Wednesday, April 28, 2010

Why teams pay equal prices for free agents, Part II

The last couple of posts, I argued about why every team should pay the same amount for a free agent win, and how teams decide how many wins to buy. In those posts, I made some simplifying assumptions about how free agents are purchased. Now, let me come back to those points and argue that they don't matter much.


1. Slaves and Arbs

Previously, I explicitly assumed that every player was a free agent. But, of course, that's not true in real life. Many players are "slaves" (where the team can set their salary) or "arbs" (arbitration-eligible players who earn more than slaves, but less than free agents). Does the existence of slaves and arbs change the price of free agent wins?

I don't think so. Look at it this way: suppose team X wants to wind up at a talent level of 42 WAR (about 89 wins). Now, suppose they already have 20 WAR in slaves and arbs. Will that change the amount they're willing to pay for free agents?

No, I don't think it will. It's true that they only have to buy 22 WAR on the market, rather than the full 42. But, at the margin, the revenue value of each of those WAR is exactly the same as before. If the 89th win nets them $4MM in revenue, it nets them $4MM in revenue, regardless of where the other 88 wins come from.

But won't they be willing to spend more on a win if they have more money to spend? Don't they have a budget?

Again, I don't think a budget enters into it. If a team can make $5 million from a win, and they can get it for $4 million, they're going to find a way to buy it, whether they have the money or have to borrow it. And no matter how much money they have burning a hole in their pocket, they're not going to buy a win for more than it's worth to them and deliberately lose money.

On the free agent market, having 20 WAR in slaves is worth up to $80 million. Having 20 WAR in slaves, for purposes of team strategy, is exactly the same as having 0 WAR and $80 million in cash.

Here's an analogy. Every week, you buy 10 gallons of gas for your car, at $3 a gallon. One day, your local gas station tells you, "you're such a loyal customer, we're going to give you one gallon a week at the "slave" price of $1, and another gallon at the "arb" price of $2."

Does that change the price of the other 8 "free agent" gallons you buy? No, it doesn't. It saves you $3 a week, for which you're grateful, but you're still going to buy the other eight gallons at $3 each. The fact that your first two gallons were cheap shouldn't affect how many gallons you buy, any more than if you saved $3 on broccoli at the supermarket it would change how many gallons you buy.

Having said all that, I should add that there's one exception: if a team has more wins from slaves/arbs than it wants in total. Suppose that the Pirates were planning on buying free agents only up to 68 wins, because the 69th win would cost $4 million but only bring in $3.9 million. But when they look at their roster, they realize that they have 73 wins in slaves without having to pick up any free agents at all.

In that case, we should expect the Pirates should sell some of their players: they have five wins that other teams value at $4 million, that the Pirates value at less than $4 million. They can make more profit by selling off a couple of players.

But maybe they can't do that, for whatever reason (maybe the fans would revolt), and they're stuck with the extra wins. What that means is that now, with five wins "wasted" in Pittsburgh, there are five fewer free-agent wins available on the market. But demand from the other teams hasn't changed. And so, the price of free agent wins will rise a little bit, in order to price five wins out of the market for the other 29 teams.

Does that actually happen, that teams wind up slave/arb wins that they'd like to get rid of but can't? I don't know, but, even if it does happen, I think the effect would be quite small.


2. Valuation

Another thing I assumed is that teams can evaluate talent perfectly. If free agent X is a 4.3 WAR player, all teams know he's a 4.3 WAR player.

Obviously, that's not true ... teams will often sign players to contracts that don't appear to make sense to others. The obvious explanation is that the signing team has a higher expectation of the player's future performance.

There's a principle called "The Winner's Curse," which suggests that a party that wins an auction will have tended to overpay. Suppose a free agent comes along, and all 30 teams try to figure out what he's worth. The average estimate is 3.0 WAR, which happens to be exactly right. But not all teams come in at 3.0 WAR. Some teams are too low, and estimate 2.5. Some teams are too high, and estimate 3.5. One team is the highest, and estimates 3.6. The team that guessed 3.6, the one that overestimated the worst, is obviously most likely to be the team that winds up signing the player, because they're the ones who will offer the most money. And so, unless teams take into account that they might be "cursed," and lower their estimates accordingly, they all will overpay.

That might very well be happening. If it is, it will manifest itself in higher prices per win. A team will want to pay $3 million a win, and so it buys what it thinks is a 2.0 WAR player for $6 million. But because of the curse, it turns out he's only a 1.5 WAR player. And so, we observe that the team paid $4MM, and we conclude that's what it thought the win was worth.

So if all teams are doing that, it would be hard to tell whether teams were really valuing wins at $4MM, or whether they were valuing wins at $3MM and just screwing up. We'd maybe have to look at the details of their business, to find out that wins really *are* only worth $3MM, and then maybe we could guess that the Winner's Curse accounts for the remaining million. But unless we work for their accounting firm, how can we know?


3. Hometown Players

Maybe some players are more valuable to only a certain team, because of some connection with the city. Suppose Joe Mauer is a 6 WAR player, so his wins are worth $24 million, but with the Twins he's worth an extra $2 million because he's a Minnesota boy and the fans love him so much.

What happens? Well, when free agent time rolls around, 29 teams bid his price up to $24 million. Then Minnesota bids $24 million and one dollar. The Twins win the auction, and make an extra $2 million minus 1 dollar in profits that year.

Well, not really ... Mauer's agent also knows about the extra $2 million, and negotiations ensue. The Twins and Mauer finally settle on an amount between $24 million and $26 million, and each of them makes more money than if Mauer had gone elsewhere.

But that doesn't affect the price of free agents in general. At least, not unless Mauer affects the Twins' revenue curve somehow. I suppose it's possible ... it might be that fans are less likely to care about wins if Mauer is on the team ... and so instead of the 88th win being worth $4 million, it's worth only $3.95 million. And so the Twins buy only up to 87 wins instead of 88, which pushes the value of a win down a little bit.

It's possible, I suppose, but I just made it up and there's no real reason to believe it's true, much less that it's significant.


4. The Yankee Effect

The Yankees spend two-and-a-half times as much on salaries as the average team. How much does that push up the price of free agents?

Well, suppose the Yankees have been buying 15 wins more than average: 49 WAR instead of 34. And suppose they suddenly decide to stop, and drop back down to average. The supply of free agents goes up. How much does their value drop?

Well, the other 29 teams had been buying 951 WAR (1000 minus the Yankees' 49). Now they're going to buy 966. That's about a 1.5% increase. How much does the price of free agents have to drop in order to increase the quantity purchased by 1.5%?

An obvious estimate is that for the quantity to rise by 1.5 percent, the price would have to have dropped by about 1.5 percent. (That is, a demand elasticity of 1, as economists would say.) That's not much: a win goes from $4 million to $3,940,000. You wouldn't even notice that.

But who says the elasticity is exactly 1? It's a reasonable guess -- it's necessarily true that if everything in the world dropped in price by 1.5%, we'd be able to buy 1.5% more stuff, on average. But instead of buying 1.5% more apples and 1.5% more oranges, we might choose to buy 2% more apples and 1% more oranges. So if you want the elasticity of wins, you have to figure it out empirically. And I don't think there's the data to do that.

But, suppose we *did* have the data. Specifically, suppose that we knew for sure that the curves I drew in the previous post are accurate. Here's the one for the "average" team:



Now, at $4MM, we see that the average team might have bought 96 wins. From the graph, the 97th win is worth $3.5 million. So the team would buy an extra win if the price dropped to $3.5 million.

But wait! Teams don't have to buy even numbers of wins: they also come in fractions. It looks like if the team bought half an extra win, that half a win would be worth about $3.9 million per win. So for this team to go from 96 wins to 96.5 wins, the price would only have to drop from $4MM to $3.9MM.

Suppose all other 29 teams are exactly average. Then, a price drop to $3.9 million would get every one of the 29 teams to buy an extra half win. That's 14.5 wins -- almost exactly the 15 wins the Yankees stopped buying!

Now, more realistically, suppose all other 29 teams are *not* exactly average. If we drew the curves for all 29 teams separately, and did the same kind of analysis, I suspect the result wouldn't be much different. I suspect the market-clearing price, the price which would induce teams to buy an average of half an extra win each, would be not that much lower than $4 million. Whether it would be $3.94 million or $3.83 million I don't know, but I'm pretty sure it wouldn't be anywhere near as low as $3.5 million.

I guess my summary is: as successful as the Yankees are, they're only buying an extra 1.5% of the total talent out there, so their effect on prices isn't as big as you might think.


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7 Comments:

At Wednesday, April 28, 2010 3:19:00 PM, Blogger Brian Burke said...

Here's one more twist--the "Dice-K maneuver."

I recall that when the Red Sox paid for the negotiating rights for Dice-K, part of the rationale for paying so much was that Boston was getting a 2-fer. They would both gain rights to the WAR provided by Dice-K himself and deny his WAR to the Yankees, their only competition in the division at the time.

In other words, truly competitive teams are not simply purchasing WAR, they are purchasing "pennant probability." Just like in a game, teams aren't worried so much about runs as they are about win probability. In a 2-team bidding war with your only rival for a particular player, pennant probability is a zero-sum game. So every bit your team gains, the other team loses. Would this double the real value of a player's WAR? Hard to wrap my head around. Does your value graph already take that into account?

Not sure if I'm making any sense, and I'm sure this wouldn't have a wide-spread effect on the FA market anyway.

 
At Wednesday, April 28, 2010 3:28:00 PM, Blogger jfpbookworm said...

I get what you're saying, but calling players who aren't eligible for arbitration or free agency "slaves" rings such a wrong note that it makes this almost unreadable.

 
At Wednesday, April 28, 2010 4:26:00 PM, Blogger John said...

Brian

I don't thats right. You assume that the WAR the two teams are bidding for are the only WAR available on the market.

However, I think the team that loses the auction should simply be able to go and purchase the WAR elsewhere once the bidding for the first player exceeds their $/WAR threshold.

In the Dice-K example NYY would simply go and but a 2 WAR pitcher + a 2 WAR corner outfielder to replace their 0 WAR outfielder (I'm making this up) to match the WAR they would have got from Dice-K.

I think the Dice-K maneuver was Boston securing negotiating rights with a high bid with the expectation they could low ball the contract, giving them a lower $/WAR than would be available on the open market.

 
At Wednesday, April 28, 2010 4:52:00 PM, Blogger Brian Burke said...

John-I'm probably way off, now that I think about it more.

Still, I think it might be useful to think in terms of 'pennant probability' and not just in win totals.

 
At Wednesday, April 28, 2010 4:57:00 PM, Blogger Phil Birnbaum said...

I agree with John on the Dice-K issue, that if the Yankees picked up 4 Dice-K wins, the Red Sox could just keep up with them buy buying 4 other wins elsewhere.

But "pennant probability" is an issue too, as Brian points out. The revenue curve spikes after 80 wins precisely because of pennant probability. If the Yankees were to sign Dice-K, the Red Sox would revisit their wins vs. revenue curve. Maybe before the Yankees improved, the Red Sox figured 92 wins would give them (say) $175 in revenue. But now, 92 wins is not as attractive as it was, and they revise their figure down to (say) $165 in revenue.

The Red Sox might find that, now, the 93rd, 94th, and 95th win have enough extra playoff probability to justify buying three more wins.

But my idea was that pennant probability (or playoff probability) was built into the revenue curve. And I agree with Brian that other teams' moves might certainly affect the Red Sox revenue curve.

 
At Monday, August 23, 2010 10:46:00 AM, Anonymous Anonymous said...

One more complication -
It's not as easy to sign an additional couple of +2 WAR players, since presumably the team is already good and has +1 and +2 WAR players at many spots. So you will have to sign the +3 WAR player and then hope you can get the 4MM for the +1 WAR player in a trade, which is not certain.
So the Bosox/Yankees Wnat to sign the +4 WAR player because its easier than signing two +2 WAR players (since they don't have two +0 WAR players on the roster) or four +2 WAR players to replace four +1 WAR players and then have to trade them and get their money back).

Because of the limited number of roster spots, its generally better to get the higher WAR players, and these are scarser than a large number of +1s and +2s, hence snagging the +4 can actually be more beneficial than one would think.

 
At Monday, August 23, 2010 4:28:00 PM, Blogger James said...

Anonymous is right, it's a brick-layer vs a gladiator problem as Brian has said before: http://www.advancednflstats.com/2009/04/draft-picks-bricklayers-or-gladiators.html

You can't fill a team with all +1 WAR players and still get the 30 to 40 wins you desire. That makes a +2 WAR player more valuable than two +1 WAR players because he's taking up less space on your line-up/rotation/roster.

That means two things: 1a) The elite players are more valuable than simply WAR*(cost of a win) because of their "WAR density", to make up a term. Just as a +2 WAR player is more valuable than two +1 WAR players, a +5 WAR player is even more valuable still. 1b) This is compounded again because a +1 WAR player is much more common than a +5 WAR player, thus scarcity also drives up a +5 player's price in addition to his natural "WAR density". This feeds back into Anonymous's point above.

2) Complimentary pieces and/or specialization also plays a role on the value of a win at some point - talent will benefit from more talent around it, and from talent that minimizes others' flaws. Taken to a theoretical extreme, Player A with a 1.000 OBP is useful to his team only if his teammates are productive. If another team had Player B that hit a home run every at bat, it would literally double Player B's value if his team acquired Player A. Similarly, it doesn't matter if a catcher can't catch a runner attempting to steal if his pitcher never lets a batter get on base. A ground ball pitcher's effectiveness is determined by how well his infield can field the grounders and turn them into outs/DPs, etc.

Ultimately I'm not sure how much Point 2 changes things but it must have some effect, even if minor. However, I'm convinced Points 1a and 1b play a significant role into costs of a win.

 

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