# An election identical to its reversal

#voters their vote
v H>A>K
v K>A>H
w H>K>A
w A>K>H

This is a (2v+2w)-voter three-candidate election where v and w are any integers with 0≤v≤w. The three candidates are named A, H, and K.

Suppose every voter reverses her preference order (i.e. they are now attempting to choose the worst candidate rather than the best). In that case the same election, with the same set of ballots, results!

Hence every voting method that does not output a perfect tie in this election, exhibits "reversal failure" where, if all ballots are reversed so voters are "trying to determine the worst candidate" rather than best, then, embarrassingly, the same "winner" is chosen. In fact, the entire finish order is the same, making that voting method look extremely stupid because it just totally contradicted itself.

Examples (with 0<v<w):

• Plain plurality voting system (also true for Carey's voting system): H wins with v+w votes. A comes in second with w votes. After reverse all ballots, still H wins with v+w votes with A second with w votes. How can the "best to worst" ordering be the same as the "worst to best" ordering?
• Bucklin voting system: K wins with A second.
• A version of the IRV (instant runoff) system in which, to break ties in some round, we agree that if candidate X was leading candidate Y before the round involving an XY tie, then we regard that round as "won by X" instead of "tie." In this system, K is eliminated in round 1, then the AH tie in the second and final round is broken in favor of H. (After all ballots reverse: same thing happens.)
• The "opposite" version of IRV in which, to break ties in some round, we agree that if candidate X was leading candidate Y before the round involving an XY tie, then we regard that round as "won by Y" instead of "tie." In this system, K is eliminated in round 1, then the AH tie in the second and final round is broken in favor of A.
• Coombs' runoff system (with similar tie-breaking conventions as we just explained for IRV).

All of those voting systems are maximally-embarrassed by this election example: they all, insanely, contend the candidates are ordered from best-to-worst in the "same" unique order as you get by ordering them worst-to-best.

However, all voting methods (including Borda and many Condorcet systems) which are based purely on the pairwise table regard this as a perfect 3-way tie, because every pairwise 2-candidate subrace is a perfect v+w:v+w tie. These include the following systems: Baldwin, Black, Borda, Copeland, Dodgson, Nanson, Raynaud, Schulze, Simpson, Small, and Tideman.

Superiority of score voting versus every ranking-based voting method: For any election in the Saari-Barney family, every ranking-based voting method contends either that

1. the election is a perfect 3-way tie, or
2. makes an insane maximally-self-contradictory "reversal failure" contention.

Both of those outcomes are unattractive. But score voting will never exhibit any kind reversal failure, and will in general not produce ties even in scenarios that ranking methods regard as ties. For example the scores A=9.14, H=7.43, K=1.12 provided by a score-voter would be regarded by a ranking method as the "same ballot" as A=5.00, H=0.01, K=0.00. Meanwhile score voting correctly regards these two ballots as quite different. With continuum scores, score voting will have "probability 0" of producing a tie, even in scenarios which ranking systems regard it as a tie. Why throw away information?

### Source

This election-family was given by

Donald G. Saari & Steven Barney: Consequences of reversing preferences, Mathematical Intelligencer 25,4 (2003) 17-31.

and then also discussed by Ivars Peterson in Science News (16 October 2003).

However, Saari, Barney, and Peterson did not draw from this the lesson that I do, which is that RANKINGS ARE BAD, SO USE SCORE-BASED BALLOTS.

Instead, the (much less profound, but also correct) lesson Saari draws from this election is that Borda's voting method is the only one among "positional weighted" ranking-based voting systems, which, in 3-candidate elections, is immune to reversal failure.