Your example of the 1 in a million chance at a trillion points is interesting. It says that you shouldn’t pursue options with positive scoring expectation if the probability of the payoff event is so small that it’s unlikely to occur within the scope of a game bounded by turns or points. By similar argument, you can (should) pursue options with negative scoring expectation if the probability of the loss event is so small that it’s unlikely to occur within the scope of the game. This starts to sound a little more like a possible Farkle situation: even if you have a very large number of points accumulated, maybe you should go ahead and roll 6 dice because the chance of a 6-die farkle over the course of a game is small.

So when you’re dealing with extremely improbable events relative to the length of a game, the strategy that (on average) lets you reach your goal quickly can deviate from the strategy that gives you a high expected score.

But what about choices associated with events that are NOT improbable relative to the game length? Do fast strategies and high expected point strategies faced with these choices tend to make the same decisions?

Clearly if the game is long enough (enough turns, or the target score is high enough) then the fastest strategy degenerates into highest expected score strategy. Even the option to take a 1 in a million shot at a trillion points becomes attractive to the fast strategy if the game doesn’t end till you reach a quadrillion points.

But when you’re dealing with only 10 turns or a smallish 10,000 point goal, it’s not so clear to me when the fast strategy will agree with the high expected point strategy.

My thinking is muddled. I should ponder this some more.

For instance, what is the threshold score you would require yourself to have in order to bank with three dice left. Then what is it for four, five, and six. Obviously it should increase with the amount of dice you have for the next throw.

I imagine you could run a simulation setting rules for each of these situations (3-6 dice thresholds), possibly always rerolling 6 dice no matter what.

I follow a similar strategy and have questioned that ‘threshold’. I ran with 1000 for a while, meaning I didn’t ever take anything less than a thousand. I set a couple new high scores, but had a bunch of lousy games as expected. Higher aggression leads to lower average, but higher upside.

So I wrote a simulation. Full on Farkle game. Then wrote in my ‘system’ for when to go and when to hold. I am using the Facebook scoring system.

Once I worked out the bugs, I started running many games – a couple thousand games at the following thresholds: 300, 500, 750, 1000, 1500.

Here’s what I learned. If I simulate only a couple hundred games. The above simple theory holds. Regardless of the system, the threshold sets your aggression to a point. So over a couple hundred games, I wouldn’t receive any big games, maybe 8k or 9k for 300 and 500 thresholds.

And the averages were in the 4k’s.

The higher the threshold, the lower the average, but there were some big hitters in there. 10, 11, 12k. But some diminishing returns on 1500.

I then figured, it was obviously, being that aggressive will only pay off after MORE games. But it’ll pay off BIG.

Not quite, and it made sense afterwards. It did pay off big, but like 12k. hmm. And the average sucked. So there seemed to be diminishing returns with only 10 rounds. I then noticed that at 2000+ games, the MAX on the 300 and 500 were also way up there.

It started to click. As mentioned above, to get to the really high scores you aren’t keeping 300 or even 1000/round. You need BIG scores. So the higher threshold ups your chance to sticking around long enough in a round to get on a lucky streak. But once on the streak it doesn’t matter what your threshold was. If the basis is only 300, you just won’t stick around long enough, but the lucky streak may hit from the first roll and again, the threshold is irrelevant.

Interesting to me.

So there’s a balance in there. Higher threshold gives you a chance of getting to a higher max score more readily. But it doesn’t mean you will get any higher than with a lower threshold. And it’ll be frustrating while you do it. You have to have confidence and keep at it.

Thoughts?

p.s. the results are on another machine or I’d post some of the numbers. Perhaps if there is interest I’ll still do it.

Knowing what score I am trying to beat definitely changes my strategy. You have to take risks to score high, and at the same time cash in your winnings.

A tale to tell: Yesterday I scored my all time high of 12000+, beating my friend’s top score by at least 2000 points. Within 8 hours he had me beaten again with a score of 14000+! Sometimes life just is not fair.

But I can answer the question you asked. If you roll n dice, the probability of being able to score them all and get a new batch of dice is given below. (Picky point: this calculation uses the facebook convention where three pairs is necessarily three different pairs, not a four of a kind and a pair.)

n=1: 1/3 (33.33%)
n=2: 1/9 (11.11%)
n=3: 1/18 (5.56%)
n=4: 13/324 (4.01%)
n=5: 59/1944 (3.03%)
n=6: 101/1296 (7.79%)

The strategy I have taken is to play for the N-of-a-kind. That is, to remove the least number of dice possible, in order to maximize the odds of the big points. So, if I roll a 1-1-5-5-3-4, I’ll pull a 1 out and roll 5 dice. I throw out 2-2-2 and sometimes 3-3-3 unless I don’t have an option.

The probabilities I’m looking for (and I’m too lazy to compute right now) are the odds of using all of the dice. IOW, if I roll N dice, what are the odds of all N of them being useful at once and I get a full cup of dice to shoot again.

The other part of this is, what is the cutoff score for rerolling N dice? What are you willing to risk to roll 1 die? I’ve farkled 2700 points off of 6 dice, but I never would have gotten that far had I not been willing to throw 2 dice. What score are you willing to risk for a 33% chance of getting at least 500, more probably 750, and possibly 2000 if you manage a straight on your next roll. 450? 300? 900?

What I have done is cooked up a spreadsheet that tracks my farkle scores. I track the min/med/max, +/-2 sigma. As for your friends’ 13950, look at my distibution, and you can see I have 3 10k games that are all on a long tail. Those games were gifts. In one, I rolled 2 straights in a row, another I rolled 3 pair, 5×1+5, and then 4×6. My 12450 I farkled twice, the 12k not once. Few games are like that. My 10500 game was the one I farkled 2700 points. I think its mostly luck but you need to have a strategy that takes advantage and plays for lucky streaks, with a conservative backup. And play a lot of games

Here is the spreadsheet.
http://spreadsheets.google.com/ccc?key=0AiCaQ3LQ24c2dFJyQTJETWNGeDV0d184SG5mUThjaHc&hl=en