The Limitations of Voting Systems
The Arrow Impossibility Theorem, named after Kenneth Arrow who won a Nobel Prize for his work in economics and political science, is a fundamental result in social choice theory that helps us understand the difficulties of aggregating individual preferences into a collective decision-making process. Its fascinating implications uses combinatorial math to understand the limitations of voting systems.
The theorem states that it is impossible for a voting system to satisfy a set of specific conditions that we would expect to see in a fair and democratic system. These conditions include things like…
Pareto Efficiency: if every individual prefers option A to option B, then the social choice function should prefer A to B (basically unanimity)
Independence of Irrelevant alternatives: the social choice function should be independent or irrelevant alternatives (basically, adding or removing an option shouldn't change the outcome of the vote)
Non-dictatorship: no individual should have too much power in the decision-making process
This theorem has important consequences for our understanding of the limitations of democratic decision-making and the trade-offs that are involved in collective decision-making. It shows that there is no perfect way to aggregate individual preferences into a collective decision, and that any social choice function will necessarily involve some form of compromise or trade-off. It also highlights the importance of transparency, accountability, and deliberation in the decision-making process, as these can help to mitigate some of the limitations of any given social choice function.
To understand the real-world implications of this theorem, let's take an example problem: imagine we have three candidates running for student council president: A, B, and C. There are ten voters in total, and we want to determine the winner using a ranked voting system (where each candidates from their most preferred to least preferred).
To solve this problem, we could use combinatorial math to create all possible voting scenarios and analyze them based on the Arrow Impossibility Theorem conditions. For example, we could create a table showing each voter's preferences and then use a mathematical algorithm to calculate the winner based on different voting systems (such as plurality voting, ranked choice voting, etc.).
However, we would quickly realize that no matter which voting system we use, it is impossible to satisfy all of the Arrow Impossibility Theorem conditions. For instance, if we use ranked choice voting, we may encounter a situation where one candidate wins in one scenario, but loses in another scenario, violating the independence of irrelevant alternatives condition.
This highlights a fundamental flaw in voting systems and shows why it is difficult to create a fair and democratic system that satisfies all of our expectations. The Arrow Impossibility Theorem, while producing such disappointing results, helps us make more informed decisions about how we structure our democratic systems and how we approach decision-making processes in our everyday lives.
