Abstract
Security solutions for two stage games, where the second stage consists of a strategic form noncooperative game, are not reachable in many problems. The aim of this paper is to investigate such solutions. Mixed extension for the second stage game is considered and existence results for approximate mixed security solutions, together with the convergences of values, are given and illustrated by significative examples. The results apply to the class of quasi harmonic games.
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Notes
A correspondence \(\Gamma : X \rightarrow 2^{Y}\) where \(2^{Y} \) is the collection of all subsets of Y, including the empty set \(\emptyset \), is said lower semicontinuous (l.s.c.) at \(x_0\in X\) if \(\Gamma (x_0)=\emptyset \) or for any sequence \(x_n\) converging to \(x_0\) and for any \(y\in \Gamma (x_0)\) there exists a sequence \(y_n\) converging to y such that \(y_n\in \Gamma (x_n)\) for n large.
A correspondence \(\Gamma : X \rightarrow 2^{Y}\) is said closed (closed graph) at \(x_0\in X\) if for every sequence \(x_n\in X\) converging to \(x_0\) and for any sequence \(y_n\) with \(y_n\in \Gamma (x_n)\) converging to some \(y_0\), one has \(y_0\in \Gamma (x_0)\).
Berge theorem (or Maximum theorem Aubin 2007) Given the function \(f: X\times Y \rightarrow \mathbb {R}\) and the correspondence \(\Gamma : X\rightarrow 2^{Y}\), consider the problem
$$v(x)= \displaystyle \sup _{y\in \Gamma (x)} f(x, y).$$Assume that \(v(x)<+\infty \) for every \(x\in X\):
(i) if f is l.s.c. and \(\Gamma \) is l.s.c., then v is l.s.c.
(ii) if f is u.s.c. and \(\Gamma \) is closed and compact valued, then v is u.s.c.
We thank one of the referees for proposing this example.
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The authors thank the anonymous reviewers for their helpful comments.
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This work has been supported by GNAMPA 2020, project:“Problemi di ottimizzazione con vincoli via trasporto ottimo e incertezza”.
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Mallozzi, L., Sacco, A. Stackelberg-Nash equilibrium and quasi harmonic games. Ann Oper Res 318, 1029–1041 (2022). https://doi.org/10.1007/s10479-020-03916-x
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DOI: https://doi.org/10.1007/s10479-020-03916-x