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We consider the problem of selecting a portfolio of entries of fixedcardinality for contests with top-heavy payoff structures, i.e. most of thewinnings go to the top-ranked entries. This framework is general and can beused to model a variety of problems, such as movie studios selecting movies toproduce, venture capital firms picking start-up companies to invest in, orindividuals selecting lineups for daily fantasy sports contests, which is theexample we focus on here. We model the portfolio selection task as acombinatorial optimization problem with a submodular objective function, whichis given by the probability of at least one entry winning. We then show thatthis probability can be approximated using only pairwise marginal probabilitiesof the entries winning when there is a certain structure on their jointdistribution. We consider a model where the entries are jointly Gaussian randomvariables and present a closed form approximation to the objective function.Building on this, we then consider a scenario where the entries are given bysums of constrained resources and present an integer programming formulation toconstruct the entries. Our formulation uses principles based on our theoreticalanalysis to construct entries: we maximize the expected score of an entrysubject to a lower bound on its variance and an upper bound on its correlationwith previously constructed entries. To demonstrate the effectiveness of ourinteger programming approach, we apply it to daily fantasy sports contests thathave top-heavy payoff structures. We find that our approach performs well inpractice. Using our integer programming approach, we are able to rank in thetop-ten multiple times in hockey and baseball contests with thousands ofcompeting entries. Our approach can easily be extended to other problems withconstrained resources and a top-heavy payoff structure.

Picking Winners in Daily Fantasy Sports Using Integer Programming