Yes, this is a sports-related post. Sorry. But it’s got some econ and math and a request for some analysis at the end. So that’s pretty fun, right?

Well, anyway, it’s fantasy football season, so fans (and non-fans) everywhere have just finished drafting their players, who will now go on to collect stats that count toward the fantasy team’s weekly points. This year, however, I did something a little different with a team-based league rather than a player-based league, and I’m trying to figure out if there’s a “right” valuation strategy. If you’re familiar with football drafts (snake and auction varieties) feel free to skip to the “But What About This?” section. If you want a little background, well…

**Background**

Most fantasy football teams allocate players through a “snake” draft, which means the people in the league are assigned a random order–1 through 12 or whatever–and then the first person picks whomever he wants as his first player, then the second person goes and picks the second player, etc., until everyone in turn has picked one player. Then the process turns around (like a snake?); for the second round the person who picked last in the first round picks first in the next round. And it keeps snaking back and forth until everyone’s team is complete.

The process is pretty simple. Yes, you need to balance out your team with quarterbacks and running backs and wide receivers and a kicker and so on. But generally, the “best” player in the NFL gets picked first, and then the second best, and then the third best, etc. Unless a person is “wrong” about how good a certain player is, there’s a somewhat standard distribution of “goodness” on each team, because player talent slowly diminishes over the course of the draft and the right to select is randomly distributed. It’s more complicated than that (because of differences in positional scarcity) but that’s about it. Good players get drafted before not-as-good players. And if you pick first, you just sit there and watch while everyone else gets to go twice before you pick again. And then you pick twice. And then you wait. (You have to wait less if you’re somewhere in the middle of the round, but there’s still waiting.)

To mix things up a bit, some leagues (like mine) use an auction to distribute players. Everyone starts with the same amount of money (generally $200) and you use that money to bid on players until your roster is full. If you want, you can spend $185 or so on one player, and then just fill out your roster with $1 players. (That’s probably not a good idea, but you get the point.) Everyone takes turns “nominating” players, but those players’ services can then be bid on by anyone in the league.

There are a few benefits to this system. First, let’s say you really really want to have two of the four best players in the NFL on your team. Well, that’s impossible with the snake draft, because even if you have the first or second pick, you can only select one of those players, but there’s no way he’s going to be available by the time everyone else picks and the draft works its way back to you. With an auction, however, you can determine the talent distribution of your team. You can have a bunch of average players. You can have a few really good players and then a bunch of nobody-knows-if-they-are-good-but-I-hope-they-pan-out players. And so on. Whats more, as players’ values are determined during the auction, you might have to adjust your strategy on the fly, as you look for players or entire positions that are undervalued. Anyway, it’s a lot of fun (especially for baseball!) and you really have to pay attention during the entire draft to see if there are bargains.

A lot of people run leagues like this, so ESPN and a bunch of other sites have programs or guides that can provide you with estimated/recommended “prices” for various players, depending on the structure of your league. Given the complexity of the auction process, it’s useful to have some sort of benchmark to anchor the bidding, whether that’s based on an analysis or just an averaging over a number of drafts. If the “normal” auction price for a given player is $X, and I think I like that player 10% more than average, then I can budget about 10% more than $X for that player. Repeat that a bunch of times and I can put together a team with the balance of players that I want.

**But What About This? A League Based on Team-Wins**

This year I started a new league based on *teams* instead of *players*. Here’s how it works: You draft a team to be a part of your portfolio. If the team gets a win, then you get a win. At the end of the year the person with the most total team-wins in his/her portfolio wins. Really, it’s simple. If I pick the Vikings, the Colts, the Bengals, and the Falcons, and at the end of the year those teams go 8-8, 10-6, 9-7, and 9-7 respectively, then my portfolio has 8+10+9+9=36 wins. If that’s more wins than other folks’ portfolios, then I win the league! And the goal is to win. There’s no bronze medal.

We used an auction to allocate the teams. There are eight people in the league, and each person was required to buy 4 teams for his/her portfolio. Once you get your 4 teams, that’s it, you’re done. With only 32 teams in the NFL, that means every team in the NFL would be selected (at some price) in the auction. Every person had $100 to spend. It’s not real money, and it doesn’t carry over to anything, so if you don’t use your money in the auction then you just sort of wasted it.

The nomination process worked like this: Each person was randomly assigned a place 1-to-8 in order. Person 1 nominated a team and the bidding began. When that team was selected by someone, then Person 2 nominated, and so on until Person 8. Because the nomination process is not a selection process, the turns didn’t snake. Instead, after Person 8 nominated a team then it was Person 1’s turn again, then Person 2, and so on. Obviously the nomination/auction process went on for four rounds.

**How to Value Teams?**

Since we sort of made up this league/auction, we couldn’t find any preset algorithms or guides for valuing teams. But we talked about it a lot before and after the draft, and this is about as far as we got. I’d love someone with more econ/modeling rigor to give a more serious go at it.

Obviously, the average team in the NFL is going to go 8-8 on the year. So the average person in the league, with 4 teams in his portfolio, will have 32 total wins on the season. With $100 to spend that breaks down to an “average” (of some sort) cost of $3.125 per win. In other words, you should be willing to pay about $25 for a team you expect to go 8-8.

At this point, I should say that I’m not really interested determining which teams will win more than other teams. Being good at picking winners is an interesting football problem, but not really an interesting mathematical/economic problem. I basically just used the Vegas lines with some modifications to get estimates on the teams’ expected wins. I made a few small adjustments based on teams I “liked” but for the most part I think everyone pretty much agreed on who was expected in win about how many games.

To the extent it matters, assume the expected win distribution of the 32 teams is something like this (from the Vegas over/unders): 11.5, 11, 11, 10.5, 10.5, 10.5, 9.5, 9.5, 9, 8.5, 8.5, 8.5, 8.5, 8.5, 8, 8, 8, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 7, 7, 7, 6.5, 6.5, 6.5, 6.5, 5, 5.

If we go based on the “average” price-per-win, then the expected 11-win team should go for $34.38, and so on down.

But that can’t be the right price for the 11-win team, because we know that the auction has to clear all the teams. If I’ve spent almost all of my money and I’ve only got $1 left at the end of the auction, the ** worst** thing that can happen to me is I get stuck with a 5-win team (i.e., the worst team in the NFL) for a buck. Using the “average” price-per-win that 5-win team should cost $12.50. So getting it for $1 is a great value!

So maybe the better tactic would be to hold about $1 in reserve for a bad high-value team, and then use the remaining $99 on three better-than-average teams, perhaps teams with an expected total of 9+ wins. With $99 left, I’d have $33 per remaining team and I should be able to get some really good teams with my remaining three picks.

But of course there are only 14 above-average teams (and really only 9 appreciably above-average teams if we don’t count the teams with 8.5 expected wins). There are not enough “good” teams to go around if everyone saves that $1 for a bad team and uses the rest to get good teams.

So what to do? Any thoughts on the right (or a right) way to model this? If ESPN ran a league like this, how would they come up with their recommended auction values for the teams? I’m at a loss.

How much do you care about a closed form formula here? You could almost certainly set up some sort of Gaussian routine in Matlab/Octave and let it crank.

Nate! I knew I could count on you! So I have no idea how to set up those things you mentioned, but regardless, what would you be trying to optimize in that routine? Roughly, what would the process be? I think you’re right that a continually adjusting system based on other people’s bids/purchases would make sense. But the same is true for the standard NFL fantasy player-based auction. And somehow ESPN comes up with an initial/average/recommended price for those sorts of auctions.

Relatedly, I was thinking more about this last night, but there must be some math/econ literature, not about this particular problem, but about the more general problem of this sort of rule-based auction with a one-to-one ration of goods and available purchasing opportunities. Right? Is there any other situation where this comes up, even as a pure thought experiment?

Yes, this class of problems has received a lot of attention in baseball and can be evaluated by sabermetric statistics. Concepts such as WARP (wins above replacement) and win shares can readily be applied.

Here’s an example of how sabermetrics can lead to some interesting insight:

If we go based on the “average” price-per-win, then the expected 11-win team should go for $34.38, and so on down.

(DISCLAIMER: contains advanced math, which may not be suitable to liberal arts majors)

WARP tells us that you are vastly undervaluing the 11-win team.

WARP(T) starts with the premise that your typical bottom of the barrel player (team) is only worthy of commanding the minimum salary ($1), and that any additional money paid out is for *how much that player contributes beyond the prototypical minimum salary player*

Here, a convenient starting point is to look at our hapless friends, the Raiders and Jaguars. These teams will each command an asking price of exactly $1 (honestly, you’d have to PAY me to pick the Raiders).

Let’s talk about the Vikings now. When you go to bid for a 6.5-win team, the sabermatrician asks “how much should I pay for the *additional* 1.5-wins I would get compared to the $1 Jags?” Your answer will probably fall in the “not much” bucket.

Likewise, the 11-win team should be evaluated based on the 6 additional wins it provides over a replacement-level team.

Doing this math across the board, there are a total of 98.5 wins above replacement available across 32 teams and $800 among the managers to spend on these teams. This comes out to roughly $7.88/win-above-replacement.

Here’s a chart of how much I would pay for each team

11.5 wins (go Hawks!): $51.23

11.0 wins: $47.29

10.5 wins: $43.34

9.5 wins: $35.47

9.0 wins: $31.53

8.5 wins: $27.59

8.0 wins: $23.65

7.5 wins: $19.70

7.0 wins: $15.76

6.5 wins: $11.82

5 wins: $1