Stable Sets in multi-good pillage games are small
AbstractIt 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.
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Bibliographic InfoPaper provided by Department of Economics, University of Birmingham in its series Discussion Papers with number 10-05.
Length: 23 pages
Date of creation: Feb 2010
Date of revision:
pillage games; cooperative game theory; stable sets; Hausdorff dimension;
Find related papers by JEL classification:
- C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
- D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies
- P14 - Economic Systems - - Capitalist Systems - - - Property Rights
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- Manfred Kerber & Colin Rowat, 2012. "Sufficient Conditions for the Unique Stable Sets in Three Agent Pillage Games," Discussion Papers 12-11, Department of Economics, University of Birmingham.
- Simon MacKenzie & Manfred Kerber & Colin Rowat, 2013. "Pillage Games with Multiple Stable Sets," Discussion Papers 13-07, Department of Economics, University of Birmingham.
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