Wednesday, March 30, 2011

How often does the best team win?

A couple months ago I asked how often the best team wins the world series. It turned out to be a surprisingly low percentage, and with the wacky final four I started asking the same questions about college basketball. Using the random chance in sports method to determine how often the best team won I calculated the probability of the best team being champion. Naturally it extrapolated itself to all sports:

NBA: 81%
This is by far the highest, and also the most wrong. Single teams don't distance themselves as often in the NBA, and some research I did in high school suggests that better teams win 60% of finals games, not 80%. Adjusting for that in the last two rounds puts the odds at around 48%.

College Football: 57%
People rag on the BCS, but I showed during BCS week that small playoffs are often better at selecting champions. Again, adjusting for using only the top teams and not the general random chance lowers the odds to 47%, which is still pretty high.

NFL: 55%
Even though this is the pro sport where the best team wins most often, single elimination hurts it. Byes help though (these odds assume the best team gets a bye). Assuming that the best team only wins .7 of the time (better for the playoffs) lowers the odds to 42%

You may be laughing now because I've corrected the random chance odds for each sport so far. It's much more accurate for the remaining sports though, either because top teams distance themselves farther (MLB and NHL) or because there is a full range of opponents in tourney play (MCBB and WCBB).

Women's College Basketball: 36%
We're probably a little thrown off because of UConn's recent success, but actually with 6 rounds of play even the women's tourney is ripe with upsets.

MLB: 33%
Series' are a necessity in a sport where even the worst teams win almost 40% of their games. The first round should probably become a best of 7 to help that out (it'd boost odds about 5%).

Men's College Basketball: 28%
We've seen this on full display in the tournament, as the clear best three (Ohio St, Kansas, Duke) have all fallen to lesser opponents. 6 rounds is fun though, so it's worth diluting it, but a 7th in the future would be a stretch.

NHL: 28%
This may seem weird with many former dynasties like the Canadiens, but in reality the modern day NHL is as full of parity as baseball, and with an extra round the best team's chances are the worst of any major U.S. sport.
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Tuesday, March 29, 2011

Final Four Odds and Analysis

Okay, I haven't fared well so far, but I still ran the bracket predictor for final four teams (both Men's and Women's). Who gonna be the champ?
Men's Odds
Kentucky 46.7
Connecticut 30.1
Butler 17.8
VCU 5.4

Kentucky was actually (yeah Dad, actually) the sixth highest rated team in the polls, and a has had a reasonable chance to win it all along. Now they should definitely be considered the favorites, but as this tournament has shown anybody can win (even VCU, although the odds are still not good).

Women's Odds
Connecticut 47.7
Stanford 26.9
Texas A&M 17.7
Notre Dame 7.7

The odds look shockingly similar to the men's odds, but the women's tournament has been light on upsets. Why are they so spread out? There are two reasons: first, the top few women's teams are comparatively better than the top few men's teams (look at the power ratings), so UConn and Stanford spread it out a little bit. The best teams are also on opposite sides of the bracket, as opposed to the men where either Butler or VCU has to make the final. Either way, both final fours look to be exciting with a clear favorite but a couple strong underdogs.
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Tuesday, March 22, 2011

The Game Theory of 21

On Sunday a great basketball rivalry was renewed. It was not the Kings and Timberwolves fighting to stay out of last place, it was Andy, Jimmy and I playing 21 (a basketball game). In general when we play I win probably two thirds of the time, Andy will win one third, and Jimmy will never win (Burn!*). Recently though, Jimmy has started winning, highlighted by a game Sunday when Jimmy had 19, I came back to 17, and then Andy let Jimmy get a free layup for 21! With two victories in the last nine games (and zero for Andy) it appears that my secret has been uncovered: don't guard Jimmy.
Turns out, this is a classic Prisoner's dilemma. In all situations it is better for both Andy and me to not guard Jimmy (example odds of winning in each case are illustrated in the table). As a result though, both of us have significantly lowered our odds of winning the game.
This sort of idea comes up often in life, and a fair bit in recreational sports too. The same phenomenon applies to capture the flag: Offense is fun, but defense wins, so both teams play defense and have no fun, but still each have 1/2 chance of winning.
Could this phenomenon happen in national sports leagues? Absolutely. Off the top of my head here are a couple examples.
Helmet hits in NFL: Knocking the other team's star player out helps you win, but both teams are better off if there are no helmet to helmet hits.
College Recruiting Violations: Paying to get the best players can help you win (SMU), but if everybody does it then everybody loses.
Steroids in Baseball: Again, it's an advantage for you if you're the only one on the juice, but once everyone does it then the playing field is level again and your health sucks.
How do you get out of a Prisoner's Dilemma? Put a negative incentive on the negative behavior, including fines, sanctions, and suspensions. The leagues are all actually controlling for it fairly well. That doesn't mean I'll guard Jimmy though.
*Jimmy actually played really well when we played today. Somewhere over the past couple years he has learned to shoot and take advantage of his athleticism on the boards. He should probably get better than 0% odds on winning. That wouldn't amuse me as much though.
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Sweet 16 Odds and Analysis

My personal bracket has been busted, and some of the ratings favorites (Texas, Purdue) didn't fare well either. Fortunately there's still four rounds left to go, and places like Yahoo! have bracket contests for them too (here's the stats bracket for the last four rounds). What? You say you need the odds for the last rounds to fill that out? Here they are:

Rd 3 Rd 4 Rd 5 Rd 6
Ohio St 0.628 0.456 0.278 0.177
Kentucky 0.372 0.228 0.111 0.058
Marquette 0.414 0.113 0.038 0.014
North Carolina 0.586 0.203 0.083 0.037
Duke 0.731 0.49 0.286 0.178
Arizona 0.269 0.115 0.04 0.016
Connecticut 0.456 0.17 0.067 0.029
San Diego St 0.544 0.225 0.098 0.046
Kansas 0.826 0.676 0.462 0.256
Richmond 0.174 0.087 0.028 0.006
VA Commonwealth 0.286 0.04 0.008 0.001
Florida St 0.714 0.197 0.074 0.019
Butler 0.281 0.091 0.025 0.005
Wisconsin 0.719 0.393 0.183 0.075
BYU 0.583 0.32 0.146 0.058
Florida 0.417 0.196 0.075 0.024

Of particular interest here is the improvement of Kansas's odds thanks to weak opponents. From the championship odds below it's easy to see that Kansas is now the favorite to win it all, in large part because they have a 67.6% chance of making the final four.
Remaining Championship Odds
Team Odds
1 Kansas 0.2559
1 Duke 0.1779
1 Ohio St 0.1774
4 Wisconsin 0.075
3 BYU 0.0581
4 Kentucky 0.0575
2 San Diego St 0.0465
2 North Carolina 0.0369
3 Connecticut 0.029
2 Florida 0.0244
10 Florida St 0.0194
5 Arizona 0.0158
11 Marquette 0.0137
12 Richmond 0.006
8 Butler 0.0055
11 VA Commonwealth 0.001

A lot of weird things happened earlier in the tournament (chances VCU makes the sweet 16: 1.26%), and I started wondering if momentum actually played a role in college basketball. I weighted the Power Rankings to give the last 4 weeks 3 times the weight, and the last 2 weeks got 5 times the weight, expecting to see that Louisville/Purdue/Texas would drop significantly. The top 15 shows otherwise though:
Top 15 with Mo'
1 Ohio St
2 Duke
3 Louisville
4 Kansas
5 Notre Dame
6 Kentucky
7 San Diego St
8 Pittsburgh
9 Texas
10 Connecticut
11 Syracuse
12 Purdue
13 Washington
14 North Carolina
15 West Virginia

There were 5 surprise losses in the first round (LV, ND, Pitt, TX, Purdue). Adding up their rankings in the regular rankings they add to 38, and here? 37. Looks like momentum doesn't explain why the ratings have fared poorly so far. I think the real reason that my computer (and other stats ratings) haven't fared well is that sometimes, these things happen.

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Tuesday, March 15, 2011

Bracket Analysis 2011

There were a lot of number in that last post, but what can you really get out of it? Here are some key bits o' knowledge.

Ten Most Likely Champions
Seed Team Chance
1 Ohio St 0.148
1 Duke 0.136
1 Kansas 0.121
1 Pittsburgh 0.088
4 Texas 0.065
3 Purdue 0.045
4 Louisville 0.039
4 Kentucky 0.039
2 San Diego St 0.033
3 BYU 0.033

This year is dominated by the one seeds and... the four seeds? The strength of the four seeds shows up throughout this bracket.

Best Final Four Odds By Region
EAST Odds WEST Odds
1 Ohio St 0.368 1 Duke 0.357
4 Kentucky 0.148 4 Texas 0.207
7 Washington 0.13 2 San Diego St 0.152
3 Syracuse 0.112 3 Connecticut 0.097
2 North Carolina 0.095 5 Arizona 0.048
SWEST Odds SEAST Odds
1 Kansas 0.318 1 Pittsburgh 0.325
3 Purdue 0.177 3 BYU 0.178
4 Louisville 0.155 4 Wisconsin 0.131
2 Notre Dame 0.14 2 Florida 0.105
6 Georgetown 0.042 5 Kansas St 0.04

Each Region's 4 seed is better than it's two seed? Maybe I was too early in praising the selection committee. Also notable are Washington's strong Cinderella chances.

Most Likely First Round Upsets
UDOG OVER FAV ODDS
1 11 Marquette 6 Xavier 0.583
2 9 Villanova 8 George Mason 0.545
3 9 Illinois 8 UNLV 0.541
4 10 Michigan St 7 UCLA 0.53
5 11 Gonzaga 6 St Johns 0.497
6 10 Forida St 7 Texas A&M 0.495
7 12 Utah St 5 Kansas St 0.489
8 10 Penn St 7 Temple 0.474
9 9 Tennessee 8 Michigan 0.469
10 11 Missouri 6 Cincinnati 0.465

I know everybody likes 12-5 upsets, and everybody wants Arizona and Vandy to lose. But that's not what's good this year, instead it's time to look at the 11-6 games. They've got the best bang for your buck.

Over/Underachievers
Compared to how their seed usually does, here are the top 5 over/under achievers by number of wins better/worse than usual, and by number of times as many wins (i.e. Belmont can expect 2.16 times as many wins as a normal 13 seed, but still not many).
5 UNDER Wins 5 OVER Wins
2 North Carolina -0.716 7 Washington 0.902
2 Florida -0.655 4 Texas 0.654
6 Xavier -0.63 4 Louisville 0.514
1 Pittsburgh -0.575 11 Marquette 0.512
1 Kansas -0.545 4 Kentucky 0.441
5 UNDER Perc 5 OVER Perc
14 St Peter's 0.483 13 Belmont 2.16
6 Xavier 0.507 7 Washington 2.06
10 Georgia 0.511 11 Marquette 2.02
13 Morehead St 0.537 11 Gonzaga 1.69
12 Memphis 0.56 11 Missouri 1.65

As we saw before, 4 seeds are good here while 2 seeds are bad. 11 seeds also show strongly, as does Washington. If you're looking for someone to make a run this tournament, Washington and Marquette may be the way to go.

There you have it, the expected trends at a glance going into the tournament. Not all of these are going to happen, and I'll probably feel like a fool, but expect at least some of these trends to show up.
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Bracket Odds 2011

Who wins the tournament? How likely is it that BYU makes the Final Four? Which 12-5 upset has the best chance of happening? Here are the odds for each team to win in each round in the 2011 NCAA Men's Basketball Tournament.
(A quick review on how to read this, if Ohio St has a .368 under Rd 4, then they have a 36.8 percent chance of winning in round 4 {making it to the Final Four}).
TEAM Rd 1 Rd 2 Rd 3 Rd 4 Rd 5 Rd 6
EAST
1 Ohio St 0.977 0.768 0.526 0.368 0.228 0.148
16 UT San Antonio 0.023 0.002 0 0 0 0
8 George Mason 0.455 0.097 0.035 0.013 0.004 0.001
9 Villanova 0.545 0.133 0.053 0.022 0.007 0.003
5 West Virginia 0.577 0.251 0.085 0.04 0.015 0.006
12 Clemson 0.423 0.156 0.044 0.017 0.006 0.002
4 Kentucky 0.884 0.571 0.254 0.148 0.074 0.039
13 Princeton 0.116 0.023 0.002 0 0 0
6 Xavier 0.417 0.16 0.054 0.014 0.004 0.001
11 Marquette 0.583 0.267 0.11 0.035 0.012 0.005
3 Syracuse 0.86 0.542 0.275 0.112 0.049 0.023
14 Indiana St 0.14 0.03 0.005 0 0 0
7 Washington 0.771 0.468 0.287 0.13 0.063 0.032
10 Georgia 0.229 0.077 0.025 0.005 0.001 0
2 North Carolina 0.866 0.435 0.241 0.095 0.04 0.018
15 Long Island 0.134 0.02 0.003 0 0 0
WEST
1 Duke 0.971 0.785 0.503 0.357 0.214 0.136
16 Hampton 0.029 0.003 0 0 0 0
8 Michigan 0.531 0.117 0.038 0.015 0.004 0.001
9 Tennessee 0.469 0.094 0.029 0.01 0.003 0.001
5 Arizona 0.776 0.296 0.101 0.048 0.018 0.007
12 Memphis 0.224 0.038 0.005 0.001 0 0
4 Texas 0.861 0.623 0.316 0.207 0.112 0.065
13 Oakland 0.139 0.044 0.007 0.002 0 0
6 Cincinnati 0.535 0.27 0.13 0.043 0.016 0.006
11 Missouri 0.465 0.219 0.099 0.03 0.01 0.004
3 Connecticut 0.839 0.478 0.256 0.097 0.04 0.018
14 Bucknell 0.161 0.033 0.006 0.001 0 0
7 Temple 0.526 0.196 0.079 0.021 0.006 0.002
10 Penn St 0.474 0.166 0.063 0.015 0.004 0.001
2 San Diego St 0.89 0.615 0.362 0.152 0.069 0.033
15 N Colorado 0.11 0.024 0.004 0 0 0
SOUTHWEST
1 Kansas 0.96 0.692 0.471 0.318 0.213 0.121
16 Boston Univ 0.04 0.004 0 0 0 0
8 UNLV 0.459 0.131 0.058 0.025 0.011 0.003
9 Illinois 0.541 0.173 0.083 0.039 0.018 0.007
5 Vanderbilt 0.626 0.259 0.085 0.036 0.016 0.005
12 Richmond 0.374 0.115 0.027 0.008 0.003 0.001
4 Louisville 0.887 0.602 0.274 0.155 0.087 0.039
13 Morehead St 0.113 0.024 0.003 0 0 0
6 Georgetown 0.584 0.241 0.116 0.042 0.018 0.006
11 USC 0.416 0.142 0.057 0.017 0.006 0.002
3 Purdue 0.921 0.606 0.367 0.177 0.1 0.045
14 St Peter's 0.079 0.011 0.001 0 0 0
7 Texas A&M 0.505 0.197 0.073 0.022 0.008 0.002
10 Florida St 0.495 0.191 0.07 0.021 0.008 0.002
2 Notre Dame 0.884 0.587 0.312 0.14 0.075 0.032
15 Akron 0.116 0.024 0.003 0 0 0
SOUTHEAST
1 Pittsburgh 0.931 0.74 0.485 0.325 0.176 0.088
16 UNC Asheville 0.069 0.015 0.002 0 0 0
8 Butler 0.562 0.147 0.053 0.02 0.005 0.001
9 Old Dominion 0.438 0.097 0.029 0.009 0.002 0
5 Kansas St 0.511 0.22 0.086 0.04 0.014 0.004
12 Utah St 0.489 0.205 0.078 0.036 0.012 0.004
4 Wisconsin 0.68 0.431 0.218 0.131 0.061 0.026
13 Belmont 0.32 0.144 0.049 0.02 0.006 0.002
6 St John's 0.503 0.206 0.1 0.036 0.011 0.003
11 Gonzaga 0.497 0.202 0.098 0.035 0.011 0.003
3 BYU 0.859 0.559 0.354 0.178 0.081 0.033
14 Wofford 0.141 0.033 0.007 0.001 0 0
7 UCLA 0.47 0.211 0.082 0.027 0.008 0.002
10 Michigan St 0.53 0.253 0.105 0.037 0.011 0.003
2 Florida 0.846 0.503 0.248 0.105 0.04 0.013
15 Santa Barbara 0.154 0.033 0.005 0.001 0 0








I'll talk more about trends here later, but there are a few things that stand out for now. The strength of the 4 seeds vs the weakness of the 2 seeds indicates they should probably switch places. Washington has an impressive push to go far. Thanks to easier schedules, all of the #1s are favorites to make the Final Four (if not by much). And THE most likely champion is... Ohio State.
How would my computer fill out a bracket? It's all here.
You'll notice it's a lot of chalk, but that's what you get with a computer trying to max out expected value.
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