A New Property and a Faster Algorithm for Baseball Elimination

Abstract

Abstract. In the baseball elimination problem, there is a league consisting of n teams. At some point during the season, team i has w i wins and g ij games left to play against team j. A team is eliminated if it cannot possibly finish the season in first place or tied for first place. The goal is to determine exactly which teams are eliminated. The problem is not as easy as many sports writers would have you believe, in part because the answer depends not only on the number of games won and left to play but also on the schedule of remaining games. In the 1960's, Schwartz showed how to determine whether one particular team is eliminated using a maximum flow computation. This paper indicates that the problem is not as difficult as many mathematicians would have you believe. For each team i, let g i denote the number of games remaining. We prove that there exists a value W * such that team i is eliminated if and only if w i + g i < W * . Using this surprising fact, we can determine all eliminated teams in time proportional to a single maximum flow computation in a graph with n nodes; this improves upon the previous best known complexity bound by a factor of n.

A hybrid constraint programming and enumeration approach for solving NHL playoff qualification and elimination problems

“…The problem of determining when a sports team has mathematically clinched a playoff spot has been well studied for several sports, including baseball (Schwartz, 1966;Robinson, 1991;Wayne, 2001;Adler et al, 2002) and soccer (Ribeiro & Urrutia, 2005). The problem is known as a winner determination problem.…”

“…However, their model could not be solved when secondary and tertiary tie-breaking rules were included. Wayne (2001) introduces the concept of a lower bound that could be used to determine whether or not a team was eliminated from the playoffs. Gusfield & Martel (2002) show how this idea can be extended to include a single wild card team.…”

An application of integer programming to playoff elimination in football championships

“…Adler et al (2002) showed how integer programming can be used to compute the GQS and the PQS in the case of the MLB. Wayne (2001) showed that there exists a number of points such that every team is eliminated from the quest for the first place in the MLB if and only if it cannot reach this threshold. Gusfield and Martel (2002) generalized Wayne's result, showing that this threshold exists for every tournament with certain characteristics, including football leagues following the {(3, 0), (1, 1)} rule.…”

Studying playoff qualification in motorsports via mixed-integer programming techniques

“…Different approaches for determining the minimum number of points guaranteeing playoff qualification have been previously applied for soccer [3], hockey [4][5][6][7], and baseball [8]. These sports feature matches between pairs of teams according to a prespecified schedule.…”