Thus, in an 32 team cup, the average group will have a team of rating 1081, one of 706, one of 512, and one of 381. We can then calculate the competitiveness of each group stage if the top n seeds make that round. Upsets in earlier rounds or qualifying would mean the actual competitiveness factor will be slightly lower, but it ought to affect each relatively evenly. I've calculated the average difference by using the standard deviation, rather than the average sum of differences for all teams (which is much harder) , but it should be fairly similar. The conversion factor from rating difference to an expected margin is 0.2. It can be seen that the average margin for most combinations sits somewhere between 0.5 and 1.0 of the standard deviation of actual margins (60 runs).
Twelve formats have been chosen for consideration. By multiplying the number of games played in each combination by the expected margin, and dividing by the total number of games in the tournament, we can calculate an average expected margin and rating quality for each format.
The format used in the 1999 world cup has the lowest average margin. Sort of anyway, the model doesn't adjust for skipped games in the super-6 stage that push it back to 30.52. The format used in the WT20 is second-best which shows the ICC is doing something right. There is very little difference between most models. Obviously competitiveness gets worse as more teams are added, but the change in expected probability of victory for a 10 team competition (0.70) to a 20 team competition (0.77) is only 7%. Until a tournament reduces to 6 teams or fewer, the probability of an uneven contest is fairly stable.
Meaning is a little more difficult. For this I am going to use a precise mathematical definition:
A game's "meaning" is the change in probability that a team will qualify for the next round based on the result of a single game.
For a knockout game between even sides, the change in probability is 0.5: each team begins with a 50% chance of progressing and ends with either 0% or 100% chance. For a four team group with two qualifiers, the probability of progressing begins at 0.5. A team needs 2 wins to progress so the probability of progressing becomes 0.75 after a win and 0.25 after a loss (again assuming even contests). In practise it is slightly more complex than this as a team might not progress with 2 wins, or progress with 1 win (in 3% of cases). The difference is miniscule and roughly even across group sizes, so for the sake of simplification it is being ignored.
Three adjustments were made however:
Needless to say there is a strong correlation between meaning and the number of games, particularly the number of knockout matches. The sooner a team faces elimination, the more likely the games are to be meaningful. Far from being the most "meaningful" format, the 10 team world cup fares particularly poorly, each match being roughly the equivalent of a knockout between two teams where one was 97% likely to win.
We can relate these two factors by comparing them to a baseline factor. 60 runs for competitiveness and 12.5% for meaning, then graphing them against each other. Better, obviously, is furthest from the xy-origin of 1.
Previous world cups are marked in green, cups that don't meet the media criteria of 48 games in red. 12 team tournaments with squares, the biggest and smallest with triangles, and serious proposals with larger boxes.
It is worth noting here that how you interpret the graph depends on what you want from a world cup. If competitiveness is the only criteria, then it runs 12,10,16,14,20. But that, perversely, rates the 2007 cup over the 2011, when it clearly wasn't, even though both had a similar number of upsets.
I think competitiveness is a valuable measure, which is why it is included in my interpretation of meaning, via the adjustment from equal, to unequal groups (the graph heavily favours knock-out heavy competitions without the adjustment). But meaning conforms much more closely with what we have seen, the 12 team cup was good, but not a lot better than the 14, or the unfortunate 16 team editions. And a 10-team cup, while competitive, would in reality be a tedious bore, as game after game would be played for small odds, and only a handful of concluding matches having a real effect on qualification.
Obviously the ICC and its media partners have an interest in maintaining the presence of certain sides in the competition. But let's not pretend it adds "meaning". By using a defintion of "meaning" that we can implicitly understand - that a game is meaningful when it affects the expectations we had of it - it can be seen that a 10-team world cup would have been the most meaningless yet contrived.
Idle Summers 5th July, 2011 15:17:11 [#]
Quantifying World Cup Formats
Quantifying World Cup Formats