VOTING PARADOXES AND HOW TO DEAL WITH THEM Hannu Nurmi University of Turku Turku, Finland

VOTING Satisfaction and justice in voting outcomes is important Every day, somebody is rackin ’ and stackin ’ Voting is a way to reach equitable consensus

PARADOXES OCCUR 1992 ELECTION –Bush and Poirot win popular election 2000 Election –Bush II loses popular vote, wins election They happen every day in the rack/stack method used in DoD

ASSUMPTIONS Equal Weight One Vote Each Independence (no gaming) Transitivity (A < B and B < C implies A < C) DEFN: An Alternative is one of the choices NOTATION: a > b means a is prefered to b

PREFERENCE PROFILE COUNT ST ABCABC 2 ND BCACAB 3 RD CABBCA

WHO WINS?

TOURNAMENT MATRIX PAIRWISE COMPARISON MATRIX for 12 voters, B>A (note: nontransitivity)

CONDORSET WINNERS AND LOSERS A < B, 13 vs. 8 A < C, 13 vs 8 B < C, 13 vs. 8 But, A wins plurality vote! A is the Condorcet loser –uniformly despised

BORDA (1770) give k points to last place give k + a points for second to last give k + 2a points for third from last etc. Borda never elects the Condorcet loser Does Not always elect the Condorcet winner

SUMMED RANK Is the usual bad? One (1) point for first place Two (2) points for second place etc. Sum the point scores Select the alternative with the lowest score

ANALYSIS Reverse the ranks k = 1 a = 1 Always selects the Condorcet winner if it exists May select Condorcet loser if it exists

VOTING PARADOXES What follows is a set of situations where the vote fails to reflect consensus. Many of these situations are famous.

NO SHOW PARADOX 26%47%2%25% ABBC BCCA CAAB Plurality run-off voting 1 st Round: Eliminate C –A wins in run-off with 51% Suppose the 47% no-show –B is eliminated, C subsequently beats A –the 47% get their second choice, not their 3rd

INCONSISTENCY PARADOX east west 35%40%25%40%55%5% ABCCBA BCBBCC CAAAAB Plurality run-off voting in each district B wins the East in run-off, wins West outright Taken as a whole, C beats B in a run-off

ALABAMA PARADOX OF 1881 Hamiltonian Apportionment Seats allocated by integer part, remainder allocated by largest fraction remaining

OSTRAGORSKI ’ s PARADOX Arises because the following two produce different winners: 1.BEAUTY CONTEST: Each voter votes for the candidate whose stand is closest to his in a majority of issues. 2.ISSUE CONTEST: For each issue, voters pick candidates. The winner is the one winning the majority of issues.

BEAUTY WINNER AXXXX BXYXX CYXXX DYYYY EYYYY ISSUE WINNERYYX

SIMPSON ’ s REPRESENTATION PARADOX Percent who favor higher in the East for both employed and unemployed Total percent in favor larger in the West EASTWESTEASTWESTEASTWEST EMPLOYED 400,00090,00080,00015,00020%17% UNEMPLOYED 100,00080,00050,00035,00050%44% total500,000170,000130,00050,00026%29% POPULATION FAVOR INITIATIVE