I may have Fark’d this up
My family loves card games, board games, and dice games. But I don’t think we play games like most families do. Most games are silent, each person studying their hand, analyzing and judging the others’ moves. The post-game analysis of a Hearts lasts longer than the actual game. That’s the fun part.
Sometimes it gets unintentionally headed
With the start of the pandemic, we started playing Farkle, or Ten Thousand. It was only a matter of time before we started discussing when is the correct time to take the points and when you should go for it. This is the definitive guide what you should do*.
The Theory:
Let’s say you have 350 points this turn and one remaining die to roll, should you risk the points for a few more?
The expected value of your next roll should dictate what you do next. What is the expected value or rolling the last die?
Half sent solution:
The expected value of rolling 1 die in Farkle is pretty simple. You can roll a 1 for 100 points, a 5 for 50 points, or anything else for Farkle.
$$ E[1 die] = 100*\frac{1}{6}+50*\frac{1}{6}+0*\frac{4}{6}=25$$
The expected value of rolling 2 dice is slightly more complicated but still easily visualized. You can roll two ones, a one and a five, two fives, or a single one or single five:
$$ E[2 dice] = 200*\frac{1}{36}+150*\frac{2}{36}+100*\frac{1}{36}+\left(\frac{8}{36}*(100+E[1 die])\right)+\left(\frac{8}{36}*(50+E[1 die])\right)$$
$$E[2 dice] = 61 $$
$$…$$
$$E[6 dice] =A lot of terms$$
Following the pattern above becomes very difficult to calculate by hand when rolling 6 dice; and harder still to account for all possible combinations of scoring rolls, when three of a kind and higher scoring rolls become possible. This is a great opportunity for a program to calculate all combinations, probabilities, and scores.
So if I can write a problem that can correctly calculate the expected values of 1 and 2 dice rolls, following the same rules, it should give the expected value of rolling 6 dice, right?
Well, according to my program, here are the expected values of rolling a set number of dice:
| Dice | Expected Value |
|---|---|
| 1 | 25 |
| 2 | 61 |
| 3 | 116 |
| 4 | 197 |
| 5 | 306 |
| 6 | 502 |
That seems … high. The program correctly calculates the expected value of rolling one or two die. From there the expected values of three through six die seem anecdotally high to me, but perhaps understandable given that the highest scoring rolls are skewing the mean to the right.
BUT WAIT! The above equations are also incomplete. After clearing all of the die, you get to roll all 6 dice again. A more complete equation would look like this.
$$ E[1 die] = \frac{1}{6}(100+E[6 dice])*\frac{1}{6}(50+E[6 dice])+0*\frac{4}{6}=??$$
To add more complexity to the problem, we can see that the expected value of rolling 1 die is dependent on the expected value of rolling 6 dice, which if we write out as a full equation, is dependent on the expected value of rolling 1 die. Cool. So my estimated expected value will be revised upward significantly.
It's hard to explicitly solve for this way so I will settle for a guess and check method by guessing and checking if the equation returns to same value. After a few (seven) iterations, I get close enough and find my "true" expected value.
| Dice | Expected Value |
|---|---|
| 1 | 261 |
| 2 | 245 |
| 3 | 289 |
| 4 | 376 |
| 5 | 485 |
| 6 | 708 |
Another way to read this is that the expected value of rolling 1 die is worth 25 points on the roll and 236 points because you may be able to start over with 6 dice.
So what seemed like an unreasonably high threshold has gotten even higher. However, I cannot think of a reason NOT to believe this result.
* You should definitively** not do this, my first reaction to expecting an average score of xxx was “that’s way too high”.
** Not definitively, maybe use it, since I haven’t heard a better strategy from you***.
*** Or maybe I made up this whole exercise to throw my family off the scent of the true optimal Farkle strategy, like I would publicly disclose such valuable information.
How Many Sends?
Since this projects seems threateningly complete, and thus a poor candidate for the first post on a website about things half done, I will admit I got this far about a year ago and have not changed or refined my strategy. Simulating turns and plotting the score may be useful. Or writing a program that plays Farkle against itself and returns the optimal strategy. Maybe one day.