Tuesday, April 20, 2010

Why teams pay equal prices for free agents

Big-market teams like the Yankees will sign more good free-agent players than small-market teams like the Royals. That's because wins attract more fans, and, also, existing fans get more excited and spend more money with the team. It seems like the Yankees, who have more fans, will get more benefit from the extra wins than (say) the Royals will. More fans to spend money means more dollars.

Now, suppose a new free agent becomes available, and he's good enough for 3 WAR (wins above replacement -- that is, a team that gets this player will win 3 more games than if they had the best available minimum-salary player in his place). Who will bid most for him?

Before I thought it through, it just seemed like it would be the Yankees, since his wins are worth more to them than to anyone else. But, after I thought about it a bit, prompted by a discussion over on Tango's site, I started to think that's not true. I realized that players go for the same price, regardless of which team signs them.

I think if you look at the empirical evidence, you'll find that's true; if you divide a free agent's salary by his projected performance, I'd bet you'd find it hovers around the same number ($4.5 million per team? I gotta ask Tango what the current number is), independent of which team signs him. There might be fluctuations, but I'd bet that there wouldn't be a huge difference between teams in the top half of the league and the bottom half of the league.

And that makes sense, if you think about it. It's not just players who are more valuable to the Yankees: it's everything. Baseball gloves, say. Obviously, Derek Jeter's glove is more important than Yuniesky Betancourt's glove: without a glove for Jeter, he can't play: and the Yankees lose a lot more money with Jeter sitting out gloveless than the Royals do with Betancourt sitting out gloveless. But both teams pay about the same amount for the actual glove. The same is true for everything a team uses: jet fuel, bus service, the food served in the clubhouse after the game, and so on. Why would wins be any different?

Anyway, once I thought about it a bit, I came to the conclusion that wins are like any other product they talk about in economics. Here's my logic, which won't be too surprising to economists, if I got it right. In fact, I'm writing out mostly to get it straight in my own mind.


I'm going to start with a bunch of simplifications, which won't affect the argument much. I'll come back to some of them later. Those assumptions are:

-- Every team is about to sign 25 players to one-year free-agent contracts
-- Every team has the same accurate projection for every player
-- Every team has good information about the revenue projections for every other team
-- Every team cares only about wins, and not about the personalities of the players
-- Every team pays $0 for replacement players

Suppose a team, say, the Phillies, is trying to figure out how much to spend on players. They start by writing down revenue projections for each number of WAR they might buy. If they buy 0 WAR, they will finish with 47.666 wins (I chose this number arbitrarily for convenience -- it's reasonably close.)

But if they buy 0 WAR, the fans will revolt -- they'll see their team not spending any money at all. There will be a big scandal, and the team will get in trouble. It would be such a bad situation that it works out as if the team got no revenue at all.

0 WAR -- $0 revenue

If they buy 1 win, or 2 wins, or 10 wins, it's the same thing. Suppose the team figures that the lowest realistic option is to buy 20 wins (to win 68 games). That will net them $100 million in revenue. (I'm making that number up, as well as all other numbers in this post.)

20 wins -- $100MM

They also figure that the 21st win will draw an additional $7 million out of the fans' pockets:

21 wins -- $107 MM

And 22 wins another $7 million, and so on, with diminishing returns. Once their business analysts have done their projections, they wind up with a chart like this. I've left off many of the rows, to keep things easier to read:

20 wins -- $100 MM
21 wins -- $107 MM
22 wins -- $114 MM
23 wins -- $120 MM
30 wins -- $158 MM
35 wins -- $180 MM
40 wins -- $206 MM
41 wins -- $210 MM
45 wins -- $225 MM
50 wins -- $235 MM
51 wins -- $237 MM
52 wins -- $238 MM
60 wins -- $240 MM

So the Phillies can buy 20 wins, and make $100 million, or they can buy 60 wins, finish at 108-54, and make $250 MM. Which option is best? Well, it depends how much it costs to buy those wins.

Suppose wins are $3 million each. Then the Phillies can make a chart, which I'm going to call that "Chart 1" because I might come back to it in a later post:

Chart 1 -- Hypothetical Phillies Business Analysis

Wins Revenue Salaries Profit
20 .. 100 ..... 60 ... 40
21 .. 107 ..... 63 ... 44
22 .. 114 ..... 66 ... 48
23 .. 121 ..... 69 ... 52
30 .. 158 ..... 90 ... 68
35 .. 180 .... 105 ... 75
40 .. 206 .... 120 ... 86
41 .. 210 .... 123 ... 87
45 .. 225 .... 135 ... 90
50 .. 235 .... 150 ... 85
51 .. 237 .... 156 ... 81
52 .. 238 .... 159 ... 79
60 .. 240 .... 180 ... 60

And so the Phillies conclude: if wins wind up costing $3 million each, we're best off if we buy 45 of them. They'll make $235 in revenues, pay $135 in salaries, and pocket $90 million profit.

But: what if wins wind up costing $4 million each? They repeat the above chart, updating the "cost" and "profit" columns, and discover that they make the most profit when they buy 40 wins: that's revenue of $206 million, salaries of $160 million, for profit of $46 million.

What if wins are $5 million? Then the best thing to do is buy 30 wins. Revenue $158MM, salaries $150MM, profit $8 MM. Any other number of wins leads to less profit.

What if wins are $1.1 million? Then the highest profit probably occurs at about 51 wins.

So the Phillies make a second chart:

Cost of Wins ... # Of Wins We Should Buy
$1MM ................ 51
$3MM ................ 45
$4MM ................ 40
$5MM ................ 30

They repeat this process for every value that makes sense ... like $3.2 million, or $2.7 million, and so on, and put the results on a graph:

Now, the Yankees do the same thing, and come up with their own curve. You can assume that the Phillies know the Yankees curve, or that they don't -- it doesn't matter much. The Yankees curve will be higher, because their higher revenue makes it desirable for them to buy more wins:

And here's a chart for how many wins will be purchased by the Phillies and Yankees combined. For this chart, all I've done is add up the numbers for the two teams. For instance, how many wins will the two teams buy in total if wins cost $3MM each? Well, looking at the above chart, the Phillies will buy 45, and the Yankees will buy 60, for a total of 105.

The Phillies now repeat the process for the other 28 teams. They take the 30 curves, and add them up to create one composite curve. That might look something like this:

So, if wins are $1MM each, the 30 teams will try to buy 1350 of them. If they're $6MM each, they'll try to buy only 400 of them. (Reminder that I'm making these numbers up; they're probably not realistic.)

But what will the price actually be? Well, it turns out that we know for sure that there are exactly 1,000 WAR available for purchase. How do we know that? We know (well, we assumed) that a replacement-level team will win 47.66 games. But an average team must win 81 games. The difference is 33.33 WAR per team. Multiply that by 30 teams, and we get exactly 1,000 wins.

So when will teams choose to buy exactly 1,000 WAR? From the graph, we see that happens when wins cost exactly $4MM each. And so, teams bid up the price of free agents exactly to 4 million dollars.

No free agents will go for $3 million a win, because then teams would want to buy 1150 WAR when only 1000 are available, and the price will be bid up. Wins can't go for $5 million, because then teams would want to buy only 800 WAR when 1000 are available, and the remaining free agents would be knocking down GMs' doors offering to sign for less.

The equilibrium price, in our example, is $4 million, and that's what wins will go for.


But still, it seems weird that every team pays the same amount per win. Aren't wins worth more to the Phillies than the Royals?

Yes, the *average* win is worth more to the Phillies than the Royals. The *Nth* win is worth more to the Phillies than the Royals. But the *marginal* win is worth almost exactly the same.

That is: the Royals may only have bought 20 wins. Why didn't they buy a 21st win? Because that 21st win cost $4 million, and the increase in revenue they'd get from it would be worth less that $4 million: maybe $3.9 million.

The Phillies *did* buy a 21st win: as you can see from the first chart, the 21st win was worth $7 million to them: it bumped their revenue up from $100 million to $107 million.

The 21st win IS worth more to the Phillies than the Royals. And the 22nd, and the 23rd, and probably almost every number.

But the big-market teams will keep buying wins even after the small-market teams have stopped. The Phillies will keep buying until they hit 40 wins -- from the chart, we see that the 41st win is worth exactly $4 million (bumping revenue from $206MM to $210MM). (Actually, you'll have to pretend that I rounded, and that win is worth only $3.95 million: that's why the Phillies didn't spend $4 million on it.)

The Phillies' marginal win -- their 41st -- is worth about the same as the Royals' marginal win -- their 21st.

And the same is true for every team. The last win they chose to buy was worth more than $4MM, or they wouldn't have bought it. And the next win they chose not to buy was worth less than $4MM, or they *would* have bought it. When you consider that teams can buy fractional wins, you can simplify to say that the teams all buy fractions of a win until the next fraction gains them revenue that exactly matches the cost ($4MM per win).

That means: suppose a 1 WAR player retires just after every team fills its roster, and another 1 WAR player comes out of retirement to replace him. Who will bid highest on the new player? It will probably be the team that lost the original player, even if it's a poor team like the Royals. The other 29 teams gain less than $4 million in revenues if they pick up the player -- we know that because they chose not to buy any more $4MM wins than they already did. But the team that lost the player, the team that's now 1 WAR short of where they want to be, values the player at more than $4 million. We know that because they chose to buy the original (now retired) player for $4 million in the first place.

Of course, in real life, teams don't go to that many decimal places. In that case, every one of the 30 teams is roughly equally likely to sign the new player. We know that the last WAR each team signed was worth $4MM to them ... so the *next* WAR probably isn't worth much less than $4MM. And so we shouldn't be surprised if the Royals are just as likely to sign the new guy as the Yankees are.


I guess I'll stop here for now ... next post I'll try to explain what I think happens when you eliminate some of the oversimplifications. What happens when not everyone is a free agent? What about arbs and slaves? What happens when some players have marquee value beyond their wins? What happens when teams can't really give a pinpoint forecast of their wins (which is always)? What about wins being worth more around the playoff bulge (85-95 wins, say)? What if teams have no idea what other teams' revenue curves look like? What if certain players have extra fan value to only one team because of special circumstances? What if teams are risk-averse? [Spoiler: I don't think all that much happens differently even when you account for all these things.]

Most of this stuff you can figure out with a bit of logic ... I make no claim to having huge expertise here. Those of you who know more economics than I do, let me know if I got anything wrong so far.

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At Thursday, April 22, 2010 9:35:00 PM, Blogger Don Coffin said...

I think that's just about correct, so long as all players are hired in a competitive market. But if some players (those not eligible for free agency) have less bargaining power, then they will be signed for less than free agents would. So there's, in effect, some "surplue value" to be haggled over by free agents and teams. In a sense, as soon as some players are signed for less than their WAR*(Marginal value of a win), there's uncertainty in the outcome. Note that the fewer players who are available as free agents, the available WAR will not necessarily by 1000...and the number might be variable over time within the same signing season.

Also, you've made some (strong) assumptions about what everyone knows. Auction theory talks about "the winner's curse," which tells us that, even in a "common value" auction, different bidders *may have* different expectations about the value of the property. So, in this case, managements that are *more optimistic* about player performance ("we have Leo Mazzone--or Dave Duncan--coaching our pitchers, so well get more value out of them"--the error here, in part, is that the value should be attributed to the coach, but let it go) will be willing to pay more.

Also, while the adding-up constrain you introduce is absolutely correct, it's correct only in an ex-post sense. Teams might cumulatively *think* they're signing more value than that (unless teams share their valuations, which will get them hit with a collusion judgement). So the Yankees (for example) might be willing to pay more for some additional wins that they *think* they can get...This works the other way, as well..."adding up" the wins teams think they can get might be an underestimate of what's actually available.

But in a competitive world with complete and accurate information, I think you are probably right.


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