main-content

Published in:

22-02-2021

# Differential Games with Incomplete Information and with Signal Revealing: The Symmetric Case

Author: Xiaochi Wu

Published in: Dynamic Games and Applications | Issue 4/2021

## Abstract

In this paper, we investigate the existence of value for a two-person zero-sum differential game with symmetric incomplete information and with signal revealing. Before the game begins, the initial state of the dynamic is chosen randomly among a finite number of points in $$\mathbb {R}^n$$, while both players have only a probabilistic knowledge of the chosen initial state. During the game, if the system reaches a fixed closed target set K, the current state of the system at the hitting time is revealed to both players. We prove in this paper that this game has a value and its value function is the unique bounded continuous viscosity solution of a suitable Hamilton–Jacobi–Isaacs equation.

### Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

• über 69.000 Bücher
• über 500 Zeitschriften

aus folgenden Fachgebieten:

• Automobil + Motoren
• Bauwesen + Immobilien
• Elektrotechnik + Elektronik
• Energie + Nachhaltigkeit
• Finance + Banking
• Management + Führung
• Marketing + Vertrieb
• Maschinenbau + Werkstoffe
• Versicherung + Risiko

Testen Sie jetzt 15 Tage kostenlos.

### Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

• über 50.000 Bücher
• über 380 Zeitschriften

aus folgenden Fachgebieten:

• Automobil + Motoren
• Bauwesen + Immobilien
• Elektrotechnik + Elektronik
• Energie + Nachhaltigkeit
• Maschinenbau + Werkstoffe

Testen Sie jetzt 15 Tage kostenlos.

### Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

• über 58.000 Bücher
• über 300 Zeitschriften

aus folgenden Fachgebieten:

• Bauwesen + Immobilien
• Finance + Banking
• Management + Führung
• Marketing + Vertrieb
• Versicherung + Risiko

Testen Sie jetzt 15 Tage kostenlos.

Literature
1.
Aumann RJ, Maschler MB (1995) Repeated games with incomplete information. With the collaboration of Richard E. Stearns. MIT Press, Cambridge MATH
2.
Bardi M, Capuzzo-Dolcetta I (1996) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, Basel MATH
3.
Bardi M, Koike S, Soravia P (2000) Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximations. Discrete Contin Dyn Syst 6(2):361–380
4.
Bernhard P, Rapaport A (1995) Étude d’un jeu de poursuite plane avec connaissance imparfaite d’une coordonnée. Automatique-productique informatique industrielle 29:575–601
5.
Buckdahn R, Cardaliaguet P, Quincampoix M (2011) Some recent aspects of differential game theory. Dyn Game Application 1(1):74–114
6.
Buckdahn R, Quincampoix M, Rainer C, Xu Y (2016) Differential games with asymmetric information and without Isaacs’ condition. Int J Game Theory 45:795–816
7.
Cardaliaguet P (2007) Differential games with asymmetric information. SIAM J Control Optim 46(3):816–838
8.
Cardaliaguet P (2009) A double obstacle problem arising in differential game theory. J Math Anal Appl 360(1):95–107
9.
Cardaliaguet P (2010) Introduction to differential games. Université de Bretagne Occidentale, Lecture notes
10.
Cardaliaguet P, Jimenez C, Quincampoix M (2014) Pure and random strategies in differential game with incomplete informations. J Dyn Games 1(3):363–375
11.
Cardaliaguet P, Quincampoix M (2008) Deterministic differential games under probability knowledge of initial condition. Int Game Theory Rev 10(01):1–16
12.
Cardaliaguet P, Quincampoix M, Saint-Pierre P (2001) Pursuit differential games with state constraints. SIAM J Control Optim 39(5):1615–1632
13.
Cardaliaguet P, Rainer C (2009) Stochastic differential games with asymmetric information. Appl Math Optim 59(1):1–36
14.
Crandall MG, Ishii H, Lions P-L (1992) User’s guide to viscosity solutions of second order partial differential equations. Bull Am Soc 27:1–67
15.
Crandall MG, Lions P-L (1983) Viscosity solutions of Hamilton–Jacobi equations. Trans Am Math Soc 277:1–42
16.
Elliott RJ, Kalton NJ (1972) The existence of value in differential games of pursuit and evasion. J Differ Equ 12(3):504–523
17.
Evans LC, Souganidis PE (1984) Differential games and representation formulas for solutions of Hamilton–Jacobi–Isaacs equations. Indiana Univ Math J 33(5):773–797
18.
Forges F (1982) Infinitely repeated games of incomplete information: symmetric case with random signals. Int J Game Theory 11:203–213
19.
Isaacs R (1967) Differential games. Wiley, London MATH
20.
Jimenez C, Quincampoix M, Xu Y (2016) Differential games with incomplete information on a continuum of initial positions and without Isaacs condition. Dyn Games Appl 6:82–96
21.
Kohlberg E, Zamir S (1974) Repeated games of incomplete information: the symmetric case. Anal Stat 2:1040–1041
22.
Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New York CrossRef
23.
Neyman A, Sorin S (1997) Equilibria in repeated games of incomplete information: the deterministic symmetric case. In: Parthasaraty T (ed) Game theoretic applications to economics and operations research. Springer, US
24.
Neyman A, Sorin S (1998) Equilibria in repeated games of incomplete information: the general symmetric case. Int J Game Theory 27:201–210
25.
Oliu-Barton M (2015) Differential games with asymmetric and correlated information. Dyn Games Appl 5(3):378–396
26.
Petrosjan LA (1993) Differential games of pursuit, volume 2 of series on optimization. World Scientific Publishing co Ltd, Singapore CrossRef
27.
Roxin E (1979) Feedback strategies with finite memory in differential games. J Optim Theory Appl 27(1):127–134
28.
Soravia P (1993) Pursuit-evasion problems and viscosity solutions of Isaacs’ equations. SIAM J Control Optim 31(3):604–623
29.
Varaiya P (1967) On the existence of solutions to a differential game. SIAM J Control 5(1):153–162
30.
Veliov VM (1997) Lipschitz continuity of the value function in optimal control. J Optim Theory Appl 94(2):335–363
31.
Wu X (2017) Existence of value for differential games with incomplete information and signals on initial states and payoffs. J Math Anal Appl 446(2):1196–1218
32.
Wu X (2018) Existence of value for a differential game with incomplete information and revealing. SIAM J Control Optim 56(4):2536–2562
33.
Yong JM (1988) On differential pursuit games. SIAM J Control Optim 26(2):478–495
Title
Differential Games with Incomplete Information and with Signal Revealing: The Symmetric Case
Author
Xiaochi Wu
Publication date
22-02-2021
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 4/2021
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-021-00376-1

Go to the issue