GAME THEORY IN ELECTIONS
The Winning Strategy: A Sequel
Why candidate most preferred by the Median Voter will be elected?
In the previous article, The Winning Strategy, we discussed how all the elections in the world are unfair and how do we conquer such contests. This time we are back with something fresh but equally astounding.
What if we tell you all the previous elections in the world could be explained by just one theorem? It is possible to predict the results of the next election using Game Theory. Don’t believe us, read ahead to find out.
The Median Voter Theorem is a proposition relating to voting based on the ranked preference set forth by Duncan Black in 1948. Two core assumptions underlie the Theorem :
- The candidates/parties may be placed along a one-dimensional political spectrum.
- Single peak preferences are prevalent among the voters, meaning voters have one alternative they prefer over the other.
The Median Voter Theorem in microeconomics and game theory states that:
“In a majority rule voting system, the candidate/party most preferred by the median voter will be elected.”
Since the median voters prefer party B, the candidate from party B wins.
Understanding the theorem through an application
The median voter theorem originates from Hotelling’s model of Spatial competition 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 sales of ice creams on a hypothetical neighbourhood block. Let us assume for simplicity that there are 10 houses and all the houses on this hypothetical block are equally spaced and are of equal size, with 10 residents living in each house. We will assume that these two ice cream vendors sell the same product at the same low price. Since both vendors sell the same product at the same price, the resident’s 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 customer’s 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 but they cannot cross each other assuming the street is single lane. They can push the truck from one end of the street to the other. The goal of both vendors is to make the most profit from the sale of their ice cream. Let us assume that the Pink vendor will start their business by placing himself at the west end of the block. Having no competitors, they have the advantage of having all the customers on that block coming to buy ice cream from their truck.
Now let us introduce the Pink vendor’s competition, Blue Vendor, who places their ice cream business right next to Pink Vendor on the east. The two vendors now split the customers in such a way that the Pink vendor can make 10% of the total sale since just one house of the block falls in its share while the Blue vendor makes 90% of the total sale as all the remaining residents will buy ice cream from the Blue vendor.
If the Blue vendor pushes their truck away from the Pink vendor in front of the 8th house, in this scenario the Pink vendor makes 15% of the total sale and the Blue vendor will be able to make 85% of the total sale.
Having said this, is there an optimal strategy for the Pink vendor to increase their sale if all else is held constant? Clearly, the best strategy is for him is to situate their truck in front of the 9th house right next to the Blue vendor so that all the customers west to their truck will buy ice cream from him and there will be an increase in their sale share from 15% to 20% and the Blue vendor will make 80% of the total sale.
Now let us loosen one of our conditions about the vendors crossing each other. If we allow the Pink vendor to cross the Blue vendor, is there something that the Pink vendor can do to better their chances of winning this competition of ice cream sales against the Blue vendor? The answer is yes, they can. Holding the Blue vendor’s place constant, the Pink vendor can simply situate their cart to the east side of the Blue vendor. The Blue vendor, on the other hand, can move their cart to the east side of the Pink vendor, and the Pink vendor 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 the Pink vendor nor the Blue vendor 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 theoretical 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 and “Nash Equilibrium” is obtained.
The Median Voter Theorem can be used to understand the General Elections 2019 in India. Narendra Modi, the candidate from BJP was right wing and the opposition, “Mahagathbandhan” was a political alliance of around 12 major political parties. Since the voters (including the Median Voter) were predominantly right leaning, BJP won with majority and formed the government.
There are some downsides to this theorem as well.
Voters are in reality not distributed evenly between liberals and conservatives, and sometimes the voting distribution is biased towards one side. Comparing the positions of more than two candidates becomes very complex, and dealing case-wise does not bring out the best outcome.
In reality, candidates are not able to deviate from the centrist political position since convincing people is a tough job.
This model exists in every single competitive marketplace where the position of all the players determines the outcome of the market. It explains why there are multiple restaurants in the same place rather than being spread out. If you have seen Burger King and McDonald’s or Pizza Hut and Domino’s Pizza across the streets’, now you know why. The neutral position is called the Nash Equilibrium, where neither player can deviate to gain more profits.
In conclusion, the Median Voter Theorem does a satisfactory job of explaining why and how the candidates choose their positions, and how the voters choose whom to vote for. So next time while voting or contesting, THINK ABOUT THE WINNING STRATEGY!
If you liked this article, we recommend reading the prequel to this one by Intellectually Yours — The Winning Strategy
Authors: Aditya Choubey and Saket Raj
References:
- Danaher, John. Game Theory (Part 4) — The Median Voter Theorem. Philosophical Disquisitions. Available at: https://philosophicaldisquisitions.blogspot.com/2011/05/game-theory-part-4-median-voter-theorem.html
- Talwalkar, Presh. Hotelling’s Game, or Why Gas Stations Have Competitors Nearby. Mind Your Decisions. Available at: https://mindyourdecisions.com/blog/2008/03/25/game-theory-tuesdays-hotelling%e2%80%99s-game-or-why-gas-stations-have-competitors-nearby/
- Veisdal, Jørgen. “The Median Voter Theorem”. Medium.
Available at: https://www.cantorsparadise.com/the-median-voter-theorem-c81630b57fa4 - Downs, Anthony. An Economic Theory of Democracy. Pearson (1997)
