HEAD-TO-HEAD: Tools of the Trade

Nervous fantasy owners actually considered trading Albert Pujols in early May. After a sluggish start, Pujols appears to be on track; however, the experience provides a valuable case study in quantifying probability and knowing when to trade players. Using these tools shows head-to-head owners not only how to adapt to change, but also make precise decisions regarding specific players.

Playing the odds

Through 72 games, Albert Pujols is currently pacing 25 home runs.  Since Pujols has never hit less than 32 home runs at the major league level, and his career high for home runs in a single season is 49, is there a simple way to calculate the probability that he will hit 30 home runs or more in 2012?

Step 1: Determine the realistic outcomes

Using his current pace of 25 home runs as the lowest possible outcome and his career high of 49 as the highest possible outcome, it’s easy to determine that Albert Pujols has a range of 24 likely outcomes for the 2012 season.

Step 2: Calculate the probability

With 24 outcomes, each outcome has a 4.2% chance of occurring (1/24=.0417). Assuming an equal distribution, Pujols has a 4.2% chance of hitting exactly 37 home runs this season, equaling his 2011 total.  However, he also has a 71% (17/24=.7083 or 70.83%) chance of hitting within his career average range of 32-49 home runs, and a 29% chance of hitting outside his career average.  Of course, these odds will change with every new at-bat.

Step 3: Establish a production baseline

Because fantasy owners generally regard home run hitters as elite when production values can be found in the 30+ and 40+ categories, let’s calculate the odds of Pujols reaching those milestones in 2012:

Production Baseline (Pujols):

HR Range     Probability
========     ===========
 25-29         16.68%
 30-39         41.66%
 40-49         41.66%

Based upon this analysis, Albert Pujols has roughly a 83.32% chance of hitting 30+ home runs in 2012. As these values change throughout the season, the percentages will indicate when trades should occur, as well as when players should be inserted into lineups or benched.

Should I trade Albert Pujols?

It depends. Suppose Joey Votto was offered for Albert Pujols. Although both players could be considered first round picks, and each has similar statistics, the initial reaction for many fantasy owners would be to reject the trade because Albert Pujols was more valuable than Joey Votto on draft day. Beginning with a current statistical comparison, let’s examine how to clarify trade decisions by using probability and expected rate of return.

Apples-to-apples comparison

Because the starting point for most trades is a statistical comparison, the following chart provides current production for both players:

2012:            G     AB    R   HR  RBI  SB  AVG.   
=============   ===   ===   ===  ==  ===  ==  ====   
A.Pujols($34)    72   282    22   8   29   2  .240   
J.Votto($30)     71   245    31   8   30   3  .329  

(Through 6/24/12)

If Albert Pujols has a 83% chance of hitting more than 30 home runs, would it benefit fantasy owners to know that Joey Votto only has a 54% chance of hitting 30+ home runs in 2012? Though currently pacing 31 homers, Votto’s career range (24-37) indicates that only seven of thirteen probable outcomes (30-37) favor hitting 30+ home runs this season.   In fact, Votto has only had one season with more than 30 home runs (2010).  Now, let’s look at the influence of variable change.

Expected rate of return

Applying dollar values to probability equations allows owners to calculate expected rates of return and account for variable change. Because Pujols’ preseason projection of 40 home runs and his preseason value of $34 will not change, accounting for variable change means recognizing that just as every home run increases the odds that Pujols will reach 40 home runs, every strikeout also decreases the odds that Pujols will hit 40 home runs.  Once again, the first step is to establish a range of probable outcomes based upon Pujols’ track record:

Career Home Runs (Pujols):

Year:      2001  2002  2003  2004  2005  2006  2007  2008  2009  2010  2011
=========  ====  ====  ====  ====  ====  ====  ====  ====  ====  ====  ====
A.Pujols    37    34    43    46    41    49    32    37    47    42    37

Assuming Pujols has an 18% chance of hitting 32-36 home runs based upon two occurrences (2002, 2007) in eleven results between 2001-2011, he would also have a 27% (3/11=.2727) chance of hitting between 37-40 home runs and a 55% (6/11=.5454) of hitting 40+ home runs.  Assigning a 5% premium for the first category (32-36 HRs), a 10% premium for the second category (37-40), and a 15% premium for the third category (41-49) allows for the calculation of his expected rate of return:

Expected Rate of Return (Pujols):

Step 1:  Calculate expected return

(.18x.05)+(.27x.10)+(.55x.15)=.1185 or 11.85%

Step 2:  Find the premium


Step 3:  Add the premium to projected value


Based upon the probability of Pujols finishing within one of these three categories, his expected rate of return is 11.85% or $4.03 on a $34 preseason value. Of course, these numbers are used for illustration purposes, and can be adjusted by fantasy owners to be as simple or complex as needed. However, the sum of all probabilities must always equal 100%. When compared to Joey Votto’s expected rate of return on $30, it’s apparent that Votto’s smaller sample size (2008-2011) and range (24-37) indicate a higher probability of 8% (1/13=.0769) per occurrence.

Career Home Runs (Votto):

Year:       2008   2009   2010   2011
=========   ====   ====   ====   ====
J. Votto     24     25     37     29

Assuming that Joey Votto will finish the season within his career range of 24-37 homers, he has 14 probable outcomes for 2012.  Because his track record isn’t as extensive, the production baseline method can be used to calculate Votto’s expected rate of return, where the probability of Votto generating a 5% premium by finishing with 24-28 home runs is 36% (5/14=.3571); followed by a 10% premium for the 43% (6/14=.4285) chance of finishing with 29-34 home runs; and a 15%  premium for the 21% (3/14=.2142)  chance of finishing with 35-37 home runs:

Expected Rate of Return (Votto):

Step 1:  Calculate expected return 

(.36x.05)+(.43x.10)+(.21x.15)=.2344 or 23.44%

Step 2:  Find the premium


Step 3:  Add the premium to projected value



Despite a higher expected rate of return for Joey Votto, the corresponding premium is not enough to justify a trade at this point in the season. Of course, the equation has been simplified for illustration purposes, and is intended to show how fantasy owners can account for variable change. Remember, the probabilities associated with these equations will change on a daily basis—with every new at-bat.

Changing fortunes

Knowing the odds significantly changes the strategy for head-to-head fantasy baseball. Although home runs were used to illustrate the role of probability and expected rates of return for a fictional trade of Joey Votto for Albert Pujols, the methodology can be easily used for evaluating trades between any two players. What if strikeouts were compared for a trade offer of Felix Hernandez for Justin Verlander? What is the probability that new Phillies closer, Jonathan Papelbon, will save at least 40 games in 2012? Although the odds may change, the success rate of owners using these tools of the trade will always remain constant.

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  For more information about the terms used in this article, see our Glossary Primer.