# American Institute of Mathematical Sciences

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

## Game theory and dynamic programming in alternate games

 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.

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
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##### References:
Full game tree of dominoes
Collapsed game tree for dominoes
Full game tree for Jehiel's example
Collapsed game tree for Jehiel's example
Counting paths
Two squared grids $5\times5$ and $10\times10$
Two rectangular grids $5\times10$ and $10\times5$
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