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Dynamic Games and Applications
Published in: Dynamic Games and Applications 2/2018

10-05-2017

Stochastic Differential Games for Which the Open-Loop Equilibrium is Subgame Perfect

Authors: Ricardo Josa-Fombellida, Juan Pablo Rincón-Zapatero

Published in: Dynamic Games and Applications | Issue 2/2018

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Abstract

It is generally admitted that a correct forecasting of uncertain variables needs Markov decision rules. In a dynamic game environment, this belief is reinforced if one focuses on credible actions of the players. Usually, subgame perfectness requires equilibrium strategies to be constructed on Markov rules. It comes as a surprise that there are interesting classes of stochastic differential games where the equilibrium based on open-loop strategies is subgame perfect. This fact is well known for deterministic games. We explore here the stochastic case, not dealt with up to now, identifying different game structures leading to the subgame perfectness of the open-loop equilibrium.
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Footnotes
1
See Başar and Olsder [1], where the issue of information in dynamic games is thoroughly analyzed. Also, Fershtman [6] provides a clever analysis of the meaning of open-loop and closed-loop rules.
 
2
The results shown in the paper remain valid for a variable discount factor \(\mathrm{e}^{-\int _t^s \rho ^i(r) \mathrm{d}r}\), where \(\rho ^i(r)\ge 0\) is a continuous function. The changes in the equations shown in the paper are straightforward.
 
3
See Fleming and Rishel [7] or Dockner et al. [4] for details about the HJB equation.
 
4
If no bequest function is imposed, then the extraction rate equilibrium becomes unbounded at the terminal time. The model is similar to that studied in Reinganum and Stokey [14]. Uncertainty does not mitigate the problem.
 
Literature
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Fershtman C (1987) Identification of classes of differential games for which the open loop is a degenerate feedback nash equilibrium. J Optim Theory Appl 55(2):217–231MathSciNetCrossRefMATH
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Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, New YorkCrossRefMATH
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Kamien MI, Schwartz NL (1991) Dynamic optimization: the calculus of variations and optimal control in economics and management, 2nd edn. North-Holland, AmsterdamMATH
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Reinganum JF, Stokey N (1985) Oligopoly extraction of a common property natural resource: the importance of the period of commitment in dynamic games. Int Econ Rev 26(1):161–173MathSciNetCrossRefMATH
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Metadata
Title
Stochastic Differential Games for Which the Open-Loop Equilibrium is Subgame Perfect
Authors
Ricardo Josa-Fombellida
Juan Pablo Rincón-Zapatero
Publication date
10-05-2017
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 2/2018
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-017-0221-y

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