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## The probabilistic concept of value

Posted by Derek Carty at 10:25pm

The concept of value seems to be a hot topic of late, so I thought I'd chime in with my thoughts on it.

Let me preface this entire article by saying that I do not believe dollar values are the end-all-be-all when deciding who to take in an auction. If you like a player, but it costs a dollar or two more than you were hoping to get him for, you should absolutely do it. I do not, however, think that you should make this a regular habit.

###### Stat projection parallel

While it is very true that a player you have projected for \$30 could easily end up anywhere from \$25 to \$35, I don't think that warrants spending \$33 or so every time a player you have projected for \$30 comes up.

I think this draws a parallel to the way a stat projection should be made (and is made, really, by most of the major systems). To take an example, let's look at batting average. A player with an expected batting average of .275 could find himself hitting anywhere from maybe .250 to .300 at the end of the season with little change in his skill set and still be within reasonable statistical variation.

There is a chance his BABIP will be lucky in the .390 range and a chance it will be unlucky in the .260 range. His contact rate could be up or down a point or two, or his home run rate could be a smidgen up or down due to some luck. When projecting him to hit .275, we need to place a weight on every single one of these possibilities — on every single possible batting average — and we eventually end up with .275 as the average expected batting average... even though we realize it could easily be higher or lower with the exact same skill set.

While we don't literally weigh every single possibility, when we look at a player's skills to project his production, this is exactly what we are doing.

If you're familiar with PECOTA, you are probably following along nicely right now. PECOTA breaks a player's possible statistical outputs into percentiles. It's 50th Percentile projection is the average of all the percentiles combined.

Here's where our discussion gets slightly abstract, so please keep an open mind. Given the information we now know about the player's percentile projections and given a universe where the batter plays out the upcoming season an infinite number of times, the player will average his 50th percentile estimate. Sometimes he will perform at the various other percentiles, but on average, the 50th percentile is where he will end up.

So if we're going to use his projections for the coming season in this universe, in which the player has just one crack at the season, which set of projections would be the best to use? Obviously, the 50th percentile projection is the safest bet. And if we use the 50th percentile projections for all players, we will undoubtedly project more of them correctly than if we used, say, the 10th percentile projections. Understand?

###### Poker parallel

Another great parallel to this concept is poker. Let's assume we're sitting at a no-limit Texas Hold 'Em table, and we're looking for the fifth card of a flush going to the turn. We know that we have roughly a 4-1 chance of catching that card to complete our flush. That means that in order to break even on this hand, our pot odds need to be better than 4-1. If the odds are better than 4-1, we call this a high equity play. This means that throughout our poker career, if we continually make this bet (drawing at a flush on fourth street) while we have better than 4-1 odds, we will make a profit.

Sometimes we will not make our flush, and we will lose the hand, despite making the mathematically proper play. It will not work out every single time; it can't. But over time, making that same exact play, logic and probabilities dictate that we will indeed make a profit. It would be ill-advised to assume that the values are different just because the outcome changes each time.

It's the same with batting average, going back to our example from before. Just because our player can hit .300 doesn't mean we should value him as such. It will happen occasionally, but more often he will hit .275, so that's what we should value him as. Consistently valuing these types of guys as .300 hitters would ultimately yield negative value, because they can't all hit .300. Collectively, on average, they will hit .275 and you will be paying too much for them. Understand?

###### Back to dollar values

Now back to the matter at hand — dollar values. A \$30 player could easily end up as a \$25 or \$35 player, but on average he will be worth \$30. So to say, "well I can pay a few extra dollars for this guy because we don't know exactly where his true value will end up" seems like it would be a mistake.

It's like in poker saying, "well, the odds are 4-1 that I'll make my flush on the next card, but it will cost me 3-1 pot odds to make the call. Oh, well, it's close enough, and we really don't know what the next card will be, so I'll call." That will be a poor call because while occasionally you will make the flush (the odds of making it never change), over time you will lose money because you are putting in more to make the calls then you will receive when you eventually catch your card and win the hand.

The same goes for our \$30 player. If you are consistently paying \$33 dollars for him (or other players you have valued as \$30), you will lose money in the long run. Sometimes he will be worth \$33, sometimes \$25, sometimes \$35, sometimes \$30, and so on. Over time, though, that player, on average, will yield \$30 in profit. Consistently paying \$33 for him will cause you to lose money. Plain and simple.

###### Spend wisely

I did say in the beginning, though, that we should not become overly obsessed with our dollar values, and I absolutely believe this. If, occasionally, you want to pay \$32 for the \$30 player, go right ahead. It can be difficult — if not outright impossible — to get excellent value with every pick, especially in a competitive league. Plus, you will almost surely have the opportunity to make up that \$2 in value later on in the draft. Remember, though: do not make this type of behavior a habit.

###### Concluding thoughts

We'll be talking more about value in February, but I think this is an excellent start. I know all of this is a little abstract and might be difficult to understand if this is the first time you're being exposed to it (I hope I explained it well enough), but if you can wrap your head around this probabilistic concept of value now, you will be in great shape going forward.

On a separate note, I just completed my first mock auction of the year tonight, so be on the lookout for a recap tomorrow.

Derek Carty, 23, has also been published by NBC's Rotoworld, Sports Illustrated, FOX Sports, and USA Today. This season, he'll be contributing to FanDuel and will be linking to all of his work at DerekCarty.com. In his three years competing in expert leagues, he has won 2 titles with 4 top three finishes, including a LABR NL title in 2009, making him the youngest person to ever win a major expert league title. Derek is a proud graduate of the MLB Scouting Bureau's Scout Development Program and is a firm believer in the importance of combining stats and scouting. He welcomes questions via e-mail, Facebook, or Twitter.

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