Rethinking the Win Curve

Editor’s Note: This piece initially was given as a presentation at the marvelous 2016 Saberseminar.

Transaction analysis has become one of the most important topics in all of baseball research. The primary tool that is used in almost all public transaction analysis is a dollars/WAR calculation. This tool has been convenient in many applications. It is especially good for predicting how much money an upcoming free agent will receive on the open market.

However, dollars/WAR also has significant limitations. In many situations, it simply does answer the questions we are interested in. As an example, let’s look at the Chris Sale trade to the Chicago White Sox. Last offseason, the White Sox traded Sale to the Red Sox in exchange for Yoan Moncada, Michael Kopech, Luis Alexander Basabe and Victor Diaz.

By a dollars/WAR calculation, this was pretty close to an even trade, seeming to be mutually beneficial. The Red Sox preferred the present wins in the form of a starting pitcher, and the White Sox preferred the prospects. A dollars/WAR calculation does not allow us to see how much each team benefitted from this deal. I will present a new framework that can help teams make decisions under uncertainty. We will be able to see how much any player is worth to any team given a team’s preferences and the player’s impact on the team’s projections.

The Win Curve

About 10 years ago, Vince Gennaro, Nate Silver and others began writing about the win curve. The win curve graphs the marginal value of an additional win given a team’s final record. Here is the win curve Silver came up with.

As you can see, the value of a win peaks around win number 90. These wins greatly increase a team’s chances of reaching the playoffs, advancing in the playoffs, and winning the World Series–many of the things teams really care about. Focusing on a team’s regular-season win total can serve as an effective proxy for these goals.

I will distinguish Gennaro’s win curve by calling it a roster value curve. Gennaro graphed the total value of a roster over the number of wins it produced. The win curve is simply a graph of the derivative of the roster value curve. The key finding Silver and Gennaro found was that, for the most part, not all wins are created equal. Teams in close contention for the playoffs see much more marginal value from additional wins.

There has been a lot of debate about what the win curve actually looks like. Many have argued that Silver significantly undervalued wins that did not affect playoff odds. For this analysis, it is important to remember the win curves are for the teams to decide. The curve simply shows how much a team values each potential outcome.

Silver and Gennaro focused on how much these win totals affected a season’s total revenue. However, other factors could shape a team’s preferences, such as how much an owner wants to win a World Series or how much one year’s win total impacts future revenues. The preferences of any team can only be decided by the team itself. I will use a roster value curve as a team’s unique utility function.

Accounting for Uncertainty

While a team could identify its preferences and build its own win curve, the curve has limited ability in helping to guide its decisions. Obviously, when teams have to make roster decisions, they are operating under uncertainty. They do not know where they will land on the win curve or exactly how their potential acquisitions will perform. Phil Birnbaum wisely pointed this out when commenting on Silver’s win curve:

Silver’s graph tells us how much an *actual* win is worth. But, before the season starts, a team can’t know how many wins it will achieve with that kind of precision. Even if it’s perfectly omniscient about how much *talent* its team has, there’s still a standard deviation of about six wins between talent and achievement. A team that’s created to be perfectly average in every respect should go 81-81–but, just by random chance, it will win fewer than 75 games about one time in six, and it’ll win more than 87 games one time in six.”

Birnbaum correctly concludes that this uncertainty will make the hump in the graph wider and shorter. However, we can be a lot more precise and end up with a much more useful result. We should leave the win curve as it is and call it an ex post win curve. The ex post win curve will simply show the value of each marginal win at the end of the season. From this we can model a preseason, ex ante win curve, which will show the value of a marginal projected win. This stochastic model will allow us to find the marginal value of adding a given player to a specific team.

Building the Ex Ante Win Curve

To start, I will use a hypothetical ex post roster value curve.

And here is its corresponding ex post win curve. I chose to use a curve that looked a lot like Silver’s.

Next, I simply used probability mass functions to come up with distributions of potential records given preseason forecasts. I created normal distributions of win total projections with means between 60 and 100, all with a standard deviation of eight wins. The uncertainty in these forecasts comes from three main sources: random variation, injuries, and uncertainty of players’ true talent. Here is the probability mass function for a team projected to win 81 games.

Using these distributions and the roster value curve, I found the expected values of the rosters projected to win between 60 and 100 games. The value of any projected record (or any asset in general ) is the sum of the probabilities of ending up in every potential state multiplied by the value of ending up in these states–our discounted expected payoff.

These expected values were easy to calculate because there are a discrete number of outcomes for any season; a team can win between zero and 162 games. Here is the formula I used.

Where x = projected final win total, w = actual final win total, and z_w = payoff given final win total.

If the summation notation is unclear, here is a quick example:

E(Projected Win Total) = … + p(65 wins)*payoff(65 wins) + p(66wins)*payoff(66 wins) + p(67 wins)*payoff(67 wins) + …

Once we have the expected values of the projections, we can plot a preseason, ex ante win curve. This win curve will show us the value of rosters given their projected win total.

From here, we can easily build a new ex ante win curve.

There is a lot to observe in these new ex ante curves. First, we can see the win curve is much flatter and has a wider hump, just as Birnbaum predicted. Any increase in uncertainty will continue to flatten the win curve. While the marginal wins on the hump of the win curve (between 85 and 95 wins) are most valuable, you do not know where you will end up on the win curve before the season. By improving your projection from 82 to 83 wins, you may end up getting yourself some of those most valuable wins.

Furthermore, we can see this team should never pay more than about $6.5 million to add a projected win to its preseason forecast. This means it should not pay the going market rate for most free agents despite the fact that this team has a $210 million payoff from winning 95 games. It is not sensible for many teams to spend significant money in free agency, especially when they are not in a high-leverage spot on the win curve. Empirically, we see teams generally recognize this. The most valuable wins are worth over $10 million to this team. However, it can’t go buy these wins with certainty.

We can model the trade deadline by decreasing the amount of uncertainty in the win curve. At the deadline, teams already have played over half of the season. Therefore, they are much more certain of the value of the wins they are acquiring. If we lower the standard deviation of the wins, we can build a trade deadline win curve.

You can see this win curve clearly has a much larger peak than the ex ante win curve. A team in contention may be willing to give up much more for a projected win at the trade deadline than it would in the offseason. (The win curve becomes a worse proxy for the outcomes that a team cares about at the trade deadline, but this is a topic that requires a separate post.) Adding a projected win at the trade deadline has a much higher probability of adding the actual wins you are hoping for.

Making a Multi-year Model

The final issue that needs to be tackled is making this single-period model into a multi-year model. Luckily, this shouldn’t be too difficult. We can still use the same shaped win curves for every year in the future; they just need to be discounted.

There are three factors to consider when discounting these future wins: baseball’s continuing salary inflation, the interest rate, and impatience. Baseball has seen consistent salary growth now for decades. We will label the inflation rate as π, and the interest rate as r. The factor for impatience will be β, where 0<β<1. We have seen many teams, most notably Mike Illitch’s Detroit Tigers, operate with very significant impatience. Depending on this unique preference, it can be very rational to sacrifice the future for an extra win now.

We can adjust the value of wins in future years by a factor of ⌈(1+π)β/((1+r))⌉^t, where t is the number of years we are in the future. This equation is simply a scalar to adjust the win curve up or down. The interest rate and β decrease the value of a future win, while (1+π) increases how much we value a future win. In the current year, where t=0, this scalar will just go to 1.

Transaction Analysis

Finally, we can now discuss using the model for transaction analysis. We need to have a team’s win curve and its projections with and without a specific player. Given those two things, we can precisely calculate how much that player is worth to a team. The fundamental concept of asset pricing theory is that price equals expected discounted payoff. We can now calculate the expected discounted payoff of any player for a given team.

For each year the player is under contract, we take the expected value of the roster with the player and subtract both the expected value of the roster without the player and the player’s salary. This will give us a net present value evaluation of any player. In simplified mathematical terms, it’s merely:

Where x = projected final win total, current year is t=0

Going back to our original example of the Chris Sale trade, we can make no declarations from here on how much each team benefitted from the deal. But if we had a win curve for both teams and their projections with and without the players, we could easily find the unique dollar value of each player in the deal to each of the two teams.

This model provides a simple framework to evaluate the payoff of every potential transaction, though it does come with a few limitations. The biggest issue is that it assumes a player will remain with the team throughout his entire contract and for no longer. However, these sorts of minor concerns can be accounted for manually. The question of how much a player is worth to a given team no longer has to be a guessing game.

References & Resources

• Dave Cameron, FanGraphs, “Valuing the 2017 Top 100 Prospects”
• Dave Cameron, FanGraphs, “Chris Sale Makes the Red Sox the AL Team to Beat”
• Vince Gennaro, The Hardball Times, “Diamond Dollars: The Economics of Winning in Baseball (Part 1)”
• Neil deMause, Baseball Prospectus, “When Is A Win Not A Win?: Improving on MP/MW”
• Wright Thompson, ESPN the Magazine, “The Mastermind”
• Phil Birnbaum, Sabermetric Research Blog, “The marginal value of a win in baseball”
• Wikipedia, “Risk-neutral measure”
• Cork Gaines, Business Insider, “The Detroit Tigers were the biggest spenders in free agency”
• Dave Cameron, FanGraphs, “An Attempt to Find the Market Price for Wins in July”
• Jeff Sullivan, FanGraphs, “Mike Ilitch Gives His Money to Justin Upton“