Jonathan Bales is the author of the *Fantasy Sports for Smart People* book series, and most recently *Fantasy Football for Smart People: How to Win at Daily Fantasy Sports*

Galileo supposedly deduced that objects fall at the same velocity, regardless of their mass, from a simple thought experiment. It went something like this…

If a light object and heavy one were tied together and dropped from a tower, the heavier one would fall faster if it were true that mass dictated velocity. In that case, the line connecting the objects would become taut, creating drag and slowing down the heavier object. Once that occurs, the two objects together should be heavier than either of them in isolation, which should make them fall faster together.

Thus, a heavy object tied to a light one should fall both faster and slower than the heavy object by itself—a clear logical contradiction. Galileo might have discovered a fundamental law of gravity while he was on the can for all we know. That’s where I recently started setting my DraftKings lineups, too, so I’m just waiting for the money to roll in. Now I’m going to attempt a DFS-related thought experiment that should have implications that rival those from Galileo’s thought experiment…

### Value in DFS

At its core, daily fantasy is a game of finding value; we all want players who are underpriced relative to how they might perform.

The way we go about defining and identifying that value, however, can be different. The most common method is a dollar-per-point system; how many dollars do I need to spend for every point I can expect a player to score? I’d say most daily fantasy players use this system or one similar to it. The dollar-per-point system utilizes a median projection for each player, which I think is ultimately a big problem at worst, and misleading at best.

### What Do We Want Out of Players?

When it comes down to it, what we want out of a lineup—and for the players in it—is to have access to specific types of scores. In tournaments, those scores obviously need to be high. It makes no sense to create a low-variance lineup with little access to a high ceiling if you’re in a tournament. Even if it deflates your average score, you need access to outliers; it’s worthless to score between 140 and 160 points consistently in a GPP. In cash games, the opposite is true. While I think head-to-heads are somewhat about point-maximization—and thus median projections can hold more value—50/50s are about risk-minimization. That doesn’t mean you should minimize risk at all costs, but overall, increasing the floor of your lineup will lead to more success. In those games, you don’t need to hit a home run; if you can consistently score between 140 and 160 points, you can make some money.

### Ceiling, Floor, and Probability

In reality, what we want to know about a player isn’t really his median score, but rather the range of potential outcomes he has in a game. If this game were to be played 10,000 times, how often would he fall at each point total?

Two players who have similar median projections can have drastically different ranges of outcomes. Someone like Tyreek Hill has access to a ridiculously high ceiling—but also a low floor—whereas a player like Travis Kelce has a much more narrow range of outcomes. Really, we’re trying to think about players in terms of probabilities. What’s the probability this player scores X points? How does that fit with my goals for this particular lineup and league? These questions are more valuable than trying to maximize median projections.

### The Effect on GPP Strategy

I’ve talked about these ideas a bit in the past, but today I was really thinking about how this might alter lineup construction in GPPs. Specifically, I was considering high/low vs. balanced techniques, i.e. using primarily stars and scrubs in a lineup versus one filled with second and third-tier players across the board.

In terms of maximizing pure value, I think you can use either approach. Cheap players almost always offer the highest dollar-per-point value because they need to score so few points to “pay off” their salaries, but then you’re “forced” to start some studs if you want to maximize the overall projection of your lineup. Those top-priced players have the most difficult time offering pure dollar-per-point value, so things tend to even out; the dollar-per-point of a high/low lineup often resembles that of a balanced one.

But I think things might change if we get rid of this faulty idea of median projections and replace it with probability-based ceiling projections. The reason I think this is the case is there seems to be really diminishing returns when it comes to upside versus salary. Here’s an example of what I mean, and this is truly an example; these numbers are for demonstration purposes only.

Anecdotally, it seems to me as though there isn’t a massive difference between a player who is $9,000 versus one who is $8,000 in terms of their upside, and then not a huge drop from $8,000 to $7,000, either. But as you drop too far in price, you start to find that the odds of a player reaching a high enough ceiling to be tournament-worthy are dramatically lower.

If we were to represent this by selecting two hypothetical six-man fantasy teams—one with three studs and three scrubs (three at $9,000 and three at $3,000) versus one with all $6,000 players—the latter team would have a 0.129% chance of all reaching X points in this completely made-up scenario, while the former would be at .0000064%—roughly 20,000x worse. This is an extreme example, but even a small non-linear diminishing return would result in a balanced GPP approach being the superior one if the goal is to maximize the probability of each player reaching a certain scoring threshold.

### Thought Experiments

A lot of daily fantasy is data-driven and mathematical. But I also think there’s a lot of room for philosophy—questioning even our most basic beliefs about what we think we know—and you can generate some really meaningful insights just by asking questions and thinking critically about problems.

In this article, I completely solved an important issue and there’s zero room left for debate. I’m just 100% right, so don’t even question anything I just wrote.