Greedy Dice has one decision in it. You have rolled a few times, there are some points on the table, and you can either bank them or roll again. Roll a one and they are gone.
The game is better known as Pig, and people have been arguing about when to stop for as long as it has been played. There is an answer. It is simple enough to work out on the back of an envelope, and it explains a lot about risk in general.
What one more roll is worth
Say you have built up some points this turn. Call the amount T. You are deciding whether to roll once more.
Five times in six, you roll a two, three, four, five or six. Those average four. So five times in six, you gain four points on average.
One time in six, you roll a one and lose all T points.
Put those together. The average gain from rolling is five sixths of four, which is about 3.3 points. The average loss is one sixth of T.
Rolling is worth it when the gain is bigger than the loss. That is, when one sixth of T is less than 3.3. In other words, when T is less than twenty.
The rule
So the classic rule of thumb is this.
- Below twenty: roll.
- Twenty or more: hold.
At exactly twenty, the two sides balance, and it does not matter.
How often will you get there?
Not as often as you would like. To reach twenty you usually need about five rolls, and every roll has to avoid a one.
- Surviving one roll: about 83 in 100.
- Two rolls: about 69 in 100.
- Three rolls: about 58 in 100.
- Four rolls: about 48 in 100.
- Five rolls: about 40 in 100.
So a player holding at twenty comes away with nothing on more than half their turns. That feels terrible. It is still the right way to play, because the turns that work are worth so much.
This is the hard part of the game, and of risk generally. A good decision can have a bad result most of the time.
Three reasons to break the rule
The rule treats every turn as if it were on its own. A real game has a score, and the score changes things.
When you are far behind. If your opponent is on forty and you are on ten, banking a tidy twenty leaves you on thirty and still losing. You need more than safe play can give you. Keep rolling past twenty.
When you are ahead. Banked points cannot be lost. If you lead comfortably, hold earlier, at fifteen or so, and let your opponent take the risks.
Near the finish. Our game is first to fifty. If you have thirty-eight banked and twelve on the table, you have won: hold. If your opponent is on forty-six and you are on twenty, holding at twenty is pointless, because they are likely to finish on their next turn. Roll until you win or bust.
It is not quite the best possible play
Two researchers, Todd Neller and Clifton Presser, worked out the truly optimal strategy for Pig by computer in 2004. It depends on both players' scores and the turn total, and it looks like a strange curved surface when drawn.
Their finding was that "hold at twenty" is good but not perfect. The optimal player wins a little more often. The differences are exactly the ones above: be bolder when behind, more careful when ahead, and go for the win when the end is close.
The mistake everyone makes
After rolling a one, people feel they are owed a good run. After several safe rolls, they feel a one is "due".
Neither is true. The dice has no memory. The chance of a one is one in six on every single roll, whatever came before.
What changes is how much you have on the table. That is the only thing the decision should depend on, along with the score.
Why this is a good game for two
Most dice games are luck with a thin coat of choice. This one has a real decision in it, small enough to think about and rich enough to disagree over.
It is also revealing. Play a few matches and you will learn whether your opponent is cautious or reckless, and whether they change when they are losing. People are remarkably consistent.
It works well between an adult and a child, too. The sums are small, and "is it worth the risk?" is a question worth getting used to.
Try the experiment
Play ten games where one of you always holds at twenty and the other plays by instinct. Keep the tally on the scoreboard at the top of the page. Ten games is not many, and luck will blur the result, but over a longer run the rule tends to come out ahead.
As HTML5 games for two go, it takes four minutes and leaves you knowing a little more about chance than you did before.