Bracket Pool Builder

[SportsDino]

Many people here have probably heard about the billion dollar bracket challenge issued by Quicken Loans on the Yahoo site. It is a reasonably safe bet that it cannot be won, even if all 15 million players (or more) worked together to create the best possible combination of pools.

At least, the challenge cannot be won purely on the use of average statistics and large numbers alone. The goal of this thread is to devise a strategy that could take down the prize, or at the very least aim for the very lucrative consolation prize of top 20 $100,000 each! The cornerstone of this strategy system will be harvesting as much expert knowledge as possible to restrict the solution space, allowing for each subsequent bracket to gain the most ground possible.

While both of these problems are mathematically hard it is possible that some of the techniques covered by this analysis can be applied to your usual bracket design or small pools where you are allowed multiple entries.

[sterlingice]

I'm curious

SI

[SportsDino]

So how likely is it to pick a perfect bracket, avoiding all the random nonsense that the odds are 9 quintillion?

Well, my baseline starting bracket which uses a sort of historical version of the 'chalk' says you can probably get your initial guesses down to about 2,657,205,000 and work from there.

Why such a massive reduction from the random odds?

Seeds themselves are not random, while not always perfectly set up by the selection committee they are, on average, a reasonable predictor of tourney performance. There also is a tournament history going back almost 30 years with the 64 team bracket, providing a sample of 120 records for each seed. Using this data you can start to formulate what an 'average' year looks like and create a basic model for how far each seed can progress.

Why are there still so many permutations?

The 2.6 billion number is based on picking rules that are valid over 75% of the time in history. Unfortunately, the odds of your average matchup are probably much closer to 55-60%, and there are a LOT of them. To get the few rules we do have there are a few stretches made, however, this gives us a baseline which we can then apply knowledge towards to map it to the particular reality of a tourney year.

In a tournament full of upsets it may seem impossible to pick anything with certainty, but without pruning out the least likely scenarios it will be impossible to even get in the ballpark of a perfect bracket.

So we will introduce the first tool in our bracket picking arsenal, the lock. A lock pick is basically stating that you are no longer considering any alternative outcomes for that pick, there will only be one outcome in every single bracket in the pool. Predicting a lock has risk, if that one upset happens the entire pool loses the perfect game. So we introduce the notion of pool survival percentage.

To calculate survival you basically take the probability that each rule will hold, according to whatever probability model you introduce to the bracket. For now we will work without the real teams in front of us using an imaginary average based on historical records. Other good models may be computer based matchups, crowd-sourced votes, or expert opinion panels, all of which need to assign a percentage that the outcome will occur.

Our first lock pick is the 1-16 first round matchup. The 16 has never won in 120 tries so lets call it a 99.2% in favor of the 1st round seed. Doing that 4 times gives us 99.2% ^ 4, or 96.8% survival.

As you can see the survival percentage is going to quickly race towards zero, still it is a good way to measure how much risk you are taking with each rule.

So what does the lock give us?

Well, the perfect bracket game is really all about combinations, a tourney involves 63 games, each one is a win or loss. This can be easily modeled as a string of binary digits (1 for A wins, 0 for B wins), and within that string we can segment it into regions that represent different collections of games. What a lock does is it will essentially 'lock the bit', no matter what that bit will always have a certain value no matter what the rest of the bits look like. This essentially removes a game from the tourney, cutting the solution size in half. Doing that 4 times means we have cut the solution down to 1/16th the previous size and only gave up 3% survival percentage to do so.

Eliminating combos from the pool entirely is the focus of this strategy, getting rid of the least likely scenarios allows our pool to cover more of the most likely scenarios.

So we have taken our first step, and we just made a no-brainer decision that everyone will agree is bracketology 101 and got rid of a lot of riff-raff brackets.

[SportsDino]

While locks are the most powerful tool for cutting down the solution space the number of historic locks is very limited and will not be enough to completely narrow the pool. The second tool is the 'ceiling' which involves a slightly more complex model of the solution space.

Since every game is a binary decision, either A or B wins, while you can represent every unique bracket with a 63-bit string it can also be represented as a binary tree of depth 6 rooted at the championship game (with that decision resulting in the winner). The binary tree model allows us to introduce the following basic equation:

The number of possible outcomes from any node in the tree is twice the product of the possible outcomes of the child nodes of that node, or:

O = 2* C1 * C2

Building up by induction, the first round game is considered to have two inputs of 1 each (each team), and there are two results, team A wins or loses. So 2 * 1 * 1 = 2 outcomes. The second round game has two child games from the first round, each has 2 outcomes. All possible combos of those two games involves a 1 or 0 from the A game, a 1 or 0 from the B game, or 4 possible combos. Regardless of how the combos came in, the second round game itself is a win/lose decision, so 4 inputs each with 2 outputs causes there to be 8 possible outcomes, or 2 * 2 * 2 = 8.

This process continues all the way to the championship which calculates to the true probability of the tourney, 9 quintillion.

What this allows for is the calculation of independent sub-trees, with each sub-tree by depth being more closely related. Whether Florida makes the championship game has very little to do with Michigan this year, but it has a heck of a lot to do with UCLA. Reduction of a sub-tree carries forward to the rest of the tree, you may not be able to predict much about the B inputs, but in the 1-16 matchup you can guarantee the A input, so the second round game looks like 2 * 1 * 2 = 4 instead of 8. This matches our earlier result, every lock cuts the field in half.

The ceiling is a natural result of this new structure. For each team there are a string of games between it and the championship, and each of these have a probability of falling in favor of that team. Also, historically there are records for how far each seed has made it in the tournament, for instance a 15 seed has never made the championship game. A ceiling is that for a particular team you predict that it cannot make it past a certain round. By doing so, although you cannot fix the inputs to that round, you can reduce the outcomes from that round (every combo where that team could have won becomes a guaranteed loss).

For instance, if you have a 3-14 matchup and you choose not to lock it so the 14 always loses (14.7%). However, you may decide that in all of history a 14 seed made it past round 2 only 2/120 times (1.7%) so you want to put a ceiling on the 14 seed. The inputs to the second round are 2 and 2, however the outcomes are that every combo the 14 seed wins the first game it will always lose the second game. Out of 4 inputs, half involve the 14 seed, and normally they would generate wins and losses. Now you can remove the win possibilities (2). So the possible outcomes of round 2 are now 2 * 2 * 2 - (2 * 2 / 2) = 8 - 2 = 6.

Ceiling reductions are smaller than locks, however, constraining options within each sub-tree gets multipled at each depth level, so those 2 eliminated possibilities can end up covering quadrillions of combos eliminated from the pool (assuming no other reductions).

It can also be considered in terms of a prime factorization, the total outcomes of the pool is now 2 * 3 * factor(rest o' tree) instead of 2^3 * factor(rest o' tree). Since powers of two are extremely common while performing these calculations by hand I usually left the tree in prime factor form for speed.

The ceiling effect changes based on locks or other ceilings involved, to calculate the effect you always subtract from the total number of outcomes the number of input strings that contain the team you are restricting. This can be done recursively by building up the number of inputs from the first round leaf of the tree to the current node where the ceiling hits, multiplying by the size of the other sub-tree.

Since this involves a subtraction it will break the prime factorization, but often it can be refactored, and an automatic tool would eliminate manual calculation as a concern anyway.

Using the lock and ceiling exclusively we can create the baseline bracket based off historical results which reduces the solution space to 2.6 billion.

[sterlingice]

A suggestion when this is all done: apply these rules to all brackets since the tournament went to 64 teams and see what we get.

There is one problem, of course, and it's that every time there is a stat like "X has never happened", it's basically a retrospective study. I appreciate that you take this into account (making a 16 seed 0.8% likely to win and not 0%) because we get a lot of nonsense this time of year about "all teams have have won the championship have these characteristics". Unfortunately, this really is a lot of correlation but not necessarily causation because of the small sample size of "NCAA tournament winning teams".

SI

[SportsDino]

Now it is time to reveal the baseline bracket, which will immediately cause a firestorm of trouble and doubt which I will then explain away a bit at a time.

Since every tree is independent, and we are not dealing with real teams, the baseline bracket focuses purely on a single region, ignoring the rest of the tree.

We can calculate the top of the tree, which represents the Final Four or outside the region, by using the earlier described relationship. The input into each Final Four is two region outcomes (R). Assuming R is identical, each Final Four match up is 2 * R * R = 2R^2. The championship is 2 * 2R^2 * 2R^2 = 8R^4.

Now we just need to figure out the value of R, trivially it represents a 16 team tournament, or 2^15 = 32,768. But that puts us back into the quintillions!

I believe the following rules generate an R value of 135 (manually calculated so I may be off slightly):

1-16 = Lock for 1 = 99.2%
2-15 = Lock for 2 = 94.0%
3-14 = Lock for 3 = 85.3%
4-13 = Lock for 4 = 78.4%
1-8/9 = Lock for 1 = 85.3%
2-7/10 = Lock for 2 = 68.7%
3 Ceiling at Round of 8 = 80%
4 Ceiling at Round of 8 = 88.3%
5 Ceiling at Round of 8 = 93.3%
6 Ceiling at Round of 16 = 83.3%
11 Ceiling at Round of 32 = 87.5%
12 Ceiling at Round of 32 = 83.3%

Survival of Pool: 4.5% (48 coin flip decisions)
Pool Size: 2.6 billion

Of course, the first thing you will say is that pool has been busted every year the tourney has existed! You can't go around saying teams always win or always lose, that is an abomination!?!?!?!

All this is trying to do is lean as many coin flips our way as possible, would you rather be betting on an 80% probability or supporting a 60/40 split 48 times (survival 0.000000002%)?

Even with all of these outright eliminations the pool size is still massive. Essentially all we have proven is that it really is impossible to cover the bracket challenge on probability alone.

It really is not intended to play the pool as it is, what this baseline is aiming to do is bias the pool away from known bad bets and encourage decisions to deviate from this norm. Those deviations are where skill becomes involved, using these equations and tree structure underneath to calculate the impact of a decision and weigh it's risk factor on the pool.

The more locks we can introduce in a particular bracket year, the better this calculation will perform. For instance, this year you can say Florida and Arizona might be your strong 1 seeds, you lock them with confidence to beat the 8/9, and if you feel they have a weak 4th matchup you may lock that game as well. That extra lock is an advantage over the baseline.

On the other hand you might consider Wichita State your weak 1 seed, while assigning it the automatic 1-16 match you remove the lock against the 8/9. This is a disadvantage against the baseline, but it adds the potential defeats back into your pool so you can live another day if it occurs. You could go the other way and lock Kentucky for 8-9 and 1-8, which would be a relatively bold prediction (especially given the score). However, this sort of thinking is relatively dangerous, ya you predicted a 1 seed getting knocked off, but what did you prove? Those two locks are no better than predicting any other two locks, and while it is slower a large number of ceiling bets may have given you a similar reduction.

Ideally you should add rules in probability order, don't lock 50/50 games, look for sure bets and large Vegas spreads and cover the coinflips with the pool.

So now that we have the baseline from here the posts will focus on adding new tools to gain an advantage, and how best to target skill picks while avoiding upsets. A common problem that will be faced is trading combos for certainty, one bad decision will cost you the entire pool, but leaving it to a probability sampling of your pool may be the more foolish choice. Particularly if you are playing a small pool for 'second place $100,000'.

[SportsDino]

Ya, historic sampling is not enough to prove anything, as I show by indicating my particular snapshot of history is only right about 4% of the time (actually none of the time, this rule set probably already fails every real tournament).

If I looked up some bracket software with historical results it might be a funny addition to say when a particular bracket set rule failed. I am not quite that dedicated though to do the data entry for 30 years of tournaments.

The only way this gets within the realm of possibility is if you have a really good game picker. And I don't mean no namby-pamby Florida will Pittsburgh 85% of the time... you need someone saying Florida will crush Pittsburgh by 15 and remove it from consideration altogether. The most likely chance to win with a pool is a year where there are overpowered veteran teams and a lot of crap auto-qualifiers. Rule out the people who just don't belong, and bank on the thoroughbreds (to a degree), then use team history to predict my next tool 'contingent matchups'... that is if Team A somehow meets Team B in the bracket they will always crush them.

Some of these tools will seem like they are really biting for scraps, but scraps can lead to thousands or millions of bits of information gain in the solution space. For instance a contingent matchup of Florida/Michigan where Florida always wins, you may just ask yourself well how much does that really help me? Well those are actually two independent sub-trees from the other two regions entirely, so that one contingency eliminating the Michigan win covers about 18,225 entries with R=135.

[SportsDino]

I guess to finish the thought on contingent matchups...

So far this pool has been dealing with outright eliminations of a team from the tree at certain points. This is very good if you consider teams roughly interchangeable in ability and that the survival of weak teams can be predicted simply by how many games they will face. This is not always true, sometimes there are early upsets leaving a bracket as a free for all, sometimes a team just matches up better against some other team, and every once in a while you have a high seed that you just can't believe in.

All of those situations are risk factors for the lock or ceiling. A ceiling is really based on historically a team will be beat up by someone sooner or later, but it is possible the team to beat them somehow got knocked out on a fluke itself.

Now these pools tend to be very fragile towards flukes to begin with, so planning to survive one is not really in the picture unless you have banked enough sure-bets to give you some wiggle room. However, the contingency is a tool to bring back some of those picks.

A contingency reduces a tree at the top, it makes no predictions about the sub-trees, but does reduce the outcomes going forward, having a multiplier effect based on the depth to the contingent matchup.

To calculate it each node along both paths to the contingency becomes a lock, with the value of the other sub-tree into the node being multiplied together. Assuming nothing else a Sweet 16 contingency would occur 4 times out of 64, by predicting the contingency you save 4 outcomes.

Contingencies are most useful when you cannot predict an early round upset comfortably, particularly of a high seed, but you feel it will lose to a higher seed up at the Sweet 16 or higher level. In the event of a couple upsets they can still meet and both outcomes get covered by your bracket pool.

It basically is a tool for hedging around wildcard teams that either have a high propensity for failure (a high seed you think will win but can also blow up like a Puke) or a little explosiveness to get the first weekend but cannot hang with a top level team.

Cross region contingencies are a slightly more likely occurrence, you can't predict who will get to the Final Four, but you can make predictions about what will happen if certain pairs do make it and narrow the field.

While contingencies and high ceilings (up at the Final Four or champ level) have the least impact on the size of the pool, they can be used to buy back a little more space for low level upsets. Contingencies are particularly risky, since the fact they occurred means the two teams battled to a meeting point and it may be closer to a historic coin-flip match. However, sometimes these matchups are easier to predict since the teams may have history or certain playing styles that can help predict a winner (a weak post team against a team with a post superstar for instance).

A contingency generally only makes sense for Sweet 16 and up, in the round of 32 unless you have an extremely odd situation you should probably have a ceiling or nothing at all. It might be used if you have a first round potential upset from a complete team mismatch, but in the second game if the preferred win from the first round gets there you know they can handle the contingency team, whereas you are unsure about the upset special from first round.

Contingencies based on their depth in the tree only impact O / 2^(2d-1) outcomes roughly, so Sweet 16 is 4/128, Elite 8 is 256/32,768, and so on. However, these savings are multiplied up the rest of the tree so again if they are all you can predict they can still provide significant savings. With R=135 a single outcome saved by Elite 8 contingency is worth 19.7 million brackets.

[Young Drachma]

Very cool

[SportsDino]

When I continue I will start to breakdown how a particular region looks as you start making decisions and calculating the resulting combos. For illustrative purposes I will use the current tourney, which will start to show how hard it is to really hedge for the real world while keeping your options covered and the value of skill picks to constrain scenarios.

[sterlingice]

What really is staggering to me is how, even if you can fill in chalk through the Sweet 16, which you can't at all, there's 32K possible combinations after that. So each upset you pick before the Sweet 16, you might as just multiple by 32K to get the number of brackets you need to increase (it's not quite that unless you think no upsets prior to the Sweet 16 will move on but the concept is there).

SI

[SportsDino]

There is no doubt this is a staggering problem given the unpredictability of college basketball. I basically keep circling back around to the same conclusion as I try to design these scenarios, you need to make aggressive predictions and accept that the whole pool cannot cover everything.

It is going to come down to having a really good way of predicting particular matchups. Unfortunately I think that is an impossible problem, it is hard to matchup two teams and predict a winner with even 70% confidence. If you could you would see a lot of large point spread predictions since to predict at that level you have to assume that one team is just going to pull away over the course of the 40 minutes.

The best we can do is target where to invest those matchup decisions as accurately as possible. To really make this happen you need a lot of pre-bracket scouting of the teams during the year so you gather a good analysis of their strengths and weaknesses. You need to predict as well as you can how a team can matchup with any other team you have scouted.

If you can pick a favorite to reach the championship game, and can restrict the Final Four to about 2 contenders per region, you can chop a lot of possibilities from the top of the tree. So after the region analysis post I will probably take a tangent off into game prediction, even though that is my weakest area it is of course the most important.

[sterlingice]

Unfortunately, in a year where a 7 plays an 8, it's almost impossible to pick that bracket as it would have required so many high ceilings that the potential pool would be ridiculous.

SI