Example of Ambiguity Resolution

To demonstrate the basic procedure for all the ambiguity resolution or non-Condorcet election methods listed in the previous two sections, consider the following example, again based on the rock-paper-scissors example mentioned in The Paradox. In brief, each voter will benefit their object beats the elected object, be harmed if the elected object beats theirs, and be unaffected if the elected object is the same as theirs.

• 45% of voters have rock, and thus their preferences are (in order) scissors, rock, paper, and they “approve” of scissors and rock (where they will not lose any money).
• 20% of voters have paper, thus their preferences are rock, paper, scissors, approving of the first two.
• 35% of voters have scissors, thus their preferences are paper, scissors, rock, approving of the first two.

A note on strategic voting

The purpose of this explanation is to demonstrate the method by which winners are determined given electoral data, not to suggest how the election might turn out. All of these computations involve tabulation of completely honest data, and as a result there is relative consistency in the results. This is generally true: given honest data, most electoral systems will find the best result, but in most cases one must also consider how the possibility of strategic voting might emerge. This is briefly discussed in some of the sections below. FILL THIS OUT WITH MORE ONCE YOU’VE WORKED OUT OTHER SYSTEMS.

The most utilitarian result: Scissors, Paper, Rock

Before considering what result various election methods will find, let us decide what result they should find, from a utilitarian perspective. Namely, we determine the difference between how many voters benefit, and how many are harmed, by each potential result.

• Rock: 20% benefit, while 35% are harmed, so the utility is -15%.
• Paper: 35% benefit, while 45% are harmed, so the utility is -10%.
• Scissors: 45% benefit, while 20% are harmed, so the utility is +25%.

Thus we see that in this particular example, the best result is for scissors to be selected, then paper, then rock. Bear this in mind as you examine the results produced by the various methods.

Plurality: Scissors, Paper Rock

In a plurality system, assuming no strategic voting, those with rock will dominate, and thus elect scissors, followed by paper, then rock, exactly in step with the utilitarian calculation (though this will certainly not always be the case, Mouse over for exampleSuppose instead 45% have rock, 35% paper, 20% scissors. In this case, the highest utility is to elect rock (35% - 20% = 15%), but plurality would elect scissors, which has utility 45% - 35% = 10%.).

Strategic Factor: It is plausible that rather than all voting for their preferred outcome, those voters with paper would perceive that the two front-runners are scissors and paper, and vote strategically for their second choice, paper. In this case, those voters with paper and those with scissors would form a 55% block of the electorate, capable of electing paper over scissors, which is overall a less desirable outcome for the society as a whole.

Condorcet: Ambiguity

As you might expect, since we consider ambiguity resolution, there is a Condorcet ambiguity. Assuming all voters vote their preferences, the following result matrix emerges.

Thus we see that we have the following majority rule cycle: the electorate prefers rock to paper to scissors to rock. An ambiguity resolution procedure is needed, such as one of the methods exemplified below. Note this in an election with more candidates, it is likely that before goign to ambiguity resolution, it would first be necessary to reduce the candidate pool to the Smith SetThe Smith set is the set of candidates on which a Condorcet ambiguity resolution procedure acts. It is formed by finding the smallest group of candidates such that each candidate in the set defeats each candidate not in the set. This is the natural second step of the Condorcet process, since all ambiguity lies exclusively within this set.. In this case there are only three candidates thus no reduction is needed.

Minimax: Scissors, Paper, Rock

The worst defeats are as follows:

• Rock: 80-20 (versus scissors)
• Paper: 65-35 (versus rock)
• Scissors: 55-45 (versus paper)

Thus the least worst defeat it scissors, which becomes the winner. The runner up is simply the winner between rock and paper, which is paper.

Ranked Pairs: Scissors, Rock, Paper

Ranking defeats by strength yields the following order: scissors over rock (80-20), rock over paper (65-35), paper over scissors (55-45). Thus the strongest defeat is “locked in,” establishing that scissors shall be ranked higher than rock. The next defeat states that rock beats paper, thus locking in the order scissors, rock, paper. The final defeat is not considered, since it violates a previously locked in ranking, scissors over paper.

Approval: Scissors, Rock, Paper

The approval votes are as follows:

• Rock: 65% (those with paper or rock)
• Paper: 55% (those with scissors or paper)
• Scissors: 80% (those with rock or scissors)

Thus the result is scissors, rock, paper.

Instant Runoff: Paper, Scissors, Rock

The electorate casts three distinct ballots in the following way:

• 45% Scissors > Rock > Paper
• 20% Rock > Paper > Scissors
• 35% Paper > Scissors > Rock

In the first runoff, Rock is the loser with 20%, and is thus eliminated from all ballots. Thus the remaining ballots are:

• 45% Scissors > Paper
• 20% Paper > Scissors
• 35% Paper > Scissors

Thus Paper wins this runoff with a majority of 55% and wins the election. Note that this is equivalent to what would happen in plurality voting if those voters with paper vote pragmatically for their preferred front-runner, but it is NOT the best outcome for the society as a whole.

Strategic factor: Those voters with rock have a rather peculiar sort of strategicA vote is called "strategic," "tactical," or "pragmatic" if it does not reflect the voter's true preference, but is rather an attempt by the voter to secure the best possible outcome, given the voter's prediction of how other voters will act. voting available to them, wherein they can in fact cause scissors to win by ranking it lower. They can benefit from instead voting “Rock-scissors-paper.” By doing this they can force a first-round defeat of paper, causing their second choice, rock to win, which is better than scissors. But there is another peculiar factor. Suppose instead that only 20% of the 45% switch their vote in this way. Then this will still force the elimination of paper in the first round, but then reveal a 35% block of votes for scissors, which could easily swing the election to scissors. As long as no more than 30% do this, this technique of lowering scissors on their choices will in fact cause scissors to win, which has a truly paradoxical feel to it, and is a stunning example of lack of monotonicityThis is technical term referring to the property that expressing your preference for a particular candidate should increase that candidates chance of winning. It is fairly obvious why this should be the case..

Borda Count: Scissors, Paper, Rock

Different weighting schemes are possible, but suppose that in this election, a first-place choice is given 2 points, and a second-place choice 1 point. Then the election results will be as follows:

• Rock: 85% (2 times 20% plus 1 times 45%)
• Paper: 90% (2 times 35% plus 1 times 20%)
• Scissors: 125% (2 times 45% plus 1 times 35%)

Thus scissors is the victor, followed by paper.

Strategic Factor Similarly to plurality voting, the 20% minority with paper might choose to compromise by placing paper first. In this case, this is not sufficient to give Paper the victory (the new count would become Rock: 65%, Paper: 110%, Scissors: 125%), but it is conceivable that a faction might achieve compromise in this way.