Auction Value Aging Curves

After extending my stay at Prospect Station, I have arrived at the next leg of my travels: Aging Curve Junction. I acknowledge that this metaphor is somewhat tortured, but I do think it gestures toward something genuine. A junction is a place where two paths of travel meet, and a dynasty ranking system requires merging the hypothetical of prospects and the real-life production of current big-leaguers over a period of future years. For our purposes, the bridge between those links is the aging curve.

Aging curves are a well-trod topic here. Several of my RotoGraphs colleagues – Jeff Zimmerman, Mike Podhorzer, and Jordan Rosenblum (apologies if I have missed someone!) – have made various contributions to the research and discourse, to say nothing of Tom Tango, J.C. Bradbury, Mitchel Lichtman, and Jonathan Judge at various other outlets over, at this point, decades of analysis.

Where I make a contribution to this field is admittedly niche, but nonetheless is novel. The existing body of work has clear implications for our game, but is predicated on real life stats. Here, I make, from what I can tell, the first attempt at fitting an aging curve on based on auction dollars explicitly.

Literature Review

Before embarking on my analysis, I studied the history of research on this topic. I rely most heavily on Tango, Judge, and Lichtman but garner influence from all the sources provided at the end of this article. The review below focuses primarily on hitter aging curves, as research on pitcher curves is somewhat more scant.

There are two key methodologies that have been used to derive aging curves: the delta method, and regressions. The delta method is arithmetically simple but computationally somewhat involved. It requires pairing off every consecutive set of player-seasons across a given sample, measuring the change (i.e., delta) of the chosen statistic between the two years, and attributing it to the age threshold (e.g., 22-23 or 34-35) which those seasons cover. If you do that over a lot players and a lot of seasons and take the average of the changes for each age pair, you can construct yourself an aging curve. Regressions are theoretically more complex but computationally straightforward. You simply need a dataset featuring your target variable (statistic) and independent variable (age) and can use one of any number of regression methodologies to fit a curved line on your data.

Tango and Lichtman are delta method pioneers, and have devoted a lot of time to untangling various selection biases that can occur when trying to filter the data. A key one is survivorship bias. In his blog post, Tango views each player’s final season as unrepresentative because to the extent that their poor performance is unlucky or predicated on managerial decision making about playing time, they are not receiving another season to regress back to the mean. Therefore, their career sample is often ending in a manner which is artificially worse than it might be. He finds that if you omit a player’s final season, the aging curve shifts to the right (i.e., players peak somewhat later). This methodology is somewhat blunt, and Lichtman notes another way survivor bias can persist in the delta method in his 2009 research. He puts it better than I could, so I will quote him directly:

Say there are 100 marginal players in MLB at any one age, and their true talent batting average is .220. Let’s say that they get 200 at-bats in Year I and half of them end up with an average of .180, and the other half, .260.Since in 200 PA, their “sample” BA will be all over the place (the standard deviation of batting average in 200 AB is 29 points), but centered on .220, this is a plausible, albeit simplified, scenario. And let’s say that only those 50 players who hit .260 were allowed to play the next year. After all, if a marginal (or old) player, talent-wise, has a very unlucky season, he is often benched or retires the following year.

So now we have 50 players remaining who hit .260 in Year I and are allowed to play and amass another 200 AB in Year II. What will they hit, on the average, in Year II, if they neither improved nor declined in true talent (say they were around age 27)?.220. After all, we already said that they were true talent .220 hitters. But what would the delta method say about them? It would say that they declined by 40 points(.260 in Year I and .220 in Year II)!

If all 100 players were allowed to play the next year, and they all hit .220, as they should, then half would decline by 40 points, and half would improve by 40 points(.180 in Year I and .220 in Year II), and the net change for all 100 players according to the delta method would be zero, as it should be. That is why survivor bias produces more decline (and less improvement) than it should at every age interval.

He addresses this bias by devising a method of simulation to assign a “phantom” second season to the non-survivors mentioned above. His analysis likewise results in an adjusted aging curve which shifts right relative to the biased sample. Rosenblum takes this one step further, addressing a pocket of survivor bias by forecasting missing seasons (i.e. a player might have a 23-24 and 25-26 pair, but not 24-25). This technique is particularly useful in an age where pitchers are often missing large chunks of at least two full seasons due to Tommy John surgery.

Judge and Bradbury sidestep bias created by pairing seasons by leveraging regression approaches that measure the relationship of age and the target variable across the full stretch of a player’s career. Judge likewise shows that survivor bias in regression approaches is generally minimal, and under circumstances when it does appear, can be addressed.

Typically, studies which leverage regression-based approaches result in both flatter curves and later peaks. This can be found especially in Judge’s fabulous work comparing the performance of the delta method against his chosen regression model. This makes intuitive sense – if efforts to address survivor bias in the delta method produce curves which are shifted right (therefore peaking later), an approach which does not fall victim to this concern should likewise show a similar result.

Auction Value Aging Curve

When creating my own curves, I experimented with both methodologies, following Judge’s similar comparison approach. I used the same assumptions and methodological choices he applies, with one main difference (other than my dependent variable): I used data from 1986-2025. This is admittedly arbitrary; the time range is an artifact of the range needed calculate the career value of all the players who appeared in the historical Baseball America prospect lists. I had the data on hand and cached to my system, so I figured I may as well use it. Judge uses 40 years of data (1977-2016) in his research and I think the same time range but on a more updated timeframe is reasonable. I have looked at some alternate time slices within my sample, but a more expanded analysis is certainly feasible and worth exploring in the future. As with my other analyses, I am using a standard 12-team, 5×5 roto setting with 23-man rosters.

Below, I have depicted the curves using the delta method and the regression (GAM, or generalized additive model) approach. The delta method curves are smoothed with a spline regression.

Here, we have a very similar broad-strokes finding: the delta method produces both a steeper and left-shifted curve relative to the regression model.

That the peak ages are earlier is somewhat interesting, and makes sense. For hitters, standard roto scoring is reliant on stats which tend to peak sooner: steals and batting average. Runs and RBI are generally downstream of batting average. The pitcher age peaks are overall consistent with the results found by Zimmerman.

The shapes, too, are noteworthy. For both the GAM and delta methods, the pitcher curve is less steep than the hitter curve and has less build-up prior to their peak level. One thing to keep in mind is that the flatter curve doesn’t mean that pitchers are better than hitters. To illustrate this, if you take a 22-year-old hitter and pitcher each worth $5, because the pitcher is closer and the slope to his peak is less steep, he will be less valuable overall than the hitter. In the column above, find age 22 for hitters, and age 22 for pitchers, and add the values in the delta columns to $5. For the pitcher, he would peak at $8. For the hitter, he would peak at $16. Same goes for the other side of the curve. That 22-year-old, $5 hitter would be forecast to still be worth $7 at age 30. Meanwhile, the pitcher would be worth $0. Broadly, the flatter curve means that you can dream on hitters more than you can pitchers, which aligns with general perception.

I also tested the performance of the delta method and GAM models in a manner similar to Judge, likewise using MAE as the performance benchmark:

For full details and explanation on the test setup, please refer to his analysis. I tested two different blind periods (2016-2025 and 2021-2025), and for each of those, two different calibration periods: one on the full data, and one with a more recent window. The Average-Only column represents the model result if you simply leveraged the age-weighted average dollar value across the entire sample. Judge does not report this in his article but does include it in his code; I thought it was useful so I replicated it and provided the result here.

Here too, as in Judge’s research, the GAM model performs better, though double-digit errors in any case are still quite high relative to the typical dollar value of a fantasy performer. In addition, these models aren’t outperforming the naïve average-only benchmark. This doesn’t necessarily mean the aging curves are bad or wrong – it just indicates that auction value is volatile and there are other aspects which can affect a player’s auction value than just age. Something, perhaps, for a future iteration of the model to address.

Aging Curves in the Dynasty Model: Delta Method

Ultimately, I am choosing to use the delta method to construct aging curves for my dynasty model. For transparency, I will note why, and also that I am not attempting to correct for any type of bias at this juncture. This is for several reasons.

One, as it relates to survivor bias, this would add yet more complexity into a model which already has a number of moving parts. I think it would certainly be a worthwhile pursuit, but in an effort to actually release the model before I myself age off of the curves above, I have to make decisions and prioritize certain aspects of model development relative to others. Based on my read of the research, survivor bias is genuine but usually small. I have bigger fish to fry.

Second, other types of bias that can occur using regular baseball statistics (year and park effects) are naturally adjusted for or at least mitigated in the construction of my target variable. Auction value for every season is already indexed relative to that season, so I feel comfortable that year effects are not materially affecting my results. The next, park effects, matter less when considering auction value than, for example, wOBA. For us, $20 is $20 whether it’s achieved in Coors or Seattle. Yes, playing in Coors might make it easier for a hitter to accrue fantasy value, but unless we expect Coors or any other home park specifically to affect how that value changes year-over-year, this will not impact our results. Now, this obviously this does not apply to players who change teams. I suspect that this effect is small and is probably leveled out within a large enough sample. This is something, though, that I do want to prioritize in a future update to the model.

Finally, even though the regression model generally performs better in the above test, I think that the shape of the delta curve is more aligned with my modeling goal. A steeper curve is going to penalize players on either side of their peak more extremely. For young players (hitters, anyway, based on the plots above; pitchers behave pretty similarly pre-peak), this means that a 19-year-old player is going to be heavily penalized relative to his peak based on his place in the aging curve. This makes sense for what we’re trying to achieve – prospects are not playing in the major leagues and tie up fantasy roster spots without contributing any production until they’re called up. Philosophically, this is something I want reflected in the rankings and this conservativeness with respect to prospect value is something that I think will differentiate the model from other industry dynasty rankings.

Likewise, on the other side of the peak, older players’ values will decrease more steeply. I will show this in further detail with the full model launch article, but using the less aggressive curve created weird edge cases where players in their 30s who are atypically good (Kyle Schwarber, Chris Sale, etc.) remain so in a manner I think is unrealistic. Additionally, the impact of the curve in outer years will be mitigated by a discounting factor. These are the types of tradeoffs you have to think about when building a model and, ultimately, I think that given the overall body of work supporting the delta method, the choice is nonetheless sound.

Fun Stuff

As promised, I also wanted to look at how these curves are behaving over time. To this end, I split my sample in half and generated the curves for both methodologies:

If you wanted to see the effects of the steroid era mapped onto fantasy performance, this is probably as clear of a picture you’ll find. For hitters, players in the 1986-2005 sample had gentler declines regardless of the method of calculating the curve. The picture for pitchers is a bit mixed, and I would be interested to poke into whether the gentler curve for the late 20s and early 30s in the more recent sample is genuine. I could think of reasons why this might be the case – better surgical techniques and overall prognoses over time could lead to older pitchers staying good for longer. Something worth future research, to be sure.

As has become a pattern for me in these types of articles, it’s time to remember some guys. For kicks, I looked at some players who had the chalkiest aging patterns. That is, their personal aging curves closely matched the overall curve.

To look at a few of these guys graphically:

Interestingly, a lot of these players are catchers. This could be random. Or, this could be something unique to catchers. In my earliest drafts of aging curve analysis, I attempted to create position specific curves – something I eventually abandoned to keep the overall exercise more reasonable in scope – but perhaps it is worth examining whether catcher aging patterns should be treated differently.

For pitchers, there is nothing particular of note that jumps out, but I still think the graphs are cool:

Conclusion

The overall results here are not altogether surprising; the peak ages and curve shapes are broadly consistent with the existing body of literature on the subject. That there seems to be a small fantasy-specific twist (earlier peaks) is interesting and worthwhile to dig out.

This is far from the final version of this analysis as well. There are numerous avenues for future analysis – hunting down omitted variables bias, diving deeper into the pitcher curve behavior in recent seasons, or seeing if there should be a catcher-specific (or perhaps other position-specific) curve as well. Surely there are more.

That said, I believe that this represents a good start and is a valuable addition to the broader canon of aging curve research. Clearer understandings of how age interacts with our fantasy target – player value – will help us on the path to robust models and valuation systems.

References

Many of these are linked in the body of the text above, but if you are looking for further reading, please see below. You may find a surprising amount of snark in some of the articles from the 2000s; there appeared to be a bizarre amount of acrimony in the debate at the time regarding how best to fit aging curves.

• http://blog.philbirnbaum.com/2009/11/bradbury-aging-study-re-explained.html
• http://blog.philbirnbaum.com/2009/11/bradbury-aging-study-re-explained-part.html
• https://www.baseballprospectus.com/news/article/9933/how-do-baseball-players-age-investigating-the-age-27-theory/
• https://www.baseballprospectus.com/news/article/59491/an-approach-to-survivor-bias-in-baseball/
• https://www.baseballprospectus.com/news/article/59972/the-delta-method-revisited/
• https://tht.fangraphs.com/how-do-baseball-players-age-part-1/
• https://tht.fangraphs.com/how-do-baseball-players-age-part-2/
• https://www.tangotiger.net/aging.html
• https://fantasy.fangraphs.com/the-stolen-base-aging-curve-depicted/
• https://blogs.fangraphs.com/component-changes-in-new-hitter-aging-curves/
• https://tht.fangraphs.com/aging-curves-revisited-damn-strikeouts/