Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semi-restricted Rock, Paper, Scissors

Sam Spiro, Erlang Surya, Ji Zeng

Abstract

Consider the following variant of Rock, Paper, Scissors (RPS) played by two players Rei and Norman. The game consists of $3n$ rounds of RPS, with the twist being that Rei (the restricted player) must use each of Rock, Paper, and Scissors exactly $n$ times during the $3n$ rounds, while Norman is allowed to play normally without any restrictions. Answering a question of Spiro, we show that a certain greedy strategy is the unique optimal strategy for Rei in this game, and that Norman's expected score is $Θ(\sqrt{n})$. Moreover, we study semi-restricted versions of general zero sum games and prove a number of results concerning their optimal strategies and expected scores, which in particular implies our results for semi-restricted RPS.

Semi-restricted Rock, Paper, Scissors

Abstract

Consider the following variant of Rock, Paper, Scissors (RPS) played by two players Rei and Norman. The game consists of rounds of RPS, with the twist being that Rei (the restricted player) must use each of Rock, Paper, and Scissors exactly times during the rounds, while Norman is allowed to play normally without any restrictions. Answering a question of Spiro, we show that a certain greedy strategy is the unique optimal strategy for Rei in this game, and that Norman's expected score is . Moreover, we study semi-restricted versions of general zero sum games and prove a number of results concerning their optimal strategies and expected scores, which in particular implies our results for semi-restricted RPS.
Paper Structure (15 sections, 18 theorems, 106 equations)

This paper contains 15 sections, 18 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Rock, Paper, Scissors
  3. A More General Setting
  4. Organization and Notation
  5. Optimal Strategies
  6. Bounding the Score
  7. General bounds
  8. Spectral bounds
  9. Oblivious Strategies
  10. Open Problems
  11. Optimal Strategies
  12. Improved Bounds
  13. Proof of Theorem \ref{['thm:3path']}
  14. Proof of Theorem \ref{['thm:randomEulerian']}
  15. Restricted Games

Key Result

Theorem 1.1

The greedy strategy is the unique optimal strategy for Rei in semi-restricted RPS. Moreover, if the game consists of $3n$ rounds with both players playing optimally, then Norman's expected score is $\Theta(\sqrt{n})$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Lemma 3.1
  • ...and 31 more