# Are Fair Voting Systems Mathematically Possible?

To American voters, it’s an all-too familiar dilemma: do you cast your lot with the candidate most likely to win, or risk spoiling the election by supporting the third-party candidate in whom you actually believe? What if, instead of choosing one candidate, voters were instead given the opportunity to rate each potential office-holder, in the same way that Olympic judges score athletes? Brian Dunning at Skeptoid takes an interesting look at the mathematics of voting systems:

In the 1969 film Putney Swope, members of the board of executives were prohibited from voting for themselves, so they all voted for the one board member they were sure nobody else would vote for. Ergo, this free, democratic election produced a chairman that no voter wanted.

In a perfect democracy, everyone gets an equal opportunity to vote, and equal representation. Therefore, we hold elections to let everyone have their say, to either vote representatives into office, or to enact certain laws. It’s a fine idea, and most countries do their level best to implement such systems. Some voters take advantage of it, and some choose apathy and don’t vote. Some try to anticipate what other voters might do, and cast a vote in an unexpected direction not to vote for a candidate, but to affect another candidate’s chances. This is what went wrong in Putney Swope: each voter cast a throwaway vote hoping to improve his own chances. In most elections, everyone has the right to do any of these things; the election is theirs, and theirs to decide. But what many of them might not know is that virtually any electoral process is flawed. Some outcomes are surprising. There are a number of different circumstances in which the candidate most desired does not win.

Democratic voting is only simple if there are just two candidates, or if it’s a Yes or No vote. In those cases, any attempt to vote tactically or to create a voting block — casting votes that don’t represent your preference — work against you. What we’re talking about today are elections where there are three or more candidates. And the idea that all the various systems for running such elections are flawed (subject to results that do not represent the group’s preference) is not just a whim or a crazy opinion of mine. It’s proven by Arrow’s Impossibility Theorem, named for the economist Kenneth Arrow, winner of the 1972 Nobel Prize in economics and the 2004 National Medal of Science. He proved it in 1951 with his Ph.D. thesis at Columbia University.

Arrow’s theorem can be simplified into one clear statement: that no fair voting system exists when there are three or more candidates. To this I ask: What do you mean by fair? That’s the key to Arrow’s theorem. It holds true, depending on a rigid definition of fair that must satisfy three criteria:

1. If every individual prefers X to Y, then the group prefers X to Y.

2. If every voter’s preference of X over Y stays the same, then the group’s preference of X to Y stays the same, even if other preferences change: such as Y to Z, or Z to X.

3. There can be no dictator, as Arrow called him; a single voter with the power to dictate the group’s preference.

Arrow’s theorem applies to election systems that require voters to rank the candidates. This is the case with most voting systems worldwide. Typically, when you vote, you mark an X in the box for one candidate. That’s a ranking; you’ve ranked that candidate first. Arrow’s theorem applies to these simple ranking systems, but its richest mathematical complexities come from systems with three or more candidates and the voters rank all candidates in order of preference. This isn’t used in many real-world elections, but it’s the theoretical basis for social choice theory.

### Haystack

#### 35 Commentson "Are Fair Voting Systems Mathematically Possible?"

1. Democracy, in its true form, does not have elections; rather, officials are chosen by lot.  See http://en.wikipedia.org/wiki/Sortition for details.

2. Democracy, in its true form, does not have elections; rather, officials are chosen by lot.  See http://en.wikipedia.org/wiki/Sortition for details.

• FunToThinkAbout | Nov 14, 2011 at 2:28 pm |

Interesting! This reminds me of an idea which Arthur C. Clark presented in his novel, Imperial Earth: all eligible citizens have their social security numbers thrown into a computer which every two years selects a number at random and whoever gets picked has to serve as President. The idea being that anyone who runs for office must have an agenda/ulterior motives and should therefore be disqualified immediately. Then again, in this future which Clarke envisioned, the citizens were all fairly educated…a far cry from today’s general lot. But I like the idea that a system like this would force people to get involved and participate…educate yourself and stay current because you could be the next number called.

3. Enronforever | Nov 14, 2011 at 12:58 am |

Are Fair Voting Systems Mathematically Possible?…the answer: no especially when hundreds of thousands of votes get lost in states like Florida ala 2000 election.  But then again was it even fair to begin with ?  The average elector we all supposedly get to vote for is about as honest as Bernie Madoff.

4. Enronforever | Nov 13, 2011 at 8:58 pm |

Are Fair Voting Systems Mathematically Possible?…the answer: no especially when hundreds of thousands of votes get lost in states like Florida ala 2000 election.  But then again was it even fair to begin with ?  The average elector we all supposedly get to vote for is about as honest as Bernie Madoff.

5. Lets make it more complicated: Everyone gets 2 votes. That way there’s no wasted vote syndrome, and if a majority of rebups and dems actually cast a second vote for the same 3rd party, then at least a larger number of people will be happy(er) with the results

6. Lets make it more complicated: Everyone gets 2 votes. That way there’s no wasted vote syndrome, and if a majority of rebups and dems actually cast a second vote for the same 3rd party, then at least a larger number of people will be happy(er) with the results

7. Anonymous | Nov 14, 2011 at 3:13 am |

I think we should be able to vote against people. Say, Ron Paul, so I can cancel out one of his dozen or so followers.

8. I think we should be able to vote against people. Say, Ron Paul, so I can cancel out one of his dozen or so followers.

9. Jin Onikoroshi | Nov 14, 2011 at 9:46 am |

Anything is better than ‘first past the post’… it is so undemocratic it makes the american system look completely rational and logical.

10. Jin The Ninja | Nov 14, 2011 at 5:46 am |

Anything is better than ‘first past the post’… it is so undemocratic it makes the american system look completely rational and logical.

11. Anonymous | Nov 14, 2011 at 2:26 pm |

Possible -yes. Probable, no.

12. GoodDoktorBad | Nov 14, 2011 at 10:26 am |

Possible -yes. Probable, no.

13. FunToThinkAbout | Nov 14, 2011 at 5:43 pm |

Eliminate the electoral college and then make it the law that you have to vote…make Voting Day a holiday…and if you don’t vote, you lose your citizenship. So now everyone has to vote, and they know that their vote will count. Guess what? Goodbye, two-party system! Whenever someone has an issue (or issues) which they feel are not being appropriately addressed  by the currently existing parties, then they have a simple choice: start their own party. That is the world  that I want to live in. But I am not holding my breath…

14. FunToThinkAbout | Nov 14, 2011 at 1:43 pm |

Eliminate the electoral college and then make it the law that you have to vote…make Voting Day a holiday…and if you don’t vote, you lose your citizenship. So now everyone has to vote, and they know that their vote will count. Guess what? Goodbye, two-party system! Whenever someone has an issue (or issues) which they feel are not being appropriately addressed  by the currently existing parties, then they have a simple choice: start their own party. That is the world  that I want to live in. But I am not holding my breath…

• I’d rather lose my citizenship.

• FunToThinkAbout | Nov 14, 2011 at 4:31 pm |

Not surprising, coming from the person who wants the right to “vote against” another candidate.

*facepalm*

• Pfft. Calling Ron Paul a candidate is like calling my cat a tiger.

• Jin The Ninja | Nov 14, 2011 at 7:07 pm |

in fairness to your cat, he MAY be a tiger in his mind…

• And as Camus said, if a post man has the conscience of an emperor in reality he is an emperor, so there you go.

• mrtastycakes | Nov 15, 2011 at 8:48 am |

First-past-the-post electoral systems ensure a two-party system because of tactical voting. Even if everyone voted, we would still have a two party system. Preferential voting (and proportional representation, although it has issues) would be the only way out.

I also think we should emulate the Soviets and give away booze at the polls to get out the vote.

• FunToThinkAbout | Nov 15, 2011 at 2:48 pm |

“First-past-the-post electoral systems ensure a two-party system because of tactical voting.” I just did some reading up on “first-past-the-post” and “tactical voting” – very interesting – but am still uncertain as to how it would ensure a  two-party system? Not trying to be argumentative…am genuinely curious about how/why you think this is the case. Nothing which I read indicates that it must be inevitable…

• mrtastycakes | Nov 15, 2011 at 7:22 pm |

It’s called Duverger’s law–and he was from a country with proportional representation. The US uses single member districts, which reinforces the two party system even more.

Duverger’s Law claims that all plurality systems
will become two-party systems over time due to tactical voting. So, for example, as Green Party voters (representing 30% of votes cast) realize they are splitting the
vote with their ideologically-comparable foe, the Democratic Party (representing 30%), they will
start voting for the more powerful Democratic Party in order to beat the
Republicans (representing 40%).

This stuff gets way more complicated, but if your interested, keep researching. You can read only about electoral methods (and the 1968 party reforms) and have a better understanding of US politics than pretty much anyone.

15. FunToThinkAbout | Nov 14, 2011 at 6:28 pm |

Interesting! This reminds me of an idea which Arthur C. Clark presented in his novel, Imperial Earth: all eligible citizens have their social security numbers thrown into a computer which every two years selects a number at random and whoever gets picked has to serve as President. The idea being that anyone who runs for office must have an agenda/ulterior motives and should therefore be disqualified immediately. Then again, in this future which Clarke envisioned, the citizens were all fairly educated…a far cry from today’s general lot. But I like the idea that a system like this would force people to get involved and participate…educate yourself and stay current because you could be the next number called.

16. Anonymous | Nov 14, 2011 at 6:30 pm |

I’d rather lose my citizenship.

17. Anonymous | Nov 14, 2011 at 6:30 pm |

I’d rather lose my citizenship.

18. FunToThinkAbout | Nov 14, 2011 at 8:31 pm |

Not surprising, coming from the person who wants the right to “vote against” another candidate.

*facepalm*

19. Anonymous | Nov 14, 2011 at 8:49 pm |

Pfft. Calling Ron Paul a candidate is like calling my cat a tiger.

20. Jin Onikoroshi | Nov 14, 2011 at 11:07 pm |

in fairness to your cat, he MAY be tiger in his mind…

21. Anonymous | Nov 14, 2011 at 11:33 pm |

And as Camus said, if a post man has the conscience of an emperor in reality he is an emperor, so there you go.

22. mrtastycakes | Nov 15, 2011 at 12:48 pm |

First-past-the-post electoral systems ensure a two-party system because of tactical voting. Even if everyone voted, we would still have a two party system. Preferential voting (and proportional representation, although it has issues) would be the only way out.

I also think we should emulate the Soviets and give away booze at the polls to get out the vote.

23. mrtastycakes | Nov 15, 2011 at 12:52 pm |

Maybe not, but instant run-off voting or the Condorcet method is way more fair than plurality.

24. mrtastycakes | Nov 15, 2011 at 8:52 am |

Maybe not, but instant run-off voting or the Condorcet method is way more fair than plurality.

25. FunToThinkAbout | Nov 15, 2011 at 6:48 pm |

“First-past-the-post electoral systems ensure a two-party system because of tactical voting.” I just did some reading up on “first-past-the-post” and “tactical voting” – very interesting – but am still uncertain as to how it would ensure a  two-party system? Not trying to be argumentative…am genuinely curious about how/why you think this is the case. Nothing which I read indicates that it must be inevitable…

26. mrtastycakes | Nov 15, 2011 at 11:22 pm |

It’s called Duverger’s law–and he was from a country with proportional representation. The US uses single member districts, which reinforces the two party system even more.

Duverger’s Law claims that all plurality systems
will become two-party systems over time due to tactical voting. So, for example, as Green Party voters (representing 30% of votes cast) realize they are splitting the
vote with their ideologically-comparable foe, the Democratic Party (representing 30%), they will
start voting for the more powerful Democratic Party in order to beat the
Republicans (representing 40%).

This stuff gets way more complicated, but if your interested, keep researching. You can read only about electoral methods (and the 1968 party reforms) and have a better understanding of US politics than pretty much anyone.