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 1/162^{nd} 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 millionValue 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.

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A fifth starter worth 6 WARP and $27 million? In what universe?

"For instance, if Halladay is worth about six wins above replacement level â€” above a typical fifth starter, in other words â€” we would say that the value of his replacing that fifth starter is about $27 million above the MLB-minimum salary of $400,000."

Matt, nice to see you on the BP roster!

If you have, say, a 40% chance of beating the Jays on the days he pitches, and a 70% chance of beating the replacement-level pitcher who replaces him, that's a net gain of 1.5 wins (from 2.0 to 3.5 expected). Taking the max value of $4.5 million/win, that's a high-end estimate of $6.75 million in value from that.

Since 1)you will probably face him fewer than 5 times and 2)the effect is less pronounced against a better team (since you're less likely to win anyway), I just can't see avoiding him being a deciding factor. The benefit is certainly dwarfed by that of simply having Halladay make a half-season of starts for you. However, it might be enough to add a few million to the incentive pot, though, bumping the Yankees up into the White Sox/Giants bracket...

I think Nate did a market size study a few years ago, but that doesn't really work either -- the Yankees and Mets play in the same market, but the value of a marginal win is clearly much higher for the Yankees.

Rather than focus on the heterogeneity from market to market-- a valid topic, for sure-- this article was generally supposed to focus on the heterogeneity with respect to playoff odds. Both are valid issues, but I think playoff odds are a larger effect. After all, you sometimes see small market teams make big moves in the thick of the playoff chase, like the Brewers making a run for Sabathia, but you never really see large market teams that are out of the race go after players like that. The larger effect is probably playoff situation, though I agree that there is more to be done on the topic of heterogeneity with respect to marginal revenue in general.

In regards to the difficulty estimating the relationship, of course that's true -- there's a lot of problems with data, but I think your particular criticism isn't so relevant conditional on proper application of statistical techniques. I haven't taken the time to actually look at this, but it seems to me that you could probably perform a fixed effects regression and estimate the team-specific effects if you could find a good instrument for either team record this year or playoff appearance the year before to avoid the cross-year correlation.

In regards to which is the bigger effect, market heterogeneity or playoff probability, isn't it to some degree likely that market heterogeneity drives playoff probabilities? Large market teams have higher marginal revenues from winning and therefore acquire more talent, which puts them in high-payoff situations (in terms of playoff probability) more often?

Revenue_t = a + b1 * (Win%_t) + b2 * (Playoffs_t-1) + (other championship variables at time t-1) + e

Is that right? What would your instruments be exactly? Would you really trust a time-series regression to give you accurate estimates? (I suppose you can estimate this for all teams simultaneously and include a year fixed effect which is constant for all teams, but I don't think that really solves the problem).

y_it = a + b1(Win%_it) + b2(playoff_i,t-1) + e_it

where "a" is a vector of every time-invariant portion of the model and you estimate with clustered standard errors. Given that playoff_i,t-1 is probably correlated with Win%_it, although it may not be, you would have to instrument one of the two. As I mentioned above, I haven't really invested a lot of time in thinking of possible instruments. One possible instrument for Win%_it would be the number of team days on DL or even injury cost...maybe that's correlated with y_it, which is a problem, but its endogeneity and relevance are testable. That's the one source of difficulty in this besides data, not the statistical technique.

Obviously this isn't a good solution due to lack of reliable data, but to come back to the original point, acknowledging that different teams have vastly different marginal revenue curves (including across only big market teams) is important. It's important enough to make the Yankees spend ~$200 million whereas other large market teams spend at most ~$130-140 million. Not all of that is the Steinbrenners being irrational.

Also, just to throw this out there, I really enjoy reading Matt's work. I guess I wish he was doing it with the same degree of rigor that he does in his day job, although I can understand why that would make the whole thing significantly less fun for lots of people, probably Matt included.

2) You would need to instrument both variables (the fact that they are or are not correlated with one another is irrelevant). I'm guessing that good instruments are hard to find here. Definitely for Win%, though possibly less so for playoff success (try to use some randomness in the outcomes?). But I'd still feel pretty uncomfortable with the results in a time-series regression.

Again, none of these problems are your fault, but I'm trying to figure out what assumptions you're willing to make. I actually think estimating separate coefficients would be very useful, but we just need to be clear on the restrictions that must be applied.

Another way of saying some of what I just wrote is...maybe write down what you think the correct specification is? What you wrote in your last post assumes the same effect for each team (which is what you're arguing against).

You want this?

Revenue_it = a_i + (X_it)'b_i+e_it

where X includes Win%, playoff success vars, etc. And you're assuming X is full rank for each team (not necessarily the case - eg. Nationals).

Right?

Also, the playoffs have effects in the current year as you make money in those games themselves and off of merchandise sales, so that would need to be heavily factored in.

I'm also not really sure how to factor in price stickiness and various other aspects to how revenue is actually generated. I don't really know that a reduced form regression would accomplish much, because it seems like the true form would have far more variables than even observations.

Perhaps this could be corrected for by games as an observation rather than teams revenue? There's also all types of other issues with the demand curve as well. I'm sure teams have a way of choosing prices that probably incorporate a lot of private information. In general, I would think it's probably impossible to discover anything terribly meaningful from a regression like that. I really enjoy hearing what you guys think about this. Thanks.

In that article, there was also a section about inflection points in wins and that, the marginal value of each win above and beyond what is needed to win the division decreases.

Maybe it was in Baseball Between the Numbers or one of the Annuals? (digs around)

I don't think using revenue/game buys you much unless you also want to identify the effect of "record in the last 10 games" (or something similar). There's no variation in "won the World Series last year" game-to-game within a season.

Oh, great article, Matt.

Actually, the more I think about it, there's a lot of problems with doing any sort of analysis here...for sure the series' aren't cointegrated since revenue is trending upwards but it's impossible for a playoff dummy or win% to trend upwards also. Maybe revenue/game, as Matt suggested, would be stationary, though? I don't know a whole lot of time-series econometrics, though, so I don't know where you'd go from there.

I would also recommend allowing the BP staff comments to be rated. It seems pretty strange that they are currently "exempt."

I actually like the system ProTrade uses a bit where article comments show the total number of +'s and -'s on a comment. There's a big difference between a +15/-20 and a 0/-5, though here on BP, both would show as a -5.

While I agree with the full economic approach (different revenue curves for different teams) for certain questions (does this move make financial sense) I don't necessarily agree it answers other questions, such as "will this move help maximize the number of World Series we win over the next ten years." For that, you care more about playoff probability and only care about unique payroll/revenue situations in the sense that locking up players are certain rates gives a team a better or worse chance at the playoffs. (That is, the Yankees should "overpay" typical FA rates a bit for studs, while the Rays and Pirates should avoid paying anything close to free agent rates for decent players.)

What I can't figure out is why the Rays are not in the running. I think Halladay would help them more than anyone.

Why would you take on that money when your rotation is already that strong (and cheap)? Already well established that there's a next-to-zero chance of Halladay staying in the AL East, and the Rays arguably still have the strongest top-to-bottom rotation in the division. Definitely the one with the most potential. Before somebody argues, a rotation with guys like Wakefield, Smoltz, Penny, or Alfredo Aceves is not superior no matter how strong your #1 starter is.

Halladay has additional value beyond his VORP in how deep he goes into games. This makes the team more likely to win games in which he is not the starting pitcher, because it gives the bullpen more rest. It also directly reduces the number of innings thrown by pitchers who are not as good as Halladay, especially pitchers who would pitch the 6th and 7th innings of games with less durable starters.

A reliever pitching the seventh inning can be nearly as effective as Halladay and his 3.25 ERA. Relievers aren't as good, but when they go one inning at a time, they're quite effective.

I actually think we OVER-rate how far into games starters can go.

However, MUCH of Halladay's IP is lost in the NL where if it is a close game, you may need to pull him earlier to pinch hit. I rarely think that one more inning for a starter is worth a pinch hitter's bat in his place.

I think it is a relatively riskier proposition for a team to satisfy the Jays asking price for Halladay, as historical marginal revenue curves may not be applicable this season. The only scenario that seems financially rationale is an owner that plans to sell the club in the near future.

Felix Hernandez

Roy Halladay

Eric Bedard

Jarrod Washburn

That's a pretty damn hot rotation to enter the playoffs with, regardless on how soft the offence has been.

Is this because Seattle is more in the market for a couple of bats and not an arm? Halladay over Garret Olson makes a HUGE difference.

Another factor that Ricardi has considered and that any analysis of Halladay's value should consider, is that it is going to cost more for an AL East rival to get him than another AL team and more for an AL team than for an NL team. As hinted at above, the AL East team could come out slightly ahead just because they wouldn't have to face him. Conversely, the Jays don't want to see him 6 times a year for the next 5 years, either. So the price goes up for the Sox and Yanks (and O's and Rays) thereby lowering Haladay's value to those teams.