When sabermetricians try to approximate the dollar value of a player’s performance, we are mostly using recent free-agent values. For instance, if Halladay is worth about six wins above replacement level (which is a good approximation for most teams’ fifth starters), we would say that the value of his replacing a typical fifth starter is about $27 million above the MLB-minimum salary of $400,000. As his current contract pays him $14.25 million in 2009, this would imply that a full 2009 season of Roy Halladay would have been worth the difference ($13.15 million). However, let’s say that Halladay gets dealt right now, with about 70 games left to go, instead of around the deadline. Our inclination would be to say that Halladay’s remaining 2009 net value would be this pro-rated, $5.7 million above his contract, but that misses a few essential parts of the analysis.

Free agency is a process by which teams bid for a player’s services in an auction format. The winner is usually the team that offers to pay the most. What that means is that the team who signs the player is probably the team who values the player the most. That may be optimism, but it also may be that the team has more value for his services than other clubs.

Estimating the value of an object at the amount of money the purchaser paid for it is somewhat inaccurate, only because not everyone values things equally. Suppose that I told you that I recently purchased a swimming pool skimmer for $15. You might think that I got a pretty good deal until you found out that I don’t own a pool. Similarly, the Cardinals did not bid on Mark Teixeira this winter, simply because it would not be worth the $22.5 million the Yankees pay him annually to have him sit on the Cardinals’ bench and watch Albert Pujols take his cuts.

Additionally, if I told you that my friends bought a used pool skimmer for $40 at midnight last night, you might think that they were being foolish, but if you found out that their son had a pool party at 11:00 AM and their pool was filled with leaves, you would probably change your mind. The gravity of the situation matters. Similarly, if I told you that instead of letting Tom Glavine pitch the last day of the 2007 baseball season against the Marlins, the Mets were somehow able to pay Johan Santana to join the staff on that fateful Sunday and pitch them to victory instead, you might value that more than 1/162nd of his contract value. That game was definitely going to make or break their season (or at least send them into a one-game playoff), since they were tied with the Phillies on the last day of the season for the division lead, and that game was definitely pivotal.

At the beginning of the season, Roy Halladay may have been worth about $27.4 million to a contender, but a contender might not end up needing those six wins above replacement level; alternatively, they may not even have been enough. We act as though 162 games is a large sample size, but a team that has the talent to win 89 games will finish with 99 wins or more about seven percent of the time, and will finish with 79 wins or less about five percent of the time, just out of sheer luck, good or bad. The odds of even a six-win pitcher like Halladay actually making the difference in a team’s season is pretty low when calculated in April. In fact, the odds of a one-win player actually adding that one crucial game that the team would not have won anyway is about 6.3 percent. However, as the trade deadline approaches, and a team knows that they are in contention but not guaranteed a playoff spot, the odds of a one-win player adding that one crucial game go up to about 9.6 percent. Therefore, even though the number of wins that Halladay adds if he joins a team 92 games late might be only 2.6 wins over the last 70 games (instead of six wins over 162 games), he still will maintain up to 66 percent of his revenue production value while only maintaining 43 percent of his win value.

Not all of this is due to the factor described above. Although playoff series are inherently random in their outcomes from a statistical standpoint, we have to assume that if Halladay adds six wins above replacement level over the course of 35 starts, then he must add about 17 percent to a team’s odds of winning an individual game he pitches as compared with a replacement-level pitcher. If we approximate that a team will win 58.5 percent of the playoff games that Halladay pitches, 41.5 percent of the playoff games that their current fourth starter pitches, and 50 percent of the games that their top three current starters pitch, then adding Halladay into the rotation and bumping everyone down a spot (eliminating the previous fourth starter) will increase a team’s odds of winning a five-game series by about 9.6 percent, and increase a team’s odds of winning a seven-game series by about 6.4 percent.

Back in 2005, Nate Silver approximated the value of a win as $750,000 if you ignore the impact of making the playoffs, and he estimated the value of a playoff appearance at approximately $30 million. To match with the current $4.5 million per expected win value typically paid to free agents, I developed a method to bump these values up by about 50 percent. The harder part was approximating the added value of reaching each stage of making the playoffs. These are somewhat debatable, but I will report my estimates for the sake of transparency. The point should hold anyway, but here they are:

Making the playoffs but losing the Divisional Series: +$25 million (versus no playoffs)
Making the playoffs but losing the Championship Series: +$45 million
Making the playoffs but losing the World Series: +$75 million
Making the playoffs and winning the World Series: +$105 million

Value of a win, ignoring playoff implications: $1.125 million

The teams that have been mentioned the most often in the Halladay hunt (and their respective PECOTA-based playoff odds through Saturday) are the Red Sox (91%), Yankees (78%), Phillies (72%), Angels (67%), Cardinals (56%), Tigers (39%), White Sox (31%), Giants (30%), Brewers (27%), and Rangers (21%). Using ballpark estimates which should be pretty close to the true value, adding Halladay would respectively increase their playoff odds by varying amounts: Red Sox (98%), Yankees (92%), Phillies (89%), Angels (86%), Cardinals (78%), Tigers (63%), White Sox (55%), Giants (54%), Brewers (50%), and Rangers (42%).

To estimate these numbers, I used the binomial theorem to estimate a distribution for the expected win total in each team’s remaining games, and then I figured out which win total matched up with their current playoff odds. Then I figured out what that distribution would look like with a team that could be expected to win 2.6 more games on average over the course of the season, and the new probability of reaching that win total with Halladay aboard. Given the numbers listed above for wins and for reaching each stage in the playoffs, I computed the revenue gain of adding Halladay to each of the ten teams above.

Without getting into the subtleties of who Halladay would replace in the regular season and playoff rotations for each team, here are the approximate percentages of Halladay’s full 2009 revenue production value ($27.4 million) that Halladay would generate for each of the possible teams that he could join:

Team        %      $
Red Sox    44%   $12.0 million
Yankees    55%   $14.9 million
Phillies   59%   $16.0 million
Angels     61%   $16.6 million
Cardinals  65%   $17.7 million
Tigers     65%   $17.8 million
White Sox  63%   $17.1 million
Giants     63%   $17.1 million
Brewers    60%   $16.5 million
Rangers    56%   $15.2 million

The Tigers, for example, may have valued Roy Halladay’s performance at $27.4 million at the beginning of the season, and with a contract that paid him $14.25 million this year, the Tigers would have been willing to give up $13.15 million of value to have him in 2009. Even though the season is more than halfway over, they would value his performance from now on at $17.8 million. With only $6.2 million still due to him (versus the $200,000 they would need to pay a replacement-level pitcher), they would now value him at $11.8 million. To put it another way, that is 90 percent of his 2009 net value still maintained with only 43 percent of the season left.

To get the full value of Halladay through the 2010 season-or the length of time that he’s already under contract for-you add in $27.4 million for 2010, and $5 million for the approximate value of free-agent compensation picks (per Victor Wang’s calculations), and with about $21.9 million still owed to him, that makes him worth anywhere between $22.5 and $28.5 million to a team that picks him up. Sky Kalkman did an excellent article to begin the process of approximating Roy Halladay’s value, and estimated that it is about $23 million. Therefore, he supposed that Halladay was worth about a Top 26-50 hitting prospect. The extra value revealed by using this method shows that he is also worth an extra Grade-B hitter as well.

Not only is it important to remember that a player’s value is different to every team, but it is also important to remember that his value is not evenly distributed throughout the season. To a team like the Blue Jays, he is worth only around $3 million for the rest of this season; to a team like the Tigers, he is worth nearly $18 million. With Halladay worth nearly $15 million more to several teams than he is to the Blue Jays, Jays GM J.P. Ricciardi should be able to meet someone somewhere in the middle. He may not make a deal, but if he is does not, he has spoiled $15 million of surplus value available in a trade.