main-content

17-02-2021

# A Mean Field Approach for Discounted Zero-Sum Games in a Class of Systems of Interacting Objects

Authors: Carmen G. Higuera-Chan, J. Adolfo Minjárez-Sosa

Published in: Dynamic Games and Applications | Issue 3/2021

## Abstract

The paper deals with systems composed of a large number of N interacting objects (e.g., agents, particles) controlled by two players defining a stochastic zero-sum game. The objects can be classified according to a finite set of classes or categories over which they move randomly. Because N is too large, the game problem is studied following a mean field approach. That is, a zero-sum game model $$\mathcal {GM}_{N}$$, where the states are the proportions of objects in each class, is introduced. Then, letting $$N\rightarrow \infty$$ (the mean field limit) we obtain a new game model $$\mathcal {GM}$$, independent on N, which is easier to analyze than $$\mathcal {GM}_{N}$$. Considering a discounted optimality criterion, our objective is to prove that an optimal pair of strategies in $$\mathcal {GM}$$ is an approximate optimal pair as $$N\rightarrow \infty$$ in the original game model $$\mathcal {GM}_{N}$$.

### 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.

Appendix
Available only for authorised users
Literature
1.
Accacio B, Backhoff-Veraguas J, Carmona R (2019) Extended mean field control problems: stochastic maximum principle and transport perspective. SIAM J Control Optim 57(6):3666–3693
2.
Ait Rami M et al (2001) Indefinite stochastic linear quadratic control and generalized differential Riccati equation. SIAM J Control Optim 40:1296–1311
3.
Bensoussan A, Frehse J, Yam P (2010) Mean field games and mean field control theory. Springer, New York MATH
4.
Carmona R, Delarue F (2018) Probabilistic theory of mean field games with applications I. Springer, Berin CrossRef
5.
Carmona R, Delarue F (2018) Probabilistic theory of mean field games with applications II. Springer, Berlin CrossRef
6.
Dynkin EB, Yushkevich AA (1979) Controlled Markov processes. Springer, New York CrossRef
7.
Elliott R, Li X, Ni YH (2013) Discrete time mean-field stochastic linear-quadratic optimal control problems. Automatica 49:3222–3233
8.
Gast N, Gaujal B (2011) A mean field approach for optimization in discrete time. Discrete Event Dyn Syst 21:63–101
9.
Gast N, Gaujal B, Le Boudec JY (2012) Mean field for Markov decision processes: from discrete to continuous optimization. IEEE Trans Autom Control 57:2266–2280
10.
Gomes DA, Pimentel EA, Voskanyan V (2016) Regularity theory for mean-field game systems. Springer, New York CrossRef
11.
González-Sánchez D, Luque-Vásquez F, Minjárez-Sosa JA (2019) Zero-sum Markov games with random state-actions-dependent discount factors: existence of optimal strategies. Dyn Games Appl 9:103–121
12.
Hafayed M (2013) A mean-field necessary and sufficient conditions for optimal singular stochastic control. Commun Math Stat 1:417–435
13.
Hernández-Lerma O, Lasserre JB (1996) Discrete-time Markov control processes: basic optimality criteria. Springer, New York CrossRef
14.
Higuera-Chan CG, Jasso-Fuentes H, Minjárez-Sosa JA (2016) Discrete-time control for systems of interacting objects with unknown random disturbance distributions: a mean field approach. Appl Math Optim 74:197–227
15.
Higuera-Chan CG, Jasso-Fuentes H, Minjárez-Sosa JA (2017) Control systems of iteracting objects modeled as a game against nature under a mean field approach. J Dyn Games 4:59–74
16.
Jaśkiewicz A, Nowak A (2006) Zero-sum ergodic stochastic games with Feller transition probabilities. SIAM J Control Optim 45:773–789
17.
Lasry JM, Lions PL (2007) Mean field games. Jpn J Math 2:229–260
18.
Laura-Guarachi LR, Hernández-Lerma O (2015) The Mitra–Wan model: a discrete-time optimal control problem. Nat Resource Model 28:152–168
19.
Le Boudec JY, McDonald D, Mundinger J (2007) A generic mean field convergence result for systems of interacting objects. In: 4th international conference on quantitative evaluation of systems, Edinburg, pp 3-18
20.
Minjárez-Sosa JA (2020) Zero-sum discrete-time Markov games with unknown disturbance distribution: discounted and average criteria. Springer, Berlin CrossRef
21.
Peyrard N, Sabbadin R (2006) Mean field approximation of the policy iteration algorithm for graph-based Markov decision processes. In: Proceedings of ECAI, pp 595–599
22.
Piazza A, Pagnoncelli BK (2015) The stochastic Mitra–Wan forestry model: risk neutral and risk averse cases. J Econ 115:175–194 CrossRef
23.
Saldi N, Basar T, Raginsky M (2018) Markov–Nash equilibria in mean-field games with discounted cost. SIAM J Control Optim 56(6):4256–4287
24.
Song T, Liu B (2020) Discrete-time stochastic mean-field linear quadratic optimal control problem with finite horizon. Asian J Control 57:1–11
25.
Wiecek P (2020) Discrete-time ergodic mean-field games with average reward on compact spaces. Dyn Games Appl 10:222–256
Title
A Mean Field Approach for Discounted Zero-Sum Games in a Class of Systems of Interacting Objects
Authors
Carmen G. Higuera-Chan
Publication date
17-02-2021
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 3/2021
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-021-00377-0

### Other articles of this Issue 3/2021 Go to the issue
{"ft_lang_label":"__label__en","ft_lang_prob":0.9175448,"math_prob":0.5164464,"size":926,"snap":"2022-05-2022-21","text_gpt3_token_len":214,"char_repetition_ratio":0.121475056,"word_repetition_ratio":0.0,"special_character_ratio":0.24190065,"punctuation_ratio":0.10465116,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9615968,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-18T03:04:56Z\",\"WARC-Record-ID\":\"<urn:uuid:0f9c4ed5-4b43-4cfc-8c89-a5801c3fe93f>\",\"Content-Length\":\"89966\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ab2d96b5-8672-4ce0-96cf-6782bb8bf945>\",\"WARC-Concurrent-To\":\"<urn:uuid:0fc25d62-5a84-4c9a-9a11-fe7f2edd8edc>\",\"WARC-IP-Address\":\"146.75.32.95\",\"WARC-Target-URI\":\"https://www.springerprofessional.de/en/a-mean-field-approach-for-discounted-zero-sum-games-in-a-class-o/18877302\",\"WARC-Payload-Digest\":\"sha1:I4GDUNBQOU24MJJRKJOEG3ICIA3KA72M\",\"WARC-Block-Digest\":\"sha1:XCFQN4OBDYQVVCXV6EZLUB4KXTB3ITL2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662521041.0_warc_CC-MAIN-20220518021247-20220518051247-00047.warc.gz\"}"}