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.