I have been following the reality TV competition So You Think You Can Dance for some time now. This is one of a couple of competition shows that I enjoy watching (the other is Bravo’s Top Chef). I think both shows are relatively unique in the genre, in that they involve people competing in an area where they are truly the best at what they do.
The format of the show is pretty simple. Each week, the dancers perform, viewers vote for their favorite dancer by calling a toll-free number, and a panel of judges selects one of the three dancers with the fewest votes to be eliminated that week. This continues until some small number of dancers remains, at which point “America votes” for their favorite again, with the dancer receiving the most votes winning the competition.
Every season there seems to be confusion and surprise as to how some particular talented dancer ends up in the “bottom three,” or worse, that the seemingly obviously least-talented performer is not in the bottom three, and so someone is forced to go home prematurely.
I am always surprised at the surprise, so to speak, since the voting format is extremely vulnerable to this sort of unexpected outcome. Consider what happens in every round except for the finals: voters provide information about only their favorite dancer… and this information is used to attempt to identify what should be the voters’ least favorite dancer among those remaining.
It seems to me that voting for your least favorite dancer, and eliminating whoever receives the most such votes, would be at least marginally less likely to yield outcomes that are surprising or inconsistent with a majority of viewers.
A very similar situation occurred in 1993, during the International Olympic Committee’s process of selecting the host city for the 2000 Summer Olympics. This process was also a series of voting rounds, with each round eliminating a candidate city with the least number of votes. The voting results are available here. In every round but the last, the voting results were consistent with the overall ranking Beijing > Sydney > Manchester > Berlin > Istanbul. Leading up to the final voting round, Beijing appeared to be the clear favorite. But in the final round, Sydney edged out Beijing in a 45-43 vote.
Neither of these examples is meant to suggest that antiplurality voting is always superior to the much more common plurality vote. But I think in the case of “runoff” procedures such as in these reality TV competitions it can be better, while still being practical in allowing a simple means of casting a ballot, so to speak, with a text message or simple automated phone call.
But I haven’t even dug into just how bad a poor choice of voting procedure can get. The economist Kenneth Arrow’s “Impossibility Theorem” is relevant here… but only when properly interpreted and applied. As “popular” as mathematics ever is in mainstream media, it is surprising how badly it can get mangled or mis-applied. I think this happens a lot, from Godel to Heisenberg to, in this case, Arrow. This post has already gotten rather long, so perhaps we can get into the theorem in more detail later if there is interest.