12) and (13) lead to the formulation of the informed and noninformed probability of travellers to use mode P, denoted $h_I(x(t), p, T, r(t))$ and $h_N(p, T)$, respectively, with $x(t)$ the states $P_p(t)$, $C_p(t)$ and $C_s(t)$:

$$h_I(x(t), p, T, r(t)) = \frac{1}{1 + e^{-(V_{C,I}(t) - V_{P,I}(t))}} \quad (14)$$

$$h_N(p, T) = \frac{e^{-V_{P,N}}}{e^{-V_{P,N}} + 1} \quad (15)$$

The total probability of travellers to choose mode P, replacing $\mu$ by the relevant parameters is given by:

$$h(x(t), \gamma, p, T, r(t)) = \gamma h_I(x(t), p, T, r(t)) + (1 - \gamma) h_N(p, T) \quad (16)$$

### D. Mobility model

Let $N_D(t)$ be the number of travellers arriving at $\mathcal{D}$, regardless of their travel mode. The full dynamic mobility model is given by the following equations:

$$\begin{aligned}
\dot{U}(t) &= -\lambda U(t) \\
\dot{P}_p(t) &= h(\mu)\lambda U(t) - \min(F_p\Pi(t), s(t)P_p(t)) \\
\dot{P}_a(t) &= \min(F_p\Pi(t), s(t)P_p(t)) \\
\dot{C}_p(t) &= (1 - h(\mu))\lambda U(t) - \min(F_c r(t), s(t)C_p(t)) \\
\dot{C}_a(t) &= \min(F_c r(t), s(t)C_p(t)) \\
C_s(t) &= \frac{1}{\eta} \min(C_{\max}, \eta C_a(t)) \\
C_d(t) &= C_a(t) - C_s(t) \\
N_D(t) &= P_a(t) + C_a(t)
\end{aligned} \quad (17)$$

This model conserves the total number of travellers going from $\mathcal{O}$ to $\mathcal{D}$, as given by (18):

$$N_0 = U(t) + P_p(t) + C_p(t) + N_D(t) \quad (18)$$

**Example 1:** Consider the following expressions for $s(t)$, $\Pi(t)$ and $r(t)$:

$$s(t) = \frac{1}{1 + e^{-b(t - t_h)}} \quad (19)$$

$$\Pi(t) = \begin{cases} 1 & \text{if } t \bmod T \le t_b \\ 0 & \text{otherwise} \end{cases} \quad (20)$$

$$r(t) = 1 - r_{\max} \left( e^{-\frac{(t-t_{p1})^2}{2\sigma^2}} + e^{-\frac{(t-t_{p2})^2}{2\sigma^2}} \right) \quad (21)$$

Here $s(t)$ is a sigmoid function portraying the increasing departure rate as time passes, with $t_h$ the time in which

half of the remaining passive travellers depart and b a parameter regulating departure window. $\Pi(t)$ is the public transit departure window described in Section II-A and $r(t)$ portrays a morning and afternoon road congestion profile, with $r_{\max} \in [0, 1)$ the maximum capacity reduction, $\sigma$ the congestion window, $t_{p1}$ and $t_{p2}$ the congestion peak times. With parameters in Table I, Fig. 2 is obtained.

<table><caption>TABLE I: Parameters related to traveller mobility model</caption><tr><td>s(t) Parameters</td><td>t<sub>h</sub> = 60min, b = 0.1</td></tr><tr><td>Π(t) Parameters</td><td>t<sub>b</sub> = 2min, T = 12min<br>(T<sub>max</sub> = 15min)</td></tr><tr><td>r(t) Parameters</td><td>r<sub>max</sub> = 0.5, σ = 20, t<sub>p1</sub> = 60min, t<sub>p2</sub> = 200min</td></tr><tr><td>Maximum flows</td><td>F<sub>p</sub> = 100, F<sub>c</sub> = 50</td></tr><tr><td>Parking capacity</td><td>C<sub>max</sub> = 6000, η = 1</td></tr><tr><td>Public transfer price</td><td>p = 3€ (p<sub>max</sub> = 3€)</td></tr><tr><td>Undecided travellers</td><td>λ = 0.015, γ = 0.5, C<sub>crit</sub> = 50</td></tr><tr><td>Initial populations</td><td>U(0) = 4000, P<sub>p</sub>(0) = 3000, C<sub>p</sub>(0) = 3000</td></tr></table>

<figure><img src="image_3.png" /><figcaption>Fig. 2: Evolution of travellers over time.</figcaption></figure>

Travellers who start in $U$ (light blue) move to $P_p$ (blue) or $C_p$ (brown) progressively over time. All travellers in $P_p$ move to $P_a$ (yellow) and travellers in $C_p$ move to $C_s$ (dark purple) until the maximum capacity is reached. Afterwards, all remaining travellers move to $C_d$ (green). Functions and parameters from Example 1 will be used in all subsequent numerical examples, unless specified otherwise.

### III. MODEL ANALYSIS

To analyse the evolution of the number of travellers (as in Example 1), several evaluation metrics are introduced, whose definitions adapt the concept of settling time in control theory. In the context of human mobility, the settling time is redefined as the total travel time $t_{\text{tot}}$, in which at least ninety eight percent of travellers have reached $\mathcal{D}$:

$$t_{\text{tot}} = \min\{t \mid N_D(t) \ge 0.98 N_0\}. \quad (22)$$

To compare different total travel times, a normalized travel time is defined using the nominal travel time observed in Example 1, where $t_{\text{tot},ex} \approx 240$ min:
$$t_{tot,norm} = \frac{t_{tot}}{t_{tot,ex}} \qquad (23)$$

The second metric evaluates the modal split of travellers who arrived at D. This metric has been widely used in existing literature to evaluate travel behaviour in a transportation network [12], [13]. In this context, the fraction of travellers that arrived at D using mode P is given by:

$$\alpha_P = \frac{P_a(t_{tot})}{0.98N_0} \qquad (24)$$

Note that the fraction of travellers using mode C is equal to 1 - α<sub>P</sub>. Accounting only for unsatisfied travellers leads to the introduction of the final metric, namely the number of travellers unable to find a parking spot at total travel time, simply expressed as C<sub>d</sub>(t<sub>tot</sub>). This expression is normalized using the total private vehicle users such that:

$$C_{d,norm} = \frac{C_d(t_{tot})}{C_d(t_{tot}) + C_s(t_{tot})} \qquad (25)$$

These metrics are used to draw conclusions regarding exogenous levers (ratio of informed travellers and ratio of undecided travellers) and endogenous ones (price and bus frequency). The results will be depicted with normalized values with respect to a nominal condition given by the parameters in Table I.

### A. Exogenous variables

The effect of informed and undecided travellers is evaluated for different values of γ and ϕ respectively, with ϕ defined as the ratio of undecided over total travellers. Results on the modal split are shown in Fig. 3. The modal split decreases with ϕ but increases with γ. In other words, public transit usage is decreased when more people choose their preferred transportation mode. However, providing travel information to more undecided travellers will positively correct this trend by leading them back to public transit.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Measure of α<sub>P</sub> for varying ratios of undecided travellers and informed travellers ϕ and γ</figcaption></figure>

Results on the number of unsatisfied travellers are shown in Fig. 4. When ϕ is low, there are no unsatisfied travellers.

This is indeed the case due to the fact that parking capacity can hold up to 60% of travellers in this example. For ϕ > 0.3, a number of unsatisfied travellers appear, which increases with ϕ and decreases with γ. This is directly correlated with modal split, since more private vehicle travellers implies more travellers unable to find a legit parking space, resulting in a large number of unsatisfied travellers.

<figure><img src="image_5.png" /><figcaption>Fig. 4: Measure of C<sub>d,norm</sub> for varying ratios of undecided travellers and informed travellers ϕ and γ</figcaption></figure>

Now let us analyse the effect on the total travel time, as depicted in Fig. 5. It can be noticed that total travel time decreases with ϕ until a minimum is reached (between 0.4 and 0.6) then it starts increasing. For large values of ϕ, an increased demand for private vehicles results in an increased total travel time due to high congestion on the road. For smaller values of ϕ, the total travel time is increased because of wait time, since the demand for public transit is larger.

The effect of informed traveller ratio on the total travel time is not significant for an extremely high or low value of ϕ. Nevertheless, we can observe that when ϕ is around 0.5, the following effect is observed: lower values of γ lead to lower total travel times. This can be interpreted as a direct effect of travellers preferring public transit.

<figure><img src="image_6.png" /><figcaption>Fig. 5: Measure of t<sub>tot,norm</sub> for varying ratios of undecided travellers and informed travellers ϕ and γ</figcaption></figure>
## B. Endogenous variables

The effect of public transit fare and frequency are evaluated for different values of p and T respectively. In particular, the effect of lower public transit fares on the three performance metrics is shown in Fig. 6. The parameters in Table I are kept constant while the public transfer is lowered. It can be noticed that lowering the price leads to an increase of public transit travellers. As a consequence, the total travel time is increased and the number of dissatisfied car users is zero. The total travel time is increased due to waiting time; public transit travellers increase while frequency and capacity of buses are kept constant. Reducing the price is an effective strategy as it helps decrease car use and limits the negative impact of unauthorized parking at the destination. However, it also increases bus wait times, which may cause dissatisfaction among public transit users.

<figure><img src="image_7.png" /><figcaption>Fig. 6: Evaluation metrics from varying nominal price $p_{nom}$</figcaption></figure>

The effect of increasing the bus frequency (i.e. lowering the period T) is shown in Fig. 7. This increase has a similar effect on public transit adoption as reducing fares. However, it also reduces the total travel time by lowering waiting time. Dissatisfaction of public transit users would be diminished as a consequence. Therefore, it is expected that a joint reduction of transportation fare and increased bus frequency would expand public transportation adoption and improve satisfaction among public transit and private cars users.

# IV. OPTIMAL PUBLIC TRANSIT FARES AND FREQUENCY

As noticed in Section III, a simultaneous variation on bus fares and frequency is a significant lever to tackle the challenge explained in the introduction. This section presents an objective function, adapted to a mobility manager's preference, and formulates an optimization problem based on the traveller mobility model to select the best incentives.

## A. Problem formulation

The objective function incorporates two factors relevant for a mobility manager: the reduction of the number of unsatisfied travellers and the operational cost of public transit. A normalized expression of unsatisfied travellers is:

<figure><img src="image_8.png" /><figcaption>Fig. 7: Evaluation metrics from varying nominal period $T_{nom}$</figcaption></figure>

$$ \text{Unsatisfied travellers} = \frac{C_d(t_{\text{tot}})}{C_{\text{spill}}} \qquad (26) $$

where $C_{\text{spill}}$ is a parameter regulating tolerance to the number of unsatisfied travellers. Operation cost involves a normalized term describing the number of trips done by buses:

$$ \text{Number of trips} = \frac{t_{\text{tot}}/T}{t_{\text{max}}/t_b} = \frac{t_{\text{tot}}t_b}{Tt_{\text{max}}} \qquad (27) $$

where $t_{\text{max}}$ is defined as the maximum allowed total travel time while $t_b$ stands for the minimum bus period. Therefore, the factor $t_{\text{max}}/t_b$ describes the maximum number of buses in a case where they are always operational during the total allowed duration. Additionally, the normalized profit obtained from bus fare tickets is subtracted from the operation cost. This term is deduced from (10) and (24):

$$ \text{Profit} = p_{\text{norm}}\alpha_P \qquad (28) $$

The objective function is expressed using a linear combination of satisfied demand and operation cost defined in (26), (27) and (28). A weighting parameter $\beta$ is introduced in the linear combination such that:

$$ J(p, T) = \beta \frac{C_d(t_{\text{tot}})}{C_{\text{spill}}} + (1 - \beta) \left( \frac{t_{\text{tot}}t_b}{Tt_{\text{max}}} - p_{\text{norm}}\alpha_P \right) \qquad (29) $$

The complete optimization problem is expressed as:

$$ \begin{array}{ll} \min_{p,T} & J(p,T) \\ \text{s.t.} & t_{\text{tot}} \le t_{\text{max}}, \quad 0 \le p \le p_{\text{max}}, \quad t_b \le T \le T_{\text{max}} \end{array} \qquad (30) $$

where $t_{\text{tot}}$ and $J(p, T)$ defined in (29) are dependent on $P_a$, $C_d$ and $N_d$. Recall that these variables are the outputs of the dynamical model (17) presented in Section II. This problem is clearly non linear and non convex. Solving such a problem requires employing heuristic methods. In particular, surrogate optimization [14] is useful when evaluating time consuming functions, such as the one presented in this paper.
## B. Results

Consider different scenarios by varying the ratio of undecided people and informed people, respectively, while keeping the other parameters in Table I fixed. For each scenario, the optimization problem, with $\beta = 0.5$, $C_{split} = 200$ and $t_{max} = 300$, is solved using the surrogateopt function from the Global Optimization Toolbox in MATLAB [15] in order to find an appropriate bus ticket price and period.

<table><caption>TABLE II: Evaluation metrics and optimal parameters for scenarios involving different ratios $\gamma$ of informed travellers.</caption><thead><tr><th>γ</th><th>t<sub>tot,norm</sub></th><th>α<sub>P</sub></th><th>C<sub>d,norm</sub></th><th>p<sub>norm</sub></th><th>T<sub>norm</sub></th></tr></thead><tbody><tr><td>0</td><td>0.8371</td><td>0.3879</td><td>0</td><td>0.8908</td><td>0.4279</td></tr><tr><td>0.2</td><td>0.8391</td><td>0.3881</td><td>0</td><td>0.9000</td><td>0.4766</td></tr><tr><td>0.4</td><td>0.8394</td><td>0.3878</td><td>0</td><td>0.9335</td><td>0.5125</td></tr><tr><td>0.6</td><td>0.8406</td><td>0.3879</td><td>0</td><td>1.0000</td><td>0.5134</td></tr><tr><td>0.8</td><td>0.8435</td><td>0.3878</td><td>0</td><td>1.0000</td><td>0.6085</td></tr><tr><td>1.0</td><td>0.9804</td><td>0.3885</td><td>0</td><td>1.0000</td><td>0.7785</td></tr></tbody></table>

1) *Scenarios involving different ratios $\gamma$ of informed travellers:* From Table II, for higher values of $\gamma$ the optimal incentive is only based on bus frequency (period), which is reduced at least by 22%. For lower values of $\gamma$, bus fares and period are reduced up to 10% and 57% respectively. The optimal solution leads to a reduction of the total travel time, a constant modal split, and no unsatisfied traveller.

<table><caption>TABLE III: Evaluation metrics and optimal parameters with varying undecided traveller ratio $\varphi$.</caption><thead><tr><th>φ</th><th>t<sub>tot,norm</sub></th><th>α<sub>P</sub></th><th>C<sub>d,norm</sub></th><th>p<sub>norm</sub></th><th>T<sub>norm</sub></th></tr></thead><tbody><tr><td>0</td><td>0.5343</td><td>0.5067</td><td>0</td><td>1.0000</td><td>0.3117</td></tr><tr><td>0.2</td><td>0.7137</td><td>0.4411</td><td>0</td><td>1.0000</td><td>0.4907</td></tr><tr><td>0.4</td><td>0.8414</td><td>0.3879</td><td>0</td><td>0.9807</td><td>0.4957</td></tr><tr><td>0.6</td><td>0.9553</td><td>0.3882</td><td>0</td><td>0.4637</td><td>0.4757</td></tr><tr><td>0.8</td><td>1.0358</td><td>0.3878</td><td>0</td><td>0.1223</td><td>0.5890</td></tr><tr><td>1.0</td><td>1.0953</td><td>0.3883</td><td>0</td><td>0</td><td>0.5805</td></tr></tbody></table>

2) *Scenarios involving different ratios $\varphi$ of undecided travellers:* From Table III, the optimal incentives mainly require reducing the period, even more aggressively than in the previous case. The optimal fare decreases with the ratio of undecided travellers $\varphi$, while the optimal bus period increases. An optimal choice of price and bus period resolves some of the negative repercussions of an increased number of undecided travellers (Section III). Unlike the previous scenario, the modal split decreases with $\varphi$ while the total travel time increases. Except for $\varphi > 0.6$, the travel time is reduced compared to the nominal case.

## V. CONCLUSION

This paper developed a traveller mobility model to describe mode choice based on current traffic information and destination capacity. In the context of large gatherings, this paper utilizes total travel time to define evaluation metrics

for traveller dynamics. Simulations showed that reducing bus fares improves private vehicle satisfaction but increases waiting time for public transit travellers, while increasing bus frequency reduces waiting times for all travellers and improves user satisfaction. Since increasing bus frequency comes with an operational cost on mobility managers, an optimization problem was formulated to select public transit incentives, namely the bus fare price and frequency, such that the number of unsatisfied travellers and operation costs of public transit are reduced. Results of this optimization problem demonstrate that incentives are required in cases where travellers are not informed and many travellers are undecided on their preferred travel mode. Further investigations include the validation of these recommendations in real scenarios.

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