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23-01-2022

Zero-Sum Stochastic Games with Random Rules of Priority, Discrete Linear-Quadratic Model

Authors: Daniel Hernández-Hernández, Joshué H. Ricalde-Guerrero

Published in: Dynamic Games and Applications

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Abstract

This paper is concerned with zero-sum stochastic games with random rules of priority; this means that at each turn one of the players is randomly selected and will be able to see the decision of the other player, before his own action is chosen. We focus on the discrete linear-quadratic model, since its tractability allows us to obtain explicit expressions for the equilibrium strategies. These controls are expressed in terms of the parameters of the model and are adapted to the information available for both players in each turn. The dependence of the noise in the solution is also analysed. Finally, an example is implemented and explored through simulations in a program coded in Python.

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Appendix
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Literature
1.
go back to reference Başar T, Moon J (2017) Riccati equations in Nash and Stackelberg differential and dynamic games. IFAC PapersOnLine 50(1):9547–9554 CrossRef Başar T, Moon J (2017) Riccati equations in Nash and Stackelberg differential and dynamic games. IFAC PapersOnLine 50(1):9547–9554 CrossRef
2.
go back to reference Başar T, Olsder GJ (1980) Team-optimal closed-loop Stackelberg strategies in hierar chical control problems. Automatica 16(4):409–414 MathSciNetCrossRef Başar T, Olsder GJ (1980) Team-optimal closed-loop Stackelberg strategies in hierar chical control problems. Automatica 16(4):409–414 MathSciNetCrossRef
3.
go back to reference Başar T, Olsder, GJ (1998) Dynamic noncooperative game theory. SIAM Başar T, Olsder, GJ (1998) Dynamic noncooperative game theory. SIAM
4.
go back to reference Bensoussan A, Chen S, Sethi S (2015) The maximum principle for global solutions of stochastic Stackelberg differential games. SIAMJ Control Optim 52(4):1956–1981 MathSciNetCrossRef Bensoussan A, Chen S, Sethi S (2015) The maximum principle for global solutions of stochastic Stackelberg differential games. SIAMJ Control Optim 52(4):1956–1981 MathSciNetCrossRef
5.
go back to reference Bismut, JM (1978) Contrôle des systemes linéaires quadratiques: applications de l’intégrale stochastique. In: Séminaire de probabilités XII. Springer, pp 180–264 Bismut, JM (1978) Contrôle des systemes linéaires quadratiques: applications de l’intégrale stochastique. In: Séminaire de probabilités XII. Springer, pp 180–264
6.
go back to reference Bismut JM (1976) Linear quadratic optimal stochastic control with random coefficients. SIAM J Control Optim 14:419–444 MathSciNetCrossRef Bismut JM (1976) Linear quadratic optimal stochastic control with random coefficients. SIAM J Control Optim 14:419–444 MathSciNetCrossRef
8.
go back to reference Bowen TR, Chen Y, Eraslan H (2014) Mandatory versus discretionary spending: the status quo Effect. Am Econ Rev 104(10):2941–2974 CrossRef Bowen TR, Chen Y, Eraslan H (2014) Mandatory versus discretionary spending: the status quo Effect. Am Econ Rev 104(10):2941–2974 CrossRef
10.
12.
go back to reference Fleming WH, Soner HM (1993) Controlled markov processes and viscosity solutions. Springer-Verlag, New York MATH Fleming WH, Soner HM (1993) Controlled markov processes and viscosity solutions. Springer-Verlag, New York MATH
14.
go back to reference Haurie A, Krawczyk JB, Zaccour G (2012) Games and dynamic games, vol 1. World Scientific Publishing Company Haurie A, Krawczyk JB, Zaccour G (2012) Games and dynamic games, vol 1. World Scientific Publishing Company
16.
go back to reference Isaacs R (1965) Differential games: a mathematical theory with applications to warfare and pur-suit. Control and Optimization. JohnWiley & Sons Ltd, New York MATH Isaacs R (1965) Differential games: a mathematical theory with applications to warfare and pur-suit. Control and Optimization. JohnWiley & Sons Ltd, New York MATH
17.
go back to reference Kar D et al (2017) Trends and applications in Stackelberg security games. In: Handbook of dynamic game theory, pp 1–47 Kar D et al (2017) Trends and applications in Stackelberg security games. In: Handbook of dynamic game theory, pp 1–47
18.
go back to reference Kolokoltsov VN, Malafeyev OA (2010) Understanding game theory. World Scientific, London CrossRef Kolokoltsov VN, Malafeyev OA (2010) Understanding game theory. World Scientific, London CrossRef
19.
go back to reference Krasovskiĭ NN, Subbotin AI (1988) Game-theoretical control problems. Springer-Verlag, New York CrossRef Krasovskiĭ NN, Subbotin AI (1988) Game-theoretical control problems. Springer-Verlag, New York CrossRef
20.
go back to reference Kreyszig E (1978) Introductory functional analysis with applications, vol 1. Wiley, New York MATH Kreyszig E (1978) Introductory functional analysis with applications, vol 1. Wiley, New York MATH
21.
go back to reference Krichene W et al (2017) Stackelberg routing on parallel transportation networks. In: Handbook of dynamic game theory, pp 1–35 Krichene W et al (2017) Stackelberg routing on parallel transportation networks. In: Handbook of dynamic game theory, pp 1–35
22.
go back to reference Magnus JR, Neudecker H (2007) Matrix differential calculus with applications in statistics and econometrics. John Wiley & Sons Ltd, West Sussex MATH Magnus JR, Neudecker H (2007) Matrix differential calculus with applications in statistics and econometrics. John Wiley & Sons Ltd, West Sussex MATH
23.
go back to reference Moon J (2021) Linear-quadratic stochastic Stackelberg differential games for jump-diffusion systems. SIAM J Control Optim 59(2):954–976 MathSciNetCrossRef Moon J (2021) Linear-quadratic stochastic Stackelberg differential games for jump-diffusion systems. SIAM J Control Optim 59(2):954–976 MathSciNetCrossRef
24.
go back to reference Moore JB, Zhou XY, Lim AE (1999) Discrete time LQG controls with control dependent noise. Syst Control Lett 36(3):199–206 MathSciNetCrossRef Moore JB, Zhou XY, Lim AE (1999) Discrete time LQG controls with control dependent noise. Syst Control Lett 36(3):199–206 MathSciNetCrossRef
25.
go back to reference Rudin W (1991) Functional analysis. McGraw-Hill Inc, New York MATH Rudin W (1991) Functional analysis. McGraw-Hill Inc, New York MATH
26.
go back to reference Simaan M, Cruz JB (1973) Additional aspects of the Stackelberg strategy in nonzero-sum games. J Optim Theory Appl 11(6):613–626 MathSciNetCrossRef Simaan M, Cruz JB (1973) Additional aspects of the Stackelberg strategy in nonzero-sum games. J Optim Theory Appl 11(6):613–626 MathSciNetCrossRef
27.
go back to reference Sorin S (1983) Some results on the existence of Nash equilibria for non-zero sum games with incomplete information. Int J Game Theory 12(4):193–205 MathSciNetCrossRef Sorin S (1983) Some results on the existence of Nash equilibria for non-zero sum games with incomplete information. Int J Game Theory 12(4):193–205 MathSciNetCrossRef
28.
go back to reference Souganidis PE (1999)Two-player zero-sum differential games and viscosity solutions. In: Stochastic and differential games (eds) Bardi M, Raghavan TES, Parthasarathy T Vol. 4. Annals of the International Society of Dynamic Games. Boston, MA: Birkhäuser. Chap. 2 Souganidis PE (1999)Two-player zero-sum differential games and viscosity solutions. In: Stochastic and differential games (eds) Bardi M, Raghavan TES, Parthasarathy T Vol. 4. Annals of the International Society of Dynamic Games. Boston, MA: Birkhäuser. Chap. 2
29.
go back to reference Szajowski K (2018) On Multilateral hierarchical dynamic decisions. In: Frontiers of dynamic games. Springer, pp 269–282 Szajowski K (2018) On Multilateral hierarchical dynamic decisions. In: Frontiers of dynamic games. Springer, pp 269–282
30.
go back to reference Wang G, Wang Y, Zhang S (2020) An asymmetric information mean-field type linear-quadratic stochastic Stackelberg differential game with one leader and two followers. Optim Control Appl Methods 41(4):1034–1051 MathSciNetCrossRef Wang G, Wang Y, Zhang S (2020) An asymmetric information mean-field type linear-quadratic stochastic Stackelberg differential game with one leader and two followers. Optim Control Appl Methods 41(4):1034–1051 MathSciNetCrossRef
31.
go back to reference Yong J (2002) A leader-follower stochastic linear quadratic differential game. SIAM J Control Optim 41(4):1015–1041 MathSciNetCrossRef Yong J (2002) A leader-follower stochastic linear quadratic differential game. SIAM J Control Optim 41(4):1015–1041 MathSciNetCrossRef
Metadata
Title
Zero-Sum Stochastic Games with Random Rules of Priority, Discrete Linear-Quadratic Model
Authors
Daniel Hernández-Hernández
Joshué H. Ricalde-Guerrero
Publication date
23-01-2022
Publisher
Springer US
Published in
Dynamic Games and Applications
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-021-00417-9

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