How to rank a million images with a crowdsourced sort
Solution 1:
As others have said, ranking 1-10 does not work that well because people have different levels.
The problem with the Pick A-or-B method is that its not guaranteed for the system to be transitive (A can beat B, but B beats C, and C beats A). Having nontransitive comparison operators breaks sorting algorithms. With quicksort, against this example, the letters not chosen as the pivot will be incorrectly ranked against each other.
At any given time, you want an absolute ranking of all the pictures (even if some/all of them are tied). You also want your ranking not to change unless someone votes.
I would use the Pick A-or-B (or tie) method, but determine ranking similar to the Elo ratings system which is used for rankings in 2 player games (originally chess):
The Elo player-rating system compares players’ match records against their opponents’ match records and determines the probability of the player winning the matchup. This probability factor determines how many points a players’ rating goes up or down based on the results of each match. When a player defeats an opponent with a higher rating, the player’s rating goes up more than if he or she defeated a player with a lower rating (since players should defeat opponents who have lower ratings).
The Elo System:
- All new players start out with a base rating of 1600
- WinProbability = 1/(10^(( Opponent’s Current Rating–Player’s Current Rating)/400) + 1)
- ScoringPt = 1 point if they win the match, 0 if they lose, and 0.5 for a draw.
- Player’s New Rating = Player’s Old Rating + (K-Value * (ScoringPt–Player’s Win Probability))
Replace "players" with pictures and you have a simple way of adjusting both pictures' rating based on a formula. You can then perform a ranking using those numeric scores. (K-Value here is the "Level" of the tournament. It's 8-16 for small local tournaments and 24-32 for larger invitationals/regionals. You can just use a constant like 20).
With this method, you only need to keep one number for each picture which is a lot less memory intensive than keeping the individual ranks of each picture to each other picture.
EDIT: Added a little more meat based on comments.
Solution 2:
Most naive approaches to the problem have some serious issues. The worst is how bash.org and qdb.us displays quotes - users can vote a quote up (+1) or down (-1), and the list of best quotes is sorted by the total net score. This suffers from a horrible time bias - older quotes have accumulated huge numbers of positive votes via simple longevity even if they're only marginally humorous. This algorithm might make sense if jokes got funnier as they got older but - trust me - they don't.
There are various attempts to fix this - looking at the number of positive votes per time period, weighting more recent votes, implementing a decay system for older votes, calculating the ratio of positive to negative votes, etc. Most suffer from other flaws.
The best solution - I think - is the one that the websites The Funniest The Cutest, The Fairest, and Best Thing use - a modified Condorcet voting system:
The system gives each one a number based on, out of the things that it has faced, what percentage of them it usually beats. So each one gets the percentage score NumberOfThingsIBeat / (NumberOfThingsIBeat + NumberOfThingsThatBeatMe). Also, things are barred from the top list until they've been compared to a reasonable percentage of the set.
If there's a Condorcet winner in the set, this method will find it. Since that's unlikely, given the statistical nature, it finds the one that's the "closest" to being a Condorcet winner.
For more information on implementing such systems the Wikipedia page on Ranked Pairs should be helpful.
The algorithm requires people to compare two objects (your Pick-A-or-B option), but frankly, that's a good thing. I believe it's very well accepted in decision theory that humans are vastly better at comparing two objects than they are at abstract ranking. Millions of years of evolution make us good at picking the best apple off the tree, but terrible at deciding how closely the apple we picked hews to the true Platonic Form of appleness. (This is, by the way, why the Analytic Hierarchy Process is so nifty...but that's getting a bit off topic.)
One final point to make is that SO uses an algorithm to find the best answers which is very similar to bash.org's algorithm to find the best quote. It works well here, but fails terribly there - in large part because an old, highly rated, but now outdated answer here is likely to be edited. bash.org doesn't allow editing, and it's not clear how you'd even go about editing decade-old jokes about now-dated internet memes even if you could... In any case, my point is that the right algorithm usually depends on the details of your problem. :-)
Solution 3:
I know this question is quite old but I thought I'd contribute
I'd look at the TrueSkill system developed at Microsoft Research. It's like ELO but has a much faster convergence time (looks exponential compared to linear), so you get more out of each vote. It is, however, more complex mathematically.
http://en.wikipedia.org/wiki/TrueSkill