The Median Voter Theorem has its origins in Harold Hotelling's spatial model of firm strategies in a competitive marketplace.
To help us understand the nuts and bolts of Hotelling's model, we will take the scenario where two parties,
in our case two ice cream vendors, compete to make the most profit
from sale of ice creams on a hypothetical neighborhood block.
Let us assume for simplicity that all the houses on this hypothetical block are equally spaced and
are of equal size with equal number of residents living in each house.
We will assume that these two ice cream vendors sell the same product at the same low price.
We will also assume that it is a hot sunny summer day and everyone on this hypothetical block
will buy ice cream from one of these two vendors.
Since both vendors sell the same product at the same price,
the residents' decision to buy ice cream from one of these two vendors
will depend on how close the ice cream truck is to their home.
The customers' decision rule will be to buy ice cream from the vendor who is closest to their home.
We will also assume that these ice cream vendors are mobile.
They can push the truck from one end of the street to the other.
They are also able to move their truck around the other as well.
The goal of both vendor is to make the most profit from the sale of their ice cream.
Let us assume that Joe will start his business by placing himself at the west end of the block.
Having no competitors, he has the advantage of having all the customers on that block coming to buy ice cream from his truck.
Now let us introduce Joe's competition, Ted, who places his ice cream business on the east side of the block.
The two vendors now split the customers down the middle.
Joe takes every customer from the center to the west end of the block while Ted takes every customer from the center to the east.
What is the winning strategy for one of these two vendors?
Holding all else constant, if Ted pushes his truck towards Joe, he can have some of
Joe's customers on the west side of the street to come and buy ice cream from Ted's truck.
Having said this, is there an optimal strategy for Ted if all else is held constant?
Clearly, the best strategy is for Ted to situate his truck right next to Joe so that all
the customers to the east side of Joe's truck will come to buy ice cream from Ted.
Now, let us loosen one of our conditions about Joe staying put.
If we allow Joe to react to Ted, is there something that Joe can do to better
his chances of winning this competition of ice cream sales against Ted?
The answer is “yesâ€? he can. Holding Ted's place constant, Joe can simply situate his cart to the right side of Ted.
Ted, on the other hand, can move his cart to the right side of Joe;
and Joe can do the same. They will continue. But will they ever stop? And when they do, where do they stop?
The answer is yes they will stop. They stop in the middle of this block.
The reason for this is that it serves neither Ted nor Joe any better to move away from the center of the block.
That is, their marginal gain in sale of ice cream from an incremental move away from the center of the block is less than zero.
They actually lose sales if either vendor moves away from the center of the block.
The actual theoretical basis for the convergence of these two players to the middle of the
street is based on a game theoretic concept of "iterative deletion of dominated strategies".
That is, for each successive deletion of the dominated strategies from either end of the street,
the strategy that leads the players to the middle of the street
allows the players to be better off no matter where the other player is positioned.
Now let's take a look at how the spatial model of strategic firm behavior
translates into party strategy in a competitive electoral democracy.
Assume now that instead of ice cream vendors, we have parties.
And instead of selling ice creams, each party espouses a particular ideology,
which can be mapped on a uni-dimensional ideological space running from extreme liberal to extreme conservative.
We will assume also that the voting population is equally distributed along this ideological plane.
We will also assume that each voter's decision to vote for either one of these two parties will
depend on how close each party's ideological stance is relative to the voter's ideology.
If the electoral rule is such that the party with the most number of votes is the winner and
we assume that voter participation rate is 100%, what is the party's winning strategy?
Similar to the ice cream vendors, each party's winning strategy will be to situate themselves
right next to each other at the middle of the ideological spectrum.
The middle will be referred to as the median.
We will also change one of the assumptions about the voting population without losing generality of our model.
Instead of assuming that the voting population is uniformly distributed across the ideological spectrum,
we will assume a normal distribution.
That is, the distribution of the voting population is single peaked and symmetric around the median.
This is not an unrealistic assumption to make since we are suggesting that there are
less ideological extremists than there are moderates.
The winning strategy still remains the same for each party. Each party will seek to seize the median voter.