# An Evolutionary View on Equilibrium Models of Transport Flows

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## Abstract

**:**

## 1. Introduction

## 2. Traffic Assignment: Problem Statement

**Beckmann model.**An important idea behind the Beckmann model is that the cost (e.g., travel time) of passing a link e is the same for all agents and depends only on the flow ${f}_{e}$ along it. We denote this cost for a given flow ${f}_{e}$ by ${t}_{e}={\tau}_{e}\left({f}_{e}\right)$. Another essential point is a behavioral assumption (the first Wardrop’s principle): each agent knows the state of the whole network and chooses a path p minimizing the total cost ${T}_{p}\left(t\right)={\sum}_{e\in p}{t}_{e}.$

**Population games dynamic for (stochastic) Beckmann model.**Let us consider each driver to be an agent in population game, where ${P}_{w}$, $w\in OD$ is a set of types of agents. All agent (drivers) of type ${P}_{w}$ can choose one of the strategy $p\in {P}_{w}$ with cost function ${T}_{p}\left(t\left(f\left(x\right)\right)\right):={\tilde{T}}_{p}\left(x\right)$. Assume that every driver/agent independently of anything (in particular of any other drivers) is considering the opportunity to reconsider his choice of route/strategy p in time interval $[\mathrm{t},\phantom{\rule{3.33333pt}{0ex}}\mathrm{t}+\mathsf{\Delta}\mathrm{t})$ with probability $\lambda \mathsf{\Delta}\mathrm{t}+o\left(\mathsf{\Delta}\mathrm{t}\right)$, where $\lambda >0$ is the same for all drivers/agents. It means that with each driver we relate its own Poisson process with parameter $\lambda $. If in moment of time t (when the flow distribution vector is $x\left(\mathrm{t}\right)$) the the driver of type ${P}_{w}$ decides to reconsider his route, than he choose the route $q\in {P}_{w}$ with probability

**Theorem**

**1.**

**Proof.**

## 3. Origin–Destination Matrix Estimation

## 4. Two-Stages Traffic Assignment Model

## 5. Numerical Experiments

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Gasnikova, E.; Gasnikov, A.; Kholodov, Y.; Zukhba, A.
An Evolutionary View on Equilibrium Models of Transport Flows. *Mathematics* **2023**, *11*, 858.
https://doi.org/10.3390/math11040858

**AMA Style**

Gasnikova E, Gasnikov A, Kholodov Y, Zukhba A.
An Evolutionary View on Equilibrium Models of Transport Flows. *Mathematics*. 2023; 11(4):858.
https://doi.org/10.3390/math11040858

**Chicago/Turabian Style**

Gasnikova, Evgenia, Alexander Gasnikov, Yaroslav Kholodov, and Anastasiya Zukhba.
2023. "An Evolutionary View on Equilibrium Models of Transport Flows" *Mathematics* 11, no. 4: 858.
https://doi.org/10.3390/math11040858