When a Physicist Needs to Consult an Economist

As a physicist, I never expected to turn to the results of economics to advance my research. And I never have. But the death this past week of Kenneth Arrrow, a Nobel-Prize winning economist, reminded me of one occasion on which I had to invoke Arrow's work in my role as department chair to settle a dispute. Or rather, to show that it could never be settled. Arrow proved some mathematical results concerning elections that are so bizarre and so disturbing that it's difficult to believe them -- and these results are still largely unknown to most people.

Consider this purely hypothetical situation. Suppose that you have a group of people who need to choose one person in their midst to receive an award. If there are only two nominees -- let's call them Alice and Bob -- then it's simple. Just take a vote, and the majority winner gets the award. But what if you throw in a third candidate, Carl? What now?

One possibility is to just take a three-way vote, and give the award to the person receiving the most votes. But that clearly isn't fair - what if Alice gets just over 1/3 of the votes, but everyone who voted for Bob would take Carl as their second choice (and vice versa)? This leads to sequential elimination voting -- drop the candidate who comes in third, and then have a runoff election between the other two candidates (in practice, this can be achieved with a single vote by having everyone rank their candidates). This is probably the most commonly-used voting system -- it's the system, for instance, used by the science fiction community to choose the Hugo Awards -- although the latter allows for a voter to choose "no award", which can lead to bizarre results.

Another possibility is to take sequential head-to-head votes -- have the voters choose between Alice and Bob, Bob and Carl, and Carl and Alice, and the person who wins both of his two elections gets the award. But a little thought reveals that this procedure isn't guaranteed to produce a result -- it's possible that Alice would defeat Bob AND Bob would defeat Carl AND Carl would defeat Alice. Yet another alternative would ask each voter to rank the three candidates, and assign, for instance, 3 votes for each first-place ranking, 2 votes for second-place, and 1 vote for third place.

The problem is that these different voting systems can produce different winners. So what is the best voting system once you get past two candidates? It seems reasonable to require that any voting system ought have the following properties:

1. If the voters favor Alice ahead of Bob in a two-person election, and then Carl is introduced, the result should be the election of either Alice or Carl, but not Bob.  Similarly, if Alice wins a three-person election by a majority vote, and one of the losing candidates drops out, Alice should still win against the remaining candidate.
2. If Alice wins the election, and then some ballots are altered in such a way that the position of Alice is raised on some of the ballots without changing the order of the other candidates, then Alice should still win.

These are common-sense requirements for any reasonable voting system -- I wouldn't trust a voting system that violated them. But Arrow proved that no voting system can satisfy both of them, a result known as “Arrow’s Impossibility Theorem”. In a nutshell, Arrow’s Theorem shows that there is no perfect voting system when you have more than two candidates -- the best you can do is to try to find the least bad system.