Part I: Intro to Daily Fantasy Sports

My ex played daily fantasy sports professionally. Not recreationally, but as a serious, full-time pursuit. I spent a lot of time watching how lineups were constructed, how models were adjusted day to day, and how results were evaluated across long horizons rather than individual slates. I also spent time around other professionals, where the language was not about teams or narratives, but about distributions, leverage, and long-run expectation.

At the time, I didn’t quite have the mathematical vocabulary to describe what I was seeing. Now it feels natural to return to it and write about daily fantasy sports as what it really is: a constrained stochastic optimization problem with partial information.

This post focuses on introducing DFS. Future posts will dive into the specific math and techniques in basketball, baseball and (American) football.

A daily fantasy slate begins with a finite player set $P=\{p_1,\dots ,p_n\}$. Each player has a salary $s_i$ and a random fantasy score ${\mathrm{Xi}}_{}$. The lineup construction problem is to choose a subset $L \subset P$ that maximizes an objective subject to budget and roster constraints. Even in its simplest form, this resembles a knapsack problem with additional structure.

The key point is that $X_i$ is not a number. It is a random variable. A serious model does not output a projection, but an estimated distribution. In basketball, this distribution is shaped by minutes played, usage rate, pace, efficiency, and opponent context. These factors influence not only the mean $\mathbb{E}[X_i]$, but also the variance and tail behavior (My ex used to say, almost as a mantra, that everything comes down to variance. He was right.)

This distinction matters because different contest types impose different objective functions. In cash games, the goal is to maximize the probability that total lineup score exceeds a threshold. This favors lower variance and tighter distributions. In tournaments, the objective shifts toward maximizing expected value in the upper tail. Lineups that are suboptimal in expectation can still be optimal under a nonlinear payout structure.

Correlation further complicates the picture. Player scores are not independent. Usage is conserved within a team, and minutes are shared. Positive correlations arise from shared game environments and pace. Negative correlations arise from role overlap. Ignoring covariance leads to lineups that are fragile under realistic variance. Incorporating it turns the problem into something closer to portfolio optimization, where the covariance matrix matters as much as individual expectations.

There is also a temporal component. Injury reports, starting lineups, and late news introduce discontinuities. The relevant quantity is not the projection itself, but the ability to update conditional distributions quickly. Bayesian updating, whether explicit or implicit, dominates static modeling approaches. Information velocity becomes a competitive advantage.

At a Bulls game

One subtle aspect that often goes unnoticed is that the entire system is adversarial. Ownership percentages couple players’ decisions. A lineup’s value depends not only on its score distribution, but on how many others share similar structures. This introduces a game-theoretic layer, where leverage and differentiation matter as much as raw performance.

What struck me most while observing this world closely is how little it resembles gambling. It is not about predicting outcomes correctly. It is about constructing systems that survive variance over time. Losing days are expected. Winning is statistical, not emotional.

I’m writing about daily fantasy sports because it sits at an unusually honest interface between mathematics and reality. Models are exposed every night. Assumptions are punished immediately. There is no room for elegance divorced from performance.