We all know that no sixteen seed has ever beaten a one seed in the NCAA tournament, right?    We all know that twelve seeds often beat five seeds. ...we all know this at the moment; don’t quiz us in six months.   But what are the odds that a #10 seed, for example, would beat a #4 seed, or that a #6 would beat a #1?   What are the odds that an eight seed would win the entire tournament?

            I have wondered about this for years, and, when a friend of mine (who will remain nameless until we are certain his data is right. . .if there is an error there it’s on me). ..when a friend recently sent me the overall first-round records of teams by seed, I decided to get off my lazy ass and actually try to construct a system to predict these outcomes.

            What I am interested in here is a “predictive heuristic”. ...a simple formula, like runs created or the Pythagorean system, which predicts the performance of each team because there is some awkward resemblence between the math and the real world.   My first assumption was that the system might work something like this:

            Take the seed of each team.

            Subtract that from 17.

            Square it.

            The results represent the relative strength of the two teams.

            In other words, when a four seed is playing a thirteen seed, that’s 13 squared against 4 squared, or 169 to 16. . .the four seed should win a little more than 90% of the time. 

            Your mind will object here quickly that this can’t be right because no sixteen seed has ever won, and this system allows a sixteen seed to possibly win, so that can’t be right.   But actually, that’s not a big problem.   This system would predict that the ratio of wins when a one seed plays a sixteen would be 256 to 1.   It’s actually 96 to 0.   No sixteen ever HAS won a game, but one WILL win a game, eventually, and one time in 257 seems like as reasonable a guess as any other. 

            It gets by that problem, but not some others. . .that system doesn’t work all that well.   Four seeds don’t really beat thirteen seeds 91% of the time; it’s actually only 79%.  That system gives far too large an advantage to the higher seeds.   These are the actual and predicted results of this system:

            1 vs. 16            1 wins .996 in theory  (256 to 1)         1 is 96-0 actual   1.000

            2 vs 15             2 wins .983 in theory (225 to 4)          2 is 92-4 actual     .958

            3 vs 14             3 wins .956 in theory (196 to 9)          3 is 81-15 actual   .844

            4 vs 13             4 wins .914 in theory (169 to 16)        4 is 76-20 actual   .792

            5 vs 12             5 wins .852 in theory (144 to 25)        5 is 65-31 actual   .677

            6 vs 11             6 wins .771 in theory (121 to 36)        6 is 66-30 actual   .688

            7 vs 10             7 wins .671 in theory (100 to 49)        7 is 60-36 actual   .625

            8 vs 9               8 wins .559 in theory (81 to 64)          8 is 45-51 actual   .469

            You can see that this model predicts far better winning percentages for the higher-seeded teams than they are actually able to achieve.          There are two ways to address that problem within this framework:

            1)  Subtract the seed from some number larger than 17, or

            2)  Use a power less than 2.0 (2.0 meaning “squared” in the description above.)

            I experimented with both options.    If you subtract the seed from a number higher than 17, the number that works best is 22.   If you raise the “value” to a number less than 2.0, the number that works best is 1.2.  

            The latter option actually works better, but

            1)  Both methods substantially understate the performance of one and two seeds, and

            2)  I don’t like futzing around raising numbers to odd powers.  

            Repeating the chart above, only substracting the seed numbers from 22, rather than 17, we get:

1 vs. 16            1 wins .925 in theory  (441 to 36)       1 is 96-0 actual   1.000

            2 vs 15             2 wins .891 in theory (400 to 49)        2 is 92-4 actual     .958

            3 vs 14             3 wins .849 in theory (361 to 64)        3 is 81-15 actual   .844

            4 vs 13             4 wins .800 in theory (324 to 81)        4 is 76-20 actual   .792

            5 vs 12             5 wins .742 in theory (289 to 100)      5 is 65-31 actual   .677

            6 vs 11             6 wins .679 in theory (256 to 121)      6 is 66-30 actual   .688

            7 vs 10             7 wins .610 in theory (225 to 144)      7 is 60-36 actual   .625

            8 vs 9               8 wins .537 in theory (196 to 169)      8 is 45-51 actual   .469

            This system works very well on half of the matchups (3 v 14, 4 v 13, 6 v 11 and 7 v 10), and not so well on the other half.   Stating the same thing another way, this system works very well except for the anomalous outcomes on 5 vs. 12 and 8 vs. 9—about which we can’t do anything—and except for the fact that it is significantly too low on the one and two seeds beating up the 15s and 16s.

            We can correct the top-end/bottom-end problem in one of two ways:

            1)  By assigning higher values to the one and two seeds, or

            2)  By assigning lower values to the fifteens and sixteens. 

            Assigning higher values to the top teams would be the appropriate remedy if the problem is caused by the distribution curve of the top teams tailing sharply to the right. . that is, if the #1 team in the country is better than the #10 team by a much wider margin than the #11 team is better than the #20 team.   

            Assigning lower values to the fifteens and sixteens would be the appropriate remedy if the problem is caused by teams that don’t really belong in the tournament getting the 15 and 16 seeds by winning league tournaments in leagues where McDonalds All-Americans are less common than Sasquatch.  

            By experimentation (meaning “Oops, that didn’t work. ..there went a day out of my life”). . .by experimentation I concluded that doing either (1) or (2), by itself, does not work, and leads to unreasonable results later in our project.  Both (1) and (2) above are true, and it is necessary to model both in order to get results we can live with for now. 

            OK, we assign each seed a “value” which is 22 minus their seed, unless they are

seeded one, two, fifteen or sixteen.   If they are seeded one, we enter their value as “30” (rather than 21).  If they are seeded two, their value is 24 (rather than 20).   If they are seeded fifteen, their value is 6 (rather than 7).   If they are seeded sixteen, their value is 3 (rather than 6).  This makes the chart above the same as it was, except for the top two lines:

1 vs. 16            1 wins .990 in theory  (900 to 9)         1 is 96-0 actual   1.000

            2 vs 15             2 wins .941 in theory (576 to 36)        2 is 92-4 actual     .958

            OK, now we have a system that essentially predicts first-round outcomes, based just on the seeds, about as well as you are realistically going to predict them.    Extrapolating from this based on the same assumptions, we can predict the percentages for any matchup.  These would be as follows:

            There are questions here demanding more research.   This chart predicts that one seeds would beat eight or nine seeds 82-84% of the time in the second round—but do they really?   I don’t know.   Intuitively it sounds about right, but I don’t really know.  I’m trying to trigger a discussion here; if I stop to research all of these things now the tournament will be over before I give anybody else a chance to participate in the discussion. 

            However, assuming that our heuristic works, we can then answer any number of questions which can’t be answered by studying the tournament itself, since the history of tournament is not thousands of years long.   For example, what is the probability that a #16 seed would make the Final Four?    According to my model, it is .000 000 433, or one in 2.3 million.  

            According to my model, the chance that a #3 seed makes the Final Four is .099 049, or essentially one in ten.   We work that out in this way.   In the first round the three seed plays the fourteen seed, and we figure (above) that the three seed wins 84.9% of the time. . .actually, 84.9411764705882%.   In the second round the three seed plays the winner of the six/eleven game.   67.9% of the time this will be the six seed, and when it is the six seed the three seed should beat them 58.5% of the time.   32.1% of the time this will be the eleven seed, and when it is the eleven seed the three seed should beat them 74.9% of the time.    Thus, the chance of the #3 seed winning a second-round game is

            .849 *  (.679 * .585    +   .321 * .749)

Which is

            .849  *  (.3972  +  .2404)

            .849  *  (.6376)

Working this out (with more decimals) we conclude that a three seed should win a second-round game 54.2% of the time—or 63.8%, if we assume that they won the first-round game.  

            It gets more complicated after that.  IF the three seed does beat the six/eleven winner in the second round, they could face, in the third round, the 2, 7, 10 or 15 seed.  In the fourth round they could face the 1, 4, 5, 8, 9, 12, 13 or 16 seed.   Still, this is what spreadsheets were invented for, and our model predicts the following:

            One thousand  #1 seeds should go

            990-10 in the first round

            823-167 in the second round (into the Sweet Sixteen)

            631-192 in the third round (into the Regional Championship Game)

            435-196 in the fourth round (into the Final Four)

            266-169 in the fifth round (into the Championship Game)

            155-111 in the Championship Game

            In other words, a #1 seed as a  43.5% chance to be in the Final Four and a 15.5% chance to win the NCAA title, according to my model. 

            One thousand #2 seeds should go

            941-59 in the first round

            707-234 in the second round (into the Sweet Sixteen)

            474-233 in the third round (into the Regional Championship Game)

            236-238 in the fourth round (into the Final Four)

            120-116 in the fifth round (into the Championship Game)

            57-63 in the Championship Game

            In other words, a #2 seed has a  23.6% chance to be in the Final Four and a 5.7% chance to win the NCAA title, according to my model.

One thousand #3 seeds should go

            849-151  in the first round

            542-307  in the second round (into the Sweet Sixteen)

            252-290  in the third round (into the Regional Championship Game)

            99-153  in the fourth round (into the Final Four)

            40-59  in the fifth round (into the Championship Game)

            15-25 in the Championship Game

            A #3 seed has a 1.5% chance (or 1 in 68 chance) to win the National Title.

One thousand #4 seeds should go

            800-200 in the first round

            471-339 in the second round (into the Sweet Sixteen)

            156-315 in the third round (into the Regional Championship Game)

            71-85 in the fourth round (into the Final Four) 

The percentage goes up here because the #4 seed plays the #1 seed, if at all, in the third round.   If they reach the fourth round they would always be playing a team that was no better than a two seed.

            27-44  in the fifth round (into the Championship Game)

            9-18 in the Championship Game

            The odds against a #4 seed winning the National Title are about 107 to 1.

            One thousand #5 seeds should go

            743-257 in the first round

            396-347 in the second round

            122-274 in the third round

            52-70 in the fourth round (into the Final Four. . .5.2%).

            18-34 in the national semi-final

            6-12 in the Championship Game

            The odds against a #5 seed winning the National Title are about 168 to 1.

            One thousand #6 seeds should go

            679-321 in the first round

            321-358 in the second round

            124-197 in the third round

            40-84 in the fourth round (into the Final Four. . ..4%)

            13-27 in the national semi-final

            4-9 in the Championship Game

            The odds against a #6 seed winning the National Title are about 250 to 1.

            One thousand #7 seeds should go

            610-390 in the first round

            192-418 in the second round (usually dropping to the #2 seed)

            87-105 in the third round

            26-61 in the fourth round (into the Final Four. . ..2.6%)

            8-18 in the national semi-final

            2-6 in the Championship Game

            The odds against a #7 seed winning the National Title are about 500 to 1.

            One thousand #8 seeds should go

            537-463 in the first round

            100-437 in the second round

            43-57 in the third round

            15-28 in the fourth round (into the Final Four. ...1.5%)

            4-11 in the national semi-final

            1-3 in the Championship Game

            The odds against a #8 seed winning the National Title, in my model, are about 1000 to 1.  

            We will pause here to note that a #8 seed (Villanova in 1985) DID win the National Title.   My model says that it’s a 1000-to-1 shot.   There have only been 96 #8 seeds since the current system (excepting the play-in game, to which I refuse to pay any attention) began.   One of those 96 did win.   Doesn’t this suggest that my model is wrong?

            Well, yes and no.   First of all, OF COURSE my model is wrong; I’m just experimenting here, offering theoretical answers to questions to which the exact answers cannot be known.   A model is by definition much simpler than real life.  I am assuming that all #1s are the same and all #8s are the same.   They’re not.   This will effect the outcomes.

            On the other hand, a sample of one is not anything for us to worry about very much.   Our model predicts that SOME lower seed, sometime, will win—a seven, an eight, a nine, a ten.   An eight has won, but no 7s, 9s or 10s.  The eight stands in for the group.  Moving on. . .

            One thousand #9 seeds should go

            463-537 in the first round

            77-386 in the second round

            30-47 in the third round

            9-21 in the fourth round (into the Final Four. . ..0.9%)

            2-7 in the national semi-final

            1-1 in the Championship Game (actually 0.5 – 1.8)

            The odds against a #9 seed winning the National Title are about 2000 to 1.

            One thousand #10 seeds should go

            390-610 in the first round

            92-298 in the second round

            32-60 in the third round

            7-25 in the fourth round (into the Final Four. ...0.7%)

            2-5 in the national semi-final

            0-2 in the Championship Game

            The odds against a #10 seed winning the National Title are about 3300 to 1.

            One thousand #11 seeds should go

            321-679 in the first round

            100-221 in the second round

            24-76 in the third round

            5-19 in the fourth round (into the Final Four. . ..0.5%)

            1-4 in the national semi-final

            0-1 in the Championship Game

            The odds against a #11 seed winning the National Title are about 6600 to 1, but here we have something interesting.   Don’t you hate it when you are rooting for a team who is 3^rd-4^th in their conference, and they draw a #9 seed?    Why’d you have to make them a #9 seed, for Christ sake?  Why couldn’t you have made them a #11 seed?  

The eleven seed gets a theoretically tougher matchup in the first round, but not really. . .a six versus an eight.   A six, in general, is not significantly tougher than an eight.   But the nine seed gets the one seed in the second round, whereas the eleven seed, if they beat the six, gets the winner of the 3 v. 15 game—no higher than a three seed.  

So the eleven seed, intuitively and according to my model, actually has a much better chance of winning multiple games in the tournament than does the nine.  The water runs uphill here for just a moment.  A #9 seed has a 7.7% chance to win two tournament games, a #10 seed a 9.2% chance, but a #11 seed a 10.0% chance.   Total expected tournament wins are higher for nine seeds than elevens, but if your goal is to win two games in the tournament, it’s better to be an eleven than it is to be a nine.

One thousand #12 seeds should go

257-743 in the first round

77-180 in the second round

11-66 in the third round

2-9 in the fourth round (into the Final Four. . .0.2%)

0-2 in the national semi-final

The odds against a #12 seed winning the National Title are about 17,000 to 1.

One thousand #13 seeds should go

200-800 in the first round

56-144 in the second round

7-49 in the third round

1-6 in the fourth round (into the Final Four. .. 0.1%)

0-1 in the national semi-final

The odds against a #13 seed winning the National Title, in my model, are about 47,000 to 1.   We can safely assume that the real-life odds are

1)  better than that, but

2)  not real good, either.

One thousand #14 seeds should go

151-849 in the first round

37-114 in the second round

5-32 in the third round

1-4 in the fourth round (into the Final Four. . .. 0.06%)

The odds against a #14 seed winning the National Title are about 137,000 to 1.

One thousand #15 seeds should go

59-941 in the first round

10-49 in the second round

1-9 in the third round

0-1 in the fourth round

The odds against a #15 seed winning the National Title are about three million to 1.

One thousand #16 seeds should go

10-990 in the first round

0-10 in the second round

The odds against a #16 seed winning the National Title, in my model, are about eight billion to one.

Expected tournament wins per seed:

#1 seed   3.30

#2 seed   2.54

#3 seed   1.80

#4 seed   1.53

#5 seed   1.34

#6 seed   1.18

#7 seed   0.92

#8 seed   0.70

#9 seed   0.58

#10 seed  0.52

#11 seed  0.45

#12 seed  0.35

#13 seed  0.26

#14 seed  0.19

#15 seed  0.07

#16 seed  0.01

            One thing that is difficult to incorporate in a model is the chance that the committee will misjudge a team, and that a team will simply be much better than their seed indicates.   One could incorporate that in a model, but it is beyond the sophistication of this model at this time.

Bill James

Brookline, Mass

March, 2008