# Median Voter Theorem

In the real world, making political decisions are important. Even the small decisions of our governments affect our lives with an immense impact. But if aggregating our preferences is hard, as mentioned before, how does a politician decide which policy to select? How can she guarantee the votes in the next voting? Let's take a look at one prominent solution to this complex problem, the median voter theorem.

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## Median Voter Theorem Definition

What is the definition of the median voter theorem?

The median voter theorem suggests that the median voter decides which policy to select from a set of preferences in a majority-rule voting system.

According to Duncan Black, within majority-rule voting systems, the results of the voting will depend on the preferences of the median voter.

To get a better grasp of the suggestion, first, we should define what the median voter is.

Let's draw a line that contains the preferences of people about a hypothetical topic. In Figure 1 below, the x-axis denotes such a line. It contains the possible policy preferences about a hypothetical topic. Now, let's say there is an agent -- a voter. We can denote how much utility she gains from a preference with the y-axis.

For example, if she chooses the policy $$P_2$$, her benefit will be equal to $$u_2$$. Since the utility of the agent from the first policy, $$u_1$$, is less than the utility of the agent gets from the second policy, $$u_2$$, the agent will prefer the second policy, $$P_2$$, over the first policy, $$P_1$$.

Fig. 1 - Utility Levels of X with Respect to Different Policies.

Nonetheless, in a society, there exist many agents with different preferences. Let's say that there are now five agents in the society $$x_1,x_2,x_3,x_4,x_5$$. We can denote their utility curves with $$u_{x_1},u_{x_2},u_{x_3},u_{x_4},u_{x_5}$$. Figure 2 below shows the combination of agents in a society. Our previous agent x can be denoted with $$x_1$$ and her utility curve will be $$u_{x_1}$$. Similar to the previous setup, we can denote the utilities of agents with the y-axis and policies with the x-axis.

Fig. 2 - Utility Levels of Society with Respect to Different Policies.

Since they are seeking the highest utility from different policies, every agent wants to maximize her utility. For example, for agent $$x_1$$, the highest utility can be gained from the first policy, which is denoted with $$P_1$$. You can see that at point $$A_1$$, the utility curve $$u_{x_1}$$ reaches its local maximum. We can take a step further and denote every agent's maximum utility with $$A_1,A_2,A_3,A_4,A_5$$ respectively.

In this scenario, the median voter is $$x_3$$. Voters $$x_1$$ and $$x_2$$ will lose utility as they move toward the third policy,$$P_3$$. Similarly, voters $$x_4$$ and $$x_5$$ will suffer as they move in the opposite direction toward the third policy. Policymakers will select the third policy for getting the highest amount of votes due to the fact that with the third policy, the combined utility of the society will be higher than with any other policy.

## Median Voter Theorem Proof

We can prove the median voter theorem with two methods. One method is logical, and the other method is mathematical. The median voter theorem can be proven from two perspectives. One is from the point of view of the voters, and the second one is from the point of view of the policymakers. Both proofs depend on the information about the other group. Here, we will focus on proof from the perspective of policymakers. Both approach follow the same rules. Thus, it is easy to grasp the other one if someone knows any of them. Now let's go over the logical proof and mathematical proof.

Let's say that a party can select five policies. This party contains a group of data analysts that surveyed the five voters, and from their answers, data analysts learned the preferences of the voters. Since the party wants to gain the maximum amount of votes, this party sets its agenda with respect to the voters. If the party selects the first policy, $$P_1$$, the fourth and the fifth agent, $$x_4,x_5$$, will not vote for the party since their utility at $$P_1$$ is zero. Similarly, for the policy $$P_2$$, the fourth agent will gain the utility $$u_1$$, and the fifth agent will still get zero utility. In the graph below, we can see the utilities of the fourth and the fifth agent.

Fig. 3 - The Utility Curves of the Fourth and the Fifth Agent.

We can imagine a similar scenario for the first and the second agent. Since the party wants to gain as many voters as it can, it will select the third policy for the interest of all. Thus, the preference of the median voter sets the agenda.

Although logical proof is enough, we can prove the median voter theorem from the political party perspective with a mathematical approach too.

We can define a society with the set $$S$$ that contains $$n$$ elements:

$$S = \{x_1,x_2...,x_{n-1},x_n\}$$

We can denote all possible policies with the set $$P$$:

$$P = \{P_1,P_2...,P_{n-1},P_n\}$$

And there exists a utility function $$u_\alpha$$ with the shape above that maps the level of utility of an agent from a policy for every element of the set $$S$$. We can denote this with the following:

∃$$u_\alpha(P_i)\ | \alpha \in S\ \land P_i \in P$$

And finally, we can denote the combined utility of the society from a policy with the function $$g(P_i)$$.

$$g(P_i) = \sum_{\alpha = 1}^nu_\alpha(P_i)$$

Since the party wants to maximize the utility of society to get the highest possible votes, the party has to maximize the function $$g$$.

Now let's denote a policy, $$P_\delta$$:

$$g(P_\delta) > g(P_i) | \forall P_i \in P$$

Since $$g$$ is a quadratic function that can be generalized as:

$$g(x) = -ax^2 + bx + c | a \in R^+ \land g(x) > 0$$

$$g^{''}(x) < 0$$, $$b^2 - 4ac > 0$$

It must have one vertical symmetry line that intersects with the point where the function reaches its maximum value:

$$g^{'}(P_\delta) = 0 \iff g(P_\delta) = g_{max}$$

Thus, $$P_\delta$$ can only be the policy in the middle that maximizes the total utility of the society.

## Median Voter Theorem Examples

Now, for the application of the median voter theorem, let's look at a real-life example to apply the median voter theorem. Let's say that you are going to elect a governor for your state. Nonetheless, there are two competitors. The first candidate is Mr. Anderson, and the second candidate is Mrs. Williams.

Nonetheless, the only debate that can be a tie-breaker is on the tax rate for constructing a state-funded swimming pool. There are 5 groups in society with respect to the amounts that they are willing to pay. The swimming pool will be designed and constructed with respect to the amount of money. Now let's check the tax rates and what the state can construct with that tax rate.

 Tax Rate Specifications of the Construction 2% Standard swimming pool with no extra functions. 4% Standard swimming pool with extra functions like a cafeteria and a gym. 6% Olympic-sized swimming pool with no extra functions. 8% Olympic-sized swimming pool with extra functions like a cafeteria and a gym. 10% Olympic-sized swimming pool with extra functions like a cafeteria and a gym, a sauna room, and a massage service.

Table 1 - Required Tax Rates for a State-Funded Swimming Pool.

Let's place our costs on the x-axis and utility from them on the y-axis.

Fig. 4 - Tax Rates and Utility Axes.

Mrs. Williams is aware that this swimming pool will be a tie-breaker. Thus, she decides to work with a data science company. The data science company conducts a survey to learn about public preferences. They share the results as follows.

Society is divided into five equal sections. One section, $$\delta_1$$, does contain citizens who don't want a swimming pool. But for the sake of society, they are willing to pay 2% since they believe if they are living in a happy society, they will be happier. Another section, $$\delta_2$$, contains agents who are willing to pay a little bit more tax, 4%, for the state-funded swimming pool. Nonetheless, since they don't think they will go there often, they don't want to invest in it that much. Furthermore, they believe that there should be a cafeteria and a gym. They don't care about the size of the swimming pool.

One section, $$\delta_3$$, contains agents who want a large-sized swimming pool. They don't need extra functions that much. So they will gain the most from the 6% tax rate. One separate section, $$\delta_4$$, wants to invest in swimming more than the previous groups. They want a large-sized swimming pool with a gym and a cafeteria. They think that 8% is the optimum tax rate. And the last section, $$\delta_5$$, wants the best pool possible. They believe that a sauna is necessary to let loose a bit and relax. Thus, they believe a 10% tax rate is acceptable and beneficial.

The company shared the following utility curves applied to our previous graph.

Fig. 5 - Utility Functions of the Sections of Society.

Now, since Mrs. Williams wants to win the election, she analyses the tax rate that will get the most votes. If she selects the 2% tax rate, then 2 sections, the fourth and the fifth will not vote for her since their utility is zero. If she selects the 4% tax rate, then one section will not vote for her. Similarly, if she selects the 10% tax rate, then the first and the second group will not vote for her since their utility is zero. If she selects the 8% tax rate, then she will lose votes that are coming from the first group. Without hesitation, she selects the median tax rate for the swimming pool.

We can be sure that if the number of preferences is odd before the swimming pool tax rate selection and if Mr. Anderson decides to select any other tax rate rather than 6%, Mrs. Williams will win this election!

## Limitations of Median Voter Theorem

You might have guessed it: there are limitations of the median voter theorem. If winning elections can be so easy, what are the purposes of election campaigns? Why don't parties just focus on the median voter?

These are rather good questions. The following conditions should be met for the median voter theorem to work.

• The preferences of the voters must be single-peaked.

• The median voter must exist, meaning that the total number of groups should be odd (This can be solved with additional methods but not without the necessary tools).

• A Condorcet winner shouldn't exist.

Single-peaked preferences mean that curves must have one positive point with its derivative equal to zero. We demonstrate a multi-peaked utility curve in Figure 6 below.

Fig. 6 - A Multi-Peaked Function.

As you can see in Figure 6, the derivative at $$x_1$$ and $$x_2$$ are both zero. Therefore, the first condition is violated. Regarding the two other conditions, it is trivial that median voter should exist. And finally, a Condorcet Winner preference shouldn't exist. This means that in pairwise comparison, one preference shouldn't win in every comparison.

Not sure what a Condorcet winner is? We have covered it in detail. Don't hesitate to check out our explanation: Condorcet Paradox.

## Median Voter Theorem Criticism

In real life, voting behavior is extremely complex. Most of the time, voters have multi-peaked preferences. Furthermore, instead of a two-dimensional space, preferences are the combined results of many policies. Furthermore, the information flow is not as fluent as in the theorem, and there may be a lack of information on both sides. These can make it really hard to know who is the median voter and what the median voter's preference will be.

Interested in how to apply economics methods to the study of politics? Check out the following explanations:

- Political Economy

- Arrow's Impossibility Theorem

## Median Voter Theorem - Key takeaways

• The median voter theorem is a part of the social choice theory proposed by Duncan Black.
• The median voter theorem suggests that the median voter's preference will set the agenda.
• A Condorcet winner will prevent the median voter's existence.

#### Flashcards in Median Voter Theorem 11

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What is the median voter theorem?

Median Voter Theorem suggests that the median voter decides which policy to select from a set of preferences in a majority-rule voting system.

What is an example of median voter theorem?

Any scenario that includes a median voter without a condorcet winner and multi-peaked preferences can be an example of median voter theorem. In this kind of scenario, the median voter's preferred policy will be chosen.

Is median voter theorem true?

In some scenarios, yes, it holds. Nonetheless, it is extremely hard to analyze real-life scenarios because the theorem's assumptions usually don't hold in real life.

What are the limitations of median voter theorem?

In real life, voting behavior is extremely complex. Most of the time, voters have multi-peaked preferences. Instead of a two-dimensional space, preferences are the combined results of many policies.

Furthermore, the information flow is not as fluent as in the theorem, and there may be a lack of information on both sides. These can make it really hard to know who is the median voter and what the median voter's preference will be.

What are the median voter theorem assumptions?

• The preferences of the voters must be single-peaked.

• The median voter must exist, meaning that the total number of groups should be odd (This can be solved with additional methods but not without the necessary tools).

• Condorcet winner shouldn't exist.

## Test your knowledge with multiple choice flashcards

Existence of a Condorcet Winner will prevent the existence of a median voter.

Which of the following is not an assumption of the median voter theorem?

What methods can we use to prove the median voter theorem?

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