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Dynamic Games and Applications
Published in: Dynamic Games and Applications 3/2014

01-09-2014

Pareto Improvements of Nash Equilibria in Differential Games

Author: Atle Seierstad

Published in: Dynamic Games and Applications | Issue 3/2014

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Abstract

Frequently controls forming Nash equilibria in differential games are not Pareto optimal. This paper presents conditions that can be used to show the existence of strict Pareto improvements of Nash equilibria in such games. The conditions are based on standard tools in control theory.
previous article The Evolution of Cooperation Through Institutional Incentives and Optional Participation

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Appendix
Available only for authorised users
Footnotes
1
This limit condition is explained in the Appendix.
 
2
In case \(\hat{u}^{(i)}(t)=\check{u}^{(i)}(t,\hat{x}(t))\in \operatorname{int} U_{i}\), when calculating this derivative, the term containing \(\partial \check{u}^{(i)}(t,x)/\partial x\) drops out due to \(H_{u^{(i)}}^{(i)}=0\).
 
3
Let \(x_{\ast \ast }=(z,x)\in \mathbb{R}^{n}\times \mathbb{R}^{n}\) be governed by \((\dot{z},\dot{x})=(g(t,x,u),g(t,x,u))=:\hat{h}(t,x,u)\), let C ∗∗ be the resolvent of \(\dot{q}_{\ast \ast }=\hat{h}_{(z,x)} (t,\hat{x}(t),\hat{u}(t))q_{\ast \ast }\), and note that \(C^{\ast \ast }(T,t) (\check{g},\check{g})=(C(T,t)\check{g},C(T,t)\check{g})\) \((\check{g}\in \mathbb{R}^{n})\) and \(C^{\ast \ast }(T,t)(\check{g},0)=\check{g}\). Letting Az=(a 1 z,…,a m z), note that \(C^{\ast }(T,t)(\check{g},\check{g})=(AC(T,t)\check{g},C(T,t)\check{g})\) and \(C^{\ast }(T,t)(\check{g},0) =A\check{g}\).
 
4
The last inclusion even holds if t in (23) is restricted to (0,T), by left continuity at T.
 
5
To give one reference, note first that for some d′′>0 small enough, \(B(d^{\prime \prime }\bar{p},d^{\prime \prime }2\varepsilon ) \subset \mathrm{clco}\{\pi q^{u_{i}^{d^{\prime \prime }}}(T):i=1,\ldots,i^{\ast }\}\), where \(u_{i}^{d}=u_{i}1_{[t_{i},t_{i}+d]}+\hat{u}(1-1_{[t_{i},t_{i}+d]})\) and q u (.), u=u(.), is the solution of
$$\dot{q}^{u}(t)=f_{x} \bigl(t,\hat{x}(t),\hat{u}(t)\bigr)q^{u}(t)+f \bigl(t,\hat{x} (t),u(t)\bigr)-f \bigl(t,\hat{x}(t),\hat{u}(t)\bigr),\quad q^{u}(0)=0. $$
The inclusion follows because \(q^{u_{i}^{d}}(T)-dq^{u_{i},t_{i}}(T) \) is of the second order in d. Then the assertion in the lemma follows directly from Theorem 2 [14] in the case n ∗=n, and the same proof holds also for n ∗<n. In fact, the latter result can easily be derived from the case n ∗=n by using suitable auxiliary controls.
 
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Metadata
Title
Pareto Improvements of Nash Equilibria in Differential Games
Author
Atle Seierstad
Publication date
01-09-2014
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 3/2014
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-013-0093-8

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