Picking Winners Using Integer Programming David Scott Hunter Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA 02139,
[email protected] Juan Pablo Vielma Department of Operations Research, Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02139,
[email protected] Tauhid Zaman Department of Operations Management, Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02139,
[email protected] We consider the problem of selecting a portfolio of entries of fixed cardinality for a winner take all contest such that the probability of at least one entry winning is maximized. This framework is very general and can be used to model a variety of problems, such as movie studios selecting movies to produce, drug companies choosing drugs to develop, or venture capital firms picking start-up companies in which to invest. We model this as a combinatorial optimization problem with a submodular objective function, which is the probability of winning. We then show that the objective function 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. We then consider a model where the entries are given by sums of constrained resources and present a greedy integer programming formulation to construct the entries. Our formulation uses three principles to construct entries: maximize the expected score of an entry, lower bound its variance, and upper bound its correlation with previously constructed entries. To demonstrate the effectiveness of our greedy integer programming formulation, we apply it to daily fantasy sports contests that have top heavy payoff structures (i.e.