# American Institute of Mathematical Sciences

October  2018, 5(4): 265-282. doi: 10.3934/jdg.2018017

## On the stability of an adaptive learning dynamics in traffic games

 1 San Diego State University, Computational Science Research Center, San Diego, CA 92182, USA 2 Universidad Adolfo Ibáñez, Facultad de Ingeniería y Ciencias, Santiago, Chile

* Corresponding author: mdumett@sdsu.edu

Computational Science Research Center, San Diego State University, California, USA.
Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile.

Received  August 2017 Revised  June 2018 Published  August 2018

This paper investigates the dynamic stability of an adaptive learning procedure in a traffic game. Using the Routh-Hurwitz criterion we study the stability of the rest points of the corresponding mean field dynamics. In the special case with two routes and two players we provide a full description of the number and nature of these rest points as well as the global asymptotic behavior of the dynamics. Depending on the parameters of the model, we find that there are either one, two or three equilibria and we show that in all cases the mean field trajectories converge towards a rest point for almost all initial conditions.

Citation: Miguel A. Dumett, Roberto Cominetti. On the stability of an adaptive learning dynamics in traffic games. Journal of Dynamics and Games, 2018, 5 (4) : 265-282. doi: 10.3934/jdg.2018017
##### References:

show all references

Computational Science Research Center, San Diego State University, California, USA.
Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile.

##### References:
Stability region in the $2 \times 2$ traffic game
Three fixed points with $\psi'(\bar w)>1$
Fixed points of $\psi$: $\psi'(\bar w)>1$ (left) vs $\psi'(\bar w)<1$ (right)
Stability region for a $2 \times 2$ symmetric game in the $\mu$-$q$ parameters
 [1] J. M. Peña. Refinable functions with general dilation and a stable test for generalized Routh-Hurwitz conditions. Communications on Pure and Applied Analysis, 2007, 6 (3) : 809-818. doi: 10.3934/cpaa.2007.6.809 [2] MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777 [3] Xiangxiang Huang, Xianping Guo, Jianping Peng. A probability criterion for zero-sum stochastic games. Journal of Dynamics and Games, 2017, 4 (4) : 369-383. doi: 10.3934/jdg.2017020 [4] Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122 [5] Henri Schurz. Moment attractivity, stability and contractivity exponents of stochastic dynamical systems. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 487-515. doi: 10.3934/dcds.2001.7.487 [6] Alexandra Rodkina, Henri Schurz, Leonid Shaikhet. Almost sure stability of some stochastic dynamical systems with memory. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 571-593. doi: 10.3934/dcds.2008.21.571 [7] Mathias Staudigl, Srinivas Arigapudi, William H. Sandholm. Large deviations and Stochastic stability in Population Games. Journal of Dynamics and Games, 2021  doi: 10.3934/jdg.2021021 [8] Saul Mendoza-Palacios, Onésimo Hernández-Lerma. Stability of the replicator dynamics for games in metric spaces. Journal of Dynamics and Games, 2017, 4 (4) : 319-333. doi: 10.3934/jdg.2017017 [9] Zaki Chbani, Hassan Riahi. Existence and asymptotic behaviour for solutions of dynamical equilibrium systems. Evolution Equations and Control Theory, 2014, 3 (1) : 1-14. doi: 10.3934/eect.2014.3.1 [10] Ivan Werner. Equilibrium states and invariant measures for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1285-1326. doi: 10.3934/dcds.2015.35.1285 [11] Alan Beggs. Learning in monotone bayesian games. Journal of Dynamics and Games, 2015, 2 (2) : 117-140. doi: 10.3934/jdg.2015.2.117 [12] Luís Tiago Paiva, Fernando A. C. C. Fontes. Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2335-2364. doi: 10.3934/dcdsb.2019098 [13] Xiaoming Wang. Numerical algorithms for stationary statistical properties of dissipative dynamical systems. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4599-4618. doi: 10.3934/dcds.2016.36.4599 [14] Daniel Lear, David N. Reynolds, Roman Shvydkoy. Grassmannian reduction of cucker-smale systems and dynamical opinion games. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5765-5787. doi: 10.3934/dcds.2021095 [15] Liming Wang. A passivity-based stability criterion for reaction diffusion systems with interconnected structure. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 303-323. doi: 10.3934/dcdsb.2012.17.303 [16] Rafael Ortega. Variations on Lyapunov's stability criterion and periodic prey-predator systems. Electronic Research Archive, 2021, 29 (6) : 3995-4008. doi: 10.3934/era.2021069 [17] Ryoji Sawa. Stochastic stability in the large population and small mutation limits for coordination games. Journal of Dynamics and Games, 2021  doi: 10.3934/jdg.2021015 [18] Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control and Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010 [19] Shu Zhang, Yuan Yuan. The Filippov equilibrium and sliding motion in an internet congestion control model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1189-1206. doi: 10.3934/dcdsb.2017058 [20] Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375

Impact Factor: