Pairing Things Off: Counting Stable Matchings, Finding Your Ride, and Designing Tournaments

Abstract

This thesis discusses three problems around the common theme of pairing off agents. While the techniques vary, in each problem we observe that careful application of classical and well-understood techniques can still lead to progress. In Chapter Two, we study the classical combinatorial problem of stable matchings. Stable matchings were introduced in 1962 in a seminal paper by Gale and Shapley. In this chapter, we provide a new upper bound on f(n), the maximum number of stable matchings that an instance with n men and n women can have. The best lower bound prior to our work was approximately 2.28^n, and the best upper bound was 2^{ n log n - O(n) }. We show that for all n, f(n) ≤ c^n for some universal constant c. Our bound matches the lower bound up to the base of the exponent. In Chapter Three, we discuss online matching. The Min-Cost Perfect Matching with Delays Problem (MPMD) has been the subject of a recent flurry of activity in algorithm design. In the problem, a series of requests appear over time in a metric space, with the locations and timing determined by an adversary. The algorithm designer is charged with pairing off all the requests, attempting to minimize the sum of the amount of time the requests have to wait (between appearance and being matched) and the distances between the matched requests. We discuss our initial work on the problem, which shows that if the requests come from an unknown probability distribution (rather than from an adversary) a simple algorithm (that we call "ball growing") achieves excellent performance. We also adapt the core idea of the algorithm to the adversarial case, and show that a slight modification of the same simple idea suffices to match the best-known algorithms for MPMD. In Chapter Four, we discuss a different type of matching -- creating matchups in sports tournaments. Our work is inspired by an incident in the 2012 Olympic Badminton tournament where both teams playing in a match were incentivized to lose that match (and attempted to do so). In 2016, the tournament was redesigned, with the stated goal of eliminating misaligned incentives; we show the redesign failed in this goal. We then describe a minimally-manipulable tournament rule which could be reasonably implemented, while maintaining many of the subtler features of the current tournament that a designer would want.