Different Forms of Imbalance in Strongly Playable Discrete Games I: Two-Player RPS Games
Itai Maimon
TL;DR
The paper investigates how imbalance and playability can be rigorously defined and maximized within discrete two-player games, focusing on $n$-RPS on unlabeled $n$-tournaments. It develops multiple imbalance metrics (including $UI_v$, $UI_e$, $N_e$, and $N_t$) and organizes them via Schur majorization classes, proving that a specific $(2n+1)$-$RPS$ (and its countably infinite analogue) is simultaneously maximally imbalanced across these metrics among playable games. The authors establish constructive least-balanced playable $RPS$ designs, proving uniqueness under several imbalance criteria, and provide kernel-based arguments alongside majorization results. They further illustrate the relevance of imbalanced playable games through ecological dynamics and competitive card-game narratives, and introduce a blow-up operation to enable scalable, modular construction of larger imbalanced games, setting the stage for multiplayer extensions in follow-up work.
Abstract
We construct several definitions of imbalance and playability, both of which are related to the existence of dominated strategies. Specifically, a maximally balanced game and a playable game cannot have dominated strategies for any player. In this context, imbalance acts as a measure of inequality in strategy, similar to measures of inequality in wealth or population dynamics. Conversely, playability is a slight strengthening of the condition that a game has no dominated strategies. It is more accurately aligned with the intuition that all strategies should see play. We show that these balance definitions are natural by exhibiting a (2n+1)-RPS that maximizes all proposed imbalance definitions among playable RPS games. We demonstrate here that this form of imbalance aligns with the prevailing notion that different definitions of inequality for economic and game-theoretic distributions must agree on both the maximal and minimal cases. In the sequel paper, we utilize these definitions for multiplayer games to demonstrate that a generalization of this imbalanced RPS is at least nearly maximally imbalanced while remaining playable for under 50 players.