A Study of an Atomic Mobility Game With Uncertainty Under Prospect Theory

Emerging mobility systems (e.g., connected and automated vehicles (CAVs), shared mobility) provide the most intriguing opportunity for enabling users to monitor transportation network conditions better and make efficient decisions for improving safety and transportation efficiency. The data and shared information of emerging mobility systems are associated with a new level of complexity in modeling and control [1]. The impact of selfish or irrational social behavior in routing networks of cars has been studied in recent years [2, 3, 4]. Other efforts have addressed how people learn and make routing decisions with behavioral dynamics [5]. The problem of how travelers often have to make decisions under the uncertainty of experiencing delays, especially when uncertainties directly affect travel time in a transportation network, has not been adequately approached yet. Hence, our problem of interest is to study in a game-theoretic setting these interactions and analyze the equilibrium of the travelers’ decisions under uncertainties [6]. We study the interactions of a finite group of players that seek to travel in a transportation network (with a unique origin-destination pair) comprised of roads with splittable traffic. A key characteristic of our approach is that we incorporate prospect theory, a behavioral model that captures the perceptions of utility under uncertainty (how likely and how much).

Some of the existing game-theoretical literature in control and transportation theory assumes that the players’ behavior follows the rational choice theory, i.e., each player is a risk-neutral and utility maximizer. This makes transportation models unrealistic, as unexpected travel delays can lead to uncertainty in a traveler’s utility. There is strong evidence from empirical experiments that show how humans’ choices and preferences systematically may deviate from the choice and preferences of a game-theoretic player under the rational choice theory [7]. For example, humans compare the outcomes of their choices to a known expected amount of utility (called reference) and make their final decision, using that reference to assess their losses or gains asymmetrically. Prospect theory has laid down the theoretical foundations to study such biases and the subjective perception of risk in the utility of humans [7, 8]. This theory has been recognized as a closer-to-reality behavioral model for the decision-making of humans in different engineering problems [9, 10, 11].

In general, one of the standard approaches to alleviate congestion in a transportation system has been managing the travel demand and supply while also taking into consideration the scarce resources. Such approaches focus primarily on traffic routing, which aims to optimize the routing decisions in a transportation network [12]. Another approach is game theory that allows us to investigate the impact of selfish routing on efficiency and congestion [13] and assign travelers routes to minimize travel time under a Nash Equilibrium (NE) [14, 15, 16, 17, 18]. A fundamental theoretical approach in alleviating congestion is routing/congestion games [19, 20], which are a generalization of the standard resource-sharing game of an arbitrary number of resources in a network.

In this paper, we use Prelec’s probability weighting function and an S-shaped value function to model how travelers perceive traffic uncertainties and their travel gains/losses. So, our first contribution is incorporating prospect theory into an atomic routing game with splittable traffic to capture a realistic version of the travelers’ decision-making regarding travel time costs. The S-shaped value function is adopted to represent the curvature of the travel time cost function and account for the travelers’ perception of gains/losses in travel time according to a reference point (defined using the US Bureau of Public Roads function). To address prospect theory’s mathematical intractabilities, our second contribution proposes a smooth approximation function that estimates the non-linear piecewise prospect-theoretic cost functions. Thus, we can estimate how travelers perceive gains/losses and probabilities in travel time costs. This work is focused on establishing the fitness of the approximation function, proving the existence of at least one NE in pure strategies.

The remainder of the paper is structured as follows. In Section II, we present the mathematical formulation of the proposed game-theoretic framework. In Section III, we derive the theoretical properties of the proposed framework, and finally, draw conclusions in Section IV.

We consider a routing game with a finite non-empty set of players $\mathcal{I}$, $|\mathcal{I}|=n\in\mathbb{N}$. Each player $i$ may represents a class of travelers that could use connected and automated vehicles (CAVs) and who control a significant amount of traffic, say $x_{i}\in\mathbb{R}_{\geq 0}$. Thus, we interpret $x_{i}$ as the representation of the flow of traffic that player $i$ contributes to a transportation network. We define traffic flow in this setting as the number of vehicles passing through each point in the network over time. This decision variable is non-negative as players (or the travelers) make trips using their vehicle over time in the transportation network. This is in contrast to non-atomic routing games, where players only control an infinitesimal amount of traffic. We also assume that traffic is splittable. Travelers seek to travel in a transportation network represented by a directed multigraph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where each node in $\mathcal{V}$ may represent different city areas or neighborhoods (e.g., Braess’ paradox network). Each edge $e\in\mathcal{E}$ may represent a road. For our purposes, we think of $\mathcal{G}=(\mathcal{V},\mathcal{E})$ as a representation of a smart city network with a road infrastructure. Any player $i\in\mathcal{I}$ seeks to travel from an origin $o\in\mathcal{V}$ to a destination $d\in\mathcal{V}$. So, all players are associated with the same unique origin-destination pair $(o,d)\in\mathcal{V}\times\mathcal{V}$. Next, each player may use a sequence of edges that connects the OD pair $(o,d)$. We define $\mathcal{R}\subset 2^{\mathcal{E}}$ as the set of routes available to any player $i\in\mathcal{I}$, where their route $r_{i}$ consists of a sequence of edges connecting the origin-destination pair $(o,d)$. We are interested in how such players may compete over the routes in the network for routing their traffic flows (this is a multiple-route traffic flow decision-making problem).

Since each player $i\in\mathcal{I}$ seeks to route their traffic represented by flow $x_{i}$ in the network $\mathcal{G}$, we define, for each $i\in\mathcal{I}$, the set of actions as

where $x_{i}=(x_{i}^{r_{i}^{1}},x_{i}^{r_{i}^{2}},\dots,x_{i}^{r_{i}^{|\mathcal{R}|}})$, $\bar{x}_{i}\in\mathbb{R}_{\geq 0}$ is the total flow of player $i$, and $r_{i}^{k}$ denotes the $k$-th route in the network.

We write $\mathcal{X}=\mathcal{X}_{1}\times\mathcal{X}_{2}\times\dots\times\mathcal{X}_{n}$ for the Cartesian product of all the players’ action sets. We also write $x_{-i}=(x_{1},x_{2},\dots,x_{i-1},x_{i+1},\dots,x_{n})$ for the action profile that excludes player $i\in\mathcal{I}$. Next, for the aggregate action profile, we write $x=(x_{i},x_{-i})$, $x\in\mathcal{X}$.

In our routing game where each player $i\in\mathcal{I}$ chooses their traffic flow vector $x_{i}$ over a common set of routes $\mathcal{R}$, if player $i$ chooses to send traffic $x_{i}^{r_{i}}$ along route $r_{i}$, then this traffic will be distributed along all the edges in this route $r_{i}$. This is because a traveler’s traffic on some route $r_{i}$ is a single quantity among all the route’s edges.

Next, we introduce a travel time latency function to capture the cost that players may experience. Intuitively, we capture the players’ preferences for different outcomes using a “cost function,” in which players are expected to act as cost minimizers. For each $e\in\mathcal{E}$, we consider non-negative cost functions $c_{e}:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}$. We assume that the cost functions at each edge $e$ are convex, continuous, and differentiable with respect to $f_{e}$. One standard way to define in an exact form $c_{e}$ is by the US Bureau of Public Roads (BPR) function, as it is a commonly used model for the relationship between flow and travel time. Mathematically, we have, for any edge $e\in\mathcal{E}$,

where $c_{e}^{0}$ is the free-flow travel time and $f_{e}^{CRT}$ is the critical capacity of traffic flow on road $e$. Note that the BPR function is non-linear, continuous, differentiable, strictly increasing, and strictly convex for $f_{e}\geq 0$.

Next, for some route $r_{i}$ of any player $i$, its cost is the sum of the costs on the edges that constitute route $r_{i}$, i.e., $c_{r_{i}}(x)=\sum_{e\in r_{i}}c_{e}(f_{e}(x))$. The total cost for player $i$ is

where $f_{e}(x)=\sum_{i\in\mathcal{I}}\sum_{r_{i}\ni e}x_{i}^{r_{i}}$.

The game is fully characterized by the tuple $\mathcal{M}=\langle\mathcal{I},(\mathcal{X}_{i})_{i\in\mathcal{I}},(c_{i})_{i\in\mathcal{I}}\rangle$. This non-cooperative routing game is a simultaneous-move game where players make decisions simultaneously and commute in $(o,d)$ of network $\mathcal{G}$. Players behave selfishly and aim to minimize their costs (e.g., travel time latencies). Naturally, players compete with each other over the available yet limited routes and how to utilize them in the transportation network. Indirectly, players make route choices that satisfy their travel needs (modeled through traffic flow). Next, we clarify “who knows what?” in $\mathcal{M}$. All players have complete knowledge of the game and the network. Each player knows their own information (action and cost) as well as the information of other players. At equilibrium, we want to ensure that no player has an incentive to unilaterally deviate from their chosen decisions and change how they distribute their traffic flows over the available routes in the network. So, for our purposes, we observe that an NE in terms of the players’ traffic flows in pure strategies is the most appropriate a solution concept for our game.

In other words, a flow profile $x^{*}$ is an NE if no player can reduce their total cost by unilaterally changing how they distribute their total traffic flow over the available routes in the network. In an NE, each player’s specific $x_{i}$ has the lowest possible cost among all possible distributions over the routes, given the choices made by other players.

In this section, we provide a formal analysis of the properties of our proposed modeling framework, characterize the coefficients of $\sigma$ function, and show that our game admits an NE in pure strategies.

It follows that it is strictly decreasing, continuous, and (continuously) differentiable with respect to the traffic flow $f_{e}\in[0,f_{e}^{\max}]$ for any edge $e\in\mathcal{E}$.

Now, we discuss the error characterization of our approximation function. Let us define the error function $\Phi$ as the squared difference between $c_{e}^{\text{PT}}(f_{e})$ and $\sigma(f_{e})$, integrated over the interval $[0,\kappa]$:

The goal is to minimize $\Phi$ with respect to the parameters $\delta_{1},\delta_{2},\delta_{3}$, and $\delta_{4}$. First, we find the critical points of $\Phi$ by setting its gradient to zero and solving the resulting system of equations: $\nabla\Phi(\delta_{1},\delta_{2},\delta_{3},\delta_{4})=0$. This results in a system of equations involving the partial derivatives of $\Phi$ with respect to each of the parameters, i.e., $\frac{\partial\Phi}{\partial\delta_{1}}=0,\quad\frac{\partial\Phi}{\partial\delta_{2}}=0,\frac{\partial\Phi}{\partial\delta_{3}}=0$, and $\frac{\partial\Phi}{\partial\delta_{4}}=0$. To compute these partial derivatives, we need to differentiate the integrand with respect to each parameter and then integrate it again, for example, $\frac{\partial\Phi}{\partial\delta_{1}}=\int_{0}^{\kappa}\frac{\partial}{\partial\delta_{1}}\left(c_{e}^{\text{PT}}(f_{e})-\sigma(f_{e})\right)^{2}\mathrm{d}x$. This process needs to be repeated for all parameters. However, due to the complexity of the function $c_{e}^{\text{PT}}(f_{e})$ (being a non-linear piecewise function), it is not possible to obtain an explicit analytical expression for these partial derivatives. For our purposes, we rely on numerical optimization techniques to find the exact best-fit parameters that minimize the error function, as these methods can easily handle complex and non-linear optimization.

In this paper, we presented our mobility game incorporating an atomic splittable routing game with prospect theory to study travel behavior in mobility systems. We modeled the overestimation/underestimation of probabilities using Prelec’s probability weighting function, and we considered the traffic uncertainties and travelers’ perception of gains/losses in travel time using a prospect-theoretic S-shaped value function. We proposed an approximation function to address the non-linear and piecewise nature of the prospect-theoretic cost functions and showed that at least one NE exists. Finally, we also derived an upper bound for the error.

In future research, we can explore how to analyze a convex-concave piecewise non-linear optimization problem using optimization techniques, such as sequential convex programming or cutting plane methods. Developing such an optimization framework can enhance our ability to predict travel decisions in mobility systems under prospect theory. Another direction is to incorporate prospect theory and a taxation mechanism and study using artificial intelligence how we can incentivize prospect-theoretic travelers and the trade-offs of efficiency in the mobility systems [25, 26, 27].