Abstract
We prove that every continuous value on a space of vector measure market games \(Q\), containing the space of nonatomic measures \(NA\), has the conic property, i.e., if a game \(v\in Q\) coincides with a nonatomic measure \(\nu \) on a conical diagonal neighborhood then \(\varphi (v)=\nu \). We deduce that every continuous value on the linear space \(\fancyscript{M}\), spanned by all vector measure market games, is determined by its values on \(\fancyscript{L}\fancyscript{M}\) - the space of vector measure market games which are Lipschitz functions of the measures.
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Notes
By abuse of notation.
If \(v\in Q\) then we may always write \(v=f^\prime \circ \eta ^\prime \) with \(f^\prime \in M^{m^\prime }\) and \(\eta ^\prime \in \left( NA^1\right) ^{m^\prime }\) for some \(m^\prime \ge 2\).
Obviously, \(\phi _i^n\) depends on our specific choice of \(\Pi _n\) for every \(n\ge 1\).
The set \(J_n\) is defined as in Lemma 1.
References
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Acknowledgments
The author would like to thank his Ph.D. advisor, Professor Abraham Neyman, whose advise, guidance and support are gratefully appreciated. The author is also in debt of Professor Ori Haimanko for his patience, his valuable remarks and suggestions, and the many invaluable discussions. The author is also grateful to an anonymous associate editor and an anonymous referee for their helpful comments.
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This work is based on Chapter 6 of the author’s Ph.D. thesis done at the Center for the Study of Rationality at the Hebrew University of Jerusalem under the supervision of Professor Abraham Neyman. This research was supported in part by the Israel Science Foundation Grants 1123/06 and 1596/10.
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Edhan, O. The conic property for vector measure market games. Int J Game Theory 44, 377–386 (2015). https://doi.org/10.1007/s00182-014-0434-x
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DOI: https://doi.org/10.1007/s00182-014-0434-x