Why SAVE?

The decision is the location of a cell phone service tower for the community. The choice of location has been simplified to just picking a coordinate point in the plane that will be accepted by the community. Each community member has an individual ideal location, and all of these locations are different. Moreover, many members of the community value their privacy and may not want to reveal their ideal tower location if they can get decent service without doing so.

The desired solution is a tower location that minimizes the average distance between the tower and the community members' individual ideal locations.

The idea of using an ad-hoc committee is that it might somehow be able to propose either a single decent location, or a set of reasonable locations from which to choose. The problem here is the committee does not know what anyone outside the committee wants.

There is no reason to think a proposal from any subset of the community would be ideal for the community as a whole. It could happen, but there are no guarantees.

The committee proposed two locations, from Alice and Bob, and would have been happy with either one. The rest of the community…not so much.

The clerk initially expected to preside over a vote between proposed locations A and B using simple majority decision. Had that occurred, the community would have been stuck with location B, which would have resulted in part of the community having poor or no service at all.

Carol's proposal of location C changed that plan from a majority decision to a single elimination tournament. Using a single elimination tournament would have changed the result from the earlier B, to location C, and a slightly different part of the community having poor or no service.

Alice's objection to the single elimination tournament (or its result) and the clerk's and community's accommodation with another vote between A and C resulted in the exposure of a Condorcet cycle.

The small size of the community and the fact that the entire community was at the town meeting made it easy to accommodate the desire for multiple votes, leading to the exposure of the Condorcet cycle.

A Condorcet cycle is not an aberration but rather a condition that arises naturally with some sets of alternatives even when the underlying collective preferences are fixed, as they are in this example.

There are two main reasons why people who have not studied voting theory would not have heard of Condorcet cycles. One is a tendency to reduce decisions to just two alternatives. The other is the voting systems being used might not be able to expose the cycles.

When there is only a choice between two outcomes, it is not possible to have a Condorcet cycle simply because a cycle needs to have at least three members. Quite often we are presented with binary choices: A or B, yes or no, purchase or not purchase, take it or leave it. And in many of those cases there are really many more options possible, and the restriction of the responses to one of two choices is artificial and can prevent even the consideration of a possible, better outcome.

When the voting system is unable to detect Condorcet cycles, the problem is more insidious. Any voting system that cannot detect or expose Condorcet cycles is subject to manipulation or strategic voting. Manipulation or strategy usually needs information about what would occur without manipulation or with only honest votes, and that type of information may be hard to get. Yet without accurate information, attempts at manipulation or strategic voting may result in an outcome that is neither the "natural" outcome (with neither manipulation nor strategic voting) nor the "desired" outcome (the goal of the manipulator or the voter applying strategy), but instead something else entirely. Moreover, the fact that the voting system results in a single winner instead of exposing the cycle means the winning result has a level of legitimacy (due to "winning") that is artificial and undeserved.

The voting methods that do detect Condorcet cycles, tend to differ in how they resolve the cycles. Nicolas de Condorcet, for whom the cycles are named, proposed a method producing one of two outcomes to any election. Either the method would find the single alternative that would win a majority decision against any other alternative (now called a Condorcet winner), or it would expose a cycle. Condorcet's suggestion when a cycle was found using his method was to decide the matter using some other method. In fact, there is now a whole family of voting systems (Condorcet methods) that return a Condorcet winner if there is one, and only differ in what they do when there is a cycle.

What the community did when faced with the Condorcet cycle is something entirely different, which was only possible because all the voters were still in one room. That novel approach to a known cycle was to allow Dave to propose D as a new location, and then see how well D performed against the three cycle members.

So, a new method to deal with a Condorcet cycle is to add to the choice set.

As mentioned earlier, there is a whole family of voting systems whose goal is to find and select a Condorcet winner if there is one, and to somehow make a choice when there is no clear Condorcet winner. None of those methods could have considered D because they are all essentially single round voting methods. They can only choose a winner from the initial set of proposals, and have no mechanism whatsoever to consider adding a proposal because they have no opportunity to do so.

A small town meeting does have that opportunity, which resulted in location D: a Condorcet winner in the set {A, B, C, D}. Yet that opportunity did not disappear when Dave made his proposal. It has been known since Condorcet's time that every finite set of proposals has either a Condorcet winner or a top cycle. (A top cycle is a Condorcet cycle with the added property that every proposal not part of the top cycle is defeated by at least one member of the top cycle. Technically, a Condorcet winner is a top cycle with a single member.)

Here is where Eve enters the discussion. While Dave's proposal is a three-way compromise over the first three proposals, Eve's proposal looks at the geometry and how the voter ideals have to be distributed in order to generate the ABC cycle. She realizes quite a few community members need to have ideal locations in the right of the diagram and proposes location E.

It turns out Eve was correct, and the Condorcet winner of {A, B, C, D} is not the Condorcet winner of {A, B, C, D, E}, which has E as its own Condorcet winner.

So, the method of adding to the choice set can also be used to improve upon a Condorcet winner.

Adding to the choice set is a good method to get better outcomes, assuming the electorate is small enough to fit in a room. But there is a limit to how good the outcomes can be. After location E had been identified as the best location so far, Frank proposed four more locations to bracket E.

The facts that E is the Condorcet winner of {A, B, C, D, E, F, G, H, I} and the underlying reasons for any votes is distance pretty much means that any added location proposal to defeat E would need to be fairly close to E.

This limitation on any large improvement over E makes it clear that E is close to the best possible location. Even though it might well be possible to do better, any new locations are not going to be all that much better.

Thus the method of adding to the choice set can have a natural end to it, once voters have evidence showing the limits of possible gains.

The final step in the small town meeting process is accepting the result. There is evidence that E is the best proposal so far, and all proposed locations since E were unsuccessful at improving upon it. So the community knows this is its best proposed option.

The remaining question is: is this "best" option actually "good enough"? It is possible that if the range of the tower were something like 10km instead of 25km, the answer would have been no.

But as it was, location E was accepted and the town meeting adjourned.