We recently went through the 2015 qualifying-offer season, the basic facts of which Five Thirty Eight’s Rob Arthur provided a helpfully summarized back in November. In that piece, Arthur asserts that “Each offer is essentially a bet that the player… will be worth 2 or more WAR in the coming year.” He adds that “the math works out so that teams tender offers to almost every remotely deserving free agent [and] [w]ithout fail, those free agents refuse them.” Arthur was writing before the deadline for players to accept their offers, and for the first time this year, there were players who accepted their qualifying offers (Brett Anderson, Colby Rasmus, Matt Wieters).

While teams may have been very liberal in giving out qualifying offers in the past, now that there is a precedence of players accepting the offers, teams will likely need to be more cautious about giving out offers in the future. This article attempts to build a model that determines at what WAR threshold it makes sense for teams to give out qualifying offers.

In order to keep this model relatively simple, we will assume that WAR is the only factor in play when determining whether a team should give out a qualifying offer to a player. In reality, WAR is likely the most important consideration, but far from the only one.

Let’s define the following terms:

PWAR = Player’s projected WAR for the upcoming season

P(accepts) = Probability the player accepts the qualifying offer

P(rejects) = Probability the player rejects the qualifying offer

P(signs | rejects) = Probability another team signs the player before next year’s amateur draft, given that the player rejects the qualifying offer from his original team

Note that P(accepts) + P(rejects) = 1, since a player must always either accept or reject the offer (these are the only possibilities).

We will now derive a formula for the expected value of change in team wins if the team gives out a qualifying offer. Under the terms of this model, team wins can change if the player accepts the offer, or if the player declines the offer and signs with another team before next year’s amateur draft, thus giving the first team a compensatory draft pick. Team wins can also change if the player declines his qualifying offer and the re-signs with the team, anyway, but this model will ignore that possibility.

Below is the formula in question. (Note that E denotes “expected value of.”)

E[change in team wins] = E[change in team wins if player accepts] + E[change in team wins if player rejects and signs with another team] = E[change in team wins | Player accepts] * P(accepts) + E[change in team wins | Player rejects and signs with another team] * P(rejects and signs with another team) = (PWAR – Qualifying offer salary) * P(accepts) + (WAR of compensation pick) * P(rejects) * P(sign | rejects) = (PWAR – 2) * P(accepts) + 1 * P(rejects) * P(sign | rejects)

For the first expectation term, we reason that the change in team wins if the player accepts is the player’s projected WAR minus 2. This is because:

The player will give the team PWAR more wins compared to a replacement-level player

The $15.8 million dollars paid to the player could otherwise be used to signed a free agent with a WAR of 2

For the second expectation term, we use the reasoning from Arthur pieces that the compensatory draft pick offers about 1 WAR of value on average.

Now we want to enumerate P(accepts), P(rejects), and P(sign | rejects) in terms of PWAR. Let’s tackle P(accepts) first. This is not a particularly easy task, but we can be confident about these two data points:

A replacement-level player with a PWAR of 0 will certainly accept the offer. By definition, a replacement-level player should be paid around the minimum salary, so if he is offered significantly more than that, he will be happy to take it.

A player with a PWAR of 3 or higher will certainly reject the offer. Other teams will be willing to pay $15.8 million for 2 WAR of production. Coupled with the additional 1 WAR hit of surrendering a draft pick, that implies that other teams will be willing to pay $15.8 million for this player, and the player is going to get an offer of at least $15.8 million from some team.

Since P(rejects) = 1 – P(accepts), these two points also apply for P(rejects).

There are two similar data points for P(sign | rejects):

Another team will certainly not sign a replacement-level player with a PWAR of 0, since they can sign another replacement-level player at around the minimum salary without surrendering a draft pick.

Another team will certainly sign a player with a PWAR of 3, based on the same reasoning as above.

Let’s model these probabilities in the simplest way possible, with a linear function between these data points:

P(accepts) = 1 – PWAR/3

P(rejects) = P(sign | rejects) = PWAR/3

Plugging these functions back into the previous formula, we get

E[change in team wins] = -2/9*PWAR^2 + 5/3*PWAR – 2

A team will only want to give out a qualifying offer if the expected change in team wins is positive. Looking at this formula and graph, this is true if PWAR is greater than 1.5.

Now, let’s try to do a better job of modeling the probabilities. We know that since the large majority of qualifying offers are rejected (because the player thinks he can get more money and/or more years as a free agent), we can lower the values across the board of P(accepts), which also increases the values of P(rejects). Similarly, since the majority of players who reject the offer sign with another team before next year’s amateur draft, we can increase the values across the board of P(signs | rejects). Let’s try modeling these probabilities with exponential functions:

P(accepts) = e^(-3*PWAR/2)

P(rejects) = P(sign | rejects) = 1 – e^(-3*PWAR/2)

Note that for a projected WAR of 3, P(accepts) = 1.1% and P(rejects) = 98.9%. This isn’t exactly equal to the 0% and 100% values that we established earlier, but it is close enough.

Our new formula for expected changed in team wins is:

E[change in team wins] = (PWAR – 2) * e^(-3*PWAR/2) + (1 – e^(-3*PWAR/2)) ^ 2

This formula is much more complicated than the previous formula, but when you graph it, you see that it’s still a monotonically increasing function:

This function has a value of 0 when projected WAR is about 0.73, which means that the model suggests that teams should give out qualifying offers to players with a projected WAR of 0.73 or greater.

To make this model more precise, we should include other factors. Such factors would include attributes specific to individual players (such as the player’s age and injury history), specific to the different teams (such as roster budget and value of draft picks), and specific to the depth of the whole free agent class, among other things.