Optimal Play of the Dice Game Pig

Abstract

The object of the jeopardy dice game Pig is to be the first player to reach 100 points. Each player\u27s turn consists of repeatedly rolling a die. After each roll, the player is faced with two choices: roll again, or hold (decline to roll again). If the player rolls a 1, the player scores nothing and it becomes the opponent\u27s turn. If the player rolls a number other than 1, the number is added to the player\u27s turn total and the player\u27s turn continues. If the player holds, the turn total, the sum of the rolls during the turn, is added to the player\u27s score, and it becomes the opponent\u27s turn. For such a simple dice game, one might expect a simple optimal strategy, such as in Blackjack (e.g., stand on 17 under certain circumstances, etc.). As we shall see, this simple dice game yields a much more complex and intriguing optimal policy, described here for the first time. The reader should be familiar with basic concepts and notation of probability and linear algebra