March Madness--the NCAA college basketball championship playoffs--is among the most popular sporting events in the US, thanks in part to the wide-ranging contest that has evolved around predicting which teams will progress through the tournament. This year, almost $10.4 million is on the line in office pools or more organized competitions, and more than 40 million Americans will fill out their own versions of the playoff brackets to take part, according to the American Gaming Association. The chances of predicting a perfect bracket, which no one has ever done, are at least 1 in 128 billion and could be as remote as 1 in 9.2 quintillion. Now machine learning is taking a shot. Kaggle, the online platform for predictive modeling and analytics competitions that was acquired by Google parent company Alphabet last year, is hosting a competition for both the NCAA men's and women's tournaments. Kaggle provides a data set with information like tournament seeds going back to the 1984-85 season; final scores of all regular season, conference tournament, and NCAA tournament games since 1984-85; and every Division I men's and women's basketball play-by-play moment since 2009.

In many multiagent environments, a designer has some, but limited control over the game being played. In this paper, we formalize this by considering incompletely specified games, in which some entries of the payoff matrices can be chosen from a specified set. We show that it is NP-hard for the designer to make this choices optimally, even in zero-sum games. In fact, it is already intractable to decide whether a given action is (potentially or necessarily) played in equilibrium. We also consider incompletely specified symmetric games in which all completions are required to be symmetric. Here, hardness holds even in weak tournament games (symmetric zero-sum games whose entries are all -1, 0, or 1) and in tournament games (symmetric zero-sum games whose non-diagonal entries are all -1 or 1). The latter result settles the complexity of the possible and necessary winner problems for a social-choice-theoretic solution concept known as the bipartisan set. We finally give a mixed-integer linear programming formulation for weak tournament games and evaluate it experimentally.