Condorcet’s Paradox

… a core concept in Governance and Institutions and Atlas100

Marquis de Condorcet, 1743-1794

Marquis de Condorcet, 1743-1794

Concept description

Condorcet’s paradox is a classic problem in democracy, first formalised by the Marquis de Condorcet at the time of the French revolution, stating that majority preferences can become intransitive with three or more options. It is possible for a certain electorate to express a preference for A over B, a preference for B over C, and a preference for C over A, all from the same set of ballots.

In university courses Condorcet’s paradox is sometimes presented as an introduction to Arrow’s impossibility theorem. See, for example, MIT professor Ben Olken’s lecture “Sometimes it Get’s Complicated: Condorcet’s Paradox and Arrow’s Impossibility Theorem” at http://ocw.mit.edu/courses/economics/14-75-political-economy-and-economic-development-fall-2012/lecture-notes/MIT14_75F12_Lec12.pdf.

Olken provides the following definitions:

  • A Condorcet Winner is an alternative such that it gains a majority of votes when paired against each of the other alternatives.
  • A Condorcet Cycle occurs when there is a violation of transitivity in the social preference ordering.
  • Arrow’s Impossibility Theorem: “There is no social ranking function > such that for any group G whose members all have rational preferences, > is a rational (transitive) ranking and satisfies the Universal Domain, Pareto Optimality, Independence of Irrelevant Alternatives, and No Dictatorship assumptions.”

Olken summarizes:

“… the problem of Condorcet Cycles and agenda setting is a very deep, fundamental problem. If you want a social ordering that has Universal Domain, Pareto Optimality, Independence of Irrelevant Alternatives, and No Dictatorship, you can’t also have transitivity – you will get cycles.”

Condorcet’s paradox and agenda setting

Where the conditions exist to create a Condorcet cycle, those who set the agenda for voting can determine the voting outcome.

Christopher Vaughen, a mathematics professor at Montgomery County Community College, has produced an accessible You Tube example of the operation of Condorcet’s paradox in a famous Congressional vote on civil rights. See https://www.youtube.com/watch?v=KqNVkPK51qQ, accessed 15 August 2016.

James Stodder, writing in the International Review of Economics Education, outlines a classroom voting game that illustrates how alliances between voting blocks can determine an agenda using the three-cornered dilemma posed by Israel’s David Ben-Gurion.

Stodder uses Thomas Friedman’s description of the “Ben-Gurion Tri-lemma”:

“In November 1947 … David Ben-Gurion, then the leader of the Zionist movement in Palestine … did not shrink from clearly laying out the choice before the Jewish people … Who were they? A nation of Jews living in all the land of Israel, but not democratic? A democratic nation in all the land of Israel, but not Jewish? Or a Jewish and democratic nation, but not in all the land of Israel? Instead of definitively choosing among these three options, Israel’s two major political parties – Labor and Likud – spent the years 1967 to 1987 avoiding a choice … not on paper, but in day-to-day reality.” (T. Friedman, 1989, From Beirut to Jerusalem, New York: Doubleday, pp. 253–4)

Stodder sets up the game as follows:

“To pose Ben-Gurion’s tri-lemma as a Condorcet cycle, define options D, J and G: a Democratic Israel, with equal rights for all its citizens; a Jewish Israel, its state having an explicitly Jewish character; and a Greater Israel, extended to its ancient boundaries. Say that all participants in this game agree that all three goals are desirable. Assume also that there can be no majority without an alliance of at least two groups, Left, Right or Centre. Their rankings over D, J and G are:

L: D > J > G

C: G > D > J

R: J > G > D

“This is a ‘cyclical’ majority because two out of three groups will vote D > J and J > G, but also G > D. Ben-Gurion’s example pushes the original Condorcet problem towards crises: the third option in any combination is always impossible, logically excluded by the first two. The original Condorcet problem did not have this difficulty – merely a voting cycle for at least three alternatives.”

Atlas topic, subject, and course

Electoral Systems and Democratic Reform (core topic) in Governance and Institutions and Atlas100 Governance and Institutions.

Source

Wikipedia, Marquis de Condorcet, at https://en.wikipedia.org/wiki/Marquis_de_Condorcet, accessed 14 August 2016.

Ben Olken (2012), “Sometimes it Get’s Complicated: Condorcet’s Paradox and Arrow’s Impossibility Theorem” at http://ocw.mit.edu/courses/economics/14-75-political-economy-and-economic-development-fall-2012/lecture-notes/MIT14_75F12_Lec12.pdf.

James Stodder (2005), “Strategic Voting and Coalitions: Condorcet’s Paradox and Ben-Gurion’s Tri-lemma,” International Review of Economics Education, Vol. 4, Issue 2, 58-72, pdf available at http://www.sciencedirect.com/science/article/pii/S1477388015301316, accessed 15 August 2016.

Page created by: Ian Clark, last modified on 15 August 2016.

Image: Cropped from Painting of Nicolas de Condorcet, Wikipedia, Marquis de Condorcet, at https://en.wikipedia.org/wiki/Marquis_de_Condorcet, accessed 14 August 2016.