The kids have been enjoying Super Mario Party (the Nintendo Switch game from 2018), and while playing I've been curious about the wildly different dice blocks each character has.
Super Mario Party is basically a board game; every turn you roll a die to see how far you move. But each character has their own special die with different values on the side, and on each turn you can use a normal die (with values 1-6), your character's die, or the die of any allies you've recruited along the way. Note that some dice have sides that will give or take away coins instead of letting you move. Here's the list of all the dice (source here - note that there are various other places online that have incorrect values; as far as I can tell these values match what's in the game):
So, let's dive in!
Just from playing the game, my first guess was that dice with greater volatility would have a higher average value. A prime example of this to me was Bowser's die, which has values . It's very volatile but the total value of the moves on the die is 28, much higher than a standard die's total value of 21! (this also sounded to me like the financial idea that riskier assets should have higher rewards on average. I realize this is reading waaaay into this video game :-) )
So my first step was to graph the sum of all the die's faces versus the standard deviation of the die's faces. To begin with, let's ignore the dice with coins on them; we'll get to them in a minute: (note that you can hover over or tap the points to see what dice block it represents)
And, bam, hypothesis busted! The regression line shows a very small relationship in the opposite direction - as the total of the dice block gets bigger, the standard deviation gets smaller. But the R2 value (or Coefficient of determination) is less than 1%, meaning this has very little explanatory value. You can see this from looking at the points on the graph above; they're all over the place. Really the only thing we can learn here is that Yoshi has the lowest total.
So, let's look at the dice blocks that have coins on them. One interesting question is About how much is one coin on a dice block worth in terms of movement spaces? And we'll look at our initial hypothesis again - maybe standard deviation matters here?
This graph shows the total of the numbers on the dice block versus the total of coins on the dice block, and each point's size is proportional to the standard deviation of the numbers on the dice block:
Here the regression line is a pretty good fit, with R2 around 77%. The slope of the line is very close to -1, meaning that each extra coin takes away about one movement point. Interestingly, the x-intercept of the line is around 21.2, whereas dice with no coins have a total value of either 20 or 21 (or are Yoshi), so on average dice with coins are a tiny bit better than dice without coins.
But the individual points are also interesting here. Anything above the line is better than "expected". Donkey Kong particularly stands out; his total moves of 20 would match a lot of the dice blocks with no coins on them, but he also has a +5 coins side! Diddy Kong is similar with a total moves of 21 and a bonus +2 coins. Bowser is also good if you can stomach the loss of coins, as his total moves of 28 is by far the highest of any dice block! These three are all on the Pareto frontier, as are Boo and Wario .
On the flip side, I think Waluigi is the worst die in the game; the total moves of 21 is no better than most dice blocks with no coins on them, and there's also a -3 coins side.
Since the previous graph has the points' sizes proportional to standard deviation, it's interesting to see that it looks like the points above the line (i.e. the ones that are better "values") are bigger on average. If this is true, this would mean our original hypothesis was true, at least for dice with coins on them!
So let's check it out by graphing standard deviation versus how far away the point is from the regression line in the above graph; distances above 0 mean the point is above the line (and therefore better than expected):
And there we go! The R2 value is a smidge under 50%, and the slope is reasonable-looking, so there does seem to be a reward for choosing dice with high volatility, as long as they also have faces with coins.
To be clear, there's more to choosing dice than just maximizing the average value. You might be in a situation where the star is only 3 steps away; in that case Daisy would be a perfect choice. But if the star is 8 steps away and you think the other players are going to get to it on their turns if you don't, Bowser is a much better choice! You can always choose the standard die, and since you can choose between the dice of your allies it probably makes sense to try to collect a variety of dice with different standard deviations.
Another consideration is whether you think you'll be able to get a lot of coins from minigames, etc. If so, Bowser is a great choice. If you're worried about getting coins, Donkey Kong or Goomba are good choices. (technically Donkey Kong is strictly better than Goomba, but only by 1 coin and 1 move, so if you'd really rather have lower variance I think Goomba is a reasonable choice!)
The Partner Party game (which is 2 players vs 2 players), there are a few more considerations. For one thing, you and your partner both roll dice and add them together, so this can make it less painful to roll a 0 on your die. Also, if you and your partner roll the same number, you get a matching bonus of a few coins, so it might be worthwhile to pick characters with that in mind, like Daisy and Mario or Peach and Shy Guy .
Source files are available on GitHub.
I considered whether the standard deviation of dice blocks with coins on them should include 0's for the faces that have coins. (to represent the fact that if you roll one of those faces you move 0 spaces) It turns out it makes very little difference in the resulting analysis.