Dodgson's method

From electowiki

Dodgson's method is a Condorcet method that ranks a candidate X according to the number of times adjacent candidates have to be swapped on a ballot (not necessarily the same ballot each time) to make X the Condorcet winner. The candidate with the least number of necessary swaps wins.

If X is the Condorcet winner, the number of swaps is zero, so the method passes Condorcet. However, determining the winner in the general case is complete for parallel access to NP,[1] and thus NP-hard.

For more information, see Dodgson's method on Wikipedia.

Criterion compliances

Dodgson's method passes the Condorcet criterion. It fails the independence of clones criterion,[2] the Smith criterion, the monotonicity criterion, and also fails the homogeneity criterion:[3] an election with 100 voters may return a different result to an election with 10 voters, even if the relative size of the factions is the same.

P. C. Fishburn proposed a variant that passes homogeneity[3] and where the winner can be found in polynomial time,[4] but the variant fails the other three criteria mentioned above.


  1. Hemaspaandra, Edith; Hemaspaandra, Lane A.; Rothe, Jörg (1997). Degano, Pierpaolo; Gorrieri, Roberto; Marchetti-Spaccamela, Alberto (eds.). "Exact analysis of Dodgson elections: Lewis Carroll's 1876 voting system is complete for parallel access to NP". Automata, Languages and Programming. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer: 214–224. doi:10.1007/3-540-63165-8_179. ISBN 978-3-540-69194-5.
  2. Brandt, Felix (2009). "Some Remarks on Dodgson's Voting Rule". Mathematical Logic Quarterly. 55 (4): 460–463. doi:10.1002/malq.200810017. ISSN 1521-3870.
  3. a b Fishburn, Peter C. (1977-11-01). "Condorcet Social Choice Functions". SIAM Journal on Applied Mathematics. 33 (3): 477–479. doi:10.1137/0133030. ISSN 0036-1399.
  4. Rothe, Jörg; Spakowski, Holger; Vogel, Jörg (2003-08-01). "Exact Complexity of the Winner Problem for Young Elections". Theory of Computing Systems. 36 (4): 375–386. doi:10.1007/s00224-002-1093-z. ISSN 1433-0490.