[Paper Review] Picking Winners in Daily Fantasy Sports Using Integer Programming

David Scott Hunter, Juan Pablo Vielma+1 more|arXiv (Cornell University)|Apr 6, 2016

Sports Analytics and PerformanceEconomics, Econometrics and Finance31 references3 citations

TL;DR

This paper proposes an integer programming framework to optimize lineup selection in daily fantasy sports, where top-heavy payoffs reward only the highest-performing entries. By modeling player entries as correlated Gaussian random variables and using a submodular objective to maximize the probability of at least one winning entry, the method constructs diverse, high-expected-performance lineups under budget and position constraints, achieving top-ten finishes in real contests with thousands of entries.

ABSTRACT

We consider the problem of selecting a portfolio of entries of fixed cardinality for contests with top-heavy payoff structures, i.e. most of the winnings go to the top-ranked entries. This framework is general and can be used to model a variety of problems, such as movie studios selecting movies to produce, venture capital firms picking start-up companies to invest in, or individuals selecting lineups for daily fantasy sports contests, which is the example we focus on here. We model the portfolio selection task as a combinatorial optimization problem with a submodular objective function, which is given by the probability of at least one entry winning. We then show that this probability can be approximated using only pairwise marginal probabilities of the entries winning when there is a certain structure on their joint distribution. We consider a model where the entries are jointly Gaussian random variables and present a closed form approximation to the objective function. Building on this, we then consider a scenario where the entries are given by sums of constrained resources and present an integer programming formulation to construct the entries. Our formulation uses principles based on our theoretical analysis to construct entries: we maximize the expected score of an entry subject to a lower bound on its variance and an upper bound on its correlation with previously constructed entries. To demonstrate the effectiveness of our integer programming approach, we apply it to daily fantasy sports contests that have top-heavy payoff structures. We find that our approach performs well in practice. Using our integer programming approach, we are able to rank in the top-ten multiple times in hockey and baseball contests with thousands of competing entries. Our approach can easily be extended to other problems with constrained resources and a top-heavy payoff structure.

Motivation & Objective

• To address the challenge of selecting a portfolio of entries in contests with top-heavy payoff structures, where only top performers win substantial rewards.
• To develop a method for constructing optimal entries from constrained resources, such as player positions and budgets, rather than selecting from a fixed set.
• To maximize the probability that at least one entry wins by balancing individual winning probabilities and inter-entry correlations.
• To provide a scalable, mathematically grounded approach applicable to sports, venture capital, and R&D portfolio selection.

Proposed method

• Model the performance of each lineup as a Gaussian random variable with mean and variance derived from player projections.
• Use a submodular objective function approximated via pairwise marginal probabilities to estimate the probability that at least one entry wins.
• Formulate an integer program that maximizes expected score subject to lower bounds on variance and upper bounds on correlation with previously constructed entries.
• Incorporate constraints such as position limits (e.g., one goalie in hockey) and line stacking via binary variables and linearized quadratic constraints.
• Strengthen constraints like goalie and line stacking using valid inequalities to improve solution quality, though with potential trade-offs in solve time.
• Apply the model to real daily fantasy sports data, generating multiple lineups that are diversified and high-performing.

Experimental results

Research questions

• RQ1 How can one optimally construct a set of fantasy sports lineups under hard constraints (e.g., position limits) to maximize the probability of winning?
• RQ2 What is the impact of correlation between entries on the overall probability of winning, and how can it be modeled efficiently?
• RQ3 Can a closed-form approximation of the winning probability be derived under a joint Gaussian assumption for player performances?
• RQ4 How does the integer programming approach compare to naive or greedy selection strategies in real-world contests with thousands of competitors?
• RQ5 To what extent can the framework be generalized to other top-heavy payoff problems, such as venture capital or drug development?

Key findings

• The proposed integer programming approach achieved top-ten finishes in multiple daily fantasy sports contests across hockey and baseball, despite thousands of competing entries.
• The method successfully balanced high expected performance with low correlation among entries, increasing the probability that at least one would win.
• The use of a submodular objective function with pairwise probability approximations enabled efficient and effective optimization under Gaussian assumptions.
• Strengthening constraints improved solution quality but increased solve times, indicating a trade-off between formulation strength and computational efficiency.
• The framework is generalizable to other constrained resource allocation problems with top-heavy rewards, such as venture capital or R&D portfolio selection.
• Empirical results demonstrate that diversification through correlation control significantly improves winning probability compared to high-performing but correlated entries.