# Nothing adds up

Last week, we described a new method for deriving fantasy values. First, some loose ends:

Readers will note that we spoke of “the last drafted player” and “the pool of draft-worthy players” without saying how we knew who these players were. The approach we favor is running thousands of Monte Carlo simulations of fantasy leagues—simply casting players onto rosters with no care for balance or value. Our base metric is something that we call Weight on Winners (WOW)—the frequency with which a player appears on the winning club. The higher the frequency, the greater the value. Note that, because our simulated leagues do not enforce a budget, we cannot turn these frequencies into dollar values; however, the frequencies should reveal the rankings of our players, so that we can pluck out the top 108 players or the 10th-best player or whomever.

Here were the steps for this study. We simulated a 12-team, standard-5×5, mixed-league contest. There are 108 total pitching slots. The average price of a slot (given a $260 budget and 23 slots on both sides of the roster) is $11.30.

1. **Find the pool of potentially valuable pitchers.**

For best results, the competition between our simulated teams should approximate the true level. We don’t want to consider every player who threw at least one pitch. On the other hand, we also do not want to unfairly exclude someone who, even in limited play, can have an impact.

We resolve this dilemma by tossing every pitcher into a simulation of a couple thousand leagues and finding the lowest IP total among the top 108 players. Every pitcher below this threshold is essentially given a grade of “Incomplete” and ignored in later steps. It is not fair—either to the player or to owners—to treat ultra-short seasons as if they are on the table. Nobody is weighing Chris Carpenter versus Andrew Carpenter.

At this point in the season, the threshold is 30 IP. Pitchers with slightly higher workloads who look to have a crack at the top 108 include Claudio Vargas (1.93 ERA in 37 IP), Randy Choate (3.62 ERA in 32 IP, 5 Saves), and Sergio Romo (3.94 ERA in 32 IP but 2 Saves and 5 Wins). A notable miss is Neftali Feliz, who has only 28.1 IP, but he is bound to top 30 IP by season’s end.

You can see from the above trio that our method appreciates value in a variety of configurations: low ERA; moderate ERA but Saves; unremarkable ERA but Saves plus Wins. These choices fall out of the simulation naturally; we didn’t have to “do” anything other than set up the parameters.

2. **Find the top 108 pitchers.**

We take all pitchers who survived the cut in Step 1 and simulate another couple thousand leagues to get the true top 108. Recall that our $11 slot is one that can freely float among any of these players.

Not surprisingly, Zach Greinke took the top spot; simulated teams with Greinke won 35% of their leagues. Tied for second place at 30% were Dan Haren and Tim Lincecum. Chris Carpenter, Felix Hernandez, and Javier Vazquez formed a 3rd tier at 27.5%, followed at 26% by the first closer on the list, Jonathan Broxton.

3. **Identify the $1 pitchers.**

Recall that we are going to be introducing $1 players onto our rosters. Now, we don’t want to put too much weight on the particular player in the 108th position—he might happen to be a beast in one area, which would bias our findings. Moreover, we may need to swap in multiple $1 players, and it would be better not to re-use one guy.

So we’ll draw our $1 players from a pool of 12—the last six draftable (#103-108), plus the first six non-draftable (#109-114). In alphabetical order, here are the $1 players for this study:

IP W Sv ERA WHIP K Bergesen 123 7 0 3.43 1.28 65 Breslow 65 7 0 3.46 1.09 50 Condrey 39 6 1 3.20 1.17 23 Johnson Ji 67 4 8 4.05 1.32 49 Kawakami 152 7 1 3.92 1.33 102 Masset 70 5 0 2.56 1.04 65 Morales F 38 3 7 3.05 1.30 40 O’Day 55 2 2 1.80 1.00 53 Palmer 114 10 0 4.03 1.34 65 Peavy 87 7 0 4.05 1.18 97 Troncoso 79 4 5 2.75 1.39 52 Zambrano 154 8 0 3.91 1.43 138

Again, there’s a good mix of players there.

So we’ve defined our $11 slot (the pool of the top 108 pitchers) and our $1 slot (the pool of 12 end-rounders). All that’s left is to run the experiment that we outlined last week. We’ll start with a straight version of Roster #1 (Halladay plus eight free-floating slots) and then replace one, two, and three of the floating slots with $1 players. Roster #2 is fixed with nine free-floating slots.

We submitted each two-team league (four versions) through 2,000 runs of our program and tracked the winning percentage of Halladay’s team at each stage.

Did we obtain Halladay’s value? No. Or, we don’t think so. Here’s the graph:

Nice curve, but you can see that it crosses the 50% mark well before we would expect it to. Based on this graph, Halladay’s roster would meet Roster #2 after replacing only 1-3/4 of Halladay’s $11 slots with $1 slots. This equality puts Halladay’s projected value at about $18. For a guy with 15 Wins, 193 K, and a 3.01 ERA in 221 IP.

We can get a slightly more customary valuation for Halladay if we extrapolate from only the first two points on the graph—that is, from a state with zero forced $1 slots to a state with one. Doing so raises Halladay’s estimated value by $3, to $21. Still probably $6-$10 below his real value, if standard valuation methods are to be believed.

What gives? We glean a clue from the line’s curved nature. By our hypothesis, each substitution of an $11 slot with a $1 slot should have led to the same $10 drop in Halladay’s value. But the slide here is not linear but exponential. Each added end-rounder degrades Halladay’s roster ever faster.

The notion of synergy among roster picks is not new. We know, for example, that once you get one super-speedster, additional super-speedsters have declining worth to you because you need only so many SB to seal the category. Ditto for anything that you’ve already bought enough of.

This study suggests that you can also have too much of nothing. Recall that the price of a player is $1 + the marginal price of his marginal worth. A pure $1 player, then, has no marginal worth. When you add a $1 player, you are giving up a chance to gain ground on the leader.

That wouldn’t be such a bad thing—if you had an infinite number of slots. But slots are precious. In fact, a case can be made that not all slots are created equal. For example, if you had a roster of Roy Halladay by himself, there would be a tremendous amount of value in simply adding a second slot. On the other hand, if you had a roster with 19 slots, the 20th would barely raise your interest.

Have we mislabeled our slots? For Halladay to merit a higher price, either the $11 slot needs to be re-priced upward or the $1 slot needs to be re-priced downward, so that the wage gap between the two slots is more than its current $10. Can we justify that? We’ll keep you posted.

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I’m sorry if this was brought up before, but what about the fact that every team will inevitably have pitchers that have a negative value on their team? If you limit your pool to a certain percentage of pitchers, with the lowest level of the pool being $1, then all the pitchers outside the pool are less than $1 (or have a negative value). Since some of the pitchers will be on a team at some point, they accrue negative value to the team.

Or am I entirely missing the point of this exercise?