# Player Values with Range of Outcomes and the Importance of Upside

The term “range of outcomes” is probably familiar to many of those who play games of chance. We do not always know precise outcomes for certain situations, fantasy football being one of those situations. Player statistical output arises from an array of random forces which we can’t control or necessarily predict. Fantasy gamers may arrive at valuable estimates, however, when looking at a range of possibilities. This article utilizes basic probability mathematics to help the reader answer questions relating to player values with a range of potential outcomes, referred to as expected player values. The article is more theoretical than data-driven so do not get too caught up in the specific numbers used. Try to think more about the methodology and how it can be used to answer your own fantasy questions.

## Expected Player Values

Before we look at the unknown, let’s examine how player values are calculated in fantasy football. This article will use the familiar value based drafting (VBD) method as a start in determining fantasy expected player values (note there are a number of similar methodologies for determining player values). Player values (V) are calculated as the difference between points scored (P) and the baseline points of a replacement level player (BL). The replacement level point level is typically taken as something similar to the next best player available after all fantasy starters for a league. For example, the 13^{th} best QB in a 12 team 1QB league would be the baseline scorer, but this may also vary according to method and application. We will use average points per game (PPG) as our points in this article for simplicity. The player value equation then is simply:

*V = P – BL*

Let’s say a player scores 14 PPG and the baseline replacement player scores 12 PPG, the player’s value is equal to 2 PPG. We should also note a player’s value has a floor of zero (no negative values). A player who scores at or below the replacement level has zero value according to this method.

Now what happens if we add a bit of the unknown and don’t know what a player will score but do have an idea of possible outcomes? We may still estimate the player’s value if a suitable set range of scoring possibilities is available. Our expected player value (E(V)) is:

*E(V) = E (P – BL)*

The replacement level scorer tends to remain relatively stable from year to year and whatever variation which happens is the same for each league and position group so we assume a constant baseline for the purposes of this article. We can then present our expected value equation in the following form:

* E(V) = sum (Prob(i) x (P(i) – BL))* for all i where Prob is the probability of a player averaging a certain point total.

For example, let’s say there is a 50% chance a player scores 14 PPG and a 50% chance the player scores 16 PPG with the same 12 PPG baseline scoring used previously. The player’s expected value would simply be:

E(V) = 0.5 (14 – 12) + 0.5 (16 – 12) = 3 PPG

Now that the methodology has been presented, we may answer a basic fantasy football related question.

## Example Problem: How Much is Upside Worth?

This is a question which garnered much interest last year, maybe most famously in Scott Barret’s Upside Wins Championships. To answer this question, the article compares players with wider range of outcomes (more upside and downside) against those with narrower range of scoring possibilities (less upside and downside).

The article assumes a simplified discrete approximation of the normal distribution going forward for fantasy points per game on various mean levels with the same 12 PPG replacement level scorer. The “Example Probability Distribution” graph below displays a player with a mean of 14 PPG and 10% chances of scoring 10 or 18 PPG, 20% chances of 12 or 16 PPG, and a 40% chance of scoring 14 PPG. Our expected value for this player would thus be:

E(V) = 0 + 0.2 (12 – 12) + 0.4 (14 – 12) + 0.2 (16 – 12) + 0.1 (18 – 12) = 2.2 PPG

Note the 10 PPG component of the equation gets no value because it is below replacement level (remember no negative values).

We can then extend the concept to examine groups of scoring ranges and associated expected values as seen in the chart below. The three boxes have 11, 14, and 18 PPG mean scores. The Narrow range of the 11 PPG box spans from 9 PPG to 13 PPG while the Broad range shows a distribution from 5 PPG to 17 PPG as possibilities for example.

**Expected Values for Sample Scoring Ranges**

There are a number of key observations and implications which may be drawn from the data. The importance of upside is readily apparent when looking at the first box with a mean scoring of 11 PPG. The Narrow range of outcomes produces almost no expected value while the Broad range produces nine times the amount. There is an intuitive explanation for this. So much of a lower-tier player’s scoring distribution is at or below replacement level that they only produce value when they produce at the upper end of the distribution. That makes the player with the wider range of outcomes far more valuable in this case even though the projected stats are equal.

Contrast the 11PPG mean players with the 18 PPG high end scorers in the 3^{rd} box. The 18 PPG mean scorer produces the same expected value no matter if the scoring distribution is in the Narrow range or Broad Range. Again this makes intuitive sense. The upper-tier player is practically always a fantasy producer scoring valuable points, even at the lower levels of production. That means he doesn’t suffer from the same issues of the lower-tier player at the lower levels of the distribution and thus doesn’t have the big parts of fantasy irrelevance in the distribution. There is another concept called “risk-aversion” in which people generally prefer the less risky option. This might actually cause individuals to select the Narrow range player (less risky) over the Broad range scorer among the upper-tier players given there is no expected value difference. An individual with similar projections between Tyreek Hill and DeAndre Hopkins, for example, might prefer Hopkins if they view him as a less risky option. The conservation may change when we are talking about big tournaments and other fantasy structures weighted heavily to a very small percentage of the top teams.

The key conclusion from the previous discussion is that upside matters but it matters a lot more for those at the lower-end of the fantasy spectrum. The importance of upside fades as we move to the higher-level fantasy assets.** **

**Bio:** Bernard Faller has degrees in engineering and economics. He currently lives in Las Vegas and enjoys athletics, poker, and fantasy football in his free time. Send your questions and comments (both good and bad) on Twitter @BernardFaller1.