Two-Dice Pig is a classic game where you throw a pair of dice over and over and sum up the points you get. When you reach one hundred, you win. Easy? Well, there is one little problem. If one of your dice show a one, all your points are lost and the turn goes over to your opponent who faces the same challenge. There is a way to keep your accumulated points though and that is if you declare that enough is enough and hand over the dice to your opponent. In that case you get to keep your points. They are transfered to your ”bank”. The points in your bank are safe there unless you happen to roll two ones (snake eyes).
Though the game is very simple (or rather because the game is very simple), soon you start wondering about the optimal strategy. For how long will you risk keeping on rolling your dice? Is there a recommended limit? Or is there a recommended number of times you should try to roll the dice before you declare that your satisfied?
The variables you can utilize are the following:
a: Points accumulated during the current round
b: Index number keeping track of how many rolls you’ve made during the current round. If its the first throw b = 1
c: Points In The Bank
d: Opponent Points In The Bank
e: PlayerIndex. If you’re the first player starting out: e = 1. Otherwise: e = 2
In this implementation of the game the second player always has a last chance to try to catch up with the first player even though one hundred is already reached. If you play one thousand games you will be player number one in exactly 500 of the games.
Example: Here’s a legal strategy:
It means that as long as the number of throws are lower than four you will continue to roll the dice. In other words; when you have rolled the dice four times you declare that it’s your opponents turn.
Here’s another legal strategy:
This means that as long as you haven’t reached 12 points during the current round you will continue to roll the dice.
And here’s yet another one:
a<12 && a+c<100
This means that there are TWO conditions that need to be meet with for your rolling to continue.
The accumulated point during the current round must be lower than 12 AND the sum of the accumulated points and your points in your bank must be lower than 100.
The simplest valid strategy would just be
1 == 0
That means that you never continue to roll. You roll the dice only once, each time it’s your turn, but that’s it.
There are two strategies that can be edited on screen. One for the ”Blue program” and one for the ”Red program”. They can compete against each other. There’s also a third pre-programmed strategy belonging to the so called ”Champ program”. That a strategy is one that I’ve coded myself that I’m pretty happy with. The ”Champ program” is, I imagine, pretty hard to beat so, as an ultimate test for your own strategy you can also challenge the champ program. Can you beat the Champ?
Note: The idea of designing the perfect strategy for this game has attracted interest from mathematicians and computer scientists. Here’s a link to a pdf – Optimal Play of the Dice Game Pig – published by the computer science faculty at the Gettysburg College. The list of academic references in the end of that publication is great if you’re interested in delving deeper into the subject.