Game Theory and Fantasy Draft Strategy

My last article (part 1, part 2) took a look at the concepts at the core of fantasy draft strategy. Scarcity and consistency are, in my opinion, the critical considerations when valuing players. I also outlined an 'opportunity cost' concept in player selection. In this article, I'll take a look at particular draft strategies and try to push the envelope a couple inches.

Replacement-Level Strategy

For those not familiar with replacement-level drafting strategy, it's based on the relative value of each player above the lowest ranked player within each position. But before you do anything, you need some sort of value assignment for each player, the most logical being their total projected fantasy point production. You can use anyone's projections, such as Yahoo's, or FFToolbox, or any of the array of websites that offer them. You can just as easily use your own, or modify a published list. It (almost) goes without saying that you need to make sure the scoring rules used for the values are the same as those for your league.

If your league starts 1 QB, 2 RB, 3WR, and a TE, and your league has 8 teams, the replacement level player at each position would be the 8th-ranked QB, the 16th-ranked RB, the 24th-ranked WR, and 8th-ranked TE. For example, using my office league's scoring rules, LaDanian Tomlinson is projected by Yahoo to produce 224 points, and the 16th ranked RB is Selvin Young who is expected to produce 117 points. This gives LT a 224 - 117 = 107 value over replacement player (VORP).

Before your draft, you would calculate the VORP for each player, and simply draft the player with the highest value until each starting roster slot is full. This is also known as value based drafting (VBD). I like this system. It's simple and it usually works very well. However, it fails to fully capitalize on the irrationality of your opponents.

Implications from Game Theory

A fantasy football draft, or any similar draft including the NFL's actual draft, is what game theorists would call an n-player zero-sum game with perfect information. N-player means more than 2, which severely complicates any analysis. It's a zero-sum game because there is a finite amount of projected value available to the entire league, and players are scrapping for the largest share. (Players can voluntarily make non-optimum choices, and so the game could be considered non-zero-sum, but game theory always starts with the assumption of rational players.) Perfect information refers to the fact that you are aware of every move made by other players up to the current point in the game.

N-player games cannot be 'solved' the same way as 2-player games can, as I demonstrated with the example of the run-pass balance problem. But in game theory, there is usually a 'minimax' solution to zero-sum games. Also known as a Nash equilibrium, the minimax solution is the strategy with the maximum assured minimum gain for a player. (Technically, it's the strategy that minimizes the opponent's maximum possible score, which is effectively the same for a zero-sum game.)

Although it's technically imprecise to call it so, the VORP strategy is akin to a minimax for a fantasy draft. It guarantees you a minimally assured value total for your starters. But like all minimax strategies, it can be extremely conservative because it assumes other players are purely rational and also play their minimax strategy.

But in many cases, most other fantasy opponents are not playing a VORP strategy. They'll be biased towards hometown favorites, or picking based on hunches, or simply following a rule of thumb such as RB-RB-QB-WR... (like I typically do). They'll leave better players on the draft board than would otherwise be there. However, chances are that other opponents will be there in front of you to snatch them first. Fortunately, there is another, slightly more aggressive approach that can get even more points.

Value Over Next Available

If you have a good idea, or any idea, about what positions will come off the board between your own picks, you can adjust your choices to capitalize on your opponents' errors while maximizing your projected values beyond the minimax.

You can get a reasonable estimate of how many QBs, RBs, etc will be taken in each round in several ways. First, guys talk football draft strategy like women talk about clothes. Every guy thinks he's smartest football expert in the room. (Gosh, I hate guys like that...) If your opponents haven't blabbed their first couple picks, then it doesn't take a CIA ops officer to get them to spill the beans. But a more reliable method might be to just look at mock drafts and historical trends. For example, I found that for an 8-team league in '08, the expert mock drafts I clicked on last night typically looked like this:

1st round: 1 QB, 7 RBs
2nd round: 5 RBs, 3 WRs
3rd round: 3 RBs, 1 QB, 4 WRs


Plus, as the draft goes on, you can refine the estimates based on which players have already been taken. Team A still needs a QB...Team B took a WR in round 1, he'll need a RB...etc. That's where the "perfect information" comes into play.

As your pick approaches, estimate the number of each position that will be taken between that pick and your subsequent pick. For example, in my recent 8-team league draft, between my 4th pick in the 1st rd and my 13th overall pick in the 2nd round, I estimated there would be 1 QB, 2 WR, and 5 RBs taken. (Before my 1st pick there was 1 QB, 1 WR, and 1 RB taken.) I calculated the difference between the best available RB and the RB 6 spots down the board, because he's the next RB available to me if I don't take a RB this round. I also calculated the difference in value between the best QB available, and the next best QB. Finally, I calculated the difference between the best WR available and the WR 2 spots down the board. These differences are the costs of not picking each position in that round. I picked the position with the highest cost.

Instead of my usual RB-RB-QB-WR... pattern, this system had me pick RB, WR, QB, TE(!), RB, WR, WR. I ended up with 950 projected points. The next highest team had 850. The next highest opponents essentially used a VORP method and ended up with 815 and 814. Using my conventional strategy in an earlier dry run with the same opponents using the same strategies, I got only 694 pts, 5th out of 8 teams. One draft does not prove the system, but it does have a solid theoretical underpinning.

For those interested, I've set up a spreadsheet with an example draft that demonstrates the system, which has been dubbed VONA for Value Over Next Available. Each round of my mock draft has its own tab at the bottom of the spreadsheet. There is a text box on each worksheet that explains each selection. I don't think my system is terribly revolutionary, but it is an improvement. It's really just a dynamic VORP system. The difference is that the replacement player at each position changes as the draft evolves. The advantage comes from being able to better capitalize on opponent error.

Some Final Thoughts on Draft Strategy

1. I really like the replacement-level concept. As long as you have an objective value for each player, it will do an excellent job at ensuring a minimum level of success. If you don't have a good idea of how many players at each position would be taken between your own picks, you'll want to stick with VORP. Here a couple additional tricks I thought of:

a. If someone picks a player below your replacement player at a certain position, your replacement player has now changed, and you should adjust your VORP numbers. For example, in a 10-team league, if someone grabs the #11 QB, your replacement player is now the #9 QB. He's now the worst-case scenario at QB for you.

b. Depending on your draft position and how many starting slots your league has, your replacement level players aren't who you think they are. If you have the 10th pick in a 10 team draft, with 6 (or any even number) starting slots, you will pick 1st in the last round. The last 9 players are inconsequential. You can safely eliminate 2 or 3 of the bottom TEs and WRs (and perhaps the 10th QB) from your list. They'll be taken after your last pick (of offensive starters). The same principle applies if you have the 9th pick or 8th pick, or the first few picks with an odd number of starting slots.

2. You may value a backup RB more than a #3 WR, or even a mid-range TE. That's up to you. If you do, just consider the #3 RB as a starting slot, and the VORP or VONA system will still work.

3. We haven't discussed kickers or defenses yet. But everything I've seen says they're the least predictable and most replaceable. Plus, defenses can easily be picked up according to weekly match-ups. Maybe that's something to be looked at in detail in the future.

4. All of this probably doesn't matter.

It's true. The vast majority of decisive factors are luck--injuries, other unforeseeable events, sudden declines of teams, or even weather (Remember the Pats-Jets game in the snow/wind storm last year--it helped get me in the playoffs). If in a 10-team league you have a 10% chance of winning, perfectly optimizing your draft gives you, what, a 12% chance? 13%?

5. Finally, if you win at fantasy football, it doesn't mean you know anything about the NFL or its players. Unless you're an NFL scout, it very probably means you are just lucky. Everyone has access to the same volumes of up-to-the-second information, projections, and match-ups for every NFL player. Even if fantasy football was as much 50% skill, you'd need to compete successfully for several years before you had really any certainty that you were any good.

Now that I've sapped every last drop of fun out of it, enjoy your draft and good luck!

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22 Responses to “Game Theory and Fantasy Draft Strategy”

  1. Shake'n'bake says:

    Just in time for my first draft, tomorrow night. Good series.

  2. Anonymous says:

    Great job man...I spend hundreds of hours every year preparing my draft charts. You have a lot of awesome work here. I really appreciate you taking the time to share that information. I'll definitely be incorporating this information in al of my future drafts.

  3. Michael says:

    Thanks for this, I really enjoyed the series. I use "value over average" for drafting which basically yields the same thing.

    This is problematic in that ranking a players performance tends to be a very difficult thing to do year-in and year-out. Players coming off of career years tend to get penciled in for those type of numbers (which rarely ever happens as there is usually equal parts luck and skill involved in a Madden-cover worthy season). There is not a surefire method for removing this subjectivity.

    Also, I think there are players that are inherently conservative and risky. Donovan McNabb is injured every year. When he plays, he's a top-shelf QB. Then there are guys who are the model of consistency, like Reggie Wayne. In drafting, I personally tend to opt for safer bets early and take big risks later on. Looking down my value-over-average sheet doesn't tell the whole story in this way. I might pass over a player because he's risky or I might target a player a little early because, even though he doesn't project as high as some, his ceiling is far higher than the rest of the pack. From my experience, fantasy drafts are won and lost on those types of picks.

    This type of analysis that you have adeptly articulated is ultimately the starting point for me, though. Great post.

  4. Brian Burke says:

    Michael- Agreed. But one great thing about expected value is that can account for risk. If you think McNabb has a 50% chance of making it through the season uninjured, then his value would be = 0.50 * (McNabb full season value), or however you choose to weight his value. Usually, injury concerns are already factored in to published values and rankings. And keep in mind, everyone is an injury risk.

  5. Omar says:

    Brian - Excellent analysis. It is surprising how many people still do not use the concept of "opportunity cost" to inform their decisions. However, if your league allows costless trading, I think that the players you have already drafted and any given point should not have a large effect on which players you subsequently pick. By that I mean if you have drafted Tom Brady, and the opportunity cost of Payton Manning is higher than that of any TE, WR, RB then you should take Manning. After the draft you can try to trade him for a TE, WR, RB of similar value.

  6. roasthawg says:

    You just gave away my exact drafting strategy! I mean to a tea, this is how I draft in fantasy sports...plug the projections into an Excel SS and find the VORP. Then use average draft position to pick the best available of the guys who will probably be gone by the time it's my turn to pick again. Like you said, it's still real tough to win...a lot of great looking teams just don't produce. So much of it depends on the player projections...that's the real meat and potatoes of fantasy. Good job.

  7. Sky says:

    Good stuff, Brian. VONA is definitely a theoretical improvement on VORP, but I wonder how well we can really predict who will be drafted? If everyone else was picking off your personal VORP ratings, your VONA picks would be exactly the same as your VORP picks right? So it comes out to predicting how your own list is different from the other teams' lists.

    One way to do that is to compare your own list to an ESPN or Yahoo average-draft position list, noting the biggest differences. For anyone you think is underrated, you can wait longer.

    Nate Silver discussed a fantasy baseball drafting spreadsheet one time where he assigned each player a probability distribution for where they'd be selected. You could then combine those distributions (by position, etc.) to pick the player you need now, while letting others slide that provide the greatest expected value for your next pick (or for all future picks). Sounds nice, but difficult.

  8. Brian Burke says:

    Sky-
    I've tested and compared both strategies, and for the most part, against live opponents VORP and VONA often recommend similar picks. If everyone were playing VORP, then I'd guess, yes, they'd come out the same. The big difference is that VORP will often recommend picking a TE too early, and a QB too late.

    They would give identical recommendations if in VONA you assumed that all players above replacement will be taken in every round. VONA is really just a round-by-round VORP. And VORP is really just a worst-case-scenario version of VONA.

    I guess VONA is a little like Nate Silver's probability distribution, except that I'm not doing a full distribution, just a central best-guess.

  9. Ken says:

    I think I disagree with your definition of replacement value. If there are 10 teams and everyone gets 2 qb's, the replacement level is the 20th, the absolute worst case scenario for you and a production level you can be guaranteed to exceed if your rankings are correct.

    I feel you are suggesting that no team would take their second qb before all other slots are filled, but if you have say the 9th qb, you might take the 10th with a later pick as well if available since you aren't to confident in your 9th ranked quarterback and et cetera.

  10. Brian Burke says:

    Ken-I think that's a good point. That can happen. But if someone does that, they're leaving a lot of differential value at another position for you. It's a net gain for you. Plus, realistically, you're not going to be stuck with the #20 QB, maybe the 11th or 12th worst case. And there isn't much difference between QBs when you get that deep.

  11. BA says:

    Great stuff! I started using a VONA-like approach last season after struggling with the exact VORP problem you describe (TE too early and QB too late). My teams all were much stronger as a result and I got lots more "you stole my next pick!" comments, which is always a good sign. It's good to get some confirmation that what I am doing makes sense. I'm wondering though if there is a way to make the analysis a MULTI-ROUND one. For example, if VONA analysis recommends a WR in round 1, that will impact how others will pick in round 2, and then ultimately what my picks will be. Since the goal is to maximize my total pick values, not just one round, is there a way to account for this multi-round impact? Or is this getting too late in the causation chain? Is the variation in what others pick so large that it makes pointless any effort to think 2-3 rounds ahead? Again, great work here. Thanks.

  12. noplasma says:

    This VONA concept is interesting... I think you're on the right track but one adjustment is needed.

    See, VONA fails to consider the relative value of a prospective player with respect to playing against those players already taken. If you select someone much worse than those already taken, that's a major opportunity cost (OC).

    Value should be estimated in terms of how a player compares to those whom he will actually be playing against day in and day out. For a 8 team league, I'd compare to the value of the top 8 (for QB).

    In your spreadsheet, let V be the sum of the difference between the projected points of your prospective pick and the projected points of each of the top 8 QBs.

    Once you have a V for each player, THEN apply your VONA logic. OC is the difference between V_available_now and V_available_next_round.

  13. Michael says:

    Just found this website. Super series of articles! I just wanted to add one extra twist, which is probably obvious. After the first 6-10 rounds (depending on the size/format of the league), there isn't much VORP or VONA to be found, at which point one should switch to players with very high variance (i.e., could be busts, or could be nuggets of gold, like rookies or Leinart/Warner for example).

  14. Mark Kamal says:

    I use a combo of similar ideas. The pieces I use in our modeling here at Protrade (each is relative to the specific position in question):

    1) value over starter, which is essentially what you are calling VORP above
    2) value over last roster - there is value in having bench players - for bye week and injury purposes. For example, a 3 RB can be very important, even if you only start 2.
    3) VONA

    For each of the "value" components, we take a measure of expected points scored, as well as consistency (beta).

    The key is - how do you weight these three things together? The weighting should slide based on what round you are in the draft (are you late, and depth is all that matters), as well as what draft slot you are (for example VONA for the 9th pick in a 10 team draft in round 1 is not as relevant as for the 5th team).

    I find the game theory elements extremely interesting for fantasy football. Even if you all have perfect, similar projections for every player, you may not win because of game theory. Throw in the variability of projections, and it is a very fun game!

    -Mark (mkamal@protrade.com)

  15. Anonymous says:

    "Value over next available" is also referred to as "Dynamic VBD." There is an article or two on the web about it.

  16. Anonymous says:

    Hi Brian,

    Would you mind sending out your complete model?

  17. Brian Burke says:

    Well, I'm not sure what else I can spell out. There's no equation or algorithm, just the method I described. If you go to the 'example draft' link in the article, you'll see how the method is applied.

    If you're referring to my game prediction model, the full description is here.

  18. Brian says:

    Partially inspired by this post, I have created an ADP-adjusted Dynamic VBD spreadsheet and made it freely available. It's similar to VONA/dVBD in principle, but accounts for the fact that players are not likely to be taken in the order which _you_ have them projected. Details here:

    http://yocto.wordpress.com/2009/08/22/adp-adjusted-dynamic-value-based-drafting/

  19. Anonymous says:

    I'm very entities by the "perfect" draft strategy. Definately you want to use an "exploitative" strategy rather than assuming opponents are perfect. But the main variables are understanding variance. You cannot simply say a player gets 150 points vs 147. Realistically you are selecting player with an expected value of 150 and some kind of "standard deviation" you can estimate the range. You instead are more likely picking a player with a 120-180 range or a 135-160 player. So drafting 3 players with higher variance rather than 2 starters at a higher scoring position may be superior, or at a position with a steeper drop off if the picks are made earlier. That's what makes the problem so complicated. But I think you can run Ttests or Ztests or Chi-test or whatever stats test is required to figure out certain probabilities.
    For example you can calculate the probability that a player with ADP of 12 and standard deviation of 6 is available at the next pick of 15, and use his value. But it becomes significantly more complicated because you are not multiplying a value, but a range,

  20. Anonymous says:

    Intrigued*

  21. Anonymous says:

    I thinking all probabilities and rangesand drafted positions and picks can be known, this problem can be solved, but I just don't know the perfect way to go about doing it. Additionally, even if it can be "solved" it still becomes a competition on who can come up with the more predictable stats and projections, but also factoring in standard deviation and such.
    A great "scouting ability" player could enhance results by also coming up with some sort of "range" there are many "streaky" players that show great potential and great upside such as the most talented players, but also many of these players may have troubles learning the plays and running routes. These players have tremendous upside and a wider standard deviation or variance. So the most dynamic ranking would have both a projection and a standard deviation of each player... The optimal drafter would use that information perfectly based on probability theory

  22. Dr Football says:

    I created a spreadsheet that performs VONA similar to yours above. The way it works is you can put in your own custom player projections by simply pasting them into the individual position tabs at the bottom. As players are drafted you go to the position spreadsheet and delete the row of that player.
    You will also want to track which position each team takes, so I have placed a roster grid on the "Draft" page.
    When it is your turn to draft, use the roster grid and knowledge of other players to guess how many of each position will be taken before your next pick. The table on the top lists the highest ranked player at each position and the VONA associated with that player for the next few picks. You should draft the position that has the highest Opportunity Cost (VONA).The tool is currently set up for ten teams, but can easily be adapted.

    I put the tool on google docs, but it works best in excel.
    Try it and please provide feedback and suggestions to improve.

    https://docs.google.com/spreadsheet/ccc?key=0Atk6eCc4kv1OdHpuQmVmTzBpMWxBcnNzMTIwaW83REE

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