Abstract
The envelope theorem is extended to cover the class of discounted and autonomous infinite horizon differential games that possess locally differentiable Nash equilibria. The theorems cover open-loop and feedback information structures and are applied to an analytically solvable linear-quadratic game. The linear-quadratic structure permits additional insight into the theorems that is not available in the general case. With open-loop information, for example, the costate variable is shown to uniformly overstate the shadow value of the state variable, but the growth rates of the two are identical.
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Acknowledgements
We thank Jacob C. Engwerda for suggesting that we apply the envelope theorem to a linear-quadratic differential game. We also thank a referee for several thoughtful comments that have resulted in a better paper.
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Ling, C., Caputo, M.R. The Envelope Theorem for Locally Differentiable Nash Equilibria of Discounted and Autonomous Infinite Horizon Differential Games. Dyn Games Appl 2, 313–334 (2012). https://doi.org/10.1007/s13235-012-0045-8
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DOI: https://doi.org/10.1007/s13235-012-0045-8