Dominance and Challenging in Abstract Systems
39 Pages Posted: 24 Aug 2015 Last revised: 7 Mar 2016
Date Written: February 29, 2016
In many theories of group choice a dominance relation expressing the ability of alternatives to "defeat" or replace others is the central object on which various solutions are based. I present and argue for the use of a particular sub-relation of the dominance relation called the ultimate challenges relation as a basis for solutions in abstract systems. This derived relation is the largest sub-relation with the property that all win-sets are "self-consistent" with the ultimate challenges relation itself. When a group collectively and iteratively considers alternatives and compares them to possible replacements, the challenges relation is a useful and appropriate device for reasoning about stable outcomes. It reflects a form of iterated reasoning about the replaceability of a status quo by another alternative. Based on this new relation I propose a new solution in abstract systems. I apply it to cooperative majority voting which results in a modification to the tournament equilibrium set, potentially addressing some issues in the original solution.
Keywords: Abstract system, Tournaments, collective choice, social choice theory, retentive
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