I modified my Zilch strategy generation software to model the scoring rules for the Super Farkle game available at Facebook. I wasn’t previously a Facebook user so I created my account just to try out my strategy. Over several days, I played about 180 games of Farkle and was winning about 55% of the time. But I’m not sure if this means much for a few reasons.
First, almost everyone I played, played very well. I guess this makes sense since in Super Farkle you play for chips; and if you don’t play reasonably well, it will be very difficult to win enough chips to play at the higher stakes tables. The people I played against rarely made strategy errors that cost more than a few points. One notable exception was the common mistake of taking two-ones on an opening roll instead of just one-one. This play returns 57 fewer points for your expected score and it occurs with enough frequency to be significant in a typical game. But in general, I was impressed with how closely people played to the strategy that maximizes expected scores — especially at low turn score states. So if my strategy offered any advantage at all, it was probably very slight and I doubt 180 games was enough to make a clear differentiation.
Second, in Super Farkle whoever forms the table rolls first. The average score for a well-played Super Farkle turn is just under 550 points. One can argue the disadvantage to the player going second is half that or about 275 points. (Why don’t they just roll to see who goes first?) To make a fair test of the strategy, I should have played half my games by forming a new table, and played the other half by joining an existing table, but my competitive nature just wouldn’t allow me to concede 275 points to my opponent. So instead I spoiled my own test by always forming my own table. I suspect that the advantage of going first may have overshadowed any advantage my strategy was offering over the high quality play of my opponents.
Finally, and perhaps most importantly, there’s almost certainly something wrong with the Super Farkle dice. I detailed why I believe this to be so on the Farkle review page at Facebook. Here’s the text from that review:
This game is quite nice; but there is a serious problem. The probability of rolling a 6-die FARKLE is exactly 1 chance in 43.2. You can find this calculation all over the web. Here’s one professor at Michigan Ann Arbor that shows the calculation: http://notaboutapples.wordpress.com/2009/07/27/multinomial-coefficients-and-farkle/
Apparently I’ve played about 180 games of Farkle, but I’ve never once thrown a 6 die Farkle. If you assume a typical game has 15 turns, then that’s 15 x 180 = 2700 6-die rolls — and that’s not even considering hot-dice rolls. The probability of not throwing even one 6 die farkle in that many rolls is exactly:
(1-(1/43.2))^2700 = .000 000 000 000 000 000 000 000 000 344
If I’m counting my zeros right, that’s less than one chance in an octillion. Yes, octillion is a real number — a very very big number. So I suggest there is something wrong with the dice. Can I be sure there’s something wrong with the dice? Of course not, but I can say this. According to wikipedia, the visible universe is about 92 billion light years across. And 1 light year is about 6 trillion miles. And there are 5280 feet in a mile. If you lined people up one foot apart (you’d have to use skinny people) across our entire visible universe; and then sat them all down in front of their own laptop playing farkle; and had them all roll 6 dice over-and-over only stopping when they had their first 6-die farkle; then you’d expect about ONE of them (yes just one) to go as far as 2700 rolls without farkling. I suppose I could be that one person….uhmmm…yeah…right.
Maybe some manager made a marketing decision that 6-die farkles just annoyed people too much and the developers were simply asked to reroll the dice one time when a 6-die farkle showed up. Or maybe they are just using a really bad random number generator for their dice rolling engine. Or maybe there’s something more insidious going on. But something is surely amiss.
Interestingly, shortly after I posted this review, I was mysteriously logged out of Facebook and subsequent login attempts were denied. Coincidence? In any case, my foray into Super Farkle play is ended. I played enough games to at least see that the strategy was doing very well — and was highly consistent with the play of seasoned Farkle addicts veterans.
Hi,
I was exploring your blog (it seems we have common interests) and couldn’t help a big smile after reading that review you posted on the Farkle review page at Facebook. Very amusing.
Cheers
I came across your blog because I was myself working on the probabilities involved in Farkle and wanted to see the existing literature. Looks like you did a pretty complete study already! I too wondered why I had never encountered a six-die Farkle and I suspect that they had the developers re-roll as many times as necessary to avoid a Farkle.
I haven’t finished reading your blog but the thing I’m struggling with now is how to model the fact that your banked points go away on a Farkle when calculating the expected value of a roll.
Hi Don,
Equation (3) from my blog entry on zilch is my workhorse for solving the problem. It represents not the expected score for the entire turn, but rather the expected change in score for the remainder of the turn given that you already have accumulated S points for the turn and are facing a roll with N dice. The term y is the loss to your banked score if you zilch. If you are playing a turn where you’ve already zilched twice you set y to 500 and the term that models a negative expected score change due to a zilch is simply -pN (S + 500), where pN is the probability of zilching when you roll N dice.
But I also use that same term y when modeling turns where you have zero or one consecutive zilches as a mechanism for examining the tradeoffs between expected score and zilch probability. Although there is no immediate penalty for zilching on those turns, there is a penalty to your expected score on future turns (since a zilch on those turns brings you one step closer to having to play a turn where you do face a penalty to your banked score for zilching). By sacrificing some expected gains on the current turn you can more strongly avoid getting into a turn where you have 2 consecutive zilches and improve your long-term averages. This is modeled by equation (9). I then find values for y0 and y1 that maximize (9) through a binary search.
Let me know if anything is still unclear after you’ve had time to study the article.
Matt
Matt,
Thanks for your detailed reply! It’ll take me a while to work through both your reply and your original entries but I certainly appreciate your help!
The main reason I’m posting right now is for the first time I actually encountered a Farkle on the first roll on Facebook. Either they’ve changed something or they have a finite number of re-rolls if you Farkle on the first and I finally got consecutive Farkles. I guess two consecutive Farkles would have a 1 in roughly 1600 chance of occurring so I could believe that you get exactly one re-roll.
–Dan
Yep, This is all pretty cool!