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I have found many ressources on alternative voting system for multiple candidates (highest median, maximal lotteries, etc..), but very few on a "two candidates situation", or a yes/no situation.

With two candidates, it's clear that cardinal voting can produce a different result than a regular yes/no vote. Consider a population vote on a motion with 45% totally against (score=0/10, they will start a riot if the "yes" wins) and 55% barely in favor (score 6/10, they are in favor but do not really care). With regular voting the "yes" wins, but picking "no" (using the average for instance since median would not help) seems a better choice since it would upset less the whole population.

I know the average is not a great choice. I tried to find ressources but I think I am using the wrong keywords. Does anyone know an alternative, well-studied, voting system for a yes/no situation? Or does anyone have a ressource that explains why trying to improve a yes/no vote is not possible without strong drawbacks.

Thank you!

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For rational agents, every reasonable rule would degenerate into majority voting.

With only two alternatives (be they yes/no or two candidates), the obvious choice for a voting rule is to simply let the majority decide. As you rightly point out, this might lead to a suboptimal situation when the majority has a much weaker opinion than the minority. However, the majority can always enforce the outcome it prefers by giving their preferred option the maximum score, and the other option the minimum. In this situation I would argue that any reasonable voting rule would side with the majority.

In the multi-candidate case, strategic voting can come with a risk for a voter that lacks complete information. For example, in cardinal voting with three alternatives, it is unclear what score the strategic voter should give to their second-preferred alternative. Rate it too high, and it risks exceeding the first option in score. Too low, and it might lose to the worst alternative. Regardless however, it's clear that the voter should give the maximum score to the best option, and the minimum score to the worst.

In the two-candidate case, there is no such risk. The only way the outcome can change when a voter opts not to use the maximal values is for their preferred outcome to not be chosen. This is necessarily bad according to the valuation of the voter, and thus a rational agent should not do this.

However...

However, the above assumes a framework of voting in which an agent has an actual preference. This is generally a good framework in which to consider voting, but it does not always apply.

Consider instead a pub quiz team, deciding on their answer to a yes/no question. To arrive at an answer, we may ask each of them to give a probability that the answer is yes. Someone who thinks the answer is yes, but is unsure, might give it 60%. Someone who is almost certain the answer is no might give 5%.

In this case, there are several voting algorithms that could be used. We could simply agree with the person with the strongest opinion. We could, assuming independence of the "votes", pick the answer with the highest product of probabilities. We could also still simply take the majority vote.

Importantly, the incentives of all the voters align, and therefore the majority has no incentive to force their answer, even though they could.

A voting rule for such a situation is generally called a "judgment aggregation rule". This as opposed to a "preference aggregation rule".

Unfortunately, I'm unaware of any serious work on ordinal judgment aggregation. Most work focusses on a situation in which "voters" have (binary) opinions on multiple issues, some of which are mutually incompatible.

The only paper I can readily find with ordinal judgment aggregation in its title is Claude Hillinger's paper 'Voting and the Cardinal Aggregation of Judgment', which seems to be more of an opinion piece than a paper, and is occasionally misleading to the point of uselessness (as opinion pieces are known to be). I don't recommend it.

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  • $\begingroup$ Thank you for the detailed answer. I think your comparison with pub quiz captures the idea. My original question came from an observation during a vote where people had voted without a true conviction, out of habit, or with limited information to support their vote on the motion (and often aware of it). Instead of a binary vote, asking them something like "what's the probability that this motion, once voted, will lead to a better situation?" and using associated metrics could be intersting. I will look for ressources on the concepts you have mentionned. $\endgroup$ May 6, 2021 at 20:44

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