Our approach rests on the assumption that the well-known Bill James runs created measure is an unbiased and accurate method for predicting baseball offense. Given this, it is clear that what we are after is to discover when stealing increases runs created. This can be expressed symbolically by use of the runs created per out per plate appearance formula. Note that we consider only steals of second base in what follows. Runs created per out per plate appearence is defined as
O T
RC= --------
1-O
whereO = on-base proportion (# between 0 and 1) or that fraction of batters which reach base (in any manner)
T = total base proportion (total bases divided by plate appearances)
In order for stealing to 'pay off' we want the runs created including stealing to exceed the runs created without stealing:
O T (O-C) (T+pS) -------- < -------------- 1-O 1-(O-C)where
C = caught stealing proportion or that fraction of batters which reach base and are subsequently caught stealing
S = stolen base proportion (steals divided by plate appearances)
p = coefficient assigned to stolen bases in runs created total base formula
Note that in O and T we are explicitly ignoring the effect of base stealing, i.e., O and T are not modified by stolen bases and caught stealing.
A little algebra yields the following:
T S
----------------- < ------
p (O-C) (1-O) C
Clearly
T T
-------------- < ----------------
p O (1-O) p (O-C) (1-O)
since O > (O-C).
Then we can state
T S
------------- < -----
p O (1-O) C
which forms the basis for a decision criteria for stealing second base. Note that we could replace the S/C
which is an 'observed' quantity with P[S]/P[C] where P[ ] indicates the probability of a steal or caught
stealing. Then P[S]/P[C] can be computed directly in a given stealing situation from the SOM ratings for
the pitcher, catcher, and base stealer involved (A formula for this will be given below). We term P[S]/P[C]
the 'steal to caught ratio' or SCR. Using p=0.5 (the 'standard' runs created steals total base weight found in
any one of several Bill James publications), a team with an OBP of 320 and a SLGP of 400 would need a
SCR of roughly 3.68 to break even. Note that since we approximated the O-C term by O in our threshold
expression, the actual break-even SCR would be a bit larger than the computed 3.68 value. Shown below
is a plot of this threshold as O goes from 0.2 to 0.4 for 3 different values of T:
Of course, this plot is a gross oversimplification since T and O are generally correlated, so that it is not possible to hold one fixed and vary the other. However, this plot does illustrate typical values for the suggested SCR as a function of O and T. In particular, T is fixed as walks are varied, so that one can see that according to this analysis, more walks given fixed power means that stealing more 'liberally' will increase overall expected offense, at least according to runs created.
It is important to note that one must estimate overall values for O and T given the opposing pitcher and defense since just considering the batter's card will give erroneous results. SRA distributes a spreadsheet product, LINEUP, that perform just such a calculation for 2 lineups in a given ballpark. LINEUP also gives victory probability estimates based on the well-known Pythagorean Theorem of baseball.
Observe that even for a low power, high on-base team an SCR of at least 2.5 is needed. This number is considerably higher than other 'analysis' (we use this term loosly, since most analysis is just unsubstantiated opinion not based on anything that can be concretely and accurately measured) that we have read, such as the recent STRAT FAN article series on base stealing by Base Creations Inc. What all this can be boiled down to is this: Unless you need a 1-run strategy, you should only steal when you have an overwhelming chance of success, i.e., at SCRs of around 3.5 or so. To be complete, it should be pointed out that p is really a function of the number of outs, and that 0.5 represents just an average value. Our research indicates that p varies between roughly 0.3 to 0.8 depending on the number of outs and the overall range of O and T.
1-A Bk 18 S1-H+Bat 3 Trat
----- ---- + --- G [ ----------- + --- ---- 2 ]
20 20 20 20 20 20
SCR = ----------------------------------------------------------
1-A 20 - S2 18 20 - (S1-H+Bat)
A + ------ ---------- + --- G [ ----------------- ]
20 20 20 20
which 'simplifies' to
(1-A) Bk 18 G (S1-H+Bat +0.3 Trat)
SCR = ----------------------------------------------------
400A + (1-A) (20 - S2) + 18 G (20 - (S1-H+Bat))
whereBk = pitcher balk rating
Bat = pitcher hold + catchers arm (clipped to + or - 5)
Trat = catcher T-rating
G = chance of a good lead (# between 0 and 1)
A = chance of an automatic caught stealing on the lead roll
S1 = First stealing #
S2 = Second stealing #
H = hold factor (2 or 0 depending on held or no respectively)
Note that the factor of 2 included for the catcher throwing error term represents the extra base that is obtained on throwing errors.
Clearly this formula only applies to non-'*' players or held '*' players, since the SCR is immediately apparent for non-held '*' players. Also note that it presumes that you will only run if you get the lead, since again if a lead attempt fails the SCR is also immediately apparent (and generally poor).
This formula is VERY important, since it illustrates the critical influence on SCR that G and A assert. As an example, consider a '5, *19-6' player being held by a +2 battery with a 0 balk and 0 T-rating. The SCR for this situation is NOT 19/1 - it is just 2.375/1 even though the player is a 19 to steal on a good lead ! Even a 20 balk, 20 T only increases this SCR to 4.375/1. The pickoff component of the stealing rules effectively destroys the stealing usefullness (with respect to contributing to average offense) of low lead players. In fact, the game is biased against such players - it is much harder for low lead players to realize their actual, 'effective SCR' (i.e., SCR considering pickoffs as part of stealing) than it should be. This is due to SOM relating pickoffs to base stealing attempts. Even though pickoffs don't count as a caught stealing statistically, their effect is the same and more importantly, by associating the two SOM penalizes the stealing impact of low lead players. Since there is a 1/20 chance for a pickoff test, one needs a lead chance several times larger than 1/20 before the 'safe steal number' represents a decent approximation to the true SCR.
Of course, you should ask "So what ! I'm not going to be able to compute SCR and OBP and SLG in my head, so what good does all this do for me ?" The answer is to get a copy of our LINEUP spreadsheet. SRA plans to upgrade LINEUP to include SCR for the 1995 season card set, as well as including an extended analysis that gives stealing thresholds as a function of the # of outs in the inning and the batting order.
SRA - we take the guesswork out of STRAT !!