Fantasy Valuation

Abstract— Currently when drafting fantasy players, managers are reliant upon the rankings as they exist in their platform (ex: Yahoo, ESPN, CBSSports), regardless of the league’s actual format or statistic categories. Managers must then use a variety of other information to pre-rank players for a customized league. As an additional analysis step to “AggPro: The Aggregate Projection System” we set out to find a way to rank hitters and pitchers across any league format, with a strict mathematical basis. The following article proposes a valuation metric applicable to any Rotisserie style league.

I. INTRODUCTION
Pre-ranking fantasy players for an annual draft is an overwhelming task. Managers can consume a wealth of information, such as ESPN’s annual draft kit[1], Yahoo’s position primers[2], CBSSports’ fantasy depth charts[3], MockDraftCentral’s Average Draft Pick (ADP) data[4], Athlon Sports’ annual baseball preview magazine[5] and countless other sources of fantasy material. These resources provide analysis and rankings of players based on projected performance. In addition, managers also consider the projection systems for the statistics themselves. As discussed in “AggPro: The Aggregate Projection System”[6], there are several projection systems available, thus bestowing managers with the task of choosing which valuation system to follow. In our development of the AggPro category weights, we expanded our research, creating a standard way to turn these projections into a single value for each player which can then be used to pre-rank players for a draft.
There are three core problems with the existing valuation systems: 1) Ranking systems are inconsistent across platforms. 2) Projections themselves need to be balanced against valuation information. 3) Subjective projections (source data systems discussed in AggPro) should be objectively weighted for valuation. Using AggPro to produce a more accurate 2010 projections, it is possible to evaluate all players objectively for any league format. Our solution as discussed in this paper is also threefold: 1) Apply a valuation metric, based on standard deviation, to determine hitter and pitcher value in each projection category. 2) Determine the “overall” value by using the customized league scoring settings. 3) Apply ADP information to suggest draft value coupled with statistical value.
With this valuation formula, managers can use any projections in tandem with the specific categories for the league, to objectively define how valuable players are for that league format. These values when ordered descending produce a pre-ranked list of players, but not necessarily in the best order to draft them. Leveraging ADP data in conjunction with these values will best prepare managers for draft day. This analysis is not new; many have contributed content across the web with articles discussing this process. The remainder of this paper focuses on how to 1) gather the information 2) calculate the valuation metric and 3) organize these values in conjunction with ADP data to have a more precise list of pre-ranked players for their specific league.

II. COLLECTING PROJECTION DATA
This research uses the AggPro projections, but any source can be used for this process, provided that it projects enough players for the draft and stats for each of the categories. Some systems neglect to project ERA, but do provide ERs and IPs, so ERA can be easily calculated. For more unique leagues that use categories such as fielding errors or grounded into double plays, finding a projection system that has this data may be more difficult. The analysis uses the following 8x8 rotisserie categories:
Hitters R HR RBI SB K Avg. OBP. SLG.
Pitchers IP W L Sv K ERA WHIP K/BB
Because this 8x8 league uses additional categories, the generic rankings that Yahoo! or ESPN provide do not provide a full picture of player value.
After finding a usable projection data set, store the values in a spreadsheet or database. The next section will detail the additional mathematical steps to creating the valuation metric. The valuation metric will change depending on the field of players considered, so it is best to create the valuation for all players regardless of league size. For this analysis, the minimum benchmarks were hitters projecting to more than 200 at-bats and pitchers with at least 100 innings pitched or at least 4 saves recorded. This provided ample data for most league formats, while also focusing on players who are fantasy relevant.

III. DETERMINING EACH PLAYER/CATEGORY VALUE
For each category, create an output column that calculates the following:
Player’s Projected Stat - Average (All Players’ Projections)
Standard Deviation (All Players’ Projections)

In Excel: =( - AVERAGE($$1:$$)) / STDEV($$1:$$)

The result of calculating how far above or below a player’s stat is from the average, then compared to the standard deviation, provides a positive or negative distance from zero, which translates to that player’s value for that statistic. The larger the value, the greater the player’s impact on that rotisserie category. Next, create this stat value for each player and stat category in the league. Summing these creates a single value that translates into a player’s value in the league. ESPN’s top five 2010 ranked players are shown below[7]:
Projections:
Player R HR RBI K SB AVG OBP SLG
Albert Pujols 114 43 122 62 11 0.337 0.452 0.647
Hanley Ramirez 113 29 91 106 34 0.324 0.4 0.546
Alex Rodriguez 110 39 125 124 21 0.3 0.402 0.563
Ryan Braun 110 37 116 125 17 0.315 0.38 0.576
Chase Utley 114 30 97 111 18 0.29 0.396 0.52

Values:
Player Run Val HR Val RBI Val K Val SB Val Avg Val OBP Val SLG Val
Albert Pujols 1.95 2.20 1.98 1.32 -0.17 2.82 3.75 3.37
Hanley Ramirez 1.88 0.71 0.41 -0.12 1.61 2.09 1.57 1.31
Alex Rodriguez 1.67 1.78 2.13 -0.70 0.60 0.75 1.66 1.66
Ryan Braun 1.67 1.56 1.67 -0.74 0.29 1.59 0.74 1.92
Chase Utley 1.95 0.82 0.71 -0.28 0.37 0.18 1.41 0.78

Totals:
Player 8x8 Value 5x5 Value
Albert Pujols 17.22 8.78
Hanley Ramirez 9.47 6.70
Alex Rodriguez 9.54 6.93
Ryan Braun 8.71 6.79
Chase Utley 5.94 4.03

In looking at the resulting data set, for the 8x8 (and a standard 5x5) league, Alex Rodriguez is slightly better than Hanley Ramirez, but four other players actually come in higher than Utley (data not shown). The most notable finding is that in the 8x8, Albert Pujols is almost twice as valuable as the next best player.

IV. RANKING THE VALUE
When evaluating this valuation process, we found that value of the projections varies in accuracy depending on 1) size of the dataset, 2) degree of range between projections for the same stat, 3) a source system’s favoring of a certain statistic. While these factors impacted comparing the calculated value, ranking players 1 to n based on this value proved to be more accurate than any system in any year. The resulting new column of data represents the order of value, which is can now be used in conjunction with the ADP data to provide the final pre-ranked list.
We also chose to add a “rank round” column which is simply the round in which that player should go according to the rank of the value. The formula to calculate a round based on some rank:

((Rank – (Rank MOD x )) / x ) + 1
In Excel: =(-MOD(, x ))/ x +1

In the above, x represents the number of teams drafting.

V. ADDING IN ADP
Like projection data, ADP data is readily available online from a variety of sources. For our research we combined the 2010 ADP from Yahoo[8], ESPN[9], and KFFL[10] to obtain an average ADP. This ADP data is most useful when comparing a ranked value. To compare the ranked rounds and the ADP rounds, calculate the ADP round using the same formula as in section IV. With the addition of this information, the full data set output includes seven columns:
Player Position Value Rank Rank Round ADP ADP Round

With this dataset, it is then easy to highlight players with a net difference in projected rank round to ADP round > 1. Those with a positive difference are good values for drafting (have a value higher than that of the ADP). Those with a negative difference are not as valuable (are being drafter sooner in ADP than their value projects).
With this indicator in place, it is thus easier to rank players in an order based on 1) his value to the league 2) how soon he will commonly be drafted in a league. For our 2010 data, it’s obvious that Albert Pujols and Ryan Braun will go in the first round. However, the interesting comparisons come in for those that meet the criteria above:
Player Position Value Rank Rank Round ADP ADP Round
Haren SP 7.85 9 1 50.2 6
Sandoval 3B/1B 5.415 18 2 33 4
Kemp OF 4.331 26 3 8 1
Roberts 2B 0.911 81 9 43 5

This table shows that Dan Haren and Pablo Sandoval are good draft values, especially at their ADP levels, but Matt Kemp and Brian Roberts are commonly taken before their value dictates. Obviously, this is due to the custom league format. In a standard 5x5, this data value data will be different (stolen bases are not as valuable in an 8x8). However, this difference is the purpose of research, to break down the barriers of generic value rankings and create a more quantitatively sound and accurate valuation applicable to any league.

VI. LIMITATIONS
There are a few important points to make regarding limitations of the process. This process does not consider health concerns, supporting teammates, or potential upside/”breakout” players. Such measures are usually applied to the projection data source. This process does not consider position scarcity. Other systems, in which analysts rank players manually, likely take this into account. Admittedly, this is likely a factor as to why Utley is ranked 4th in section III but projects slightly lower. This process does not necessarily dictate a manager’s draft. The use of this indicator still requires subjective choices. For example, managers should not draft Dan Haren 9th overall because he projects that high in terms of value. However, using this process, a manager can make an educated decision in drafting him in the fifth round, just before most draft him in the sixth. Thus, that manager gets first round value in the fifth round. Likewise, waiting until the ninth round to select Brian Roberts will result in him not being available. Managers may choose to draft him sooner because of position scarcity, this research provides an indicator of what his true value is in this league format, rather than assuming it is high because common platforms say so. Even though the process thus far has focused on custom formats, it can be repeated for a standard 5x5 league to provide a sanity check against the default rankings of a drafting platform.

VII. CONCLUSION
By starting with a set of projection data, managers can evaluate players based on the statistics relevant to any custom league using a series of valuation metrics. Ranking players according to a common value metric provides useful insight for fantasy drafts. This ranking is further enhanced by including a check against ADP data to highlight players who project above or below their ADP.

VIII. REFERENCES
[1] http://games.espn.go.com/frontpage/flbdraftkit.
[2] http://sports.yahoo.com/fantasy/mlb/news?slug=be-posprimer08-rp
[3] http://cbssports.com/mlb/depth-charts
[4] http://www.mockdraftcentral.com/report_adp.jsp?period=0&sport=1&type=218&color=1
[5] http://www.athlonsports.com/store/index.php?cpath=33_274
[6] http://aggpro.blogspot.com
[7] At time of authoring, AggPro data for 2010 was not yet available. ESPN projections available: Valuation calculations may differ due to size of sample set.
[8] http://baseball.fantasysports.yahoo.com/b1//draftanalysis
[9] http://games.espn.go.com/flb/livedraftresults
[10]http://www.kffl.com/static/programs/baseball/compilations/mlb_adp_report.php

2010 AggPro Weights

For those of you playing at home, your AggPro weights for 2010 are as follows:

Bill James = .4266

CHONE = 0.0

Marcel = .2851

PECOTA = .2833

ZiPS = 0.0

To review, in 2009 the weights were:

Bill James = .37

CHONE = 0.0

Marcel = .35

PECOTA = .28

ZiPS = 0.0

And in 2008 the weights were:

Bill James = .56

CHONE = 0.0

Marcel = .15

PECOTA = .29

ZiPS = 0.0

We adapted a polynomial time matrix least-squares minimization algorithm to solve the AggPro problem this year instead of simply performing a brute force search across the search space. The new algorithm has allowed for more precise weight sets (4 digits vs. 2 digits) and a two order of magnitude improvement in speed (seconds vs. hours).

We are experimenting with finer grained weight sets this year. For example, instead of only allowing one weight for all the categories what if we allowed a separate weight set for each category? for each player? for each category for each player? Our instinct is that the most accurate results will come from allowing each system to have a separate weight set for each category for the upcoming year but at this point we don't have any empirical evidence to support it. Results will be posted here as they come in.

Official Writeup of AggPro

Tomorrow Cameron and I will be presenting AggPro at SABR 39. The official writeup of AggPro complete with evaluation of the AggPro projections for the 2006, 2007, 2008 and 2009 MLB seasons is availalbe here. The slides we are going to present are available here. AggPro has been a fun project but like most of my other research once I hit the final revision stage of the writeup I really start to HATE the project.

AggPro Final Results

The final AggPro results are listed in the table below. They are significantly better than I expected. In each year AggPro is more accurate than any of the constituent systems. Furthermore, each year it more accurate than a theoretical projection system that would be composed of the best constituent projection for each statistical category. Finally, AggPro's projection in every category for every year is more accurate than the any constituent system's projection for a given category in a given year except for one category in one year. (As of June 29th PECOTA's projections for this year (2009) for triples are less than 1% more accurate than AggPro's 2009 projections for triples.) These observations seem to show that these different constituent projection systems work in a complementary nature (i.e. the overall aggregate projection is more than just the sum of its best parts.)

Recall, the AggPro projections were formed by determining the weights for the constituent systems that minimized the standard error from the actual Major League Baseball data for 2007 and 2008. We were able to evaluate the AggPro projections for 2006 because AggPro does not weight the CHONE projections at all. The CHONE projections were the only constituent projections missing for 2006.

Year	AggPro Improvement over BCPS	AggPro Improvement over TPS
2006	5.25%	5.00%
2007	4.25%	2.65%
2008	5.90%	4.60%
2009 (through 06/29/09)	2.4%	2.0%

Table Legend: BCPS = Best Constituent Projection System; TPS = Theoretical Projection System

AggPro Early Results

I will be finishing the AggPro Projection Research this week. Most of the work has been completed and the results look good. The weights for all constituent projection systems (Bill James, CHONE, Marcel, PECOTA and ZiPS) are as follows: 0.56 = Bill James, 0.00 = CHONE, 0.21 = Marcel, 0.23 = PECOTA and 0.00 = ZiPS. Recall, these are the weights that when applied to the respective constituent projection systems for the years 2007 and 2008 mimize the standard error of the resulting aggregate projection from the actual 2007 and 2008 Major League Baseball data.

How well did we do?

The previous post described how AggPro would be measured. In short, AggPro will be considered successful if 1) the resulting aggregate projections are more accurate than any of the constituent projections for 2007, 2008, and 2009 and 2) if AggPro projections for each statistical category, for each year are more accurate than any constituent projection for the given category in the given year. The first evaluation criterion is almost trivial. Obviously, AggPro can be as accurate as the most accurate constituent projection system simply by giving a weight of 1.00 to the most accurate projeciton system. I was slightly skeptical if AggPro would be able to meet the second measure of success. It seemed possible a constituent projection system might be very accurate for predicting a statistic in a single statistical year but might mispredict almost every other category in that year or other years.

It seems as if AggPro has met this measures of success. The AggPro projections are more accurate in every statistical category for 2007 and 2008 MLB seasons than any of the constituent projection systems. In 2007 the AggPro projections are a 4.25% improvement over the leading constituent projection system (Bill James). In 2008 the AggPro projections are a 5.9% improvement over the leading constituent projection system (again Bill James). Furthermore, if one was to form a theoretical projection system by taking the most accurate constituent projection for a statistical category each year AggPro would still be more accurate in 2007 and 2008. This system is described in the AggPro baselines table in the previous post. In 2007 AggPro would be a 2.65% improvement over the theoretical system. In 2008 AggPro would be a 4.6% improvement over the theoretical system.

I have several theories as to why the Bill James predictions were so accurate (given AggPro's measures of accuracy) for these years that I may go into in a seperate post. I'll be doing a pro-rated evaluation for the 2009 season later this week. This will be a true test of AggPro's accuracy as the optimization which chose the weights did not consider any 2009 data.

Projected Categories, Players and Measures of Success

Cameron and I have spent the last several weeks acquiring and formatting the projections from Bill James, CHONE, Marcel, PECOTA and ZiPS. We are going to be using the 2007 and 2008 projections from these systems. I will refer to these projections as the constituent projections. Originally, we had planned on using the 2006 projections from these systems too, but as Chone Smith nicely explained in this comment the 2006 projections from CHONE were more experimental than anything else and probably would not be particurally useful. Since we need projections for all systems for all the years that we include in the AggPro analysis we had to eliminate the 2006 projections.

Projected Categories

We will be using the following player performance categories:

Hitters: At Bats, Hits, Runs, Doubles, Triples, Home Runs, RBIs, Strikeouts, Walks and Stolen Bases.

Pitchers: Innings Pitched, Earned Runs, Strikeouts, Walks and Hits.

This is the set of hitter and pitcher performance categories that is common to all the constituent projection systems.

Projected Players

The following lists contain the players AggPro included in the AggPro analysis: 2007, 2008.

The player list for a given year is the list of players that are common to all the constituent projection systems for the year.

AggPro Measures of Success

I also have determined the standard error of the 2007 and 2008 constituent projections from the actual Major League Baseball player performance data in each of the categories. The standard error, as a percentage of the actual population of each category, for each system, is listed below. Double click on the table to enlarge it.

AggPro will be successful if it identifies a single weight to apply to each projection system such that the resulting AggPro projections have less standard error for a given statistical category in a given year than the best constituent system projection for that category, for that year. For example, each category in the AggPro projections for 2007 must have less standard error than corresponding value for the category identified in the righthand most column in the top half of the table. Based on the early results from the simulated annealing optimization it looks like an AggPro cocktail consisting of 2 parts Bill James, 1 part Marcel and 1 part PECOTA comes very close to meeting this goal.

Welcome to AggPro

Welcome to the home of AggPro: The Aggregate Projection System. I have tried to anticipate commonly asked questions about AggPro and provide answers to them below. As AggPro is developed, status updates and results will be posted on this blog. If you have any questions regarding AggPro feel free to contact me. Cameron Snapp will be assisting me with the development of AggPro.

What is AggPro?

AggPro is a proposed Major League Baseball (MLB) player projection system. Currently there exist many different projection systems that predict the performance of MLB players in a variety of different statistical categories. The goal of AggPro is to aggregate the MLB player predictions from the existing systems into a single more accurate prediction. AggPro will use the following projection systems, Bill James Handbook, CHONE, Marcel, PECOTA and ZiPS, for the years 2006, 2007, and 2008 to form the aggregated projections.

Why do we need yet another projection system?

It is important to note that AggPro is not just another projection system. Instead it is a methodology for aggregating effective projections from different systems into a single more accurate projection. That said, we probably don't need another projection system. It appears we are reaching the limit of the accuracy that can be expected from projection systems. While AggPro may improve the accuracy of existing systems I do not expect it to improve the state of the art significantly. However, Greg Rybarczyk believes paradigm shifts that will improve the accuracy of projection systems are on the horizon. If paradigm shifting projection systems are developed, the AggPro methodology will be applicable to improve these systems as well.

My interest in aggregating predictions from effective projection systems to form a better prediction stems from BellKor, the leading solution to the Netflix Prize. In October, 2006 Netflix released a dataset of anonymous movie ratings and challenged researchers to develop systems that could beat the accuracy of its recommendation system, Cinematch. A grand prize, known as the Netflix Prize, of $1,000,000 will be awarded to the first system to beat Cinematch by 10%. The BellKor prediction system, with 8.26% improvement over Cinematch, is the leading solution. BellKor employs 107 different models of varying approaches and uses mathematical optimization methods to weight each prediction. From the weighted predictions of the 107 models BellKor forms its aggregate recommendation. I was curious how well this strategy would work when applied to MLB player projection systems, thus AggPro was born.

How will AggPro work and how will it be evaluated?

AggPro will employ a combination of mathematical optimization methods including Hill climbing, Genetic algorithms, and Simulated annealing to determine the weight for each projected statistic in each system that yields the AggPro projections with the least root mean square error (RMSE) from the MLB players’ actual performance in the 2006, 2007, and 2008 seasons. Using the same weights AggPro will provide projections for the 2009 MLB season.

All the projection systems (AggPro, Bill James Handbook, CHONE, Marcel, PECOTA and ZiPS) will be evaluated against the root mean square error from the actual player performance for the 2006, 2007, 2008 and 2009 MLB seasons.

When will AggPro be presented?

AggPro has been selected to be presented at the 2009 Society for American Baseball Research Convention (SABR 39) in Washington, D.C. on July 29th - August 2nd. The abstract proposing AggPro is available here.