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This lesson reviews Arrow’s Impossibility Theorem, which states that there is no preferential voting method that adheres to reasonable fairness principles.

An example is used to illustrate his theorem.

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Arrow’s Impossibility Theorem

When you think of the impossible, what do you think of? Many things are just impossible. You can’t have windows that open on a submarine (well, at least not that open when the sub is underwater). Years ago, man thought that human flight was impossible, and it certainly is impossible to actually ride on a bird’s back and fly as they fly. But what else is impossible?What would you say if I told you that a fair voting system is impossible? Would you agree? That is exactly what a researcher named Kenneth Arrow set out to show with his Arrow’s Impossibility Theorem.

Also called ‘Arrow’s Paradox’, the theorem states that it is not possible to obtain a preferential result in an election while also adhering to principles of fair voting.

Brief Explanation of Preferential Voting

Wow, so this guy really thought that there was no fair way to vote? Not exactly, he showed there was no ideal means of preferential voting. Preferential voting is a method of voting (which Arrow called a social welfare function) that allows voters to rank each candidate in order of preference instead of just choosing the most preferred candidate.

For example, if you were having a dinner party, you might want to serve dessert. Instead of just informing your guests what you will serve, maybe you want to be democratic and let everyone vote on it. If the choices are cookies, cake or ice cream, you might have your guests submit their choices in order. One guest might respond with, ‘I like cake, then ice cream, and last cookies.’ That is one vote; each vote is listed in order of preference.Here’s a chart showing the results from all nine guests’ preferences:

example chart of data

It shows that four of your guests prefer cookies over ice cream over cake.

Three would rather have ice cream than cake or cookies. And only two would prefer cake, but would want ice cream before cookies.Arrow might call this chart the ‘Will of the People’, and he would believe that the voting method used to obtain it should have at least three reasonable properties.

Properties of Fair Voting

Arrow said that it would be reasonable to assume that a fair voting method has:

  • No Dictators, meaning that no individual person’s ranking should determine the rankings outcome each time.

    Basically, no single person should have the power to sway the vote every time. This doesn’t mean that a vote can’t be decided by ONE ballot, but it just can’t be the same individual person’s ballot that decides it each time. That would be a dictator.

  • Pareto Efficiency, meaning if voters prefer Option 1 to Option 2, then the outcome should show Option 1 ranked highest.
  • Independence of Irrelevant Alternatives, meaning if Option 1 is ranked higher than Option 2, removing Option 3 should not alter the relative rankings of Options 1 and 2. This property relates to the removal of the losing option. It makes sense that if you remove an option that has already lost, the results for the other options should not change.

Amazingly, Arrow’s Impossibility Theorem postulates that when there are three or more options, there is no preferential voting method that can satisfy all three of these reasonable fair voting assumptions.

Example of Theorem at Work

Let’s return to the desired desserts from the dinner party you are planning.

In the results, cookies have received the highest number of first place votes, so cookies would win, right?But, wait! Look at those numbers again. Only four people would actually prefer cookies over everything else. The other five (the actual majority in this case) would prefer ANYTHING to cookies. So, is it fair that cookies will win when the majority of people actually don’t want them?This violates the Pareto Efficiency property because an option other than cookies is preferred, but cookies still wins.Now, what if you looked in your cabinet and found that you were all out of cake mix? Uh oh, you have to remove that option from the list. What happens? It shouldn’t matter because cake was the least preferred option anyway (it was the loser).Let’s have a vote again to find out.

Our new results show that five guests prefer ice cream over cookies and four prefer cookies over ice cream. Wow, look at that; it did matter. By removing the losing option, you actually change the results so that ice cream now has the most first place votes.This violates the Independence of Irrelevant Alternatives property because removing the loser changed the overall outcome of the vote. So you can see, with just three options, what seemed like a perfectly fair way to decide on dessert is not all that fair after all.

Excluded Voting Methods

You may think that if the vote can’t be fair, then there is no point in voting. But, don’t give up just yet.

Arrow’s Impossibility Theorem does not apply to all voting methods. It doesn’t apply to:

  • Straight plurality methods where voters only select their most preferred candidate.
  • Or, to votes between only two candidates/options.

Lesson Summary

Whew; that was a lot.

In this lesson, we talked about Arrow’s Impossibility Theorem, which states that voting with preference cannot be accomplished while adhering to principles of fair voting. We used an example to illustrate the three properties that Arrow determined should be assumed in an election:

  • No Dictators, meaning that no individual person’s ranking should determine the rankings outcome each time.
  • Pareto Efficiency, meaning if voters prefer Option 1 to Option 2, then the outcome should show Option 1 ranked highest.

and

  • Independence of Irrelevant Alternatives, meaning if Option 1 is ranked higher than Option 2, then removing Option 3 should not alter the relative rankings of 1 and 2.

And we discussed the two main voting methods that do not apply to Arrow’s theorem: plurality voting and elections with only two options.

Thanks for joining me. See you next time. Bye!

Learning Outcomes

Once you’ve completed this lesson, you should be able to:

  • Explain Arrow’s Impossibility Theorem
  • Summarize the process of preferential voting
  • Describe Arrow’s three properties of fair voting
  • Identify the two main voting methods where Arrow’s theorem does not apply

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