The Team Order Problem: Maximizing the Probability of Matching Being Large Enough

2026-05-20

AI Summary

This study addresses the team ordering problem in competitive settings where an opposing team’s fixed lineup and pairwise win probabilities among players are given, and the goal is to determine one’s own player sequence that maximizes the overall probability of winning. The work presents the first polynomial-time approximation scheme (PTAS), which efficiently computes a near-optimal ordering with a guaranteed loss in winning probability of at most ε for any desired precision ε > 0. By integrating techniques from combinatorial optimization, probabilistic analysis, and graph-theoretic matching models, the paper not only resolves the tractability of several special cases but also establishes a theoretical performance bound between the maximum-weight matching solution and the true optimum, thereby revealing inherent limitations in the achievable approximation quality.

Abstract

We consider a matching problem, which is meaningful in team competitions, as well as in information theory, recommender systems, and assignment problems. In the competitions which we study, each competitor in a team order plays a match with the corresponding opposing player. The team that wins more matches wins. We consider a problem where the input is the graph of probabilities that a team 1 player can win against the team 2 player, and the output is the optimal ordering of team 1 players given the fixed ordering of team 2. Our central result is a polynomial-time approximation scheme (PTAS) to compute a matching whose winning probability is at most epsilon less than the winning probability of the optimal matching. We also provide tractability results for several special cases of the problem, as well as an analytical bound on how far the winning probability of a maximum weight matching of the underlying graph is from the best achievable winning probability.

Problem

• team order
• matching
• winning probability
• assignment problem
• optimal ordering

Innovation

• Team Order Problem
• Polynomial-Time Approximation Scheme
• Matching Probability
• Optimal Ordering
• Maximum Weight Matching