# How much should Ben Sheets get?

### Background

The free agent pitching pool this year is relatively large, with pitchers such as CC Sabathia, A.J. Burnett, Ryan Dempster and Derek Lowe available. Included among these pitchers is Ben Sheets, who has become slightly overshadowed with Sabathia arriving in Milwaukee. Sheets is a very talented pitcher in his own regard, but he has suffered through inconsistent playing time in the past. I feel his contract negotiations will be among the most interesting to watch in the offseason.

I am interested because of the risk, perceived and actual, involved with Sheets. While Sheets is obviously a very good pitcher when healthy, we’re not quite sure how often he’ll be able to pitch. This could make giving him a long-term contract very risky. So what should a team offer Sheets in the offseason? To answer this question I’m going to combine a couple of the tools I have introduced in past articles. The first of these tools will be the utility curves I introduced in this article, and the second is player risk analysis I introduced in this article. We’ll need a couple of steps to estimate how much Sheets should get. These steps are:

1. Estimate Sheets’ true talent.
2. Find a team’s risk preference based on their utility curve.
3. Use Sheets’ risk profile to adjust his true talent into a range of outcomes.
4. Convert Sheets’ range of outcomes into a risk adjusted win figure and that into a dollar figure.

### Step 1: Estimate Sheets’ true talent

While most baseball websites do not have projections out for 2009, Colin Wyers ran an early set of 2009 marcel projections here. He projects Sheets for a 4.02 ERA. Sheets’ PECOTA projection from 2008 had him slated for about 143 innings pitched in 2009. However, considering that Sheets exceeded his pitching projections this year by throwing 198.3 innings, that projection will likely be higher for next year. I’ll adjust this upwards slightly and have him projected for 160 innings pitched.

Using these two figures, I will calculate Sheets’ Wins Above Replacement (WAR). More details into the methodology can be found here. I get Sheets as being worth 2.7 WAR. Factoring in expected inflation in the offseason, Sheets would be worth \$13.5 million per year. This may seem low to you right now, but later on we’ll see how a team’s approach to risk can change that valuation.

### Step 2: Find a team’s risk preference

If a team is risk neutral, meaning it does not care about the range of a player’s performance, then they should value Sheets as being worth 2.7 WAR. They would then offer him a long-term deal based on that value, adjusting for inflation in the contract offer. However, most teams will likely have a preference to risk, either being risk averse or risk seeking. This can change based on the makeup of the team, the free agent budget, and other factors like those. I am going to create utility curves for two generic teams, one that is risk averse and one that is risk seeking.

Risk Averse Utility Curve

Risk Seeking Utility Curve

On the following charts, the x-axis represents a player’s WAR while the y-axis represents the utility a player gives a team. Utility is measured in utiles. From the charts, we can see what kind of teams might fit each profile. A risk-averse team may be a team that believes to be in playoff contention and is looking to add the final piece of the puzzle. In the risk-averse chart, the largest gain in utility is from zero WAR to one WAR. A risk-seeking team could be a team that is farther away from the playoffs and needs relatively more help to make it. Of course, teams don’t have to fit these profiles to be risk averse or risk seeking. For example, a general manager may only have \$5 million to spend on the offseason and decide that he wants a safer return on that investment, so he takes a risk averse attitude towards his signings.

### Step 3: Use risk profile to create range of outcomes

I did most of the dirty work and came up with Sheets’ risk profile here. Basically I found that Sheets’ skill set is pretty low risk, but his injury risk raises his overall risk level to a “high yellow” or in between medium and high risk. So while Sheets is projected for 160 innings pitched and 2.7 WAR, there is a relatively high amount of variance around that mean. While I don’t have any formula to use Sheets’ risk to create a range of outcomes, I will qualitatively adjust Sheets’ performance to reflect his risk. Given that he is a high yellow risk, let’s say Sheets has the following range of outcomes:

Chances of being 0 WAR pitcher: 5 percent
Chances of being 0.7 WAR pitcher: 15 percent
Chances of being 1.7 WAR pitcher: 20 percent
Chances of being 2.7 WAR pitcher: 25 percent
Chances of being 3.7 WAR pitcher: 15 percent
Chances of being 4.7 WAR pitcher: 10 percent
Chances of being 5.7 WAR pitcher: 10 percent

### Step 4: Create a risk-adjusted dollar figure

With the range of outcomes being created, we can now create a risk-adjusted WAR valuation for Sheets. Here is how you do it. Find each WAR outcome on the x-axis on a team’s utility curve. Go upwards from that WAR point until you hit the actual utility curve. From that point, go across to the left until you reach the y-axis. That is the equivalent utility for a WAR outcome. From there, you weight each WAR utility equivalent by percentage chance of that WAR outcome occurring.

Let’s show an example using the risk-averse utility curve. The first WAR outcome is zero, and the utility of that is zero so it doesn’t matter what the percentage is. The second WAR outcome is 0.7 WAR. We go across the x-axis until we reach approximately 0.7 WAR. From that point we go up until we hit the utility curve. From there, we go left until we reach the y-axis. The point we hit is about .23 utiles, so the utility equivalent of 0.7 WAR is .23 utiles. We then weight that by the chances of that outcome occurring, which is 15 percent. From that we get .0345 utiles. We do that for all the range of outcomes and add it up. Once we get the total utility, we find that point on the utility curve and go down to the x axis to find the risk adjusted WAR. After doing that for both risk preferences, we end up getting this: