Comment by tzs
4 hours ago
Interesting. For 3 players this set of 3 6-sided dice would work:
#1: 1 2 3 4 17 18
#2: 5 6 7 14 15 16
#3: 8 9 10 11 12 13
But if you had those 3 dice but only 2 players you could not just have each player grab one of them and roll. If one of them happened to grab #1 they would only win 1/3 of the time instead of the desired 1/2.
With 2 players they would have to use just #2 and #3.
That's because the way I came up with those numbers is as follows.
1. Number the players 1, 2, and 3. We want #1 to win exactly 1/3 of the time. We could do that by given them a 3-sided die 1 1 H1, where all the numbers on the other dice are lower than H1 and higher than 1.
2. In the cases where #1 rolls 1, we want #2 to win half the time. Give them a 2-sided die 2 H2 where the remaining die has all numbers between 2 and H2.
3. Assuming the remain die is also 2-sided we will need a total of 6 different numbers. Using 1-6 our set of dice is (1 1 6), (2 5), (3 4).
4. Most people would probably prefer that they all have the same number of sides instead of 3, 2, 2. LCM of those is 6, so double the 3-sided and triple the two 2-sided: (1 1 1 1 6 6), (2 2 2 5 5 5), (3 3 3 4 4 4).
5. People might object to having the same number more than once on a die. We have 18 total sides so lets renumber from 1-18. Our 4 1s become 1-4, our 3 2s become 5-7, and so on, given the set of 3 6-sided dice at the start.
It seems pretty clear that this generalizes to more than 3 players, with the more players the more sides the dice will have. But all of those suffer from that annoyance of needed to exclude specific dice when you are trying to decide the starting order for less than the maximum number of players.
Do the dice in the article avoid that annoyance? I have no idea how I would go about making something like that.
Also note that my dice only determine who goes first. It would be really nice if they could be used to determine complete order. Mine fail for that because #1 is always either the highest or the lowest.
It would be possible to use #1s number on a losing roll to give their place: 1 2 means they go second and 3 4 they go third. You could even print something on the dice saying that, but I think most people would find it more elegant if it was a simple highest goes first, second highest second, and so on.
Do the dice in the article do that, or are they also just solving the who goes first problem?
A set of permutation fair dice work for any subset of dice and players. So a 4-dice perm-fair set allows any three dice to be used and is guaranteed to be perm-fair for 3 as well.
The “Go First” name is catchy for laypeople, but permutation fairness is the strongest and most interesting property.
There are sets we call “all subset place fair” which means any subset of the dice can be used and can fairly choose 1st, 2nd, and so forth, but this property is slightly weaker and doesn’t always make every ordering equally likely for every subset.
there seems to be no need to match number of dice with number of players - for example in 2 players the second player throwing second die (2,3) is really a noop as outcome is decided by the first player's die (1,4)
Yes, that bothered me. I think I’d want to add a rule that says the final order isn’t completely decided until all players have rolled.
I suspect all the other rules plus that one would make it unsolvable at least in some cases like probably in the case of 2. It also brings to mind distributed consensus of cryptocoins (pardon for mentioning crypto :)