Looking deeper at trades and uncertaintyby Victor Wang
August 29, 2008
In the past month or so, I have written on various trades that involved the exchange of established big league players for prospects and/or players still under six years of service time. Thhis analysis involves evaluating the surplus value of all players involved, evaluating how the trade affects a team's playoff odds, and evaluating the value of any draft picks.
However, this analysis has left little room for the flexibility of team preferences. For example, suppose I were to offer you this gamble: I flip a coin and if it lands heads you get double your yearly salary. If it lands tails you get nothing.
Now, according to expected value, a person would be indifferent between that proposal and receiving his yearly salary. Of course, in real life not many people would take this gamble without receiving a payment up front first. Evaluating trades without first thinking of a team’s preferences is sort of like assuming people are indifferent between taking the coin flip gamble and their yearly salary. In this article, I am going to look at what kind of preferences teams may have and how they can use these preferences when analyzing a trade.
Teams are going to favor different types of players depending on the state of their major league team and farm system. Teams can account for this by varying their discount rate when measuring future surplus value. In past trade analysi, I used a 10 percent discount rate, which is what salary inflation has been for about the past 20 years. However, teams may want to use a different rate based on their goals.
We can use the Rich Harden trade as an example. Oakland made it pretty clear with this trade that it was looking to rebuild this year, while the Cubs were responding to Milwaukee acquiring CC Sabathia. Because of their preferences, it would make more sense for the A's to use a lower discount rate; Chicago would use a higher one. Tis means Oakland is putting more emphasis on future value while the Cubs want a more immediate return from their acquired players. Let’s give Oakland a 5 percent discount rate and Chicago 15 percent and see how this affects the surplus value analysis.
To keep things simple I’m going to evaluate Eric Patterson, Matt Murton and Sean Gallagher as 1 WAR (Wins above Replacement) while I’ll have Josh Donaldson as worth .15 WAR. I’ll have Rich Harden evaluated as 3 WAR and Chad Gaudin as 2 WAR.
Surplus value with original 10 percent discount rate
Eric Patterson: $16.45 million
Sean Gallagher: $16.45 million
Matt Murton: $8.93 million
Josh Donaldson: $1.62 million
Total: $43.45 million
Rich Harden: $10.6 million
Chad Gaudin: $9.1 million
Total: $19.7 million
Difference: $23.75 million
Surplus value using 5 percent discount rate
Eric Patterson: $18.92 million
Sean Gallagher: $18.92 million
Matt Murton: $9.83 million
Josh Donaldson: $1.94 million
Total: $49.61 million
Rich Harden: $10.9 million
Chad Gaudin: $9.5 million
Total: $20.4 million
Difference: $29.21 million
Surplus value using 15 percent discount rate
Eric Patterson: $14.49 million
Sean Gallagher: $14.49 million
Matt Murton: $8.16 million
Josh Donaldson: $1.4 million
Total: $38.54 million
Rich Harden: $10.34 million
Chad Gaudin: $8.75 million
Total: $19.09 million
Difference: $19.45 million
You can see that changing the discount rates produces the desired affects for each team. When using a lower discount rate, Oakland sees a greater difference in the surplus value it trades. When using a higher discount rate, Chicago sees a greater difference in the surplus value it trades. While Oakland gains greater surplus value no matter the discount rate used, time preference is important when we visit the next preference a team might have.
Teams also may have different attitudes toward the quality they receive. For example, in its rebuilding process, Oakland's main goal may be to build its depth. If so, the A's would prefer players who have a high certainty of reaching and contributing in the majors and be willing to give up a little upside. Meanwhile, the Cubs are trying to win a World Series. so they want high quality players who give them the best shot of winning it all. Their main preference is for high quality players, and they are willing to face higher risk in return.
We can use a utility curve (http://en.wikipedia.org/wiki/Utility) to model this preference. A utility curve basically measures how much satisfaction one receives as a certain variable changes. For example, we can look at possible utility curves for the A’s and Cubs:
Example Oakland A’s utility curve
Example Chicago Cubs utility curve
The X-axis shows a player’s surplus value/year while the Y-axis measures utility. Surplus value/year is equal to a player’s surplus value divided by a player’s cost-controlled years. Cost-controlled years are the years a team can renew a player’s contract or go to arbitration with a player. On this curve utility is measured from 0-1, with $0 surplus/yr and $12 surplus/yr (in millions) as reference points.
This means that when a team gains no surplus value from a player, it will gain no utility. When a team gets $12 million in surplus per year, it achieves full utility. Note that players can have a negative surplus value. In this case, we could model this with negative utility.
Also, a team gains about $12 million/year in surplus value when it has a 4 WAR for six cost-controlled years. I chose this as a reference point because I usually use 4 WAR as a cutoff point to represent superstar players. Also note that a 1 WAR player will have $3 million/year in surplus for six cost-controlled years, a 2 WAR player will have $6 million/year in surplus value, and a 3 WAR player will have $9 million/year in surplus value. These are approximations, but they differ by only a few million if at all. Utility is measured in utils.
Looking at the A’s utility curve, we can see that they achieve relatively high utilities once a player gets in the $2-4 million/yr mark. The Cubs require much higher quality back to reach the same point as the A’s. We can use these curves to measure which package of players each team would prefer.
Using the 5 percent discount rate for the A’s, they received $49.61 million in surplus value over 22 cost-controlled years, which equals $2.26 million/yr in surplus value. Meanwhile, they sent away $20.4 million in surplus value over four cost controlled years, which equals $5.1 million/yr in surplus value. If we go from $2.26 from the X-axis on the A’s utility curve, we see the A’s get around .5 utils. Multiplying that by the 22 cost-controlled years, the A’s gain 11 utils. Doing the same for the players they traded away, we find that the A’s gave up about 3.1 utils. So, since 11 > 3.1, the A’s prefer the package they received. Doing the same analysis for the Cubs discount rate and utility, we find they gain about 1.4 utils and give up 1.1 utils so they prefer their package as well.
You can easily factor in risk using this sort of analysis as well. Let’s use Eric Patterson as an example. Let’s say he has a 60 percent chance of becoming a 1 WAR player (a solid bench player), a 20 percent chance of becoming a 2 WAR player (an everyday regular), and a 20 percent chance of becoming a 0 WAR player (a Triple-A lifer). As stated earlier, a 1 WAR player earns about $3 million/yr in surplus value and a 2 WAR player earns about $6 million/yr in surplus value. Knowing this, we can multiply the utils of each possible outcome by the chances of each possible outcome to get a player’s utility. Using Patterson we get: .6 * .6 + .85 * .2 + 0 * .2 = .53 utils. When we multiply that by his six cost-controlled years, we get 3.2 utils in total. We can do this kind of analysis for each player involved to model the risk involved.
I hope this article showed how we can move from rigid, expected value based evaluation of trades to more closely model the analysis tams actually have to make when making trades. Now, I doubt any teams actually use this kind of quantitative analysis when making trades, but they are thinking about these aspects when they decide to make a trade.
Teams ask themselves when they want to contend, what kind of players they want back, and how much risk they can handle. This sort of analysis does not need to be limited to trades; it also can be used for free agent signings and draft picks. This type of evaluation also means that it’s not as easy to say who won or lost a trade; a lot depends on a team’s preferences. However, that does not mean we shouldn’t debate whether a team made a good move or bad move as long as we rank the quality of a GM’s decision rather than the outcome.
References and Resources
Anyone who has been in a decision analysis class or read its literature will see common elements in this article. A lot of my ideas were taken from concepts in decision analysis. Anyone interested in reading more on this subject can find a good reading list here. Yes, you can also use this for fantasy baseball.
Victor Wang's work on OPS has been featured in SABR's By the Numbers magazine, and was the 2007 recipient of SABR's Jack Kavanagh Memorial Youth Baseball Research Award. He can be reached via email here.
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