The Margarita Model For Two-Party Systems

This is a fun thought experiment to trot out the next time someone tries to convince you that influencing voters can change party positions. Let's pretend there is only one political axis, and imagine voters are sitting at different places along a beach. Suppose there are two vendors of cool, refreshing margaritas serving this beach. Where will they be located?

The answer is that both will be at the median point on the beach, which is the point where exactly half the people are on either side. Why? Well, suppose that they are in some other location, with some gap in between the two vendors. Either vendor can always increase his sales by moving towards the other. He moves farther away from some people, but he's still their closest purveyor of icy refreshment - and he picks up some of those in the middle. So the vendors must be right next to each other. This location must be in the exact median, because otherwise the vendor with less customers always gains by hopping to the opposite side of his competitor. The only way for there to be no gain from moving is at this equilibrium position, with both in the center (and a tiny gap between them).

Unwrapping our metaphor, we can see why Republicans and Democrats are both so centrist. A move towards the middle picks up more voters, while still leaving the party as the best choice for the extreme it's moving away from. Even worse, the location of the center is determined by the median of voter positions - not the average. This means that if voters change their position without crossing the center, it has zero effect on the positions of the parties. The median location has not changed.

It is intuitive to think of party locations as being an average of constituent's views, which would mean that any voter move has some tiny effect on the position of the party. I think this is the implicit model which many people have. However, the logic of the situation dictates otherwise. Major parties do not choose positions at the center of mass of their voters, they end up well towards the center.

We already knew that voting loses information - its inevitable when you are collapsing many opinions into a single result. This is just another example. However, there is a more optimistic way to look at this. It suggests that whichever party wins, it will have a central position. This means that the electorate as a whole is always getting a result close to the center, rather than reflecting the average belief of the winning party's constitutents. This probably makes for a more stable democracy - although one in which minority viewpoints are utterly disenfranchised.


*) I haven't addressed why there are only two vendors. It has to do with it being a winner-take-all election, which means if there are 3 parties with a chance of winning, its always in the interests of two of them to join forces.
*) If we use multiple political dimensions, the point still holds, but more weakly. A voter who moves without crossing a median line still does not affect the position of the center. But with a line for each dimension, it becomes more likely that a move will cross one.
*) Actual parties have more separation than this model suggests. One likely reason is that there is value in being able to tell the parties apart, which requires some distance. However, note that this is likely to be more reflected in rhetoric than policy.
*) The beach model is not mine, I have no idea who first came up with it.

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Hmm. There are other cases

Hmm. There are other cases though. For example, if the people on the "beach" are not evenly distributed, but clustered strongly to one end and the other, the "vendors" will move away from the middle and towards the clusters. They don't dare sit in the middle in such cases for fear of a vote-splitter setting up shop directly inside the cluster.

The original source for this

The original source for this is the economist Harold Hotelling who developed what is now known as the Hotelling line model in, I believe, the 1920s. For example:

I think credit for the model

I think credit for the model goes to Hotelling. At least I've always heard it called the Hotelling Model. It's a little more complicated in presidential races because of the electoral college.

The other interesting case

The other interesting case that requires a more comlex analysis is if there can be more than two vendors. In that case, the third vendor could either set up shop near the other two, or could move a little ways down the beach in either direction. There's a reasonable incentive to not be in the central cluster, but instead to stake out a position a little ways down the beach.

If there's no expectation of a fourth entrant, and if it's expensive for the parties that already have a spot to move, then the third vendor wants to be only a short distance out from the center. That way he picks up most of the buyers on that end of the beach easily, and gets some of the people in the middle. Then the two who started in the middle end up leapfrogging out from the center, trying to pick up all the drinkers on the other wing.

The implication for partisian politics are more in the direction of institution design than strategies for politicians. The american system has some inherent features that lead it to be a two-party system, and the incumbents of all stripes have worked to make those tendencies stronger. It's unlikely that any changes will be effective in the here-and-now. But theorists (or those setting up smaller organizations) can think about what kind of institution designs will lead to stable multi-party interactions and whether those will better represent the voters.

BTW, I've been in favor of transferable votes for a while as well. It would seem that in smaller polities (cities, e.g.) it ought to be possible to automatically track voters' assignment of a proxy over time, and have city councils include rotating memberships as the issues caused the voting blocks to change. Companies' shares are usually voted by proxy, but most have made sure that little of substance comes before the public for consideration, so it doesn't really matter.

I agree that the potential

I agree that the potential for a new entrant provides additional incentive for the parties to separate a little. However, we must remember that there is an important difference between this model and an election, in that the election is winner take all. With the beach vendor, we can just view each as trying to maximize market share. With an election, they are actually trying to maximize the chance they win an election.

This has a profound effect on the multi-party system. On the beach, a new vendor can come in and get 1/3 of the business. In politics, a minority party gets nothing. And even if there is a potential location where it would get a majority (say a little towards the extreme from one of the main parties central position), there is a huge barrier to entry. It takes time to get the word out. So a new entrant will probably get nothing.

Thanks for the additional analysis, Chris.

The thought experiment only

The thought experiment only works if the people are willing to walk however far it takes to reach the closest vendor. If instead it's the case that people more than a certain distance beyond the vendor just do without, it's not necessarily best to locate at the center anymore.

While your analysis would be

While your analysis would be generally correct if (to continue the beach anology) the vendors were fairly close to all the beachgoers, there is another effect that comes into play as the beach goers become significantly spread out. No one will walk ten miles on a hot beach for a margarita. Thus one can imagine that at some distance from the vendors customers will stop going. Taking this into account the equilibrium position will have the two vendors seperated by a distance, since any move towards the center by the left vendor will lose more him more customers on the far left because of distance than he will gain by stealing customers from the right vendor (and vice-versa).
Returning from the beach analogy to the political reality, when either of the parties move toward they lose voters on their extreme (either to fringe parties or to non voting). If this effect is signifcant then there will be a noticeable distance between the parties. In fact, since voters in the extreme are likely to be highly attentive and sensitive to their parties platform it might not be unreasonable to expect a significant effect. And indeed there is always talk of parties playing to their respective bases and there is a noticeable distinction between the parties platforms (although how large distinction is up for debate and derision).
Thus, it seems that party members may affect their parties idealogy, pulling it away from the center by maximising the aforementioned effect. If a group of extreme voters (by which I which mean further distant from the center than their party, not attaching any moral judgement) wishes to pull their party towards them they must simply have it be known that they will not vote for their party if it is signifcantly distant from their position. In reality this generally becomes apparent after a lost election or two
It appears to me at least that this tactic can and does have a significant effect.
There are likely several other effects that you could come up with. Playing with some back-of-the-envelope quantitative models (not apparently your favorite thing in the world, but I am a Physics major, so humor me) the net result leads to some interesting situations which deserve further thought.
In sum, the model you have posited can, when augmented with a relatively resonable and apparent distancing effect, present a mechanism by which party members can in fact exert signficant control over their parties platform, a mechanism which does appear, to me at least, to be borne out in reality.

I had not noticed when

I had not noticed when writing the previous that Stormy Dragon had made a similar point - and more concisely at that. No plaigarism intended.

"On the beach, a new vendor

"On the beach, a new vendor can come in and get 1/3 of the business. In politics, a minority party gets nothing" - one of the main reasons why force-backed government is worse than free markets.

Although, even in party politics, it isn't entirely true. There's a nice living to be made running a minor party off to one extreme side or the other. You can pick up all the wackos that no center party cares about ('cause they'll never cross the median). Hence Nader.

Dragon - you are absolutely

Dragon - you are absolutely right. But I still think that this Hotelling effect reduces the degree to which a party reflects the center of mass of its constituents.

Julian - That is definitely true if one is going for something other than winning elections. Which is clearly true of the Greens and the LP, but outside the scope of this model. And yes, part of the point of this type of argument is to gently introduce people to the idea that a non-winner-take-all system might be better...before they realize that that way lies anarchy :twisted:.