# Approximations of dynamic Nash games with general state and action spaces and ergodic costs for the players

Applicationes Mathematicae (1997)

- Volume: 24, Issue: 2, page 195-202
- ISSN: 1233-7234

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topBielecki, Tomasz. "Approximations of dynamic Nash games with general state and action spaces and ergodic costs for the players." Applicationes Mathematicae 24.2 (1997): 195-202. <http://eudml.org/doc/219162>.

@article{Bielecki1997,

abstract = {The purpose of this paper is to prove existence of an ε -equilib- rium point in a dynamic Nash game with Borel state space and long-run time average cost criteria for the players. The idea of the proof is first to convert the initial game with ergodic costs to an ``equivalent" game endowed with discounted costs for some appropriately chosen value of the discount factor, and then to approximate the discounted Nash game obtained in the first step with a countable state space game for which existence of a Nash equilibrium can be established. From the results of Whitt we know that if for any ε > 0 the approximation scheme is selected in an appropriate way, then Nash equilibrium strategies for the approximating game are also ε -equilibrium strategies for the discounted game constructed in the first step. It is then shown that these strategies constitute an ε -equilibrium point for the initial game with ergodic costs as well. The idea of canonical triples, introduced by Dynkin and Yushkevich in the control setting, is adapted here to the game situation.},

author = {Bielecki, Tomasz},

journal = {Applicationes Mathematicae},

keywords = {two-person game; ε-Nash equilibrium; sequential game; long-run average criterion; existence of an -equilibrium point; dynamic Nash game; Borel state space; discounted Nash game},

language = {eng},

number = {2},

pages = {195-202},

title = {Approximations of dynamic Nash games with general state and action spaces and ergodic costs for the players},

url = {http://eudml.org/doc/219162},

volume = {24},

year = {1997},

}

TY - JOUR

AU - Bielecki, Tomasz

TI - Approximations of dynamic Nash games with general state and action spaces and ergodic costs for the players

JO - Applicationes Mathematicae

PY - 1997

VL - 24

IS - 2

SP - 195

EP - 202

AB - The purpose of this paper is to prove existence of an ε -equilib- rium point in a dynamic Nash game with Borel state space and long-run time average cost criteria for the players. The idea of the proof is first to convert the initial game with ergodic costs to an ``equivalent" game endowed with discounted costs for some appropriately chosen value of the discount factor, and then to approximate the discounted Nash game obtained in the first step with a countable state space game for which existence of a Nash equilibrium can be established. From the results of Whitt we know that if for any ε > 0 the approximation scheme is selected in an appropriate way, then Nash equilibrium strategies for the approximating game are also ε -equilibrium strategies for the discounted game constructed in the first step. It is then shown that these strategies constitute an ε -equilibrium point for the initial game with ergodic costs as well. The idea of canonical triples, introduced by Dynkin and Yushkevich in the control setting, is adapted here to the game situation.

LA - eng

KW - two-person game; ε-Nash equilibrium; sequential game; long-run average criterion; existence of an -equilibrium point; dynamic Nash game; Borel state space; discounted Nash game

UR - http://eudml.org/doc/219162

ER -

## References

top- [1] D. Bertsekas and S. Shreve, Stochastic Optimal Control: The Discrete Time Case, Academic Press, New York, 1979. Zbl0471.93002
- [2] E. B. Dynkin and A. A. Yushkevich, Controlled Markov Processes, Springer, New York, 1979. Zbl0073.34801
- [3] W. Whitt, Representation and Approximation of Non-Cooperative Sequential Games, SIAM J. Control Optim. 18 (1980), 33-48. Zbl0428.90094

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