Abstract
This chapter discusses, in both continuous time and discrete time, the issue of certainty equivalence in two-player zero-sum stochastic differential/dynamic games when the players have access to state information through a common noisy measurement channel. For the discrete-time case, the channel is also allowed to fail sporadically according to an independent Bernoulli process, leading to intermittent loss of measurements, where the players are allowed to observe past realizations of this process. A complete analysis of a parametrized two-stage stochastic dynamic game is conducted in terms of existence, uniqueness and characterization of saddle-point equilibria (SPE), which is shown to admit SPE of both certainty-equivalent (CE) and non-CE types, in different regions of the parameter space; for the latter, the SPE involves mixed strategies by the maximizer. The insight provided by the analysis of this game is used to obtain through an indirect approach SPE for three classes of differential/dynamic games: (i) linear-quadratic-Gaussian (LQG) zero-sum differential games with common noisy measurements, (ii) discrete-time LQG zero-sum dynamic games with common noisy measurements, and (iii) discrete-time LQG zero-sum dynamic games with intermittently missing perfect state measurements. In all cases CE is a generalized notion, requiring two separate filters for the players, even though they have a common communication channel. Discussions on extensions to other classes of stochastic games, including nonzero-sum stochastic games, and on the challenges that lie ahead conclude the chapter.
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Notes
- 1.
They are taken to be constant for simplicity in exposition; the main message of the paper and many of the expressions stand for the time-varying case as well, with some obvious modifications.
- 2.
We are using “T” to denote the number of stages in the game; the same notation was used to denote transpose. These are such distinct usages that no confusion or ambiguity should arise.
- 3.
We have taken α 1 not to be dependent on β 1, because when β 1=0, y 1≡0, making α 1 superfluous.
- 4.
One can take any form here, since \(\hat{\gamma}\) had annihilated v, but we take linear-plus-Gaussian in anticipation of \(\hat{\gamma}\) to be in equilibrium with v.
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Acknowledgements
This work was supported in part by the AFOSR MURI Grant FA9550-10-1-0573, and in part by NSF under grant number CCF 11-11342.
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Başar, T. (2014). Stochastic Differential Games and Intricacy of Information Structures. In: Haunschmied, J., Veliov, V., Wrzaczek, S. (eds) Dynamic Games in Economics. Dynamic Modeling and Econometrics in Economics and Finance, vol 16. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54248-0_2
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