Game Theory in Elections
The Winning Strategy
Is it possible that election results are manipulated?
“The ballot is stronger than the bullet” — Abraham Lincoln
Ever voted in the legislative assembly’s election? Or the election for the class representative in your college? The concept of voting is not limited to this era, history suggests that in the ancient Vedic period, the rajas (chiefs) were elected by the people of the gana (tribal organizations). Elections were also observed in the ancient and medieval Roman and Greek Empires. In the modern world, elections serve as the backbone of any democracy.
The piece began with thought-provoking words by Abraham Lincoln on the strength of elections. But are these elections always fair? Are the outcomes of these elections the true will of the people? Can a candidate strategize and win the election even if they were not the choice of the majority?
Let’s see how game theory can help us in finding the answers to these compelling questions. For this purpose, we would be discussing an enthralling paradox called the Arrow’s Paradox. Let’s start with the origin and statement of this paradox.
In 1951, an economics student, Kenneth Arrow (who went on to become a Nobel prize-winning economist), gave a theorem in his PhD thesis. The Arrow’s Impossibility Theorem, better known as the Arrow’s Paradox, states that:
“A clear order of preferences cannot be determined while adhering to mandatory principles of fair voting procedures.”
What are these fair voting procedures? Let’s see a few of these.
- Independence from irrelevant alternatives: If a candidate is preferred over another candidate, this order of preference should not change even if another candidate is added or withdrawn from the elections.
To understand this better, let’s take an example of a race where there are three participants.
In this example, we can note that even if Runner 2 stops in between, Runner 1 (assuming the other runners would maintain their respective speed) is going to win the race. Hence, the presence of Runner 2 does not affect the results, which makes this contest a fair one.
- Non-Dictatorship: All votes should be equal and no voter should be given preference over others.
For instance, you must have heard about the debate of giving preference to the votes of educated people. In this voting system, an educated individual’s vote will be equal to an uneducated person's multiple times. People for the motion would say that educated people better understand governance and elect a better government. Those against would say that this would give more powers in the hands of a particular class and they will act as dictators. This type of voting is unfair according to the paradox as it invokes authoritarianism in the system.
In other words, according to the paradox, it is impossible to decide a clear winner if all the fair voting procedures are followed. This means according to Arrow’s Theorem, the current voting system (FPTP system, in the case of this article) is not completely fair. It doesn’t necessarily express the will of the majority. Unbelievable, right? You might be thinking about how this can be possible. Well, that’s what makes it a paradox.
Let’s consider a scenario.
There is an election for choosing the President of a political party. Let us assume that 100 people have got a vote and every vote is equal. The elections will be conducted using the First-Past-The-Post (FPTP) voting system, in which the candidate with the simple majority wins the election. This type of voting system is used in democracies like India and the US. For simplicity, we will consider that every person votes independently and there is no lobbying for votes. So, there are 3 candidates, A, B, C, who desire to contest in the elections. There are people with different preference orders, all such cases (a total of six) are shown in the table below.
In the last column of the illustration, we see that the number of voters who preferred A over B over C are 20 and the number of voters who preferred A over C over B are 20 again. If we sum these up the total votes for candidate A are 40. Similarly, we calculate the total number of votes for B and C which comes out to be 35 and 25 votes respectively.
If we assume that this system of voting is completely fair and this really expresses the opinion of the people, we can see that A is clearly the choice of the voters and C has been completely rejected. But if this is true, removing any candidate from the election should not affect the results. For example, if B would have withdrawn from the election, the results should have not changed. Will that happen? Let’s check.
If we analyze the same table now, we see that A has 45 votes, on the other hand, C has 55 votes. This makes C clearly the winner. Withdrawal of a candidate changed the winner, this means that A was not really the choice of the majority, most people preferred C over A. Due to the presence of B, people preferred B over C which led to the reduction of C’s votes and eventually to the victory of A. Ideally the addition or withdrawal of a candidate should not have mattered. This proves that the system of voting is not fair and can be influenced by a well-thought strategy by any candidate.
If we extend the situation to real elections with many more voters and candidates, we see that to win you don’t necessarily have to be the choice of the majority, just placing a few candidates who can reduce the votes of the candidate stronger than you might secure your win! Possibly a perfect strategy to defy (or manipulate?) the will of the majority.
We see a lot of real-life examples of candidates using these strategies to manipulate election results. This puts the voter in a bit of a dilemma. Suppose there are two strong candidates in an election, and one of them is using this strategy. Now as a voter, should we vote for the candidate which we feel is the best (whose chances of winning are bleak), or should we think strategically and vote for the best out of the two strong candidates?
So is it okay to speak the truth only when it doesn’t matter? It is safe to conclude yes in the matter of elections. Vote with your hearts when you expect the decision to be in your favour or you are not affected by the results. But if it’s a tie, it’s time to don your game-theoretical hat and vote strategically.
References
- Liberto, Daniel. “Arrow’s Impossibility Theorem Definition.” Investopedia. Available at: https://www.investopedia.com/terms/a/arrows-impossibility-theorem.asp.
- Morreau, Michael. “Arrow’s Theorem.” Stanford Encyclopedia of Philosophy. Available at: https://plato.stanford.edu/entries/arrows-theorem/.
- Dixit, Avinash K., and Barry J. Nalebuff. The Art of Strategy. W.W. NORTON & CO., 2008.
