There has been a decent amount of discussion in Cardinal Nation about how long John Mozeliak should wait out Matt Holliday. I’ve gone on the record as saying that Holliday is the best available option and that I’d wait him out and live with Craig in LF if it came to that. I’ve done a decent amount of analysis to support the first claim (not exhaustive mind you), but was just going with my gut on the second.
I’d like to replace the gut feeling part with something that has a little substance, so we’ll start with that goal and see where we end up.
The rest is after the jump
I’m going to use a simple expected value (in this case WAR) comparison.
Generically the expected value for any course of action will be the sum over all of the possible outcomes of the probability of that outcome times the value of the outcome.
So now we just have to determine the outcomes there are for the two courses of action (COAs) in question (wait vs. not wait), the probabilities of those events, and the corresponding values (using WAR).
Expected value of not waiting = E(NW) = P(signing player(s) other than Holliday)*V(said player(s)) + (1-P(signing player(s) other than Holliday))*V(internal options)
Expected value of waiting = E(W)=P(signing Holliday)*V(Holliday)+P(signing player(s) other than Holliday |Holliday signs elsewhere)*V(said players)+P(no Holliday and no other external players)*V(internal options)+V(internal options*)
A couple of explanations are probably necessary. The last term is in attempt to even out the ledger if in the not waiting COA multiple players are signed (for example a Smoltz and Damon combination would have to be evened out with Jaime Garcia).
The second term in that equation is what most of the fuss is about. The feeling is that if the Cardinals wait too long then none of the decent external options will be left. That statement is factual in that there is a point in time where all of the other options will be signed. The larger question is, how long is too long?
The assumptions about values and probabilities are going to drive the analysis, so clearly this section is extremely important. I’m going to try and parameterize as many variables as possible to give a more robust solution, but that only goes so far as well. I’m just going to throw out an itemized list:
P(Holliday) will be a parameter
V(Holliday) conservatively set at 4.5 WAR (based off of simulation runs including the injury adjustment)
P(signing player(s) other than Holliday |Holliday signs elsewhere)=0.3
P(signing player(s) other than Holliday)=1
V(internal options) = Craig = 1 WAR, Garcia = 1 WAR
I’m going to look at two primary scenarios
- One spot filled (think Damon, Bay, or other higher dollar alternative)
- Two spots filled (think Smoltz and Derosa)
First scenario has the alternative assumed at 2.5 WAR, and the FA market gets worse every ten days (losing 0.5 WAR)
The chart reads chronologically left to right with expected WAR on the vertical axis. The blue line represents the declining FA market. The parallel lines are for different assumptions about the probability the Cards have to sign Holliday (P(H)) and represent the expected WAR of waiting. Clearly the longer the cards wait the lower the expected value gets. This is due to the declining nature of the FA market. One way to draw conclusions from this and the ensuing graphs is to extend the initial FA level out and see where the various lines cross the imaginary extension (for this graph it would be 2.5 WAR extended on out). For example the “break even” point between the green line (prob. of signing Holliday of 40%) and signing a free agent right away is waiting ~10 days. The red line in fact flattens out and never falls to 2.5, while the purple line is never above, which would indicate that FA alternatives need to be looked into immediately.
The seconds scenario assumes the ability to acquire some combination of 4.5 WAR on the FA market. Here’s the chart
Here the “break even” point on the 50% line is ~12 days, and the 40% line is below the “break even” mark the whole time. As an excursion I looked at having the probability of Holliday signing increasing over time (5% every 10 days). Here’s the chart for that
The increasing probability makes Holliday a better option with the 40% initial probability than it was with constant probability.
Clearly there are a great deal of embedded assumptions in this analysis, any of which could alter any potential conclusions. That being said, I think the probabilities displayed are probably on the conservative side. With that in mind, I still think the Cards should wait Holliday out, now I have at least a little analysis instead of just my gut.