Sports.ws Knowledge Base .: The Math Behind Drafting (Advanced)

Your league is ready to draft and you're looking for a leg up on the competition. Draft time is special because it's the only time where everybody sees something of value in another team. Certain rivals might never obtain a player you want, but when the draft starts, everybody has 12 reserved picks.

Because of the level playing field, it's the best time of year to objectively analyze the "value" of a player. With a few trivial assumptions, potential player value can be derived with a few mathematical formulas. You might a few scenarios surprising, and if the rest of your league doesn't, you may be able to capitalize on them via trades and savvy roster moves.

Overview

There are two things in a league size that factor into a player's worth:

Player efficiency is more important in smaller leagues, while larger leagues favor players that play a lot of minutes.
Larger leagues have smaller free agent pools, vastly affecting multiple-pick comparisons.
Certain late-round picks can be considered less valuable if it is less likely they will use up actual fantasy minutes. League size determines this likelihood.

FPPG versus FPPM

There have been 29 teams in the NBA the past 5 years. This means there are only 12 x 29 players (players per team x 29 teams) playing on any given night and no more than 5 x 48 x 29 minutes (players on court x minutes in game x 29 teams) being used.

The larger your Sports.ws league gets, the more using up all your player minutes becomes a priority. In a league of 32 for instance, only there are only enough NBA minutes (5 x 48 x 29 = 6960) to fill 90.4% of fantasy roster minutes in any given game. If you use up all your player minutes in a league this big, you're already more efficient than most in your league. For this reason, fantasy points per game (FPPG) are important because they indicate how much a player can produce in a game, regardless of how efficient they are each minute they're playing.

On the other end of the spectrum, a 6 team league only requires 5 x 48 x 5 = 1440 minutes to fill all teams' fantasy player minutes. This means it should be less likely that minutes go unused, so more emphasis needs to be put on player production and efficiency when evaluting. For this reason, fantasy points per minute (FPPM) are important because if two teams both use all their minutes, the team with the highest FPPM will yield the highest total score.

There are 30 teams this year, but the same principles apply.

Real world example:

Let's say there is an aging center (player A) that plays about 15 minutes a game. He's very effective when he's in the game and pulls down a lot of rebounds and points, but his knees and lungs give out pretty quickly.

Player A's Stats

Average minutes per game:	15
Average FPPM:	0.85
Average FPPG:	0.85 x 15 = 12.8

Compare him to a young small forward (player B) who can play a full 48 minutes without getting winded. He's used often because he doesn't tire, but he is usually a role player and doesn't get a lot of points.

Player B's Stats

Average minutes per game:	30
Average FPPM:	0.65
Average FPPG:	0.65 x 30 = 19.5

Who's better? It depends on the size of the fantasy league.

In a 6 team league (smallest offered by Sports.ws), it would be easy to find plenty of players that play 30 minutes a game and do better than 0.65 FPPM, so player B is almost useless. 0.85 FPPM is very high, so it would likely be better to take player A because the rest of the team would likely fill up the minutes not played.

In a 32 team league (largest offered by Sports.ws) either player would be a good pick because choices are slim, but player A only plays 15 minutes a game. Since many teams won't fill their fantasy minutes, it would probably be worth taking a player that puts up 19.5 FPPG instead of 12.8 FPPG (player B instead of player A) because minutes will probably be left over.

Of course, taking 12 players based only on FPPM in a small league might be a bad idea because you still want to fill up minutes. Likewise, taking 12 players based only on FPPG in a large league might not be good because you could end up with too many unused minutes. In any league, it is a good idea to weigh both.

The math:

Determining how much to weigh FPPG againt FPPM for a given player is hard to gauge. Is there a mathematically way to determine how much to consider each based on the size of the league? Nothing is perfect because there is a large number of unpredictable variables (the variables that make the game fun), but based on a variety of patterns from history (statistical and owner-behavioral), a basic mathematical rubric of sorts can be derived.

First, it is easiest to make a few assumptions. Obviously, these assumptions will almost never hold completely true, but they are the standard for which owners strive.

1) Every owner in your league will pick the best player available at every draft pick.

2) Every owner in your league will plan far enough ahead that they don't have to take a less valuable player because they lack depth at a certain position.

3) Every owner will maintain the perfect balance of players that are highly efficient vs. players that play a lot.

Based on these assumptions and historical statistical trends, we can estimate how the average player would perform in a fantasy league based on its size.

League Size	NBA Players Needed to Fill Teams (12 per)	Average Drafted Player Peformances
League Size	NBA Players Needed to Fill Teams (12 per)	Avg. FPPG	Avg. FPPM	Avg. Games Played	Avg. Minutes Played
6	Top 20%	28.6	0.82	81.6	2847
8	Top 27%	26.7	0.79	81.0	2716
10	Top 33%	25.1	0.77	80.3	2600
12	Top 40%	23.8	0.75	79.5	2488
14	Top 47%	22.5	0.74	78.5	2374
16	Top 53%	21.4	0.72	77.5	2278
18	Top 60%	20.3	0.71	76.2	2174
20	Top 67%	19.3	0.70	74.7	2071
22	Top 72%	18.5	0.68	73.2	1977
24	Top 80%	17.6	0.67	71.6	1881
26	Top 87%	16.8	0.66	69.7	1784
28	Top 93%	16.1	0.65	67.6	1694
30	100%	15.3	0.64	65.2	1607
32	100% divided by 107%	14.7	0.62	62.6	1522

League Size	NBA Players Needed to Fill Teams (12 per)	Average Free Agent Performances
League Size	NBA Players Needed to Fill Teams (12 per)	Avg. FPPG	Avg. FPPM	Avg. Games Played	Avg. Minutes Played
6	Top 20%	22.6	0.72	80.3	2457
8	Top 27%	19.7	0.70	78.5	2241
10	Top 33%	17.7	0.67	76.8	2036
12	Top 40%	15.7	0.65	74.0	1808
14	Top 47%	14.3	0.64	71.0	1633
16	Top 53%	12.9	0.62	68.3	1477
18	Top 60%	11.7	0.60	65.0	1271
20	Top 67%	10.5	0.59	59.5	1077
22	Top 72%	9.5	0.57	55.5	893
24	Top 80%	8.5	0.56	50.8	732
26	Top 87%	7.5	0.54	44.0	593
28	Top 93%	6.4	0.52	35.5	448
30	Top 100%	5.5	0.50	27.5	301
32	100% divided by 107%	4.6	0.47	20.0	174

Given this formula, we can estimate an ideal weighing of FPPG vs. FPPM by league size:

League Size	% FPPG	% FPPM
6	24%	76%
8	26%	74%
10	28%	72%
12	30%	70%
14*	33%	67%
16	36%	64%
18	39%	61%
20	42%	58%
22	46%	54%
24	49%	51%
26	53%	47%
28	57%	43%
30	61%	39%
32	65%	35%

*Real world example:

In a league of 14 teams, factoring a player's FPPM is about twice as important as the FPPG, all other factors being equal.

If we continue to analyze historical trends, we can find that the median player's FPPG is about 28 times the median player's FPPM in a relatively small league, and as low as 26 in a large league. This means we can actually add a equalizing coefficient to a formula involving FPPM to compare it directly against FPPG. Instead of guessing what "twice as important" might mean, we can determine exactly what it means.

At this point, we can come up with a crude rating for a player based on the size of the league:

League Size	Rating (R) Formula
6	R = (0.23 * FPPG + 21.6 * FPPM)
8	R = (0.25 * FPPG + 20.8 * FPPM)
10	R = (0.27 * FPPG + 19.9 * FPPM)
12	R = (0.30 * FPPG + 19.0 * FPPM)
14	R = (0.33 * FPPG + 18.2 * FPPM)
16	R = (0.36 * FPPG + 17.3 * FPPM)
18	R = (0.39 * FPPG + 16.4 * FPPM)
20	R = (0.42 * FPPG + 15.5 * FPPM)
22	R = (0.45 * FPPG + 14.5 * FPPM)
24	R = (0.49 * FPPG + 13.2 * FPPM)
26	R = (0.53 * FPPG + 12.0 * FPPM)
28	R = (0.57 * FPPG + 10.8 * FPPM)
30	R = (0.61 * FPPG + 9.6 * FPPM)
32	R = (0.65 * FPPG + 8.3 * FPPM)

This may help in comparison, but it leaves out one variable: the amount of games a player plays in a season. A player isn't worth anything if he has a high R if he only plays a few games a year. Injuries aside, star players (conveniently the same ones that have high FPPG and FPPM) play more games per year than others. Sometimes a bench player isn't needed; other times, NBA rosters are moved around so that a player might get waived and picked up on and off throughout the year, even if they're healthy.

Since the average percentage of games played in our model only variables between league sizes, we're not going to be adding any value to our ratings formula.

But, that's because we're only comparing only two players at a time. A common question is "What's the number one pick worth?". Often, a number one pick will get traded for two or three other picks. Can we use our ratings formula to determine what multiple players are worth?

Well we know one thing: two players are not always better than one. In fact, it's not even close. There are a limited amount of minutes in a fantasy game, so two players need to be pretty useful if they're going to burn up the minutes twice as fast as one player. Also, the biggest thing overlooked in multi-player trades is the value of the best free agents.

Real world example:

Let's someone wants to give you their 10th, 11th, and 12th round picks for your 5th. While this might sound tempting, consider the true value of a player picked in rounds 10, 11, or 12. If every owner is picking well, these players should only be a fraction better than the best free agent available when the draft is done. It doesn't matter if the 10th, 11th, and 12th picks' FPPG or FPPM add up to that of the player picked in the 5th. There won't be enough minutes to go around, and a free agent of comparable value might be available.

Comparing Multiple Picks

If you're making free agent pickups throughout the year, you usually stop when you feel all the players on your team are better than all free agents. At that point, free agents are worth zero to you. Let's extend that notion and add another piece to our formula. We subtract the best free agent rating R, which varies dependent on league size.

Since we can also now include the average number of games played and have it make an effect on multi-player comparisons, we do that as well.

League Size	Rating (R) Formula
6	R = (0.23 * FPPG + 21.6 * FPPM) * GP % - 20.8
8	R = (0.25 * FPPG + 20.8 * FPPM) * GP % - 18.7
10	R = (0.27 * FPPG + 19.9 * FPPM) * GP % - 17.1
12	R = (0.30 * FPPG + 19.0 * FPPM) * GP % - 15.5
14	R = (0.33 * FPPG + 18.2 * FPPM) * GP % - 14.2
16	R = (0.36 * FPPG + 17.3 * FPPM) * GP % - 12.7
18	R = (0.39 * FPPG + 16.4 * FPPM) * GP % - 11.1
20	R = (0.42 * FPPG + 15.5 * FPPM) * GP % - 9.6
22	R = (0.45 * FPPG + 14.5 * FPPM) * GP % - 8.3
24	R = (0.49 * FPPG + 13.2 * FPPM) * GP % - 6.7
26	R = (0.53 * FPPG + 12.0 * FPPM) * GP % - 5.1
28	R = (0.57 * FPPG + 10.8 * FPPM) * GP % - 3.5
30	R = (0.61 * FPPG + 9.6 * FPPM) * GP % - 2.2
32	R = (0.65 * FPPG + 8.3 * FPPM) * GP % - 1.2

GP % = Average Percentage of Games Played

Let's graph pick number against the average player rating based on stats from the last five years for three differently sized leagues (8, 16, and 24 teams).

Filling Minutes and Predictable Owner Behavior

Superstars play more minutes per game than others. They play because they're good - there is no mathematically correlation or reason. But, it doesn't mean we can't analyze the trends to help predict upcoming player value.

Consider this: in a 6-team league draft, if every pick was perfect, each team would have enough players to use all their minutes by pick 45, early in round 8. In a 24-team league draft, team minutes could be filled by pick 255, mid-11th round.

This means that if the top players were all selected in order and nobody got hurt and played as expected, you wouldn't need any of the players past this "minutes-filled" pick. Neither would ever happen, but it brings up an interesting and possible fruitful idea. Does the value of picks after the "minutes-filled" pick diminish?

First, let's look at when each league size hits this magic pick:

League Size	"Minutes-Filled"
6	Pick 45
8	Pick 63
10	Pick 81
12	Pick 102
14	Pick 125
16	Pick 147
18	Pick 169
20	Pick 198
22	Pick 218
24	Pick 255
26	Pick 278
28	Pick 288
30	Pick 299
32	Pick 308

Second, let's figure out if the likelihood of a perfect draft happening is worth investigating this concept further. During the past five fantasy seasons (same seasons that these stats are based on), my own league (18 or 20 teams) has correctly drafted 74% of players up until the "minutes-filled" pick. By "correct", I mean a player finished the season ranked above the "minutes-filled" number and was also drafted above that number. Sleepers, surprise free-agents, injuries, trades, abnormal rookie behavior, and yachting vacations (among other things) prevented us from guessing perfectly. While I'd like to think we're pretty good, we're probably average for a competitive league.

During the same time, I calculated how much time was unused for all the players picked before the "minutes-filled" number, regardless of team, position, etc. We totaled about 70% usage, which in some places would deserve a C-.

Can anything be done? Let's add a factor to picks that occur after the "minutes-filled" pick based on these numbers. Let's say the worst we'll punish a pick is 70%, since that's what our usage was. Since we only got 74% right, we'll need to have a window from the "minutes-filled" pick to 1.35 x "minutes-filled" pick for each league size (1 / 74% = 1.35). Using a similar exponential slope to determine this negative/punishment coefficient, we can slowly subtract value from each pick after the "minutes-filled" pick until that 1.35 is reached. At the point and beyond, only 30% of the stats are counted.

This logic tends to affect smaller leagues much more than larger leagues (if it affects them at all), which makes sense. If you've got an all-star lineup in a 6-team league, the 11th and 12th players are not only worth less for all the reasons derived above, but also because you'll likely never use their minutes, even with a few injuries and bad picks.

To give you a feel for how your league size compares, here are all the even-sized leagues, they're graphs (value against pick number). The thinner line represents a graphical representation of the formula in practice, without the last "minutes-filled" logic. The thicker line includes the logic.

Summary

You've read this far, maybe you've made sense of this and actually agreed with all of it. How does it help you?

Here are some great comparisons (based on these formulas) that you might be able to use in trade proposals before any players are taken:

What does trading the number 1 pick do for me?

If you're looking for two or three picks close together in exchange, you should get:

League Size	Two Close Picks	Three Close Picks
6	Picks 9, 10	Picks 19, 20, 21
8	Picks 12, 13	Picks 28, 29, 30
10	Picks 14, 15	Picks 37, 38, 39
12	Picks 18, 19	Picks 46, 47, 48
14	Picks 24, 25	Picks 56, 57, 58
16	Picks 28, 29	Picks 68, 69, 70
18	Picks 33, 34	Picks 78, 79, 80
20	Picks 38, 39	Picks 88, 89, 90
22	Picks 44, 45	Picks 97, 98, 99
24	Picks 50, 51	Picks 110, 111, 112
26	Picks 57, 58	Picks 121, 122, 123
28	Picks 63, 64	Picks 132, 133, 134
30	Picks 68, 69	Picks 138, 189, 140
32	Picks 71, 72	Picks 144, 145, 146

A classic question: how much better is first pick in the first round and last in the second than last in the first and first in the second?

League Size Picks Worth
6 Picks 1, 12 Picks 6, 7 + 25
8 Picks 1, 16 Picks 8, 9 + 38
10 Picks 1, 20 Picks 10, 11 + 45
12 Picks 1, 24 Picks 12, 13 + 51
14 Picks 1, 28 Picks 14, 15 + 58
16 Picks 1, 32 Picks 16, 17 + 71
18 Picks 1, 36 Picks 18, 19 + 88
20 Picks 1, 40 Picks 20, 21 + 100
22 Picks 1, 44 Picks 22, 23 + 116
24 Picks 1, 48 Picks 24, 25 + 133
26 Picks 1, 52 Picks 26, 27 + 152
28 Picks 1, 56 Picks 28, 29 + 165
30 Picks 1, 60 Picks 30, 31 + 175
32 Picks 1, 64 Picks 32, 33 + 182

Use the charts to identify unintuitive trends in your league's size. If something doesn't seem right, that's a good thing - chances are it may be worth looking at and proposing a trade that fools your league.

A few last reminders:

These are all estimates. Half the fun of the game is drafting your favorite players when possible. Keep in mind that all of these do not factor into any part of any formula above:

Potential - Some players peak in the last half of the season, especially rookies.
Position - certain positions attract certain FPPG/FPPM ratios. Multi-position players are valuable in a totally different way.
Injuries - A player might be injured for a long time, and worth more later. Even at 0 FPPG and 0 FPPM, they might be worth something.
Less than Perfect Owner Behavior - If the rest your league picks poorly, little of this will matter. Using these formulas in an uncompetitive league is like counting cards in blackjack without knowing basic strategy.
Biases - We all have our favorite players that we'd like on our teams if it makes sense. Likewise, you might be able to get more through trades if you know other owners' weaknesses.
Explosiveness / Fluke Factor - Some players are consistent, others are not and the overall measure of success is the Win-Loss record. Depending on your team and lineup order, deviant behavior can boost your Win-Loss record significantly.

The Math Behind Drafting (Advanced)

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