Quantum TatersThis will be of no interest whatsoever to anyone, but it needs to be published somewhere, so tough luck. I submitted the essay to an online publication called
Baseball Prospectus, where--given the nerdiness of their typical article--I actually thought I had a good chance of seeing print, but no dice. To quote from my rejection letter:
I'd sat [sic] first thing you need to do is make the writing more accessible.
[BP Writer] may use complicated stats, but he doesn't obfuscate the text with
words that seem complicated for the sake of being complicated. Write it
like you're describing the concept to a bseball fan friend who doesn't
fully understand it, not as if you're writing your master's thesis.
Oh well. Fuck you guys! I'm happy with it the way it is, so here goes...
Quantum Taters: Marginal Utility in Head-to-Head Fantasy Play
Suppose you’re in a head-to-head fantasy league and you’re losing the steals category more often than not. If you have the opportunity to pick up Dave Roberts at what seems like a minimal cost to your team, should you do it?
Intuition might provide the correct answer to this question, but it could just as easily lead you astray. To solve the problem in an analytical manner, you need to determine the marginal utility of the transaction. In other words, you must calculate the incremental competitive impact of adding a virtual Dave Roberts to your ballclub while simultaneously subtracting the player(s) that are traded for him or are relegated to the bench to make room for Roberts in your lineup.
Let’s take a closer look at the concept of utility in a head-to-head fantasy league. Since head-to-head success is measured in terms of wins and losses (as opposed to raw statistical dominance), utility must be measured in the same way. All of the simulations below are based on actual weekly data for a 12-team, AL/NL, head-to-head fantasy league I participated in last year. Though the weekly averages might differ significantly in an AL-only or NL-only league, the general concepts introduced here should be valid for any head-to-head situation.
The following chart shows a team’s performance level (with respect to the league average) on one axis and the resulting utility (or winning percentage for each category) on the other axis for 500,000 simulated head-to-head matchups using data that is statistically identical to the league’s actual 2004 results*. Also displayed in the chart legend are the league’s average weekly performance levels and standard deviation for each category.
To make use of the chart, you must first measure or predict a team’s performance level in a given category. For example, if I had assembled a 2004 lineup with an aggregate OBP that was just 10% above the league average (and indeed I did feature The Great One on my team), I would have won the OBP category in nearly 80% of my weekly matchups over the long term. At the other extreme, I might conceivably have put together a speedy squad with 30% more steals than the league average on draft day and still ended up winning the steals category only about 60% of the time. How can this be?
The explanation is twofold: first, the categories featuring high standard deviations compared to their means—such as stolen bases—offer a consistently low marginal utility for a single percentage point of statistical improvement. Even if I have a tremendous team in terms of steals, I might crush an opponent 9 to 2 the first week only to lose the following week by a 5-4 count. The high variance makes stolen bases an inefficient category to dominate, and this explains the consistently low slope of the SB line in the chart above.
But what about the curious stair-step features that appear in the lines for steals and home runs, such as the flat spot in steals between 120% and 130%? This turns out to be a consequence of the non-Gaussian distribution of the data. As a thought experiment, you could imagine an extreme “bimodal” league in which teams post either above-average weeks (5 steals) or below-average weeks (2 steals) 80% of the time, and only infrequently clock in right in the middle (3,4) or at either extreme (1,6, etc.). Below is a simulation of this league for steals only:
The further the actual statistical distribution strays from a Gaussian bell curve, the more pronounced the step-like utility results will be, because the skewed distribution creates certain “tipping points” of performance between which small fluctuations will not change the outcome very much. As a result, fantasy participants must unwittingly contend with what I call “quantum taters” and “quantum steals”—categories with flat spots where small changes in performance are nondeterministic and can lead to large changes in winning percentage for the category, or to no change at all. Other offensive categories such as R, RBI, BA, and OBP are typically much more Gaussian in their performance distribution.
Given the small sample sizes seen in a typical fantasy season, individual team performances will deviate significantly from the long-term expectations shown above. However, fantasy owners looking for every edge might consider checking into their leagues’ scoring patterns in these “quantum” categories to see if any statistical advantage can possibly be eked out.
Let’s turn back to Dave Roberts. A projected total of 45 steals works out to about 1.80 per fantasy week over a 22-week regular season, assuming that three weeks are set aside for playoffs. If he replaces a slugger with essentially no steals in my 2004 league, Roberts would single-handedly raise his team’s stolen base output by over 50% of the league average! If the team collectively started out at the league-average level in steals, then the addition of Roberts would bump up its scoring by 0.162 wins per week, or about 3.6 wins over the course of a full season—a significant number.
However, on the negative side of the Roberts ledger is…just about everything else, unfortunately. The good news is that Roberts’s impact on a team’s averages in the other offensive categories will be relatively small compared with that of steals, but what about his overall net impact in terms of wins and losses? If we use Dave’s 2005 PECOTA projections to replace a 2004 league-average player on an otherwise league-average team, here is the estimated impact in SB, R, HR, RBI, and OBP:
| Category | Dave | Δ Tm Avg | Δ % Lg Avg | Δ Wins/Wk |
| SB | 45 | +1.41 | +40.5% | +0.138 |
| R | 79 | -0.390 | -1.22% | -0.016 |
| HR | 6 | -0.716 | -8.32% | -0.068 |
| RBI | 41 | -1.76 | -5.74% | -0.064 |
| OBP | .337 | -0.00244 | -0.681% | -0.020 |
| TOTAL |
|
|
| -0.030 |
In the final analysis, the likely impact of switching out an average player for Roberts in this league is to subtract about 0.030 wins per week from a team’s total—a fairly negligible impact, especially given the possibility of a flat spot in the league’s distribution of homers or steals, but a result that nevertheless does not endorse the trade. You would almost certainly get a negative return on the Roberts transaction if you had to bench or trade an above-average performer in order to make the deal happen.
The moral of the story here is not to avoid Dave Roberts, but more generally to investigate your league’s performance characteristics to provide an analytical underpinning for your roster decisions. We’ve demonstrated here that the stolen base provides significantly less return on a percentage point of improvement than most other offensive categories, because the variance of the steals category is extremely high compared with its league average. This result will likely hold for all fantasy leagues. However, since there is as yet no Fantasy Baseball Prospectus to do the number-crunching for all conceivable fantasy universes, you are hereby encouraged to do your own investigation of the statistical distribution patterns in your own league. If you’re lucky, you may stumble upon a few quantum taters of your own to capitalize on!
* Notes on simulated matchups:
- The opponent’s score in a given category for each matchup was randomly selected from the 264 observed data points comprising the entire 2004 season for that category.
- The measured team’s score was determined using the same procedure, with the following adjustments:
- After a series of scores was generated, it was first scaled by the desired performance factor (0% - 200%), then contracted about its new mean by a ratio of the average team standard deviation to the league-wide standard deviation for that category. These ratios varied from 0.83 to 0.99.
- The resulting score sets were compared pair-wise to determine the expected value resulting from a given performance level versus the league.
