Monday, October 29, 2012

Playoff Predictions

Elo Ratings
Chance to Win

Rtng Team (Chng)
Team WC QF SF Cup
1625   Los Angeles Galaxy  (---)
Sporting Kansas City 1.000 0.612 0.368 0.213
1624   Sporting Kansas City  (---)
San Jose Earthquakes 1.000 0.536 0.296 0.202
1621   San Jose Earthquakes  (---)
Seattle Sounders FC 1.000 0.550 0.263 0.114
1600   Seattle Sounders FC  (---)
New York Red Bulls 1.000 0.512 0.229 0.112
1565   Real Salt Lake  (---)
Los Angeles Galaxy 0.842 0.426 0.238 0.111
1557   New York Red Bulls  (---)
DC United 1.000 0.488 0.214 0.102
1555   Chicago Fire  (---)
Real Salt Lake 1.000 0.450 0.193 0.078
1549   DC United  (---)
Houston Dynamo 0.648 0.246 0.118 0.035
1539   Houston Dynamo  (---)
Chicago Fire 0.352 0.142 0.071 0.031
1515   Columbus Crew  (---)
Vancouver Whitecaps FC 0.158 0.038 0.011 0.002
1488   Montreal Impact  (---)

1480   FC Dallas  (---)

1443   Colorado Rapids  (---)

1433   Philadelphia Union  (---)

1424   Vancouver Whitecaps FC  (---)

1418   New England Revolution  (---)

1409   Portland Timbers  (---)

1335   Toronto FC  (---)

1320   Chivas USA  (---)

West Average: 1498.55

East Average: 1501.3

Using my Elo Ratings, I can determine the expected value of a given match-up. I can then use Bayes' Theorem to determine all possible playoff match-ups and come up with an expected outcome of the playoffs as a whole. Above is the probability - given by my Elo Rater - that each team has to advance to a specific round of the playoffs. For instance, the numbers above say that Sporting Kansas City has a 100% chance to make it through the Wild Card (because they don't play one), a 61.2% chance to make it through the quarterfinal, a 36.8% chance to make it through the semifinal, and a 21.3% chance to win MLS Cup 2012. On the opposite end of the spectrum, Vancouver has a 0.2% chance to win the playoffs.

These values will of course change game-to-game as new information is added to the rater, so I will update these after each round of the playoffs.

A few technical details and an example: For the wildcard and MLS Cup, I treat the game as a single game with home field advantage. For the quarters and semis, I treat the match-up as a single game with no home field advantage. After the first game in the series, I will treat the second leg as a single game match-up with home field advantage with a certain result (determined by the score of the first leg) necessary to move on. Here is an example:

Los Angeles will play host to Vancouver. Using their current ratings (LA at 1625 and Vancouver at 1424) and giving LA home field advantage, we figure out (through the methods used in my Elo Rater) that LA has an expected outcome of 0.842. This value is shown above under the WC column, indicating that LA has a 84.2% of making it through the wild card. Conversely, Vancouver has an expected outcome of 1 - 0.842 or 0.158. which is also shown above. Then, San Jose will play either LA or Vancouver in the next round. To come up with San Jose's probability of winning their quarterfinal, we use Bayes' Theorem. First, San Jose has a 49.4% chance of beating Los Angeles in a home/away series, and a 75.7% chance of beating Vancouver in a home/away series. Then, San Jose has a 84.2% chance of facing LA and a 15.8% chance of facing Vancouver (determined by the expected outcome of the LA/Vancouver game) so San Jose's chance to make it through the quarterfinal is 0.842 * 0.494 + 0.158 * 0.757 = 0.536, or 53.6%. LA, on the other hand, has a 1 - 0.494 = 0.506 expected outcome against San Jose, so their expected outcome for getting through both the wildcard and the semifinal are 0.842 * 0.506 = 0.426. We then continue this process through all the match-ups and get values for every stage through the cup for each team.

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