Optimal Play of the Dice Game Pig

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

Pigtail: A Pig Addendum

The object of the jeopardy dice game Pig is to be the first player to reach 100 points. Each turn, a player repeatedly rolls a die until either a 1 is rolled or the player holds and scores the sum of the rolls (i.e., the turn total). At any time during a player’s turn, the player is faced with two choices: roll or hold. If the player rolls a 1, the player scores nothing and it becomes the opponent’s turn. If the player rolls a number other than 1, the number is added to the player’s turn total and the player’s turn continues. If the player instead chooses to hold, the turn total is added to the player’s score and it becomes the opponent’s turn. In our original article [Neller and Presser 2004], we described a means to compute optimal play for Pig. Since that time, we have also solved a number of Pig variants. In this addendum, we review the optimality equations for Pig, show how these equations change for several Pig variants, and show how the resulting optimal policies change accordingly. [excerpt

Practical Play of the Dice Game Pig

The object of the jeopardy dice game Pig is to be the first player to reach 100 points. Each turn, a player repeatedly rolls a die until either a 1 is rolled or the player holds and scores the sum of the rolls (i.e., the turn total). At any time during a player’s turn, the player is faced with two choices: roll or hold. If the player rolls a 1, the player scores nothing and it becomes the opponent’s turn. If the player rolls a number other than 1, the number is added to the player’s turn total and the player’s turn continues. If the player instead chooses to hold, the turn total is added to the player’s score and it becomes the opponent’s turn. In our original article [Neller and Presser 2004], we described a means to compute optimal play for Pig. However, optimal play is surprisingly complex and beyond human potential to memorize and apply. In this paper, we mathematically explore a more subjective question: What is the simplest human-playable policy that most closely approximates optimal play? While one cannot enumerate and search the space of all possible simple policies for Pig play, our exploration will present interesting insights and yield a surprisingly good policy that one can play by memorizing only three integers and using simple mental arithmetic. [excerpt

Pedagogical Possibilities for the Dice Game Pig

Simple examples are teaching treasures. Finding a concise, effective illustration is like finding a precious gem. When such an example is fun and intriguing, it is educational gold. In this paper, we share the jeopardy dice game of Pig, which has extremely simple rules, engaging play, and a complex optimal policy. We describe its historical uses in mathematics, and share ways in which we have used the game to teach basic concepts in CS1, and intermediate concepts in introductory artificial intelligence, networking, and scientific visualization courses. We also describe the rich challenges Pig offers for undergraduate research in machine learning