Taking the risk out of RISK: Conquer Odds in the Board Game RISK

TL;DR

This paper analyzes the probabilities in the game RISK, generalizing dice odds calculations to various configurations, and identifies the critical army ratio for a better than even chance of conquest.

Contribution

It extends previous dice odds research by considering different dice and side counts, providing a precise threshold for conquest success.

Findings

Attacker needs over 86% of defending armies for >50% chance to conquer.

Odds transition sharply around the 86%+2 threshold.

Generalized dice odds for various configurations.

Abstract

Dice odds in the board game RISK were first investigated by Tan, fixed by Osbourne, and extended by Blatt. We generalized dice odds further, varying the number of sides and the number of dice used in a single battle. We show that the attacker needs two more than 86% of the defending armies to have an over 50% chance to conquer an enemy territory. By normal approximation, we show that the conquer odds transition rapidly from low chance to high chance of conquering around the 86%+2 threshold.