[Paper Review] Playing Tennis without Envy

Josué Ortega|arXiv (Cornell University)|Aug 25, 2016

Game Theory and Voting Systems3 references3 citations

TL;DR

This paper investigates the fairness and efficiency of algorithms for assigning double tennis games among friends based on availability, showing that no assignment can simultaneously be efficient and strongly envy-free when players have varying availability. It proves that for any sport requiring q ≥ 2 players per game, there exist problem instances where no solution satisfies both efficiency and strong envy-freeness, challenging the fairness of commonly used assignment heuristics.

ABSTRACT

A group of friends organize their tennis games by submitting each their availability over the weekdays. They want to obtain an assignment such that: each game must be a double tennis match, i.e. requires four people, and nobody plays in a day he is unavailable. Can we construct assignments that will always produce efficient, fair, and envy-free outcomes? The answer is no, and extends to any sport that requires any group size.

Motivation & Objective

• To analyze whether efficient and fair tennis game assignments can be constructed when players have different availability schedules.
• To define and formalize strong envy-freeness in the context of group sports scheduling, where more flexible players should not receive fewer games.
• To investigate whether an assignment can be both efficient (maximizing total games) and fair (leximin-optimized) while avoiding envy from more flexible players.
• To generalize the impossibility result beyond tennis to any q-player sport (q ≥ 2).
• To challenge the assumption that commonly used heuristic algorithms (e.g., maximizing games per player) produce truly fair outcomes.

Proposed method

• Formalizes the tennis scheduling problem as a binary matrix A where a_ik = 1 if player i is available on day k.
• Reduces the problem to an irreducible form by removing days with fewer than 4 available players and players with zero availability across all days.
• Defines a solution as a binary matrix X satisfying: (1) x_ik = 0 if a_ik = 0 (availability constraint), and (2) sum of x_ik divisible by 4 per day (group size constraint).
• Introduces three solution properties: efficiency (Pareto optimality), fairness (leximin-optimality over game counts), and strong envy-freeness (more available players get at least as many games as less available ones).
• Constructs a counterexample with 11 players and 5 days, where 4 highly available players (a–d) must play 2 games each, and 7 less available players (e–k) must share 5 games across 3 days, making at least one of them receive only 1 game despite 3 availabilities.
• Proves via contradiction that in such configurations, at least one less-available player must be envious of a more available player, violating strong envy-freeness.

Experimental results

Research questions

• RQ1 Can an assignment be both efficient and strongly envy-free in a double tennis scheduling problem with heterogeneous player availability?
• RQ2 Is it possible to design a scheduling algorithm that ensures more flexible players are never disadvantaged compared to less flexible ones, while maximizing total games?
• RQ3 Does the impossibility of strong envy-freeness extend beyond tennis to other sports requiring fixed group sizes (q ≥ 2)?
• RQ4 How do common heuristic approaches—such as maximizing games per player—compare to fairness criteria like leximin optimality in real-world scheduling scenarios?
• RQ5 What are the structural constraints in group scheduling problems that prevent the simultaneous achievement of efficiency and strong envy-freeness?

Key findings

• There exist tennis scheduling problems for which no solution is both efficient and strongly envy-free, demonstrating a fundamental incompatibility between efficiency and strong fairness.
• In the constructed counterexample (Table 2), four players with 2 available days each must play 2 games, while seven players with 3 available days each must share 5 games across 3 days, forcing at least one to play only 1 game.
• The player with 3 available days who plays only 1 game is envious of players with only 2 available days who play 2 games, violating strong envy-freeness.
• The impossibility result generalizes to any q-player sport (q ≥ 2), meaning that no such assignment can be both efficient and strongly envy-free in all cases.
• Even though the algorithm proposed by Maher (2016) produces efficient and fair outcomes in practice, it fails to ensure strong envy-freeness, as demonstrated by George StC receiving only one game despite being available on 4 days.
• The study shows that players may have rational incentives to misreport availability to gain more games, undermining trust in the fairness of the system.