As news of the arbitration offers (or lack thereof) trickled out on Tuesday, I found myself growing increasingly disappointed with the number of teams failing to make offers to their players. Of the 21 Type-A free agents who could be offered arbitration, 11 were not. That number is stupefying. Of the 53 Type-B free agents, 38 were not offered arbitration. That number is staggering.

Faced with the prospect of a mystifying reality, my first recourse is to quantify. Sure, my desk is covered with envelopes onto the backs of which I’ve scribbled chicken scratch. But at least I experienced a degree of certainty. Let me share my back-of-the-envelope Theory of the Arbitration Offer.

The decision that teams face in determining whether to offer a player who has filed for free agency involves several variables. In order to think systematically about this decision, it is helpful to be explicit about the inputs, whether or not they are easily estimated. Put simply, if the expected surplus a player offers to the team plus the value of draft picks received if he signs elsewhere is greater than zero dollars, the team should offer him arbitration.

However, that statement oversimplifies the decision a little bit. For example, if a team has no use for a player but he is extremely unlikely to accept arbitration, it may still be advisable to offer arbitration. Let’s dig our hands into the loamy soil and see what we can unearth.

First, a team must estimate the value (V) in marginal wins a player will offer. Next, they must estimate their value of a marginal win (which of course is a function of their expected season wins). Call this W. The product of these two variables (P*W) is the price a team should be willing to pay for a player’s services (that is, it represents the point at which the marginal cost equals the marginal benefit).

If a player accepts a team’s offer of arbitration, either the team and player will reach an agreement to avoid arbitration or they will go to arbitration. It is much more likely that they will reach an agreement than actually go before an arbitrator, but either way, the salary a free agent receives is likely to be similar to that which he received the year before (and, under the CBA, it may not be less than 80 percent of his previous year’s salary). Call the expected arbitration salary S. A team must also determine the likelihood that the player will accept arbitration, which is a function of a player’s expected value, which we have already called V. We’ll call this probability P(a).

Finally, a team must estimate the combined value of the compensatory draft picks received, which is a function of, among other things, the Elias ranking of the player. This can be difficult process, as the Blue Jays learned last year with A.J. Burnett, but let’s call the variable D.

Expressed as an inequality, if (P*W-S)*P(a) + D*(1-P(a)) > 0, then the team should offer arbitration. Translated into plain language, a team should offer arbitration if the probability a player accepts arbitration times the expected surplus he offers (which can be negative) plus the probability the player does not accept (assuming he does not retire) times the value of the draft picks received is greater than zero.

That is a bit cumbersome, but there are really only three important moving parts: the probability of acceptance, the expected surplus and the value of the draft picks. Let’s apply the framework to an actual decision and see how it works.

The Dodgers declined to offer arbitration to Randy Wolf, who is a Type-A free agent. (It may be objected that the Dodgers divorce drama has made them cost-conscious, but that’s no license for bad business decisions.) In his last three seasons, Wolf has been worth 1.8, 1.3, and 4.6 wins by WARP3. The fact that he is coming off of a peak year cuts both ways. On the one hand, he will likely command more through arbitration than he would in other years. On the other hand, he will also command more money on the free agent market. The former fact lowers the surplus he would offer the Dodgers (and potentially drives it into the red), while the latter fact decreases the probability he would accept arbitration.

In fact, I think the likelihood that Wolf would have accepted arbitration is extremely low-certainly not more than 25 percent. He stands to be offered a multi-year deal on the open market, and the guaranteed money is sure to be more attractive to a pitcher with a spotty health history than a one-year deal at a higher yearly salary.

Back in 2005, Nate Silver estimated the value of the first-round pick compensation a team would receive for a Type-A free agent signing elsewhere at around $9 million. Before this season, Victor Wang valued the pick at $6.5 million. He also estimated the supplemental-round pick that a team would receive in addition at about $1 million. Assuming the team that signed Wolf would not sign a Type-A free agent ranked ahead of him, the floor estimate of the value of draft picks received is $7.5 million.

Take my completely fabricated 25 percent estimate that Wolf accepts arbitration. Plug it into the inequality and solve for the surplus figure. The result implies that the Dodgers must believe that they would suffer more than a $5.5 million loss if Wolf accepted in arbitration. Does that sound likely given the Dodgers expected number of wins next season (85-90) and need for starting pitchers (hey, who doesn’t?).

Take another example from the senior circuit: Chan Ho Park, to whom the Phillies declined to offer arbitration. Park made $2.5 million last season as he split time between the rotation and the bullpen. By WARP3, he has been worth -0.3, 1.9, and 1.2 WARP3 each of the last three seasons, listed chronologically. He was a Type-B free agent, which means the Phillies only stood to gain a sandwich-round pick, which as we know is worth about $1 million in surplus value.

Let’s assume Park had about a 50 percent chance of accepting arbitration (though the number is probably lower, since Park continues to express his desire to start). Again solving the inequality, the Phillies needed to expect that Park would earn $500,000 more through arbitration than he would be worth to them in order to justify the decision not to offer arbitration.

That doesn’t sound like much, and Park would no doubt point to his 2.52 ERA in 50 IP as a reliever in any negotiation. However, given the Phillies’ bullpen needs and expected win level just barely ahead of that of a resurgent Braves club, their willingness to pay for wins is at a high.

Viewed through this lens, some of the arbitration decisions make very little sense. Some, like the Astros‘ decision not to offer arbitration to Miguel Tejada, are much easier to justify. Nevertheless, progressing under a clear framework should simplify evaluation of the decision of whether or not to offer arbitration to a free agent.

Tommy Bennett is a contributor to Baseball Prospectus. He can be reached here.

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I like that you limited the scope of this. You could've gotten bogged down and tried to estimate each of the component parts, but you kept it simple and just described it conceptually.

Even conceptually this could be difficult to fully grasp, so this is great for a (relatively) short article. I'd look forward to quantification of each of those three moving parts though!

Great stuff.
What do you think about the relievers being offered arbitration? The Red Sox obviously made out well offering Wagner arbitration, but do you think it was wise for Atlanta to offer arbitration to Soriano and Gonzalez?
I think the likelihood that Soriano or Gonzalez accept arbitration is close to zero. There are plenty of teams looking for relief help and they are the top guys on the market. Offering arbitration to them ought to be a great call, especially since they are both Type A.
Soriano accepted arbitration.
Cool beans. Any idea *why* teams are not offering arbitration to these guys that they don't really want to keep? Is there a greater (real or perceived) loss in that 20%-50% chance that he takes arbitration and ends up costing you an extra couple million, beyond what you might be willing to pay or for a return in a trade? Are there any rule in place, for example, that restrict a team's right to trade such a player?

Your economic analysis is sound, but I wonder if there are any other factors at play?
The 80% limit on salary decline at arbitration limits teams in some cases. The economy plays a role because the comps at arbitration are historical.

Teams are also leery of being in a position where other teams know they have to make a deal (like when Placido Polanco unexpectedly accepted from the Phils and they only got Ugueth Urbina and Ramon Martinez from the Tigers).

Foregoing draft picks is also less visible. Teams tend not to get trashed on talk radio for losing a supplemental draft pick. But these things add up to actual wins in the aggregate.
I'm confused. I thought the 80% rule did not apply to free agents who accept arbitration. Is this a case where the rule doesn't apply but is adhered to in practice?
The way the CBA is written, you are correct. (See Art. XX(B)(3)). However, as you have suspected it is extremely rare to see a salary reduction less than that, since the arbitration process is all-or-nothing.
This is a nice analysis. However a general comment is in order about all of these kinds of calculations -- which I can express in a single word: uncertainty.

Given the uncertainty about some of the assumptions (e.g., the probability that a given player will accept arbitration), instead of assuming a given figure (e.g., .25), you should put a bracket around the probability (say .20 to .30), and do the same with other critical variables.

Then estimate whether the inequality that you have works under a range of assumptions about those probabilities -- the uncertainty -- including applying a range of expected WARP outcomes, reflecting the uncertainty about performance in future years.

Maybe, under the range of assumptions, the inequality "solves" as positive only 51% of the time, 0 or negative 49% of the time. In that case, you can turn to looking at how risk averse a given GM might be, but you shouldn't judge the decision as irrational. Maybe large market teams can afford to be less risk averse. Maybe the position of hte player (starting pitcher, closer, catcher, etc.) makes teams more or less willing to take a chance. Maybe the previous year's W-L record affects the willingness to take risk. And so forth.
Yes, all of this is completely correct. I assumed fixed values for the sake of simplicity in elucidating the framework. As I refine the model I will certainly take a more Bayesian approach.
I believe that under EBWilliams the Orioles never went to arbitration, although if I remember correctly they weren't above offering it to buy time if a settlement was expected.
Management said that the process forced them to undercut the player's reputation and attack his own self-worth, while forcing the player to discount his team-mates contributions to winning.
I agree. I would never go to arbitration with a player I wanted to keep, and there is no reason for offering to one you don't want.
A draft pick isn't a reason?
Clubs have more incentive give risk-averse offers, yet they still have same winning percentage (~60%) as decades ago. The system of actually going to arbitration is rigged to favor the players, and this is an important consideration.

There can also be psychological drawbacks to going into arbitration, as both sides try to discredit the opposition (it can get nasty and personal). Try to tear down a former, and potential future, employee is never easy. Just look at the absurdity of the Todd Walker / Padres hearing in '06... every team wants to avoid that.
Doesn't the model also need to take account the possibility of trading the player in the event that he opts for arbitration? For example, I would think that Randy Wolf on a one-year deal would fetch something positive on the trade market, if not in the Winter, then certainly in the Spring once a team in a big market suffers a hit to the rotation.
I agree with you on this on principle, but how is LA not a big market?
I don't think that he is implying that LA is a small market, just that a true small market team is unlikely to make a move to acquire a high-priced pitcher in this scenario.
Also, I believe that teams may release a player on an arb-specified contract by paying 1/6th the value of that contract. I.e., an arb-specified contract is not guaranteed to the same extent as FA contracts (at least in past CBAs). This complicates the math a bit, but it's another reason that Tommy's basic point (that teams should be offering arbitration far more than they do) is correct.
What about the important possibility, discussed in Sheehan's article, that the very same player (or an equivalent) can be had in free agency for a price significantly less than the player would receive in arbitration? Isn't this an important factor that's omitted from your analysis of the offer/don't offer calculus?

In other words, I should NOT offer arbitration to Player A, even if he's worth $20 million in marginal revenue to me, if (a) I expect him to make $16 million in arbitration, but (b) I expect his value on the free agent market to be approximately $5 million.