## Thursday, February 23, 2017

### Ken Arrow and Oscars Voting

Kenneth Arrow, the Nobel Prize winning economist known for his work on social choice and general equilibrium, passed away Tuesday at the age of 95.

I can't cover Arrow's broad influential work in this blog post even if I were an economist but I would like to talk about Ken Arrow's perhaps best known work, his impossibility theorem for voting schemes. If you have at least three candidates, there is no perfect voting method.

Suppose a group of voters give their full rankings of three candidates, say "La La Land", "Moonlight" and "Manchester by the Sea" and you have some mechanism that aggregates these votes and chooses a winner.

Now suppose we want a mechanism to have two fairness properties (for every pair of movies):
• If every voter prefers "Moonlight" to "La La Land" then the winner should not be "La La Land".
• If the winner is "Moonlight" and some voters change their ordering between "La La Land" and "Manchester by the Sea" then "Moonlight" is still the winner (independence of irrelevant alternatives).
Here's one mechanism that fills these properties: We throw out every ballot except Emma Watson's and whichever movie she chooses wins.

Arrow shows these are the only mechanisms that fulfill the properties: There is no non-dictatorial voting system that has the fairness properties above.

Most proofs of Arrow's theorem are combinatorial in nature. In 2002 Gil Kalai gave a clever proof based on Boolean Fourier analysis. Noam Nisan goes over this proof in a 2009 blog post.

Arrow's theorem that no system is perfect doesn't mean that some systems aren't better than others. The Oscars use a reasonably good system known as Single Transferable Voting. Here is a short version updated from a 2016 article.
For the past 83 years, the accounting firm PricewaterhouseCoopers has been responsible for tallying the votes, and again this year partners Martha Ruiz and Brian Cullinan head up the operation. The process of counting votes for Best Picture isn't as simple as one might think. According to Cullinan, each voter is asked to rank the nine nominated films 1-9, one being their top choice. After determining which film garnered the least number of votes, PWC employees take that title out of contention and look to see which movie each of those voters selected as their second favorite. That redistribution process continues until there are only two films remaining. The one with the biggest pile wins. "It doesn’t necessarily mean that who has the most number one votes from the beginning is ensured they win," he added. "It’s not necessarily the case, because going through this process of preferential voting, it could be that the one who started in the lead, doesn’t finish in the lead."
So if you’re a big fan of “Moonlight” but you’re scared that “La La Land” could win, you can help your cause by ranking “Moonlight” first and “La La Land” ninth, right?
Wrong. That won’t do a damn thing to help your cause. Once you rank “Moonlight” first, your vote will go in the “Moonlight” stack and stay there unless “Moonlight” is eliminated from contention. Nothing else on your ballot matters as long as your film still has a chance to win. There is absolutely no strategic reason to rank your film’s biggest rival last, unless you honestly think it’s the worst of the nominees.
Arrow's theorem says there must be a scenario where you can act strategically. It might make sense for this fan to put "Fences" as their first choice to potentially knock out "La La Land" in an early round. A similar situation knocked out Chicago from hosting the 2016 Olympics.

Maybe the Oscars should just let Emma Watson choose the winner.

1. "After determining which film garnered the least number of votes, PWC employees take that title out of contention and look to see which movie each of those voters selected as their second favorite. That redistribution process continues until there are only two films remaining."

I suppose in the i-th step, the sum of all votes ranking a film at positions <=i is considered?

2. Seems not necessarily true. For a given movie F, there could be voter(s) who gave F a rank j < i but one or more of their movies ranked k < j are still in contention and therefore their ranking of F at j has not yet become relevant.

3. Note that Arrow's theorem assumes that the voting mechanism is deterministic.

An alternative, which satisfies both fairness properties but somehow feels "less dictatorial", would be

"Everybody's ballot is placed in a pile, from which one person's is chosen uniformly at random. The Oscar goes to whoever that person's first choice is.

4. This comment has been removed by the author.

5. Lance, I blame you for what happened at the Oscars. You confused everyone, so the presenters announced the wrong Best Picture; Moonlight was the winner, not La La Land.

6. I think STV leads to a most acceptable result, i.e. a result where largest number of people can live with. An easier way to explain it to layman is to say we have n rounds of voting and each time we will eliminate the candidate with the least vote. That is much easier to understand than ranking and transferring votes.

>Maybe the Oscars should just let Emma Watson choose the winner.

Never!