Roger Eastman implemented a Pig dice game with optimal play as
part of his iDice 2.0 dice game compilation for the Macintosh.
Michael Plasmeier implemented a Pig dice game PLAYPIG for TI-83+/84+ graphing calculators.
Justin Boyan and Andrew Moore applied reinforcement learning to a
variation of Pig where a roll of 1 scores one point. In this acyclic
variation, you can't find yourself in the same situation twice. Such
a game can be solved with dynamic programming (computing the best
moves from the end of the game backwards to the beginning). This Pig
variation description is available as HTML,PDF and PostScript
(see reference below).
partial analysis of two dice Pig. - A hold value analysis is
performed assuming one is trying to reach 100 in N turns.
However, the analysis doesn't provide a policy for what N should be.
"In practice (in a game) you as a player have to determine how many
turns you should use to try to get to 100." This analysis is more
complex than Knizia's odds-based analysis. Although there are regions
in our Pig state space where the optimal player is clearly trying to
reach the goal in one or two turns, the true optimal policy is more
complex than "win in N turns" over much of the state space. We
conjecture that the solution for this version of Pig would be
qualitatively similar to our own, and thus more complex than this
is like NRICH's Pig variant but limited to 5 turns with the maximum
score winning. This page presents classroom teaching activities.
Note: We have also solved this variation of Pig. The number of dice
rolls is irrelevant to optimal play or score maximization.
In R.C. Bell's Discovering Dice and Dominoes (ISBN
0-85263-532-X, 1980), a variant is described where double 1's simply
ends the turn with no score. That is double 1's are like rolling a
Kyle Bradshaw's Pigalator applet
features complex four-player tournament rules with doubles doubled as in Frey's variant (but with re-roll required), double 1's eliminating all points and doubling game stakes, and all points eliminated if 100 is reached exactly.
Two Dice, Doubles are Bad: Doubles lose the turn total and end the turn. All other rolls accumulate their sum to the turn total. This variant, "Piggy", appears in "Kids Travel: A Backseat Survival Kit" by the editors of Klutz Press, 1994, p. 11.
Two Pig-Dice:Pass the
Pigs® Milton Bradley / Winning Moves game played with "pig" dice.
Also known as PigMania®
(David Moffat Enterprises, 1977). Rolls are scored depending on
whether pigs land on their feet, sides, backs, etc. in various
Five Dice, 1 is Bad: This pig dice game
from Koplow Games, is not to
be confused with Parker Brother's Pig Dice®. The goal score is
200. One 1 reduces a player's turn total by 10. Two through four 1's loses
the turn total and ends the turn. Five 1's loses the turn total and
score, and ends the turn.
Brutlag, Dan. 1994. Choice and chance in life: The game of "skunk". Mathematics
Teaching in the Middle School 1 (1): 28-33. Describes a pig variant
as a tool for teaching probabilistic concepts at the middle school
level. Unfortunately incorporates the "gambler's fallacy" in
Diagram Visual Information Ltd. 1979. The Official World Encyclopedia of Sports
and Games. London: Paddington Press. Additional source for our Pig
Falk, Ruma, and Maayan Tadmor-Troyanski. 1999. THINK: A game of choice
and chance. Teaching Statistics 21 (1): 24-27. Partial analysis of
Brutlag's SKUNK game. We have performed a more complete analysis,
computing optimal scoring play, and demonstrating that single-decision
odds-based analysis does not yield optimal scoring play.
Frey, Skip. 1975. How to Win at Dice Games. North Hollywood, CA: Wilshire
Book Co. 1997. Reprint. Source for a 2-dice Pig variant
Knizia. Dice Games Properly Explained. Elliot Right-Way Books,
Brighton Road, Lower Kingswood, Tadworth, Surrey, KT20 6TD U.K., 1999.
The rules for Pig and many other dice games can be found here.
Knizia makes an odds-based argument for Pig tactics which I call the
"Hold at 20" policy. He then goes on to point out that there are
situations where one should deviate from this policy. Knizia made me
curious about what optimal Pig play would look like.
M. Mitchell. Machine Learning. McGraw-Hill, New York,
1997. A good advanced undergraduate or graduate text on the machine
learning in the field of artificial intelligence. Chapter 13 gives an
overview of Reinforcement Learning and covers convergence properties
relevant to value iteration.
Parlett, David. 1991. A History of Card Games. New York: Oxford University
Press. Excellent reference for card game history.
Parlett, David. 1999. The Oxford History of Board Games. New York: Oxford University
Press. Excellent reference for board game history.