A computational stochastic dynamic model to assess the risk of breakup in a romantic relationship

  1. de la Cruz, Jorge Herrera 2
  2. Rey, José‐Manuel 1
  1. 1 Department of Economic Analysis Complutense University of Madrid Madrid Spain
  2. 2 Department of Mathematics and Data Science San Pablo‐CEU University Madrid Spain
Revista:
Mathematical Methods in the Applied Sciences

ISSN: 0170-4214,  1099-1476

Año de publicación: 2023

Páginas: 1-18

Tipo: Artículo

DOI: 10.1002/MMA.9292 GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Mathematical Methods in the Applied Sciences

Resumen

We introduce an algorithm to find feedback Nash equilibria of a stochastic differentialgame. Our computational approach is applied to analyze optimal policies to nurture aromantic relationship in the long term. This is a fundamental problem for the appliedsciences, which is naturally formulated in this work as a stochastic differential gamewith nonlinearities. We use our computational model to analyze the risk of maritalbreakdown. In particular, we introduce the concept of "love at risk" which allows usto estimate the probability of a couple breaking up in the face of possible unfavorablescenarios.

€ Ver financiación

Información de financiación

Financiadores

Referencias bibliográficas

  • Basar T, Olsder GJ. Dynamic Noncooperative Game Theory, 2nd Edition. 23. Society for Industrial and AppliedMathematics . 1998.2.
  • Friedman A. Stochastic differential games. Journal of Differential Equations 1972; 11(1): 79–108.3.
  • Bensoussan A, Siu CC, Yam SCP, Yang H. A class of non-zero-sum stochastic differential investment and reinsurancegames. Automatica 2014; 50(8): 2025–2037.4.
  • Deng C, Zeng X, Zhu H. Non-zero-sum stochastic differential reinsurance and investment games with defaultrisk. EuropeanJournal of Operational Research 2018; 264(3): 1144–1158.5.
  • Engwerda J. LQ Dynamic Optimization and Differential Games. John Wiley & Sons . 2005.6.
  • Josa-Fombellida R, Rincón-Zapatero JP. New approach to stochastic optimal control. Journal of Optimization Theory andApplications 2007; 135(1): 163–177.7.
  • Mannucci P. Nonzero-sum stochastic differential games with discontinuous feedback. SIAM Journal on Control andOptimization 2004; 43(4): 1222–1233.8.
  • Marín-Solano J, Shevkoplyas EV. Non-constant discounting and differential games with random time horizon. Automatica2011; 47(12): 2626–2638.9.
  • Huang J, Leng M, Liang L. Recent developments in dynamic advertising research. European Journal of OperationalResearch 2012; 220(3): 591–609.10.
  • Prasad A, Sethi SP. Competitive advertising under uncertainty: A stochastic differential game approach. Journal ofOptimization Theory and Applications 2004; 123(1): 163–185.11.
  • Sethi SP. Deterministic and stochastic optimization of a dynamic advertising model. Optimal Control Applications andMethods 1983; 4(2): 179–184.12.
  • Han J, Hu R. Deep fictitious play for finding Markovian Nash equilibrium in multi-agent games. In: Proceedings of MachineLearning Research, vol. 107. ; 2020: 221–245.13.
  • Han J, Hu R, Long J. Convergence of deep fictitious play for stochastic differential games. Frontiers of Mathematical Finance2022; 1(2): 287–319.14.
  • Nikooeinejad Z, Heydari M. Nash equilibrium approximation of some class of stochastic differential games: A combinedChebyshev spectral collocation method with policy iteration. Journal of Computational and Applied Mathematics 2019;362: 41–54.15.
  • Zhang K, Yang Z, Başar T. Multi-agent reinforcement learning: A selective overview of theories and algorithms. Handbookof reinforcement learning and control 2021: 321–384.16.
  • Dockner EJ, Jorgensen S, Van Long N, Sorger G. Differential Games in Economics and Management Science. CambridgeUniversity Press . 2000.17.
  • Goudon T, Lafitte P. The lovebirds problem: why solve Hamilton-Jacobi-Bellman equations matters in love affairs. ActaApplicandae Mathematicae 2015; 136(1): 147–165.18.
  • Rey JM. A mathematical model of sentimental dynamics accounting for marital dissolution. PLoS One 2010; 5(3): e9881.19.
  • Herrera J, Ivorra B, Ramos ÁM. An Algorithm for Solving a Class of Multiplayer Feedback-Nash Differential Games.Mathematical Problems in Engineering 2019; 2019.20.
  • Bokanowski O, Falcone M, Ferretti R, Grüne L, Kalise D, Zidani H. Value iteration convergence of"-monotone schemesfor stationary Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems-Series A 2015; 35(9): 4041–4070
  • Powell WB. Approximate Dynamic Programming: Solving the curses of dimensionality. 703. John Wiley & Sons . 2007.22.
  • Tassa Y, Erez T. Least squares solutions of the HJB equation with neural network value-function approximators. IEEETransactions on Neural Networks 2007; 18(4): 1031–1041.23.
  • Berinde V, Takens F. Iterative Approximation of Fixed Points. 1912. Springer . 2007.24.
  • Coontz S. Marriage, a history. New York, Viking . 2005.25.
  • Kazdin AE. Encyclopedia of Psychology. American Psychological Association and Oxford University Press . 2000.26.
  • Gottman JM, Murray JD, Swanson CC, Tyson R, Swanson KR. The Mathematics of Marriage: Dynamic Nonlinear Models.MIT Press . 2005.27.
  • Rey JM. Sentimental equilibria with optimal control. Mathematical and Computer Modelling 2013; 57(7-8): 1965–1969.28.
  • Herrera J, Rey JM. Controlling forever love. PLoS One 2021; 16(12): e0260529.29.
  • Duffie D, Pan J. An overview of value at risk. Journal of Derivatives 1997; 4(3): 7–49.30.
  • Bauso D, Mansour DB, Djehiche B, Tembine H, Tempone R. Mean-field games for marriage. PLoS One 2014; 9(5): e94933.31.
  • Bressan A, Shen W. Small BV solutions of hyperbolic noncooperative differential games. SIAM Journal on Control andOptimization 2004; 43(1): 194–215.32.
  • Falcone M. Numerical methods for differential games based on partial differential equations. International Game TheoryReview 2006; 8(02): 231–272.33.
  • Higham DJ. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Review 2001;43(3): 525–546.34.
  • Fasshauer GE, Zhang JG. On choosing “optimal” shape parameters for RBF approximation. Numerical Algorithms 2007;45(1-4): 345–368.35.
  • Fasshauer GE. Meshfree approximation methods with MATLAB. 6. World Scientific . 2007.36. MATLAB . Version 7.10.0 (R2010a).
  • Natick, Massachusetts: The MathWorks Inc . 2010.37. Stevenson B, Wolfers J. Marriage and divorce: Changes and their driving forces. Journal of Economic Perspectives 2007;21(2): 27–52.38.
  • Amato PR, James SL. Changes in Spousal Relationships over the Marital Life Course.
  • Alwin, Duane F.; Felmlee, Diane H.;and Kreager, Derek A. Editors (2018) Social Networks and the Life Course. Integrating the Development of Human Livesand Social Relational Networks 2018: 139-158.39.
  • Van Laningham J, Johnson DR, Amato P. Marital Happiness, Marital Duration, and the U-Shaped Curve: Evidence from aFive Wave Panel Study. Social Forces 2001; 79(4): 1313-1341.40.
  • Kulu H. Marriage Duration and Divorce: The Seven-Year Itch or a Lifelong Itch?. Demography 2014; 51(3): 881-893.41.
  • Zhou H, Zhu H, Zhang C. Linear quadratic Nash differential games of stochastic singular systems. Journal of SystemsScience and Information 2014; 2(6): 553–560.42.
  • Haurie A, Krawczyk JB, Zaccour G. Games and Dynamic Games. 1. World Scientific Publishing Company . 2012