Skip to main content
Erschienen in: Dynamic Games and Applications 2/2021

27.07.2020

Upper and Lower Values in Zero-Sum Stochastic Games with Asymmetric Information

verfasst von: Dhruva Kartik, Ashutosh Nayyar

Erschienen in: Dynamic Games and Applications | Ausgabe 2/2021

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

A general model for zero-sum stochastic games with asymmetric information is considered. In this model, each player’s information at each time can be divided into a common information part and a private information part. Under certain conditions on the evolution of the common and private information, a dynamic programming characterization of the value of the game (if it exists) is presented. If the value of the zero-sum game does not exist, then the dynamic program provides bounds on the upper and lower values of the game.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

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

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

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

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

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




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

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




Jetzt Wissensvorsprung sichern!

Anhänge
Nur mit Berechtigung zugänglich
Fußnoten
1
Note that we do not impose Assumption 2 of [26].
 
2
For example, see the delayed sharing information structure in Sect. 2.2.
 
3
Our notation is different from [23, Section IV.3].
 
4
We refer the reader to [23, Chapter III] for a detailed discussion on the universal belief space.
 
5
Note that the belief \({{\mathbb {P}}}^{({\tilde{\chi }}^1,{\tilde{\chi }}^2)}[x_t, p_t^{1:2} \mid c_t,\gamma _{1:t-1}^{1:2}] = {{\mathbb {P}}}^{({\tilde{\chi }}^1,{\tilde{\chi }}^2)}[x_t, p_t^{1:2} \mid c_t,\gamma _{1:t}^{1:2}]\) because \(\gamma _t^i = {\tilde{\chi }}_t^i( c_t,\gamma _{1:t-1}^{1:2})\), \(i=1,2\).
 
Literatur
1.
Zurück zum Zitat Alpcan T, Başar T (2010) Network security: a decision and game-theoretic approach. Cambridge University Press, CambridgeCrossRef Alpcan T, Başar T (2010) Network security: a decision and game-theoretic approach. Cambridge University Press, CambridgeCrossRef
2.
Zurück zum Zitat Amin S, Litrico X, Sastry S, Bayen AM (2012) Cyber security of water scada systemspart I: analysis and experimentation of stealthy deception attacks. IEEE Trans Control Syst Technol 21(5):1963–1970CrossRef Amin S, Litrico X, Sastry S, Bayen AM (2012) Cyber security of water scada systemspart I: analysis and experimentation of stealthy deception attacks. IEEE Trans Control Syst Technol 21(5):1963–1970CrossRef
3.
Zurück zum Zitat Amin S, Schwartz GA, Cardenas AA, Sastry SS (2015) Game-theoretic models of electricity theft detection in smart utility networks: providing new capabilities with advanced metering infrastructure. IEEE Control Syst Mag 35(1):66–81MathSciNetCrossRef Amin S, Schwartz GA, Cardenas AA, Sastry SS (2015) Game-theoretic models of electricity theft detection in smart utility networks: providing new capabilities with advanced metering infrastructure. IEEE Control Syst Mag 35(1):66–81MathSciNetCrossRef
4.
Zurück zum Zitat Aumann RJ, Maschler M, Stearns RE (1995) Repeated games with incomplete information. MIT Press, Cambridge Aumann RJ, Maschler M, Stearns RE (1995) Repeated games with incomplete information. MIT Press, Cambridge
5.
Zurück zum Zitat Başar T (1981) On the saddle-point solution of a class of stochastic differential games. J Optim Theory Appl 33(4):539–556MathSciNetCrossRef Başar T (1981) On the saddle-point solution of a class of stochastic differential games. J Optim Theory Appl 33(4):539–556MathSciNetCrossRef
6.
Zurück zum Zitat Basar T, Olsder GJ (1999) Dynamic noncooperative game theory, vol 23. SIAM, PhiladelphiaMATH Basar T, Olsder GJ (1999) Dynamic noncooperative game theory, vol 23. SIAM, PhiladelphiaMATH
7.
Zurück zum Zitat Bondi E, Oh H, Xu H, Fang F, Dilkina B, Tambe M (2019) Using game theory in real time in the real world: A conservation case study. In: Proceedings of the 18th international conference on autonomous agents and multiagent systems, pp. 2336–2338. International Foundation for Autonomous Agents and Multiagent Systems Bondi E, Oh H, Xu H, Fang F, Dilkina B, Tambe M (2019) Using game theory in real time in the real world: A conservation case study. In: Proceedings of the 18th international conference on autonomous agents and multiagent systems, pp. 2336–2338. International Foundation for Autonomous Agents and Multiagent Systems
8.
Zurück zum Zitat Fang F, Nguyen TH, Pickles R, Lam WY, Clements GR, An B, Singh A, Schwedock BC, Tambe M, Lemieux A (2017) Paws-a deployed game-theoretic application to combat poaching. AI Mag 38(1):23–36 Fang F, Nguyen TH, Pickles R, Lam WY, Clements GR, An B, Singh A, Schwedock BC, Tambe M, Lemieux A (2017) Paws-a deployed game-theoretic application to combat poaching. AI Mag 38(1):23–36
9.
Zurück zum Zitat Fang F, Stone P, Tambe M (2015) When security games go green: Designing defender strategies to prevent poaching and illegal fishing. In: Twenty-fourth international joint conference on artificial intelligence Fang F, Stone P, Tambe M (2015) When security games go green: Designing defender strategies to prevent poaching and illegal fishing. In: Twenty-fourth international joint conference on artificial intelligence
10.
Zurück zum Zitat Filar J, Vrieze K (2012) Competitive Markov decision processes. Springer, BerlinMATH Filar J, Vrieze K (2012) Competitive Markov decision processes. Springer, BerlinMATH
11.
Zurück zum Zitat Fudenberg D, Tirole J (1991) Game theory. MIT Press, CambridgeMATH Fudenberg D, Tirole J (1991) Game theory. MIT Press, CambridgeMATH
12.
Zurück zum Zitat Gensbittel F, Oliu-Barton M, Venel X (2014) Existence of the uniform value in zero-sum repeated games with a more informed controller. J Dyn Games 1(3):411–445MathSciNetCrossRef Gensbittel F, Oliu-Barton M, Venel X (2014) Existence of the uniform value in zero-sum repeated games with a more informed controller. J Dyn Games 1(3):411–445MathSciNetCrossRef
13.
Zurück zum Zitat Gensbittel F, Renault J (2015) The value of Markov chain games with incomplete information on both sides. Math Oper Res 40(4):820–841MathSciNetCrossRef Gensbittel F, Renault J (2015) The value of Markov chain games with incomplete information on both sides. Math Oper Res 40(4):820–841MathSciNetCrossRef
14.
Zurück zum Zitat Hansen EA (1998) Solving pomdps by searching in policy space. In: Proceedings of the fourteenth conference on Uncertainty in artificial intelligence, pp 211–219 Hansen EA (1998) Solving pomdps by searching in policy space. In: Proceedings of the fourteenth conference on Uncertainty in artificial intelligence, pp 211–219
15.
Zurück zum Zitat Haywood O Jr (1954) Military decision and game theory. J Oper Res Soc Am 2(4):365–385MATH Haywood O Jr (1954) Military decision and game theory. J Oper Res Soc Am 2(4):365–385MATH
16.
Zurück zum Zitat Hernández-Lerma O, Lasserre JB (2012) Discrete-time Markov control processes: basic optimality criteria, vol 30. Springer, BerlinMATH Hernández-Lerma O, Lasserre JB (2012) Discrete-time Markov control processes: basic optimality criteria, vol 30. Springer, BerlinMATH
17.
Zurück zum Zitat Kartik D, Nayyar A (2019) Stochastic zero-sum games with asymmetric information. In: 58th IEEE conference on decision and control. IEEE Kartik D, Nayyar A (2019) Stochastic zero-sum games with asymmetric information. In: 58th IEEE conference on decision and control. IEEE
19.
Zurück zum Zitat Li L, Shamma J (2014) LP formulation of asymmetric zero-sum stochastic games. In: 53rd IEEE conference on decision and control, pp. 1930–1935. IEEE Li L, Shamma J (2014) LP formulation of asymmetric zero-sum stochastic games. In: 53rd IEEE conference on decision and control, pp. 1930–1935. IEEE
20.
Zurück zum Zitat Li X, Venel X (2016) Recursive games: uniform value, tauberian theorem and the mertens conjecture. Int J Game Theory 45(1–2):155–189CrossRef Li X, Venel X (2016) Recursive games: uniform value, tauberian theorem and the mertens conjecture. Int J Game Theory 45(1–2):155–189CrossRef
21.
Zurück zum Zitat Loch J, Singh SP (1998) Using eligibility traces to find the best memoryless policy in partially observable markov decision processes. In: Proceedings of the fifteenth international conference on machine learning, pp. 323–331 Loch J, Singh SP (1998) Using eligibility traces to find the best memoryless policy in partially observable markov decision processes. In: Proceedings of the fifteenth international conference on machine learning, pp. 323–331
23.
Zurück zum Zitat Mertens JF, Sorin S, Zamir S (2015) Repeated games, vol 55. Cambridge University Press, CambridgeCrossRef Mertens JF, Sorin S, Zamir S (2015) Repeated games, vol 55. Cambridge University Press, CambridgeCrossRef
24.
Zurück zum Zitat Morrow JD (1994) Game theory for political scientists. Princeton University Press, Princeton Morrow JD (1994) Game theory for political scientists. Princeton University Press, Princeton
25.
Zurück zum Zitat Nayyar A, Gupta A (2017) Information structures and values in zero-sum stochastic games. In: American control conference (ACC), 2017, pp. 3658–3663. IEEE Nayyar A, Gupta A (2017) Information structures and values in zero-sum stochastic games. In: American control conference (ACC), 2017, pp. 3658–3663. IEEE
26.
Zurück zum Zitat Nayyar A, Gupta A, Langbort C, Başar T (2014) Common information based Markov perfect equilibria for stochastic games with asymmetric information: finite games. IEEE Trans Autom Control 59(3):555–570MathSciNetCrossRef Nayyar A, Gupta A, Langbort C, Başar T (2014) Common information based Markov perfect equilibria for stochastic games with asymmetric information: finite games. IEEE Trans Autom Control 59(3):555–570MathSciNetCrossRef
27.
Zurück zum Zitat Nayyar A, Mahajan A, Teneketzis D (2010) Optimal control strategies in delayed sharing information structures. IEEE Trans Autom Control 56(7):1606–1620MathSciNetCrossRef Nayyar A, Mahajan A, Teneketzis D (2010) Optimal control strategies in delayed sharing information structures. IEEE Trans Autom Control 56(7):1606–1620MathSciNetCrossRef
28.
Zurück zum Zitat Nayyar A, Mahajan A, Teneketzis D (2013) Decentralized stochastic control with partial history sharing: a common information approach. IEEE Trans Autom Control 58(7):1644–1658MathSciNetCrossRef Nayyar A, Mahajan A, Teneketzis D (2013) Decentralized stochastic control with partial history sharing: a common information approach. IEEE Trans Autom Control 58(7):1644–1658MathSciNetCrossRef
29.
Zurück zum Zitat Osborne MJ, Rubinstein A (1994) A course in game theory. MIT Press, CambridgeMATH Osborne MJ, Rubinstein A (1994) A course in game theory. MIT Press, CambridgeMATH
30.
Zurück zum Zitat Ouyang Y, Tavafoghi H, Teneketzis D (2017) Dynamic games with asymmetric information: common information based perfect bayesian equilibria and sequential decomposition. IEEE Trans Autom Control 62(1):222–237MathSciNetCrossRef Ouyang Y, Tavafoghi H, Teneketzis D (2017) Dynamic games with asymmetric information: common information based perfect bayesian equilibria and sequential decomposition. IEEE Trans Autom Control 62(1):222–237MathSciNetCrossRef
31.
Zurück zum Zitat Ponssard JP, Sorin S (1980) The lp formulation of finite zero-sum games with incomplete information. Int J Game Theory 9(2):99–105MathSciNetCrossRef Ponssard JP, Sorin S (1980) The lp formulation of finite zero-sum games with incomplete information. Int J Game Theory 9(2):99–105MathSciNetCrossRef
32.
Zurück zum Zitat Renault J (2006) The value of Markov chain games with lack of information on one side. Math Oper Res 31(3):490–512MathSciNetCrossRef Renault J (2006) The value of Markov chain games with lack of information on one side. Math Oper Res 31(3):490–512MathSciNetCrossRef
33.
35.
Zurück zum Zitat Rosenberg D, Solan E, Vieille N (2004) Stochastic games with a single controller and incomplete information. SIAM J Control Optim 43(1):86–110MathSciNetCrossRef Rosenberg D, Solan E, Vieille N (2004) Stochastic games with a single controller and incomplete information. SIAM J Control Optim 43(1):86–110MathSciNetCrossRef
36.
Zurück zum Zitat Rudin W et al (1964) Principles of mathematical analysis, vol 3. McGraw-hill, New YorkMATH Rudin W et al (1964) Principles of mathematical analysis, vol 3. McGraw-hill, New YorkMATH
39.
Zurück zum Zitat Teneketzis D (2006) On the structure of optimal real-time encoders and decoders in noisy communication. IEEE Trans Inf Theory 52(9):4017–4035MathSciNetCrossRef Teneketzis D (2006) On the structure of optimal real-time encoders and decoders in noisy communication. IEEE Trans Inf Theory 52(9):4017–4035MathSciNetCrossRef
40.
Zurück zum Zitat Vasal D, Sinha A, Anastasopoulos A (2019) A systematic process for evaluating structured perfect bayesian equilibria in dynamic games with asymmetric information. IEEE Trans Autom Control 64(1):78–93MathSciNetCrossRef Vasal D, Sinha A, Anastasopoulos A (2019) A systematic process for evaluating structured perfect bayesian equilibria in dynamic games with asymmetric information. IEEE Trans Autom Control 64(1):78–93MathSciNetCrossRef
41.
42.
Zurück zum Zitat Wu M, Amin S (2018) Securing infrastructure facilities: When does proactive defense help? Dyn Games Appl 9:1–42MathSciNetMATH Wu M, Amin S (2018) Securing infrastructure facilities: When does proactive defense help? Dyn Games Appl 9:1–42MathSciNetMATH
43.
Zurück zum Zitat Zheng J, Castañón DA (2013) Decomposition techniques for Markov zero-sum games with nested information. In: 52nd IEEE conference on decision and control, pp. 574–581. IEEE Zheng J, Castañón DA (2013) Decomposition techniques for Markov zero-sum games with nested information. In: 52nd IEEE conference on decision and control, pp. 574–581. IEEE
44.
Zurück zum Zitat Zhu Q, Basar T (2015) Game-theoretic methods for robustness, security, and resilience of cyberphysical control systems: games-in-games principle for optimal cross-layer resilient control systems. IEEE Control Syst Mag 35(1):46–65MathSciNetCrossRef Zhu Q, Basar T (2015) Game-theoretic methods for robustness, security, and resilience of cyberphysical control systems: games-in-games principle for optimal cross-layer resilient control systems. IEEE Control Syst Mag 35(1):46–65MathSciNetCrossRef
Metadaten
Titel
Upper and Lower Values in Zero-Sum Stochastic Games with Asymmetric Information
verfasst von
Dhruva Kartik
Ashutosh Nayyar
Publikationsdatum
27.07.2020
Verlag
Springer US
Erschienen in
Dynamic Games and Applications / Ausgabe 2/2021
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-020-00364-x

Weitere Artikel der Ausgabe 2/2021

Dynamic Games and Applications 2/2021 Zur Ausgabe