It’s hard not to think about the upcoming election looming just three weeks away. I spend hours each day, talking about it in history class, refreshing 538, the best election blog, or just checking out Donald Trump’s twitter feed. But, instead viewing this as a distraction, I remembered an intriguing way to tie game theory into the upcoming election. So today, I’m going to take a break from the Nash equilibrium to introduce the Median Voter Theorem.
The Median Voter Theorem is a game that shows the importance of the moderate position on the outcome of elections between two candidates. This theorem illuminates a candidate’s incentive to move toward the middle during a general election. Unlike an actual election, this model uses a one-dimensional political spectrum. The various complex stances have been stripped away and boiled down to whether a candidate leans more liberal or more conservative. This can then be represented on a number line going from one to ten (pictured below).
The left side, closer to one, represents a more liberal position, and the right side, closer to ten, represents a more conservative position. For this example, we assume that the opinions of the voters are equally distributed along the spectrum with each number representing 10% of the group as a whole. So, if one candidate runs with a position at three (meaning a moderately liberal position), and the other candidate runs at four (a slightly more conservative position), the first candidate would get all of the voters in sections one through three. That totals thirty percent of the vote. If there is a tie, then the votes will split fifty-fifty. So, if one candidate runs with a position of three, and the other candidate runs at five, that player would get all of the voters in sections one through three, and half of the voters at four. That totals thirty five percent of the vote. The goal of the game is for each candidate to maximize their share of the vote. One important detail in this model is that candidates can choose where on the political spectrum their position will be. The Median Voter Theorem is meant to logically show that a candidate closer to the majority opinion will be favored.
Solving this game is a great example of dominated strategies. For those who forgot, a strictly dominated strategy is any strategy where there is another strategy that will always have a higher payoff, no matter the actions of others. To solve this example, we will also use iterative deletion, which is eliminating the dominated strategies from the game one by one. This process will leave only the best possible choices for each player to make.
When checking for a dominated strategy you will need to choose two different strategies to compare. Let’s start with the most extreme positions available first, since those will be easier to check for a dominating strategy. First, we will compare the strategies of one and two for player one. If player two chooses one, and player one also chooses one, then the votes will be split, so each player gets fifty percent. If player two chooses one, and player one chooses two, then player one will get all of the votes from two to ten. That would give player one ninety percent of the vote. We can continue to repeat this process for all of the choices available for player two. Below is a table that compares player one’s choice of one or two with the first five options of player two.
|Player one chooses one||Player one chooses two|
|Player two chooses one||Payoff of 50%||Payoff of 90%|
|Player two chooses two||Payoff of 10%||Payoff of 50%|
|Player two chooses three||Payoff of 15%||Payoff of 20%|
|Player two chooses four||Payoff of 20%||Payoff of 25%|
|Player two chooses five||Payoff of 25%||Payoff of 30%|
The first two rows are a bit unusual because the votes are split between the players. But after that section, a pattern emerges, in which choosing two will always result in five percent more of the vote than choosing one. This proves that one is strictly dominated by two. Since this is a symmetrical game, then ten will also be a dominated strategy, since it will have identical payoffs to one. A rational player has no reason to ever choose a strictly dominated strategy, since there exists an option that always has a better outcome. Therefore, we can remove choices one and ten entirely from the game, and it will look like this.
Now we can go through the same process again, looking for a strictly dominated strategy within this new game. This time, let’s compare player one’s choices of two and three when player two chooses two. If player one chooses two, then the votes will be split equally again. If player one chooses three, then they will get all of the voters from three to ten. That would give player one eighty percent of the vote. Once again we will repeat the process for all of the choices of player two. Here is the table with player two’s first five choices.
|Player one chooses two||Player one chooses three|
|Player two chooses two||Payoff of 50%||Payoff of 80%|
|Player two chooses three||Payoff of 20%||Payoff of 50%|
|Player two chooses four||Payoff of 25%||Payoff of 30%|
|Player two chooses five||Payoff of 30%||Payoff of 35%|
|Player two chooses six||Payoff of 35%||Payoff of 40%|
Again, it is clear that two will be strictly dominated by three. So once again we can eliminate the dominated strategies, two and nine. This iterative deletion will repeat until all that is left is two options, five and six. Reducing the game to what is below.
While it may have been intuitive when you first saw the game that the answer was in the center, game theory provides clear, logical steps to show exactly what that happens. By eliminating every irrational decision, we have found the two best options. Choosing five or six guarantees that no matter what the other player does, you will get at least fifty percent of the vote. And while this model does work off a few oversimplifications, at its core, it shows the power a moderate candidate can have. And when compared to this current election, it illuminates one reason Clinton, a moderate left candidate, is polling better than Trump, a more extreme right candidate.
I hope you learned something today, and thanks for reading. Don’t forget to vote on election day.
For anyone who wants to follow their own interest in game theory, here are a few resources: