Inspired by the upcoming elections, I spent a little time yesterday trying to think up an example in which people could potentially have logically consistent beliefs individually but as a whole produce logically inconsistent outcomes. The result of that effort follows.

It’s election time, and there are three propositions on the ballot to spend a budget surplus. Proposition 1 is to increase funding for education. Proposition 2 is to increase funding for the healthcare. However, if both Propositions 1 and 2 pass, the tax rate needs to increase to 8% to avoid a budget shortfall. Proposition 3 is designed to do just this.

There are three voters in the town. Alice is for education only, so she supports Proposition 1 but not 2 or 3. Bob is for healthcare only, so he supports Proposition 2 but not 1 or 3. Cindy wants both education and healthcare, so she supports Propositions 1, 2, and 3. While everyone believes in something logically consistent, in this scenario, both Propositions 1 and 2 pass, but Proposition 3, the tax increase, is defeated, leading to a budget shortfall.

I posed the problem above to Justin Bledin, a graduate student in the Logic Group at UC Berkeley, to find out if the idea made sense or not.

“If you’d stumbled upon this ten years ago, it would have made a nice paper,” Justin responded before pointing me to the judgment aggregation paradox, something that he had come across in a decision theory seminar.

In 2002, Christian List and Philip Pettit‘s “Aggregating Sets of Judgments: An Impossibility Result” was published in Economics and Philosophy. The paper starts with an example similar in flavor to the one above and goes on to prove that a voting function will produce logically inconsistent output for certain logically consistent profile of inputs if the voting function satisfies the following three conditions:

1. The voting function should accept any individual’s voting profile if it satisfies certain conditions for logical consistency.
2. The output of the voting function should be the same for any permutation of the individual voting profiles.
3. If two propositions have the same votes in favor, then their outcome should be the same.

The paper concludes with strategies that could produce one a consistent voting function if one of the rules were relaxed. One idea that comes out of the second theorem of the paper is a median-based voting method, so long as there is a way to order individual voting profiles. It would be interesting to think about how one might construct such voting systems in practice.