Posted by Chase Stuart on December 19, 2007
I've got a good friend that is a big college football fan and a pretty snazzy programmer. Let's call him "Doug". Every year, "Doug" conducts a college bowl pool that's half for fun, and half so he can create a nerdy webpage. Here at PFR we'll be running the bowl pool for you guys, too, so feel free to join in on the fun and geekiness. The prizes are:
75 sponsorship bucks for 1st place
30 sponsorship bucks for 2nd place
20 sponsorship bucks for 3rd place
As for the rules, well, here's the e-mail "Doug" sent out every year:
There are 32 bowl games. For each game, one team will be designated the favorite and one team will be designated the underdog. In some games, the underdog will be designated a "longshot." The list is below, along with a sample entry.
Step #1: pick a winner for each game (straight up).
Step #2: group your picks into groups of 1, 2 or 3, subject to the condition that a group of two must have at least one underdog and a group of one must be a longshot.
To get credit for a group, ALL teams in the group must win. Whoever gets the most groups wins. Tiebreaker is whoever picks the most total games correctly (ignoring groups). [NOTE: I think there's a decent chance that the tiebreaker will come into play; that's part of the strategy.]
THE GAMES (listed chronologically):
The favorite is listed first in each game, and note that the favorite/underdog/longshot designations are set in stone. Whatever happens between now and gameday is irrelevant.
Utah / Navy
Florida Atlantic / Memphis
Cincinnati / Southern Miss <---- USM is a longshot
New Mexico / Nevada
BYU / UCLA
Boise State / East Carolina <----- ECU is a longshot
Purdue / Central Michigan
Texas / Arizona State
TCU / Houston
Boston College / Michigan St.
Oregon State / Maryland
Wake Forest / UConn
Central Florida / Miss. St.
Penn State / Texas A&M
Alabama / Colorado
Cal / Air Force
South Florida / Oregon
Georgia Tech / Fresno St.
Kentucky / Florida State
Oklahoma State / Indiana
Clemson / Auburn
Tennessee / Wisconsin
Missouri / Arkansas
Texas Tech / Virginia
Florida / Michigan <------ Michigan is a longshot
USC / Illinois <------ Illinois is a longshot
Georgia / Hawaii <------ Hawaii is a longshot
Oklahoma / West Virginia
Virginia Tech / Kansas
Rutgers / Ball State
Tulsa / Bowling Green
LSU / Ohio State
Here is what a sample entry should look like:
Air Force / Kentucky (AF is a dog)
Auburn / Missouri (Auburn is a dog)
Boise / Ok.St. / Tulsa
West Virginia / LSU (WVU is a dog)
You should have exactly 32 teams listed, one from each game. This entry would get a point if Air Force AND Kentucky win. It would get a point if Boise, Ok. State, and Tulsa ALL win. It would get a point if Illinois wins. And so on.
If you wish to participate in the ***Official PFR College Bowl Pool***, please do the following:
- Be extremely clear on who your picks are. There are three OSUs, two Techs, and lots of other possibilities for ambiguity. Any team that is not identified in a 100% crystal clear manner will be counted as a loss.
- We will not be auditing the entries very carefully as they come in. Winners will of course be audited, but it is your responsibility to make sure your entry is legal.
- Thursday at 9PM EST tomorrow is the first game. All entries must be submitted by then.
With 64 possible winners and a near infinite number of permutations, I don't think you can ever come close to finding the optimal strategy. Tomorrow I'll list my actual picks and reasoning, but today I want to do some meta-analysis. What sort of strategy should we have, roughly?
I'll start off by giving scathing criticism of "Doug". I don't agree with his first premise, which is "pick a winner for each game straight up". Let me give my proof:
Let's suppose there are six bowl games, and twelve teams with potential chances to win a game. If a team has a 40% chance of winning, we'll name the team .40U, and its opponent, .60F. Let's assume the twelve teams in the game are: 0.20U, 0.25U, 0.3U, 0.35U, 0.4U, 0.45U, 0.55F, 0.60F, 0.65F, 0.70F, 0.75F and 0.80F. "Doug's" strategy would have us taking the six favorites in two groups -- which two groups?
If you put the three strongest favorites (0.80F, 0.75F and 0.70F) in one group, they have a 42% chance of winning all three games and earning you a point. Your other group would have 21% chance of winning all three games and earning you a point. Compare that to say, putting together 0.80F with 0.60F and 0.55F -- you'd have a 26% chance of winning all three games in that group, and a 34% chance of winning the other group. The odds of both groups winning, however, is identical no matter how you divide the groups. (Do you see why?)
But what about the odds of getting one group to win?
0.55 0.55 0.60 0.80 0.65 0.60 0.70 0.75 0.75 0.70 0.80 0.65 2 win 0.09 0.09 1 win 0.45 0.43 0 win 0.46 0.48
This should make an important strategy clear: a small, but clear, edge can be found by placing your strongest favorites together. So once you decide (ignoring the longshots for now) which favorites you want, you now know how to group them.
Now, on to why I think "Doug" is being shortsighted at best, and downright crooked at worst. Picking the teams you think to win isn't a winning strategy. Why? Let's pick the three obvious underdogs -- 0.35U, 0.40U and 0.45U -- and match them with the three obvious favorites. Once again, we have the question of how should we match them up?
Group 1 0.45 0.45 0.80 0.70 Group 2 0.40 0.40 0.75 0.75 Group 3 0.35 0.35 0.70 0.80 Grp1 % 0.3600 0.3150 Grp2 % 0.3000 0.3000 Grp3 % 0.2450 0.2800 3 win 0.026 0.026 2 win 0.190 0.187 1 win 0.445 0.441 0 win 0.338 0.345
The difference here is almost infinitessimal, and certainly too close considering our degree of precision in predicting the games. But as a mathematical principal, we'd be maximizing our chances of winning by placing our strongest favorite with our strongest underdog, our second strongest favorite with our second strongest underdog, and so on.
Note that we now have two rules, neither of which are complicated, despite their complicated proofs. Match up strong favorites with strong favorites, and match up strong favorites with strong underdogs. You might implicitly think matching up a strong favorite/weak underdog and strong underdog/weak favorite might make you more likely to win both, but that's not true.
While the strong underdog/strong favorite strategy produces minimal benefit over a random pairing of underdogs/favorites, note how it compares to the other strategy. If you pick all winners first -- which would be the six favorites, of course -- you have a 9% chance of winning 2 groups and a 45% chance of winning one group. By pairing up strong underdogs with strong favorites, the contest rules allow you to have a 2.6% chance of winning 3 groups (vs. a 0% chance), a 19% chance of winning two groups (vs. a 9% chance) and a 44.5% chance of winning 1 group (vs. a 45.4% chance of winning 1 group). So it's quite clear: picking strong favorites/strong underdogs is a dominant strategy over picking all favorites.
With that out of the way, we have to deal with the pesky issue of "longshots". The value of getting an entire group if you correctly pick a longshot is huge. And really, the cost isn't that high: you miss out on a win by a favorite, but unless two other favorites or the underdog attached to the favorite wins, no harm, no foul. (Do you see why?) But just as obviously, the strategy can't be "pick all longshots". There has to be some line where the reward is not justified by the risk. Where is that line?
Assume three bowl games. Longshot-BigFavorite (LS-BF), Underdog-Favorite and Underdog-Favorite. We have two choices: two groups (longshot; underdog/favorite) or one group (three favorites). Let's assume that both underdogs have a 40% chance of victory. What percent chance would the longshot need to have to make the two group (longshot) option a good one? Let's start off with say, 20%:
LS BF F F U F 0.20 0.80 0.60 0.60 0.40 0.60 2 wins 0.05 0.00 1 win 0.34 0.29 0 win 0.61 0.71
It's clear that the longshot is going to to better even at 20%, because it is more likely to get one win and has a small (but non-zero) chance of obtaining two. What about at 10%?
LS BF F F U F 0.10 0.90 0.60 0.60 0.40 0.60 2 wins 0.02 0.00 1 win 0.29 0.32 0 win 0.68 0.68
That one's pretty close -- we get rough equilibrium if the longshot has only a 10% chance, and the other two favorites have 60% chances. If we bump the other two favorites up to 70%...
LS BF F F U F 0.10 0.90 0.70 0.70 0.30 0.70 2 wins 0.02 0.00 1 win 0.27 0.44 0 win 0.71 0.56 1
Now it's clear that you should choose the three favorites. So our choice of whether we want to go with the longshot or the favorites is not going to be an easy one. If we bump the longshot back up to 20% but keep the other two favorites at 70%, picking the favorites is still more likely to help you avoid a goose-egg.
Hopefully this exercise gets you thinking a bit about how to make your picks. Please make them in the posts below, and all entries submitted before 9PM tomorrow night will be valid.