What is Arrow's Impossibility Theorem?

Alright, let's discuss the rather profound-sounding "Arrow's Impossibility Theorem" in plain language.

Arrow's Impossibility Theorem

Imagine you and a few friends need to decide what to eat for dinner. You have three options: pizza, burgers, hotpot.

You want to find an absolutely "fair" voting method to make this decision, one that perfectly reflects everyone's preferences. Sounds simple, right?

However, in 1951, economist Kenneth Arrow proved that this is fundamentally impossible.

The core idea of Arrow's Impossibility Theorem is this: when there are three or more options, no voting system can simultaneously satisfy all the conditions we consider "fair." You must abandon at least one of them.

It's like saying you can't build a car that's as fast as a sports car, as fuel-efficient as an electric scooter, and as cheap as a bicycle. You have to make trade-offs between performance, cost, and energy consumption. Voting systems are the same.

What are these "fair" conditions we want?

Arrow proposed several "good rules" that seem self-evident, which any ideal voting system should satisfy. I'll explain the three most important ones in the simplest way possible:

1. No Dictator

• In plain language: There can't be one person who always gets their way. The voting outcome cannot always reflect only one person's opinion while completely ignoring everyone else's.
• Example: If no matter how everyone votes, the outcome always matches whatever John wants to eat, then this voting system is a dictatorial one and unfair.

2. Unanimity / Pareto Efficiency

• In plain language: If everyone prefers A over B, then the final voting outcome must rank A above B.
• Example: If all of you believe "pizza is better than burgers," then in the final ranking, "pizza" must be ranked higher than "burgers." No one should object to that, right?

3. Independence of Irrelevant Alternatives (IIA)

• In plain language: Our preference between A and B should not be affected by an irrelevant option C.
• This is the most subtle and critical condition, so let me give an example:
  • Let's say in a vote between "pizza" and "burgers," the result is pizza wins.
  • Then, I suddenly add another option: "hotpot."
  • Logically, the addition of "hotpot" shouldn't change the fact that "pizza is more popular than burgers." If, because of the "hotpot," the voting result instead becomes burgers win, then there's a problem with this voting system.
  • This is like a spoiler. The appearance of a "spoiler" option, which no one originally intended to vote for, somehow overturns the outcome between the two original popular choices. This is clearly unreasonable.

Where exactly does the "impossibility" lie?

Arrow rigorously proved with mathematical logic that:

As long as you have more than two options, you cannot design a voting rule that simultaneously satisfies these three basic criteria (and a few other more technical ones). You must sacrifice at least one.

• Do you want to satisfy unanimity and independence of irrelevant alternatives? Then a "dictator" is likely to emerge.
• Do you want to satisfy no dictator and unanimity? Then your system might be disrupted by "irrelevant alternatives."

Let's go back to our initial example to see why problems arise.

Suppose three people (A, B, C) have the following preferences:

• A: Pizza > Burgers > Hotpot
• B: Burgers > Hotpot > Pizza
• C: Hotpot > Pizza > Burgers

Let's test this:

• Pizza vs. Burgers: A and C choose pizza, pizza wins. (2 vs 1)
• Burgers vs. Hotpot: A and B choose burgers, burgers wins. (2 vs 1)
• Hotpot vs. Pizza: B and C choose hotpot, hotpot wins. (2 vs 1)

You see, this creates a cyclic preference: Pizza > Burgers > Hotpot > Pizza... This is called the "voting paradox" or "Condorcet paradox," and it is a vivid illustration of Arrow's Impossibility Theorem. We cannot derive a clear "optimal solution" using a fair rule.

What does this mean for us?

You might feel a bit disheartened, thinking this implies that democracy, voting, and such all have inherent flaws, right?

Yes, but it's not entirely a bad thing. The significance of Arrow's Impossibility Theorem lies in:

1. Reminds us there's no perfect system: It tells us not to fantasize about finding a "one-size-fits-all" perfect voting system. Every system has its weaknesses and scenarios where it might produce unreasonable outcomes.
2. Forces us to make trade-offs: We must clearly recognize which "fairness" principle we value more and which we are willing to sacrifice when designing an an election or decision-making system.
   • For example, the US presidential election (winner-take-all) is simple, but often heavily influenced by "spoilers" (irrelevant alternatives).
   • In contrast, some more complex voting systems (like ranked-choice voting) attempt to reduce the "spoiler" effect but might make sacrifices in other areas (e.g., computational complexity).
3. Explains real-world complexities: It mathematically explains why many committee decisions and parliamentary votes often lead to deadlocks or seemingly strange outcomes. This isn't necessarily due to manipulation but rather the inherent logical dilemmas within the rules themselves.

In summary, Arrow's Impossibility Theorem is like Gödel's Incompleteness Theorems in social choice theory; it draws a boundary, showing us the theoretical limits. It doesn't say that voting is useless but rather reminds us that the path to achieving "fairness" and "public will" is always fraught with compromises and trade-offs.