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Fun with probability – part I

2012/04/17

A friend of mine just posted on my Facebook wall, linking to this YouTube video about “Grime Dice,” a set of five dice with numbered faces chosen to have some interesting non-transitive properties; the first is that each of the dice will statistically beat two of the other dice, forming two “A beats B beats C beats D beats E beats A” loops, like Rock-Paper-Scissors-Lizard-Spock. The second, more remarkable property, is that if you roll two dice at a time instead of one, and add the totals, one of these loops remains unchanged, while the other reverses in order (so that E beats D beats C beats B beats A beats E).

After writing my last post, about how risk-reward decisions are affected by a game in which the goal is achieving an all-time high score, I got to thinking about more general cases of risk-reward decision-making in games, and how that is, like these Grime dice, a non-transitive thing. If you have the opportunity to see what kinds of risks your opponents are taking, you’re usually going to want to gamble either just a little bit bigger, so as to come out slightly ahead if you both succeed, or – if you feel your opponent’s strategy is too high-risk, play as safely as possible and count on them failing.

Having been reminded of this by the Grime dice, I decided to invent an extremely minimalist dice game to take a closer look at this idea in the abstract.

Here’s how the game works:

We play with a single N-sided die. N could be any integer, and as we’ll see, there are actually qualitative differences in the game depending on what kind of die we choose. We could also play with two players, or more.

First, we determine a player order randomly. Then, in turn, we each have to pick a number, ranging between 1 and N. Everyone has to pick a different number; it is forbidden to take the same number as a previous player. Once we’ve all picked a number, everyone rolls the die, trying to roll at least as high as the number they picked. If everyone rolls under their number, we roll again, repeating as needed until at least one person succeeds. Out of those who succeeded, whoever picked the highest number wins.

Example: Playing with a 6-sided die, Alice picks 5 and rolls 2, Ben picks 2 and rolls 6, and Chris picks 4 and rolls 4. Chris wins.

I’ll have to write a little computer program to investigate what the probabilities look like with more than two players, but first I worked out the math for two players on my own, and found some interesting things. But rather than just spill the beans right away, it might be more fun if I allow you the chance to either work out the results yourself, or test your intuition by taking a guess.

In ascending order of difficulty (both mathematical and intuitive), here are the questions:

  1. In general, is it better for for the first player if the die has more sides, or fewer?
  2. Is there a rule of thumb for perfect play? How should the two players decide which number to take?
  3. Are there any die sizes for which it’s better to pick first? How many, and which one(s)?
  4. Are there any die sizes for which it doesn’t matter who picks first? How many, and which one(s)?
  5. In general, is it better for the first player if the die has an odd number of sides, or an even number?
  6. Assuming perfect play, what’s the limit to how big an advantage the second player can have (for any die size)?

The answers will be given tomorrow, along with whatever results I come up with for the multiplayer case.

Related: Fun with Probability – Part II

Related: Fun with Probability – Part III

Related: Fun with Probability – Conclusions

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