Optimal Defensive Strategies in One-Dimensional RISK

RISK is a popular board game invented by Albert Lamorisse and released in 1957. The board depicts a stylized political world map divided into territories, each occupied by one or more of a player’s army units, which we will refer to simply as armies. The bulk of play consists of a turn-based series of attacks between player armies occupying adjacent territories in an effort to occupy the entire world. In the late game, it is common for one player to attempt to eliminate another player along a chain of territories. In this note, we consider the problem of how a defensive player should distribute his armies to maximize the probability of survival. In particular, we will consider a onedimensional version of the game, which takes place on a chain of m + 1 consecutive territories, as depicted in FIGURE 1. We now describe the rules of our version of the game. Experienced RISK players will note that this is a significantly simplified version of the game, but we believe that we have captured most of the spirit of the original game. At the end of the paper, we will discuss some ways in which the differences in the actual game of RISK might affect our proposed strategies.