## Discussion Forum

### When No One's Best is Best

When No One's Best is Best

In last week's Brain Potion (found here) we discussed various voting strategies. Special thanks to Darren, who correctly identified them as "Plurality", "Approval Voting", and "Score Voting" and giving a great write-up. Check out his response!

This week we examine the idea of score voting in a little more detail.

To remind us of the concept of score voting, we consider the following example:

Suppose $8$ voters want to decide a "Best Movie" among $5$ different movies, labeled $A$, $B$, $C$, $D$, and $E$. They will each rank the movies best to worst and then every first place vote gets $1$ point, every second place vote gets $2$ points, etc., and thus the movie with the LOWEST score is the "Best Movie". Assume every voter ranks all $5$ movies, and includes no ties in their individual ranking.

As explored last week, this voting mechanism can lead to a movie that is labeled "Best Movie" even though none of the $8$ votes had it as their favorite.

• Give an example of votes where the above happens.
• As a challenge:
• Give an example where most of the voters in fact prefer two movies to the "Best Movie".
• Show that it is impossible for all of the voters to prefer two movies to the "Best Movie".

(Although the voting system for the Oscars is slightly different, this type of problem is discussed in the LA Times article found here.)

In general, think of scenarios where voting, rankings, competitions, etc. happen. What are some scenarios where the idea of score voting can be helpful? Are there scenarios where it definitely is not helpful? What might be used instead?

Please share any thoughts or questions you have below. We'll monitor the responses and give our thoughts as well!

Have your own request, idea, or feedback for the Brain Potion series? Share with us in our Request and Idea Thread available here.

Re: When No One's Best is Best

It happens sometimes and it is actually highly likely when there are split opinions.

Voter one might put a 1 2 3 4 5, voter two might put a 3 2 5 4 1, voter 3 might put 4 2 5 3 1, voter 4 might put 5 2 3 4 1, voter 5 might put 4 2 1 3 5, voter 6 might put 1 3 2 4 5, voter 7 might put 5 2 3 4 1, and voter 8 might put 5 2 3 1 4. (the spaces separate each movie: 1 2 means movie one got a one from the voter and movie two got a two from the same voter)

Movie A had 28, Movie 2 had 17, Movie 3 had 25, Movie 4 had 23, and Movie 5 had 23. Movie 2 had the smallest number which meant that it was the best but didn't get any one's.