My Math Forum Arrow's theorem and social choice

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 October 23rd, 2007, 06:42 AM #1 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Arrow's theorem and social choice So now that we have an economics forum, I'm wondering if anyone else is familiar with Arrow's famous Impossibility Theorem. Consider a 'voting system' where each voter submits a ranked list of alternatives* and the system aggregates votes in some deterministic (nonrandom) way into an ordering of candidates from best to worst. Also consider these properties: Independence of irrelevant alternatives: Any two alternatives are ranked entirely based on individual votes; adding or removing other alternatives doesn't change their order. Non-imposition: Voters can choose any of the n! possible orders on the alternatives. Pareto optimality: If no voters prefer B to A, and some voters prefer A to B, then A is ranked higher than B. The only voting system that adheres to the above properties is a dictatorship, in the sense that one voter's preferences determine (and match) the social outcome. * It's not important whether the votes allow ties or not. The system can't output ties, though; this would lead to the theorem of Duggan and Schwartz instead. If ties are allowed than the dictator sets all preferences except between those where he is indifferent, and other 'sub-dictators' choose between those.
 January 11th, 2008, 08:05 AM #2 Senior Member   Joined: Dec 2007 Posts: 687 Thanks: 47 I found this (http://www.encyclopedia.com/doc/1O87-Ar ... heorm.html): Arrow's impossibility theorem n. A theorem introduced in 1951 by the US economist Kenneth J(oseph) Arrow (born 1921) and modified in 1963 to correct an error that had been found in it, showing that no voting system or any other social choice function could ever guarantee to aggregate the individual preferences of a group into a collective preference ranking so as to satisfy the following four seemingly necessary but mild conditions of fairness. (U) Unrestricted domain: the social choice function must generate a collective preference order from any logically possible set of individual preference orders; (P) Pareto condition: whenever all individuals prefer an alternative x to another y, x must be preferred to y in the collective preference order; (I) Independence of irrelevant alternatives: the collective preference order of any pair of alternatives x and y must depend solely on the individuals' preferences between these alternatives and not on their preferences for other (irrelevant) alternatives; (D) Non-dictatorship: the collective preference order must not invariably correspond to the preferences of any single individual, regardless of the preferences of the others. The proof of the theorem relies on the profile of individual preferences that gives rise to Condorcet's paradox of intransitive preferences and shows that any social choice function that satisfies U, P, and I necessarily violates D and is therefore dictatorial. Also called Arrow's paradox. I'll translate the text carefully to understand, but what is your idea on this issue? Or do you want to just have a discussion about it?

 Tags arrow, choice, social, theorem

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