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Dynamic Games and Applications
Published in: Dynamic Games and Applications 2/2021

19-05-2020

Dynamic Equilibrium with Randomly Arriving Players

Authors: Pierre Bernhard, Marc Deschamps

Published in: Dynamic Games and Applications | Issue 2/2021

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Abstract

There are real strategic situations where nobody knows ex ante how many players there will be in the game at each step. Assuming that entry and exit could be modeled by random processes whose probability laws are common knowledge, we use dynamic programming and piecewise deterministic Markov decision processes to investigate such games. We study these games in discrete and continuous time for both finite and infinite horizon. While existence of dynamic equilibrium in discrete time is proved, our main aim is to develop algorithms. In the general nonlinear case, the equations provided are rather intricate. We develop more explicit algorithms for both discrete and continuous time linear quadratic problems.
previous article Coupling a Power Dispatch Model with a Wardrop or Mean-Field-Game Equilibrium Model
next article Modelling Evolution in Structured Populations Involving Multiplayer Interactions

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Appendix
Available only for authorised users
Footnotes
1
In accordance with the above note, in the notation \(p^m\), m is an upper index just meaning that this probability may depend on m, not an exponent denoting a power.
 
2
We use prime for transposed and \(\Vert x\Vert _Q^2 = x^{\prime }Qx\), \(\Vert s\Vert _R^2=s^{\prime }Rs\).
 
3
Notice that contrary to what we did in the nonlinear case, we include the factor r in \(W^m\), to simplify later expressions.
 
4
As in the discrete time case with the \(p^m\), the m in \(\lambda ^m\) is an upper index, not an exponent.
 
5
Where we use Dieudonné’s notation \(\mathrm {D}_kJ\) for the partial derivative of J with respect to its kth variable.
 
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Metadata
Title
Dynamic Equilibrium with Randomly Arriving Players
Authors
Pierre Bernhard
Marc Deschamps
Publication date
19-05-2020
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 2/2021
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-020-00354-z

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