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Different Forms of Imbalance in Strongly Playable Discrete Games II: Multi-Player RPS Games

Itai Maimon

Abstract

Classic Rock-Paper-Scissors, RPS, has seen many variants and generalizations in the past several years. In the previous paper, we defined playability and balance for games. We used these definitions to show that different forms of imbalance agree on the most balanced and least balanced form of playable two-player n-object RPS games, referred to as (2,n)-RPS. We reintroduce these definitions here and show that, given a conjecture, the generalization of this game for m<50 players is a strongly playable RPS game. We also show that this game maximizes these forms of imbalance in the limit as the number of players goes to infinity.

Different Forms of Imbalance in Strongly Playable Discrete Games II: Multi-Player RPS Games

Abstract

Classic Rock-Paper-Scissors, RPS, has seen many variants and generalizations in the past several years. In the previous paper, we defined playability and balance for games. We used these definitions to show that different forms of imbalance agree on the most balanced and least balanced form of playable two-player n-object RPS games, referred to as (2,n)-RPS. We reintroduce these definitions here and show that, given a conjecture, the generalization of this game for m<50 players is a strongly playable RPS game. We also show that this game maximizes these forms of imbalance in the limit as the number of players goes to infinity.

Paper Structure

This paper contains 6 sections, 8 theorems, 67 equations, 1 table.

Table of Contents

  1. Introduction: Constructions of Imbalance, Playability and Blow-ups
  2. Playablity and Imbalance
  3. Constructions of Multiplayer Blow-Ups
  4. An Imbalanced $(m,3)$-RPS Construction
  5. An Important Theorem for the $(m,2k+1)$-$RPS$
  6. The Imbalanced $(m,2k+1)$-$RPS$ is Strongly Playable

Key Result

Lemma 2.2

The imbalanced $(m,3)$-$RPS$ is strongly playable.

Theorems & Definitions (32)

  • Definition 1.1: Alternative form of $(m,n)$-$RPS$
  • Definition 1.2: $k$-Playable
  • Definition 1.3: Uniform Imbalance of Variance
  • Definition 1.4: Uniform Imbalance of Entropy
  • Definition 1.5: α-Thiel Entropy Imbalance
  • Definition 1.6: Minimal Nash Equilibria Entropy Imbalance
  • Definition 1.7: Maximal Nash Equilibria Ties Imbalance
  • Definition 1.8: Multiplayer Schur-Uniform Class for $RPS$ games
  • Definition 1.9: Multiplayer Schur-Distributional Class for $RPS$ games
  • Definition 1.10: Symmetric Blow-up
  • ...and 22 more