Drafting QBs 2

My last post looked at how likely NFL quarterbacks are to become Pro Bowl selectees based on their draft order. This post will directly analyze QB performance by draft round and order based on passing stats. Ultimately, the passing stats will be converted into average expected team wins through linear regression. The result reveals how many more wins per year a 1st round pick could be expected to deliver than a 2nd round pick, or the 1st QB taken over the 2nd.

How QB Performance Was Measured

Passing performance is measured by career average yards per attempt and interception rate. These stats are combined into a single measure, commonly referred to as Adjusted Yards Per Attempt (AdjYPA). AdjYPA basically turns every interception into a -45 yard pass play. This is a commonly accepted equivalence based on research going back to a 1988 book called The Hidden Game of Football. It's also somewhat intuitive--An interception basically precludes the possibility of a punt, which is typically about 45 yds before the return. Additionally, 10 yards of passing is added for every touchdown thrown. This adjustment is intended to compensate for the truncation of the field and the added difficulty in the compressed field near the goal line.

The Data

Data were obtained from Pro-Football-Reference.com. Quarterbacks drafted from 1980 through 2000 in the first 7 rounds were studied. Players from round 8 or later were excluded because the current draft is limited to 7 rounds, and the point of this analysis is prescriptive. Supplemental picks were not included because the focus here is on draft day itself.

Adjustment for Era

Because the passing game in the NFL has evolved over the years (it's become steadily easier), an adjustment was made for year. Average AdjYPA has increased signficantly between the draft classes of 1980 and 2000. The 1980-1983 draft classes averaged 5.6 AdjYds per attempt compared to an average of 6.0 AdjYds per attempt for the 1998-2000 draft classes. The increase overall in the NFL was a fairly steady 0.13 yds per year. The midpoint of each QB's career was used to adjust all QB's stats as if they played in 2004.

Draftees Without Many Attempts

One big problem about this kind of draft analysis is how to score drafted QBs who never played. If they are assigned zero yards per attempt, an unrealistically low record, it would severely weigh down the averages of all but the very top picks. If they are excluded from the data, then the true expected value of QBs drafted in later rounds would be severely biased upward. QBs who played but did not have many pass attempts are also problematic. They can have erratic stats, some having an insanely high 15 yards per attempt or as low as -20 yds per attempt.

My solution to the problem of QBs with too few attempts was to assign them the stats of the 5th percentile qualifying QB. I chose 200 attempts as the qualifying level because it's where the the stats settle down to reasonably steady and apparently representative levels. The 5th percentile makes sense because it would be unfair to say every QB that didn't get their shot would be as bad the very worst to play. Although many of them would undoubtedly be pretty bad, some of them were simply bottlenecked behind slightly better QBs.

Thus we now have the Wuerffle Line, football's very own version of the Mendoza Line. The 5th percentile falls between Danny Wuerffle and Akili Smith. Even though I think Danny is a good guy, the "Smith-Line" just doesn't resonate.

Wins Added (per year)

Now we have a good measure of QB performance that considers passing efficiency, interceptions, and career era, but we're left with an abstract, indirect stat--AdjYPA. The bottom line for every NFL team is wins and not passing stats, so I converted the AdjYPA stat into expected team wins per season.

I ran a regression of all teams from the 2002-2006 seasons that weighs offensive and defensive passing stats, running efficiency, turnover rates, and penalty rates. The model was very similar to the one I ran here. AdjYPA was highly significant, the residuals were randomly distributed, and the model's overall r-squared was 0.72. By holding equal all the factors other than the passing stats, we can see how much passing contributes to team wins. For every 1 yd increase in AdjYPA by a QB, a team can expect to win an additional 1.4 wins each season.

A reference point is necessary when comparing QB to QB, so I borrowed an idea from baseball sabermetrics--the replacement player. But since there is no replacement QB across 21 years of draft picks, I used the 5th percentile again as the reference point. The number of wins added above the 5th percentile was calculated for each QB draft pick. Instead of "Wins Above Replacement," I termed this stat "Wins Above Basement (WAB)." The reference point itself is arbitrary, but it's needed to compare QBs from one round to another.

The first graph below illustrate how many Wins Above Basement a QB could be expected to have based on draft round. The average first round pick earns about 0.3 wins per year more than the average second round pick.
As you can see, the relationship between round and expected QB performance is very linear. But as the next graph shows, not all first round QBs are created equal. The next graph shows wins added by QB draft order. For example, the first QB taken in each draft earns about 2.4 wins per year (more than the worst QBs), and the second QB taken earns about 2.0 wins per year.
Again, we see a very linear relationship. But there may be something more. There are large drop offs in performance from the 1st QB taken to the 2nd, and from the 2nd to the 3rd. Then from there until the 9th or 10th QB taken, it's pretty random. It appears that if your team doesn't get one of the first two QB picks, it might as well take a chance on a later pick. Chances fall off quickly after the first two QBs that a team will find a franchise player.

Overconfidence

Teams often jockey their draft position, even by just a few spots, to ensure they can pick a particular player. Part of the reason they do this may lie in their overconfidence in their ability to identify the better player, a point made in the Massey-Thaler paper. Sometimes players taken later in the draft turn out to be superior to the players taken early.

The table below lists the probability that each QB taken will end up better than the next QB taken in the draft (based on AdjYPA). For example, the 1st QB taken in the draft has turned out to be better than the second QB taken 81% of the time, and the 2nd QB taken has been better than the 3rd 38% of the time. (Note that some of the lower probabilities are due to the high number of consecutive QBs in the later rounds who did not have enough qualifying attempts. They are considers "ties.")

QB Pick	Pr(Better)
1	0.81
2	0.38
3	0.52
4	0.38
5	0.50
6	0.28
7	0.31
8	0.54

It appears that after the first QB taken, there isn't much certainty among GMs in predicting who will turn out to be the better passer. Teams are jockeying draft position, and paying a price to do so, for very small marginal probability of picking the better player. Perhaps there are times when the quality level drops off sharply between a QB and the subsequent one on the draft board, and trading up makes some sense. But this table should give GMs pause when considering whether to trade away next year's second round pick to move up 7 picks in the first round.

The data really tells two stories. Overall, when taking all 21 years of picks into account, the higher picks tend to have significantly better performance compared to subsequent picks. But when you look at the data in any one year, the differences aren't so clear, especially after the second QB taken.

Years as Primary Starter

Another way to grade draft picks, aside from Pro Bowl appearances or actual performance stats, is their years as their team's primary starter. The table below breaks out the average number of years a QB draft pick will serve as their team's starter by draft round.

Round	Yrs as Starter
1	5.6
2	4.0
3	1.8
4	1.0
5	1.7
6	1.6
7	0.2

Here is the same data, this time broken out by draft order.

QB Pick	Yrs as Starter
1	6.4
2	3.9
3	2.3
4	2.6
5	2.0
6	1.2
7	1.4
8	1.8
9	1.2
10	0.2
11	0.1
12	0.0
13	0.5
14	0.0

Years as starter may be a flawed comparison, however. Opportunity is everything, especially for QBs in the NFL. Top picks would certainly benefit from disproportionate opportunities. The years as starter figures for 1st round picks may be inflated because there are nearly no zeros. Teams might also stick with poorly performing QBs who were top picks longer than they deserved because of sunk costs. I think the better measure is what players do once they get their chance on the field. (Now that I think about it a little more, a better idea may have been to compare "years as starter beyond the first year.")

Drafting QBs

If a team is in the market for a young quarterback, should it trade up and grab someone a spot or two higher on their draft board? Or should they stand pat and take the best QB that falls to their own pick? There are many considerations to weigh, including:

• What will they have to trade to move up in the draft?
• How much cap space will they have?
• How likely is it that a higher rated QB will turn out to be better than another further down the draft board?

Perhaps the last question is the hardest to quantify. As fans, we're frequently reminded by commentators how Tom Brady was a 6th round pick and how Ryan Leaf was rated as highly as Peyton Manning. Ok, but what about the dozens of low round quarterbacks who stunk up the league, or the many first round picks who are now in the Hall of Fame?

Quantifying career-long performance is more difficult than I expected. For starters, QBs playing over different NFL eras need an adjustment to their stats. Second, how do we measure a QB's total performance? Career totals? Yards per attempt and interception rates? Both methods give different results. If we use career totals, a slightly above average QB with a long career might outshine a great QB who retired sooner. But then again, longevity should count for something. If we use rate stats, there are a number of QBs with outlier performance stats simply due to small numbers of pass attempts.

We'd need to use a minimum number of qualifying attempts. But then that defeats the purpose of the analysis. QBs who aren't even good enough to win a starting job should be counted when we're comparing draft outcomes. If they're excluded we'd have a severe selection-bias in the data because only the "diamonds in the rough" of later rounds will remain, obscuring the true worth of a late-round QB.

I'll address those issues in a later post, but for now I'm taking a cheap and easy solution--I'll rate QB draft picks by how many Pro Bowls they've been selected to. This method avoids the problems of normalizing stats across eras and figuring out how to handle QBs without a significant number of pass attempts.

With data from Pro-Football-Reference.com (which has a great draft database) I compiled a list of all QB draft picks from 1980-2000. I chose those years to limit the QBs to the modern era with the post-1978 passing rules, and to give enough time for the class of 2000 to develop and be assessed. I counted how many times each QB had been to at least one Pro Bowl (PB), at least two PBs, and at least three PBs. One PB might indicate a flash-in-the-pan guy, such as Gus Frerotte, but two PBs probably indicates sustained excellence. Three or more PBs would make any GM happy.

The table and graph below show the likelihood that a QB drafted in each round will pan-out and be selected to at least 1 Pro Bowl (1+ PB), at least 2 PBs (+2 PB), and at least 3 PBs (+3 PB). The data set is not large (only 193 QBs total), so the graph contains smoothed lines for a more realistic estimate of future expected performance from each round.

Round	1+ PB	2+ PB	3+ PB
1	0.38	0.28	0.23
2	0.32	0.21	0.16
3	0.13	0.04	0.00
4	0.06	0.03	0.03
5	0.10	0.10	0.10
6	0.14	0.11	0.06
7	0.03	0.00	0.00

Historically, moving up from the second round to the first round for a QB buys you a 16% to 23% improvement in the chance he'll go to at least three PBs. It would also buy you a reduction from a 68% to 62% chance he'll never go to a single PB.

Looking at draft picks by round may not be the best approach. What round someone is selected in may have more to do with team needs and other factors aside from how much potential a prospective QB has. Looking at QBs in terms of which QB pick they were in their draft class might be helpful. For example, although Tom Brady was taken in the 6th round of 1999 and Brock Huard was taken in the 3rd round of 2000, they were both the 7th QB taken in their respective years.

The table and graph below list the same information as above--how often a drafted QB is selected to a PB. This time the QBs are broken out by which QB pick they were in their draft year. Again, smoothed lines are added.

QB Pick	1+ PB	2+ PB	3+ PB
1	0.38	0.38	0.33
2	0.29	0.14	0.05
3	0.14	0.10	0.10
4	0.19	0.10	0.10
5	0.19	0.10	0.00
6	0.11	0.06	0.06
7	0.17	0.17	0.17
8	0.19	0.06	0.00
9	0	0	0
10	0	0	0
11	0	0	0
12	0	0	0
13	0	0	0
14	0	0	0

There appears to be a very large drop-off in expected performance from the first QB taken to the second. A third of the first QBs taken went to three or more PBs, but only 5%-10% of the second through sixth QBs taken went to three or more.

I interpret these results as an indication that unless a team can get the very first QB or at least the second taken in a given year, it shouldn't expect to find a franchise player. It does happen, but it's very rare. From the 3rd QB taken through the 7th, there's no apparent difference. If a team doesn't get one of the first two, it might as well wait until later rounds and take a chance on a later pick or set its sights on a free agent. Meanwhile, it can fill other needs.

Of course, Pro Bowl selection is not the best measure of a QB's performance--it's subjective, somewhat arbitrary, and it's an all or nothing measure. But it does help answer a the question all GMs ask when drafting a QB--how likely is it this guy going to pan out? Next, I'll attempt to quantify just how much better top picks are in terms of team wins.

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Signal vs. Noise in Football Stats

In 2007, the Detroit Lion defense began the first half of the season with 13 interceptions, the most in the NFL. The next best teams had 11. It's reasonable to expect that the Lions would tend to continue to generate high numbers of interceptions through the rest of the season, notwithstanding calamitous injuries.

I wouldn't expect them to necessarily continue to be #1 in the league, but I'd expect them to be near the top. And I'd be wrong. It turns out they only had 4 interceptions in their final 8 games, ranking dead last. So halfway through the season, if I were trying to estimate how good the Lions are in terms of how likely they are to win future games, I might be better off ignoring defensive interceptions.

Although turnovers are critical in explaining the outcomes of NFL games, defensive interceptions are nearly all noise and no signal. Over the past two years, defensive interceptions from the first half to the second half of a season correlate at only 0.08. In comparison, offensive interceptions correlate at 0.27. As important as interceptions are in winning, a prediction model should actually ignore a team's past record of defensive interceptions.

You might say that if defensive interception stats are adjusted for opponents' interceptions thrown, then the correlation would be slightly higher. I'd agree--but that's the point. Interceptions have everything to do with who is throwing, and almost nothing to do with the defense.

This may be important for a couple reasons. First, our estimations of how good a defense is should no longer rest on how many interceptions they generate. Second, interception stats are probably overvalued when rating pass defenders, both free-agents and draft prospects.

I've made this point about interceptions before when I looked at intra-season auto-correlations of various team stats. That's a fancy way of saying how consistent is a stat with itself during the course of a season. The more consistent a stat is, the more likely it is due to a repeatable skill or ability. The less consistent it is, the more likely the stat is due to unique circumstances or merely random luck.

The table below lists various team stats and their self-correlation, i.e. how well they correlate between the first half and second half of a season. The higher the correlation, the more consistent the stat and the more it is a repeatable skill useful for predicting future performance. The lower the correlation, the more it is due to randomness.

Variable	Correlation
D Int Rate	0.08
D Pass	0.29
D Run	0.44
D Sack Rate	0.24
O 3D Rate	0.43
O Fumble Rate	0.48
O Int Rate	0.27
O Pass	0.58
O Run	0.56
O Sack Rate	0.26
Penalty Rate	0.58

In a related post, I made the case that although 3rd down percentage tended to be consistent during a season (0.43 auto-correlation), other stats such as offensive pass efficiency and sack rate were even more predictive of 3rd down percentage. In other words, first-half-season pass efficiency predicted second-half-season 3rd down percentage better than first-half-season 3rd down percentage itself.

But what about other stats? Are there other examples where another stat is more predictive of of something than that something itself? Below is a table of various team stats from the second half of a season and how well they are predicted by other stats from the first half of a season.

For example, take offensive interception rates (O Int). Offensive sack rates (O Sack) from the first 8 games of a season actually predict offensive interception rates from the following 8 games slightly better than offensive interception rates (0.28 vs. 0.27).

Predicting	With	Correlation
D Fum	D Fum	0.33
D Fum	D Sack	0.15
D Fum	D Run	0.12
D Int	D Sack	0.08
D Int	D Int	0.08
D Int	D Pass	0.01
D Pass	D Pass	0.28
D Pass	D Sack	0.26
D Run	D Run	0.44
D Sack	D Sack	0.24
D Sack	D Pass	-0.07
O 3D Pct	O Sack	-0.53
O 3D Pct	O 3D Pct	0.43
O 3D Pct	O Int	-0.42
O 3D Pct	O Pass	0.42
O 3D Pct	O Run	0.08
O Fum	O Fum	0.48
O Fum	O Sack	0.24
O Int	O Sack	0.28
O Int	O Int	0.27
O Int	O Run	0.06
O Int	O Pass	-0.37
O Pass	O Pass	0.49
O Pass	O Sack	-0.33
O Pass	O Run	-0.10
O Run	O Run	0.56
O Run	O Pass	0.00
O Sack	O Pass	-0.40
O Sack	O Sack	0.26
O Sack	O Run	0.03
Pen	Pen	0.58
Pen	D Pass	-0.23
Pen	O Sack	-0.08

There are a thousand observations from this table. I still see new and interesting implications whenever I look it over.

• Having a potent running game does not prevent sacks.
• The pass rush predicts defensive pass efficiency as well as defensive pass efficiency itself.
• Running does not "set up" the pass, and passing does not "set up" the run. They are likely independent abilities.
• Offensive sack rates are much better predicted by offensive passing ability than previous sack rates.
• Defensive sack rate predicts defensive passing efficiency, but defensive passing efficiency does not predict sack rate.
  

We see that many stats, such as passing and running efficiency predict themselves fairly well. But even those stats might be better predicted by using a combination of themselves and related stats. For example, in my previous post I noted how accurately offensive 3rd down percentage could be predicted using passing efficiency, sack rate, and interception rate.

The implications of these auto-correlations are numerous. Team "power" rankings and game predictions (both straight-up and against the spread) rely on a very simple premise--past performance predicts future performance. We now know that's not necessarily true for some aspects of football.

Lions head coach Rod Marinelli might be banging his head against the wall trying to understand how his defense was able to grab 13 interceptions through game 8, but only 4 more for the rest of the season. He's wasting his time. The answer is that in the first half of the season, the Lions played against QBs Josh McCown (2 Ints), Tavaris Jackson (4 Ints), and Brian Griese twice (4, 3 Ints).

Safe Leads in NCAA Basketball

Bill James takes a look at when leads become insurmountable in college basketball. In other words, when should CBS cut away from the UNC-Mt. Saint Mary's game to show us the barn-burner between Vanderbilt and Siena?

James' formula uses the lead in points, who has the ball, and seconds remaining to tell us if the lead is completely insurmountable. Here it is in a nutshell:

• x= (Lead - 3 +/- .5) ² -- [+.5 if winning team has possession, -.5 if not]
  
• If x > time remaining in sec, the lead is insurmountable

Pretty cool. This is the kind of thing James is really good at. Unfortunately, I think he buys into a logical fallacy later in his article. He says that if a team is deemed to be "dead," that is to say too far behind, but it is able to climb back inside the limits of "insurmountability," it doesn't matter. The losing team is still dead.

I'd agree that it is highly unlikely that such a team would win, but I think James has been taken in by the gambler's fallacy. He writes "The theory of a safe lead is that to overcome it requires a series of events so improbable as to be essentially impossible. If the "dead" team pulls back over the safety line, that just means that they got some part of the impossible sequence—not that they have a meaningful chance to run the whole thing."

It seems to me that if a team climbs back into contention, it's in contention. If a sequence of events are independent, it doesn't matter how lucky or how impossible previous events were. They're water under the bridge. For example, (from Wikipedia) the probability of flipping 21 heads in a row, with a fair coin is 1 in 2,097,152, but the probability of flipping a head after having already flipped 20 heads in a row is simply 0.5

The only thing that matters is the current situation. It's like saying, "There's no way they'll hit another 3-pointer. They just hit five in a row. They're due to miss."

What does this have to do with football? It would be interesting to look at something similar in the NFL. When is a lead so safe that a team should stop throwing? Or when is it so safe a team should only throw on 3rd down? And so on. Basically, when should a winning team stop trying to gain a bigger lead and start trying to simply prevent big mistakes?

The Office Pool 3

You might be wondering why I'm interested in NFL pick 'em pools in the middle of March Madness. Well, there are already plenty of statistical analyses on the NCAA basketball tournament. Here are a couple of sites to get your bracket filled out scientifically.

But for now, I've got the luxury of time before the NFL season starts, or even draft season, when I can think through these things. In the last post I looked at using the point spreads as a baseline for picking winners. I looked at the accuracy with which the point spread correctly favored straight-up winners. Over the past six seasons, the spread was accurate about 67% of the time, and no single week showed any statistically significant deviation from the overall average. In other words, the spread is no more or less accurate in early or late weeks than throughout the season.

The reason I analyzed spread accuracy by week was because when you're behind in a pick 'em pool, you'll probably have to gamble on some upsets in order to catch up. I wanted to know if it was to your benefit to go against spread favorites in any particular week. The answer is no.

But what about spread amounts? It certainly makes sense that games with +1 or -1 spreads will be less predictable than games with +14 or -14 spreads. But how much less predictable? Is there a point of inflexion when it never makes sense to go against the spread? Are there situations when it's basically a toss up and the spread is no more accurate the flip of a coin?

Below are the answers. The graph shows the accuracy of the spread in terms of predicting the straight-up winner for each spread amount. Data is from all regular season games from 2002-2007.


As expected, the spread predicts winners more accurately with increasing spread amounts. Note the fan-shaped dispersion of the data points. Some spread amounts are far less common than others. For example an 8-point spread is less common than a 7-point spread. At the least common spread amounts, there are fewer cases and therefore a wider range of accuracies.

Also notice how games with spreads less than 3 points are no better than 50% accurate. I wouldn't expect much better than 50% - 55%, but less than 50% is surprising.

The 'home underdog' phenomenon has been established in previous research. This may be due to many observers underestimating the home field advantage due to weather conditions late in the season. But whatever the reason, the home underdog effect clearly exists. The graph below breaks up the spreads into home underdogs and home favorites.


Notice the 40% accuracy of low-spread home underdog games. When the spread is below +2.5 points, the home underdog is not only likely to cover, but will probably win.

It makes sense to pick upsets in low-spread home underdog games. However, there have only been 70 such cases in the past 6 years, averaging about 11 games per year. If you go for the underdog in all 11, on average that gives you a 2-game edge over someone picking all favorites. It's not nearly enough of an advantage to guarantee you bragging rights around the office, but every little bit helps.

And if you need to catch up in the later weeks, games with low spreads and home favorites aren't much better than 50%. Going with upsets in such games would make sense because it's not a bad gamble and your leading opponent might be playing it safe by picking all favorites.

You'll want to be very careful picking against the favorite in games with spreads more than 3.5 points, for either home underdog or home favorite games. The odds against you climb rapidly beyond that point.

The Office Pool 2

When picking winners in an office pool, I'd guess that most people start with the point spreads, or at least look at the records of each opponent, when making their predictions. Most people have some sort of baseline even if it's not Sagarin, DVOA, or the regression-based predictions on this site.

So I thought it would be interesting to look at the spreads, and how often they're correct in identifying winners. If someone needs to correctly pick a few upsets to win a pool, it might be good to know that some weeks are less predictable than others. You'd ideally want to pick upsets in weeks where the spread is less accurate.

In this installment I'll look at how well the spread does in picking winners by week. My theory was that the spread would be relatively less accurate in the early weeks of the the season when there is less information about team performance. There may also be a high degree of bias towards teams expected to be strong in the pre-season. Week 17 may be inaccurate too, due to the uncertainty of some playoff-bound teams resting their better players. Additionally, late season weather may also contribute to higher uncertainty and less predictability.

Using point spread data from the 2002-2007 seasons obtained here, I analyzed how often the spread was correct. Overall, the point spread favorites win 66.2% of the time. Weekly accuracy ranges from 59.0% in weeks 4 and 9 to 72.6% in week 12. The graph and table below list the weekly averages.

Week	Accuracy
1	63.8%
2	62.5%
3	69.8%
4	59.0%
5	71.3%
6	69.5%
7	61.4%
8	64.6%
9	59.0%
10	61.6%
11	76.3%
12	72.6%
13	62.8%
14	71.6%
15	69.5%
16	63.5%
17	65.3%
Total	66.2%

Although there appears to be substantial differences between some weeks, they are most likely random. The only statistically significant difference between any one week and the season average of 66.2% was week 12 with p=0.04. However, there are 17 weeks, so we should not be surprised to see a week or even two appear significant when there really is no systematic connection (a type I error).

The bottom line here is that no week can be viewed as particularly favorable for picking upsets. If you're behind in your office pool toward the end of the season and need to pick some upsets to make up ground, one week is as good as any other to start getting aggressive.

Next, I'll look at point spreads from a different angle and see how accurate they are at picking winners by the size of the spread. I'll also break it down into two types of games, home-underdogs and home-favorites, to see if there are any inefficiencies in how the spread accounts for home field advantage.

The Office Pool 1

Say I'm in an office pool pick 'em contest. My 10 buddies and I pick NFL winners each week, and the guy with the best record at the end of the season wins. My office mates aren't particularly good at handicapping football games, so I figure that if I pick the consensus favorite in every game (the team favored by the spread), I'll have a great chance to come out on top over the long haul.

My office buddies have access to point spreads too. They tend to look at the spread (or at least look at each team's respective record, which is just as accurate) and then pick a couple upsets each week. Over the past five years, the spread identifies winners correctly 66.2% of the time. So, normally, their upsets would be correct on average 33.8% of the time (100%-66.2%), but they won't be picking upsets in lopsided match-ups. (Although we can't always assume rationality, we will assume sanity.) So in their 34 chosen upsets (2 per week), my buddies will be right 45% of the time on average. I would have a 10% accuracy advantage in those games.

After doing the math, each of my buddies would average a 63.3% accuracy rate (66.2% * 232 games + 45% * 34 games). And I'd average 66.2% accuracy. Man, I can't wait to collect my winnings!

But wait. Because of luck, some would be slightly more accurate, and some would be less accurate. In fact, the only thing that really matters is how well each of them do on the 34 games they deviate from picking the published favorite. In the other 232 games, we'd have identical picks. Of the 34 games in question, each game that one of my buddies gets right, I must have been been wrong. One of my 10 friends needs to be correct greater than 50% of the time in his 34 games to beat me.

The mathematical bottom line is, "How often is someone correct at least in 18 out of 34 trials with a 0.45 probability of being correct in any given trial?" The binomial distribution gives us the answer--it's 22.4% of the time. That's pretty good, right? I have a 77.6% chance of beating any one of my opponents. The only problem is that there are 10 of them.

The chance I would beat all 10 of my buddies is the conjunctive probability of beating one of them. It's 77.6% * 77.6%... and so on, for however many opponents I have. In this case, it's:

0.776 ¹⁰ = 0.079

In other words, my chances of winning the office pool are just 7.9%--significantly less than a fair chance of 1 in 11. That's why just picking the favorites is a bad strategy. I'd actually be better off choosing the less accurate strategy of my buddies. At least then I'd have fair chance at 1 in 11.

I realize that it is counter-intuitive that a strategy that is less accurate overall is better than a more accurate strategy. But in a contest against several opponents, the more risky strategy--with a greater deviation of outcomes--may be best.

Note: Phil Birmbaum points out that the odds of the opponents are not independent of one another, and therefore the simple compound probability I calculated here is far too low. If one opponent happens to beat you, then the other opponents may be more likely to beat you as well, and vice versa. In the end, picking all favorites may be the better play. See his comments for an explanation.