User:CRGreathouse/Arrow

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[edit] Extensions of Arrow's theorem

Arrow's theorem sparked a number of similar impossibility theorems. These theorems can work on different frameworks (nondeterministic, or a single-winner procedure instead of a SWF), or they can replace a condition of the theorem with a weaker condition.

  • Sen[1] shows that Arrow's theorem holds for cardinal as well as ordinal rankings, as long as interpersonal utility is incomparable (even if intrapersonal differences are comparable).
  • Igersheim[2] shows that if the dictatorship condition is strengthened to include anti-dictators as well, the Pareto principle can be replaced by the slightly weaker condition of Pareto neutrality, following Xu's definition[3].
  • Wilson[4] shows that universality, non-imposition, and IIA imply an authoritarian outcome (which included dictatorial and near-dictatorial results). This effectively removes Arrow's assumption of Pareto (since Pareto efficiency implies non-imposition). Malawski and Zhou[5] show that that IIA and non-imposition implies weak Pareto or inverse weak Pareto, which proves the same result.
  • Tanaka[6] is able to weaken Wilson's IIA to a particular form of monotonicity by strengthening non-imposition to strict non-imposition (for every pair of alternatives there is some set of profiles such that the first alternative is strictly preferred to the second).
  • Taylor[7] extends Arrow's theorem to voting rules (selecting a single winner or a nonempty set of winners), as well as social choice functions generated by SWFs.
  • Gibbard (1969; cited in Sen [1] and in Campbell[8] who extended the result) shows that replacing voters' transitive preferences with quasitransitive preferences and replacing Pareto with the weaker non-imposition results in oligarchy (generalized dictatorship).
  • Shelah[9] replaces Arrow's assumption that individual preferences are weak orders with the weaker assumption that they "belong to an arbitrary non-trivial symmetric class of choice functions".
  • Murakami[10] improved a result of Blau[11] on domain restrictions. Although Arrow's theorem is not true under domain restrictions (the pigeonhole principle makes non-imposition too strong a condition), a reformulation with a slightly stronger form of monotonicity and a looser form of dictatorship (a voter who can dictate between one pair of alternatives) make the theorem true for all domain restrictions. Additionally, under this reformulation, it is sufficient to have independence of one irrelevant alternative.
  • Pattanaik and Peleg[12]show that if nondeterministic methods are allowed, the only one satisfying suitable reformulations of Arrow's criteria with at least 4 alternatives is a random weighted dictatorship.

[edit] References

  1. ^ a b Amartya Sen, Collective Choice and Social Welfare (1970) San Francisco: Holden-Day, ISBN 0444851275. Chapters 8 and 8*.
  2. ^ Harrade Igersheim, "Extending Xu's results to Arrow's Impossibility Theorem", Economics Bulletin, 4, No. 13 (2005), pp. 1–6.  
  3. ^ Yongsheng Xu, "The Libertarian paradox: some further observations", Social Choice and Welfare 7 (1990), pp. 343–351.
  4. ^ R. B. Wilson, "Social choice theory without the Pareto principle", Journal of Economic Theory 5 (1972), pp. 478–486.
  5. ^ M. Malawski and L. Zhou, "A note on social choice theory without the Pareto principle", Social Choice and Welfare 11 (1994), pp. 103–107. Cited in Tanaka 2003.
  6. ^ Yasuhito Tanaka, "A necessary and sufficient condition for Wilson's impossibility theorem with strict non-imposition", Economics Bulletin, 4, No. 17 (2003), pp. 1−8.  
  7. ^ Alan D. Taylor, Social Choice and the Mathematics of Manipulation, Cambridge University Press, 1st edition (2005), ISBN 052100883. Sections 1.3 and 3.4.
  8. ^ Donald E. Campbell, "Implementation of social welfare functions", International Economic Review 33 (1992), No. 3, pp. 525–533.
  9. ^ Sharon Shelah, "On the Arrow property", Advances in Applied Math 34 (2005), pp. 217–251.  
  10. ^ Yasusuke Murakami, "A note on the general possibility theorem of the social welfare function", Econometrica 29, No. 2. (1961), pp. 244–246.
  11. ^ Julian H. Blau, "The existence of social welfare functions", Econometrica 25, No. 2. (1957), pp. 302–313.
  12. ^ Prasanta Pattanaik and Belalel Peleg, "Distribution of power under stochastic social choice rules", Econometrica 54, No. 4 (1986), pp. 909–922.