Stable Sets in multi-good pillage games are small
It is known that, in one-good pillage games, stable sets are finite. For m goods, it has been conjectured that the stable sets have measure zero. We introduce a class of sets, termed pseudo-indifference sets, which includes level sets of utility functions, quasi-indifference classes associated with a preference relation not given by a utility function, production possibility frontiers, and Pareto efficient sets. We establish the truth of the conjecture by proving that pseudo-indifference sets in Rp have p-dimensional measure zero. This implies that stable sets in n-agent, m-good pillage games have m(n - 1)-dimensional measure zero. We then prove that each pseudo-indifference set in Rp has Hausdorff dimension at most p - 1, a much stronger result than measure zero. Finally, we establish a stronger version of the conjecture: stable sets in n-agent, m-good pillage games have dimension at most m(n-1)-1.
|Date of creation:||Feb 2010|
|Date of revision:|
|Contact details of provider:|| Postal: Edgbaston, Birmingham, B15 2TT|
Web page: http://www.economics.bham.ac.uk
More information through EDIRC
When requesting a correction, please mention this item's handle: RePEc:bir:birmec:10-05. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Colin Rowat)
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.