American Institute of Mathematical Sciences

Advanced
July  2017, 4(3): 205-216. doi: 10.3934/jdg.2017013

Game theory and dynamic programming in alternate games

Eduardo Espinosa-Avila 1,, , Pablo Padilla Longoria 2, and  Francisco Hernández-Quiroz 3,

1. 

Graduate Program in Computer Science, Institute of Applied Mathematics and Systems Research Annex Building, Third Floor, Circuito Escolar s/n, Cd. Universitaria, 04510, Mexico City, Mexico, National University of Mexico (UNAM)
2. 

Institute of Applied Mathematics and Systems Research, Circuito Escolar s/n, Cd. Universitaria, 04510, Mexico City, Mexico, National University of Mexico (UNAM)
3. 

Faculty of Sciences, Circuito Exterior s/n, Cd. Universitaria, 04510, Mexico City, Mexico, National University of Mexico (UNAM)

* Corresponding author

Received  December 2016 Revised  April 2017 Published  April 2017

Fund Project: The first author is supported by the National Council of Science and Technology (CONACyT) and the National University of Mexico (UNAM).

We present an analysis of different classes of alternate games from different perspectives, including game theory, logic, bounded rationality and dynamic programming. In this paper we review some of these approaches providing a methodological framework which combines ideas from all of them, but emphasizing dynamic programming and game theory. In particular we study the relationship between games in discrete and continuous time and state space and how the latter can be understood as the limit of the former. We show how in some cases the Hamilton-Jacobi-Bellman equation for the discrete version of the game leads to a corresponding HJB partial differential equation for the continuous case and how this procedure allow us to obtain useful information about optimal strategies. This analysis yields another way to compute subgame perfect equilibrium.

Keywords: Game theory, economics, social and behavioral sciences.
Mathematics Subject Classification: Primary: 91A25, 91B06; Secondary: 91B50.
Citation: Eduardo Espinosa-Avila, Pablo Padilla Longoria, Francisco Hernández-Quiroz. Game theory and dynamic programming in alternate games. Journal of Dynamics & Games, 2017, 4 (3) : 205-216. doi: 10.3934/jdg.2017013
References:
[1]

A. Baltag and S. Smets, Keep changing your beliefs, aiming for the truth, Erkenntnis, 75 (2011), 255-270.  doi: 10.1007/s10670-011-9294-y.  Google Scholar

[2]

E. Espinosa-Avila and F. Hernández-Quiroz, Bounded rationality in a dynamic alternate game, Proceedings of TARK'13 Theoretical Aspects of Rationality and Knowledge, 2013, 71{ 77. Google Scholar

[3]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Second Edition, Graduate Studies in Mathematics, Volume: 19, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[4]

D. Fernández-Duque, E. Espinosa-Avila and F. Hern´andez-Quiroz A logic for bounded rationality, Proceeding of the short presentations of Advances in Modal Logics, AiML'14, 2014, 34-38. Google Scholar

[5]

IBM DeepBlue, What are the differences between last year's rendition of Deep Blue and this year's model?, IBM DeepBlue faq https://www.research.ibm.com/deepblue/meet/html/d.3.3a.html IBM. Retrieved Aug 29,2016. Google Scholar

[6]

IBM DeepQA, What kind of technology is Watson based on?, DeepQA faq https://www.research.ibm.com/deepqa/faq.shtml IBM. Retrieved Aug 29,2016. Google Scholar

[7]

P. Jehiel, Limited horizon forecast in repeated alternate games, Journal of Economic Theory, 67 (1995), 497-519.  doi: 10.1006/jeth.1995.1082.  Google Scholar

[8]

T. A. Marsland and Y. Björnsson, From minimax to manhattan, Proceedings of Deep Blue Versus Kasparov: The Significance for Artificial Intelligence, Collected Papers from the 1997 AAAI Workshop, (1997), 31-36.   Google Scholar

[9]

V. Mirrokni, N. Thain and A. Vetta, A theoretical examination of practical game playing: Lookahead search, SAGT'12 Proceedings of the 5th international conference on Algorithmic Game Theory, Universitat Politècnica de Catalunya, 7615 (2012), 251-262. doi: 10.1007/978-3-642-33996-7_22.  Google Scholar

[10]

R. Parikh, L. S. Moss and C. Steinsvold, Topology and epistemic logic, Handbook of Spatial Logics, (2007), 299-341.  doi: 10.1007/978-1-4020-5587-4_6.  Google Scholar

[11]

D. K. Smith, Dynamic programming and board games: A survey, European Journal of Operational Research, 176 (2007), 1299-1318.  doi: 10.1016/j.ejor.2005.10.026.  Google Scholar

show all references

References:
[1]

A. Baltag and S. Smets, Keep changing your beliefs, aiming for the truth, Erkenntnis, 75 (2011), 255-270.  doi: 10.1007/s10670-011-9294-y.  Google Scholar

[2]

E. Espinosa-Avila and F. Hernández-Quiroz, Bounded rationality in a dynamic alternate game, Proceedings of TARK'13 Theoretical Aspects of Rationality and Knowledge, 2013, 71{ 77. Google Scholar

[3]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Second Edition, Graduate Studies in Mathematics, Volume: 19, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[4]

D. Fernández-Duque, E. Espinosa-Avila and F. Hern´andez-Quiroz A logic for bounded rationality, Proceeding of the short presentations of Advances in Modal Logics, AiML'14, 2014, 34-38. Google Scholar

[5]

IBM DeepBlue, What are the differences between last year's rendition of Deep Blue and this year's model?, IBM DeepBlue faq https://www.research.ibm.com/deepblue/meet/html/d.3.3a.html IBM. Retrieved Aug 29,2016. Google Scholar

[6]

IBM DeepQA, What kind of technology is Watson based on?, DeepQA faq https://www.research.ibm.com/deepqa/faq.shtml IBM. Retrieved Aug 29,2016. Google Scholar

[7]

P. Jehiel, Limited horizon forecast in repeated alternate games, Journal of Economic Theory, 67 (1995), 497-519.  doi: 10.1006/jeth.1995.1082.  Google Scholar

[8]

T. A. Marsland and Y. Björnsson, From minimax to manhattan, Proceedings of Deep Blue Versus Kasparov: The Significance for Artificial Intelligence, Collected Papers from the 1997 AAAI Workshop, (1997), 31-36.   Google Scholar

[9]

V. Mirrokni, N. Thain and A. Vetta, A theoretical examination of practical game playing: Lookahead search, SAGT'12 Proceedings of the 5th international conference on Algorithmic Game Theory, Universitat Politècnica de Catalunya, 7615 (2012), 251-262. doi: 10.1007/978-3-642-33996-7_22.  Google Scholar

[10]

R. Parikh, L. S. Moss and C. Steinsvold, Topology and epistemic logic, Handbook of Spatial Logics, (2007), 299-341.  doi: 10.1007/978-1-4020-5587-4_6.  Google Scholar

[11]

D. K. Smith, Dynamic programming and board games: A survey, European Journal of Operational Research, 176 (2007), 1299-1318.  doi: 10.1016/j.ejor.2005.10.026.  Google Scholar

Figure 1.  Full game tree of dominoes
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Collapsed game tree for dominoes
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Full game tree for Jehiel's example
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Collapsed game tree for Jehiel's example
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Counting paths
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Two squared grids $5\times5$ and $10\times10$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Two rectangular grids $5\times10$ and $10\times5$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Serap Ergün, Bariş Bülent Kırlar, Sırma Zeynep Alparslan Gök, Gerhard-Wilhelm Weber. An application of crypto cloud computing in social networks by cooperative game theory. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-15. doi: 10.3934/jimo.2019036
[2]

Grace Gao, Sasank Maganti, Karen A. Monsen. Older adults, frailty, and the social and behavioral determinants of health. Big Data & Information Analytics, 2017, 2 (3&4) : 1-12. doi: 10.3934/bdia.2017012
[3]

Nicola Bellomo, Livio Gibelli, Nisrine Outada. On the interplay between behavioral dynamics and social interactions in human crowds. Kinetic & Related Models, 2019, 12 (2) : 397-409. doi: 10.3934/krm.2019017
[4]

Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. Kinetic & Related Models, 2011, 4 (1) : 187-213. doi: 10.3934/krm.2011.4.187
[5]

Anna Lisa Amadori, Astridh Boccabella, Roberto Natalini. A hyperbolic model of spatial evolutionary game theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 981-1002. doi: 10.3934/cpaa.2012.11.981
[6]

Rod Cross, Hugh McNamara, Leonid Kalachev, Alexei Pokrovskii. Hysteresis and post Walrasian economics. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 377-401. doi: 10.3934/dcdsb.2013.18.377
[7]

Peng Liu, Jinfeng Wang, Lijie Feng. The game of government-industry-university-institute collaborative innovation based on static game theory. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020276
[8]

Tinggui Chen, Yanhui Jiang. Research on operating mechanism for creative products supply chain based on game theory. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1103-1112. doi: 10.3934/dcdss.2015.8.1103
[9]

Marianne Akian, Stéphane Gaubert, Antoine Hochart. A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 207-231. doi: 10.3934/dcds.2020009
[10]

Eunha Shim, Beth Kochin, Alison Galvani. Insights from epidemiological game theory into gender-specific vaccination against rubella. Mathematical Biosciences & Engineering, 2009, 6 (4) : 839-854. doi: 10.3934/mbe.2009.6.839
[11]

Hideo Deguchi. A reaction-diffusion system arising in game theory: existence of solutions and spatial dominance. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3891-3901. doi: 10.3934/dcdsb.2017200
[12]

Wai-Ki Ching, Sin-Man Choi, Min Huang. Optimal service capacity in a multiple-server queueing system: A game theory approach. Journal of Industrial & Management Optimization, 2010, 6 (1) : 73-102. doi: 10.3934/jimo.2010.6.73
[13]

King-Yeung Lam. Dirac-concentrations in an integro-pde model from evolutionary game theory. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 737-754. doi: 10.3934/dcdsb.2018205
[14]

Benjamin Contri. Fisher-KPP equations and applications to a model in medical sciences. Networks & Heterogeneous Media, 2018, 13 (1) : 119-153. doi: 10.3934/nhm.2018006
[15]

Vikram Krishnamurthy, William Hoiles. Information diffusion in social sensing. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 365-411. doi: 10.3934/naco.2016017
[16]

Werner Creixell, Juan Carlos Losada, Tomás Arredondo, Patricio Olivares, Rosa María Benito. Serendipity in social networks. Networks & Heterogeneous Media, 2012, 7 (3) : 363-371. doi: 10.3934/nhm.2012.7.363
[17]

Pradeep Dubey, Rahul Garg, Bernard De Meyer. Competing for customers in a social network. Journal of Dynamics & Games, 2014, 1 (3) : 377-409. doi: 10.3934/jdg.2014.1.377
[18]

Yuki Kumagai. Social networks and global transactions. Journal of Dynamics & Games, 2019, 6 (3) : 211-219. doi: 10.3934/jdg.2019015
[19]

Carlo Brugna, Giuseppe Toscani. Boltzmann-type models for price formation in the presence of behavioral aspects. Networks & Heterogeneous Media, 2015, 10 (3) : 543-557. doi: 10.3934/nhm.2015.10.543
[20]

Jianquan Li, Xiaoqin Wang, Xiaolin Lin. Impact of behavioral change on the epidemic characteristics of an epidemic model without vital dynamics. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1425-1434. doi: 10.3934/mbe.2018065

 Impact Factor: 

Tools

Metrics

  • PDF downloads (49)
  • HTML views (42)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences