The Underlying Theory

We start from the premise that a player's value is proportional to the difference between the number of fantasy points your team would score with him, and the number of fantasy points your team would score without him.

If your team would score an extra 15 points with Michael Vick (above what it would score without him), and an extra 30 points with Antonio Gates, then Antonio Gates is worth twice as much to you as Vick is, and you should be willing to pay twice as much for him.

While that is a relatively simple principle, it is not easy in practice to project how many marginal points a given player will be worth to your team. Estimating Michael Vick's potential value to my team requires that I project: (a) how many games Michael Vick is likely to start for my fantasy team, (b) how many points he is likely to score in the games he starts, and (c) how many points I would be able to get from the QB position in those games without Vick.

Let's take a shot at answering each of those questions, starting with (c), which will be our baseline. After my entire roster is set, I'll have a better idea of how many points my backup QB is likely to be able to get me. But I don't have the luxury of waiting until after my roster is set to formulate my list of auction prices. So instead of using my actual backup QB as the baseline, I will use the typical average backup QB.

In a 12-team league that starts one QB per team, you might figure that the league-average backup QB would be the projected QB18 - and you'd pretty much be right, although possibly not for all the right reasons. In any given week, several QBs will be unavailable due to byes or injuries, so the 18th-best quarterback available might be more like the 22nd-best quarterback overall. Perhaps, then, we should use QB22 as the baseline. On the other hand, as busts are cut by their fantasy owners and surprise sleepers picked up and added in their place, the actual top 18 quarterbacks are likely to outscore the projected top 18, such that the actual 18th-best quarterback during the season will out-produce the projected 18th-best quarterback. So perhaps we should use QB15 as the baseline. As you can see, there are various factors pulling in opposite directions; and it turns out that, in practice, they roughly offset each other.

Based on actual data from the last four years in a competitive fantasy league, the true average backup player at each position tends to score about the projected points for the xth-ranked player at that position where x is 1.5 times the number of starters.1 The exception is at kicker, where expert owners do not generally carry backups, so the baseline would be simply x + 1 rather than x * 1.5.

Working backwards to (b), the projected number of fantasy points per start is slightly different from the projected number of fantasy points per game. If I project Drew Bledsoe to average 18 fantasy points per game, that projection includes the chance that he'll actually average 22 points per game, as well as the chance that he'll average only 14 points per game. But if he averages 22 points per game, he will likely start more games at QB for my fantasy team than if he averages only 14 points per game. So his projected fantasy points per start will be higher than his projected fantasy points per game.

I do not know, empirically, what the relationship is between projected points per game and points per fantasy start. It's on my list of things to explore next offseason. For now, I will guess that a player is likely to score 10% more per start than per game. So we will take every offensive player's projected points per game, and multiply by 1.1 to get projected points per fantasy start.2 Subtract the baseline from this figure, and we get projected value per fantasy start.

The last step is to multiply this number by projected starts. How many fantasy games will a player start? I don't know of any way to get the correct answer by doing philosophy, so I instead looked at the past four years of a competitive fantasy league (No Mercy - Great White) and counted. It turns out, as you might expect, that fantasy starts are strongly (and linearly) correlated with draft position - or, what is the same thing - preseason ranking. The top-ranked QB started an average of 10.75 out of a possible 13 games in the fantasy regular season, on average. The ninth-ranked QB started an average of 7.25 games, the fourteenth-ranked QB started an average of 5.75 games, and the eighteenth-ranked QB started an average of 4.0 games.

The general formula is that the xth-ranked quarterback in a 12-team league that starts one QB per team can be expected to start 10.827 - 0.376(x) out of 13 games. (Correlation coefficient = 0.82.) (I'm using thirteen games because my data come from the fantasy regular season of the Great White league, which comprises weeks 1-14 of the NFL season, during which period each team has one bye. It would not make a difference, for purposes of calculating auction prices, whether we use a denominator of 13 games, 16 games, 17 games, 100 games, or any other number. I will keep using 13 games as the denominator for purposes of this article.)

The xth-ranked running back in a 12-team league that starts two RBs per team can be expected to start 11.3 - 0.1985(x) games. (Correlation coefficient = 0.90.)

The xth-ranked wide receiver in a 12-team league that starts three WRs per team can be expected to start 12.468 - 0.1686(x) games. (Correlation coefficient = 0.89.)

The xth-ranked tight end in a 12-team league that starts one TE per team can be expected to start 11.498 - 0.4619(x) games. (Correlation coefficient = 0.83.)

The xth-ranked kicker in a 12-team league that starts one PK per team can be expected to start 13.379 - 0.6488(x) games. (Correlation coefficient = 0.88)

The xth-ranked team defense in a 12-team league that starts one TD per team can be expected to start 10.433 - 03.466(x) games. (Correlation coefficient = 0.59)

To sum up before going further, a player's auction price should be proportional to his projected value; and his projected value is equal to his projected number of fantasy starts times his value-per start over the baseline. The baseline is the player whose positional ranking is equal to 1.5 times the number of league-wide starters at that position.

All of this may be a bit confusing, so let's actually do the calculations.

Putting Theory Into Practice

In the table below, "Pts" means projected fantasy points. I am using my own projections for this exercise. "P/G" is fantasy points per game, and is generally projected fantasy points divided by 14, which is the expected number of games played (not fantasy starts) for most players. The exceptions in my projections are Carson Palmer (12 games) and Kurt Warner (13 games). (Chad Pennington [11 games] and J.P. Losman [11 games] are not listed. Players not listed are worth no more than $1.) I use 14 games instead of 16 because, on average, players tend to miss about two games per season due to injury. Most miss fewer than that, and a few miss many more than that; but two games is about the average. It would not matter, however, if we projected everyone to play 16 games instead of 14 - the points per game figure would be the same.

"P/S" is points per fantasy start - i.e., points per game plus a ten percent premium. "V/S" is value per start, or the projected points per start above the baseline at that position. (The baselines are the "P/G" of QB18, RB36, WR54, TE18, PK13, and DT18, assuming a 12-team league that starts 1/2/3/1/1/1.) Since no owner would start a player who is projected to score fewer points than his potential replacement, the values in the "V/S" column cannot be negative. Any negative numbers are therefore converted to zeroes.

"S" is expected number of fantasy starts. "V" is the player's total projected relative value, calculated by multiplying projected value per start by projected starts. I call it relative value rather than absolute value because it only tells us the ratio of one player's value to another's; it does not tell us how many salary-cap dollars a player is worth.

Once we determine each player's relative value, however, the exercise of converting it into an auction price is fairly straightforward. We first determine how many discretionary dollars are available to use in bidding on players. Consider a 12-team auction league with 18-player rosters, a $300 salary cap, and a minimum bid of $1. Twelve times $300 is $3600, but since each team is obligated to spend $18 before it even starts bidding, there are really $3384 remaining that may be freely spent.

We divide this $3384 by the sum of the relative values to find out how much each point of relative value is worth in terms of salary-cap dollars. In this case, the sum of all the numbers in the "V" column below is 4667, which means that each point of value is worth $3384 / 4667 = $0.7251.

The final column contains each player's auction price, calculated by multiplying column "V" by $0.7251.

The following list is for a 12-team league with 18-player rosters (starting 1/2/3/1/1/1) and a $300 cap. For leagues of other sizes and structures, the auction prices will of course be different, but the method of calculating them would be the same.