Fantasy Theory: How many should you draft per position?

How many RBs should you draft if you can play only 2? How many WRs if you can play 3? How many QBs and TEs where you play 1? Can we approach this from a purely theoretical point of view?

Throwing dice

Let's look at a different situation which has some interesting parallels to drafting backup RBs: let's throw some dice. If we throw with 2 dice, we pretty much know the curve we will get: it will be more or less pyramid shaped, with a peak at 7:

But what happens if we take more dice, say 4 or 5, and then we take the highest 2? Now the curve will look a lot different. The peak should be higher if we take more dice. Eg let's compare 2 out of 2, 2 out of 5 and 2 out of 10:

As you might expect, when you throw more dice and are allowed to pick the best 2, you will get better results; if you throw 10, you will almost always be able to find at least a 5 and a 6. The same of course is true if you select more RBs and can pick the best 2 out of 7; this will give you better values than 2 out of 3. 

Makes sense; but adding more dice will at some point give less benefit. What we are interested in, is deciding on what number gives the most benefit.

The benefit of logarithms

RBs are not dice however. One important different is that the later you pick your RB (or WR, TE, etc), the less benefit you expect. For instance your first rounder may have an expected value for the whole season of 250 points, whereas a 14th rounder might be only 100. In order to simulate that, we need to make each successive "die" of a lesser value; so if we start with a die worth 6, then number 7 (your 12-15th rounder) should be worth maybe 2-2.5.

Let's see how this affects our graphs:

Now that is a quite different picture. From 2 to 5 there is a clear added value; however from 5 to 10, the value is minimal. The faster the curve drops, the less the added value of more dice.

Or let's translate that to player value: the steeper the curve of the players, the less expected value from your 7th RB.

Tuning positions

All this is still fairly obvious, but now we would like to know where the added value stops. For instance if your league plays 2 RBs and 4 WRs, then how big should your bench be (ignoring the effects of injuries and surprise RBs off the practice squad)?

Obviously we are still throwing dice here. It is not that easy to simulate real drafts (although I will try to do that later).

Let's compare the following three scenarios:

- you pick the best 1 

- pick the best 2

- pick the best 4

The chart for these scenarios looks like this:

- blue represents the scenario where you play only 1 player, but you draft 2, or 3, or more (say your average QB/TE situation)

- orange represents the scenario where you have to play 2, and draft multiple (say RBs without flex)

- grey represents the scenario where you play 4 (say WRs including flex)

There seem to be 2 interesting conclusions:

- if you play only 1, drafting a backup is relatively favorable (*)

- if you play 2 or more, drafting a couple backups is interesting, but not many more than 2 or 3

- in all cases, the value of the 5th backup is almost zero.

(*) an important point you don't see in the graph above, is that it matters how "flat" the curve is. For K/Def the situation is different than for QBs/TEs.

Let's tune the above to a couple specific scenarios.

Scenario 1: Yahoo standard

Say we are in a .5 PPR, and we play QB/2RB/2WR/Flex/TE. We have already drafted 2 RB, 2 WRs, 1 QB and 1 TE. What would now be the most valuable position to draft (ignoring the Flex, K/D) in round 7?

Let's take a look at the values per position:

In this format the expected value of RB/WR/TE is somewhat similar, with the TE being worth slightly less; QB has a separate curve, which in fact appears to be more linear.
- Say we have one QB worth 450, and the next is expected to be worth 400
- We have one TE worth 180, and the next is expected to be worth 100
- we have 2 RBs worth 250 and 200, and the next would be 150
- we have 2 WRs worth 220 and 165, and the next would be 135
Which position should we pick?

Interestingly, in this scenario, with these dice, we see that:
- The added QB would be the biggest benefit (which is counter intuitive)
- the difference between the 3rd RB/WR and the 2nd TE is minimal.

Something we haven't incorporated into our model though is that QBs have far more stable results on a weekly basis than RB/WR/TE. Therefore, we may not be able to compare these positions; but a comparison between RB-WR-TE makes more sense.

(to be continued)