Go First Dice

I love that they put so much work into trying to solve this when there is an equally trivially simple mechanical solution that works just as well and easily scales up to any number of users:

1. Put consecutive numbered tiles in an opaque bag.

2. Each person reaches in and blindly grabs a tile.

3. Play proceeds in the order of tiles.

You can think of dice rolling as simple random sampling with replacement. It's naturally good at generating combinations. Drawing from a bag is random sampling without replacement. It's naturally good at generating permutations.

Using a system that generates combinations to generate uniformly distributed random permutations is a super fun mathematical exercise and I'm sure this brought plenty of joy to the authors. But if you're just trying to design a physical mechanism to generate permutations, it'll be an easier starting point if you do sampling without replacement.

In fact, if you do want to generate permutations with dice, a simple (but laborious) way to do it is:

1. Choose an arbitrary ordering for the players.

2. For each player, roll a die. If the value rolled is the same as any value rolled by a previous player, re-roll. Continue to re-roll until a unique value is rolled.

3. Play proceeds in the order of values each player rolled.

You'll probably want to use d20 or some dice with a large number of faces relative to the number of players in order to minimize collisions and re-rolling.

I'm not 100% certain, but I believe this will generate an unbiased ordering where the order of players in step 1 has no effect on the resulting order determined in step 3.

Reiner Knizia uses distributions that are meant to have this quality (or get close enough to make people happy), most notably in Ra. Ra is an auction game where each player gets the name number of bidding tokens. Each token has a unique number on it, and you can only use a single token to bid, ensuring that there are never ties.

Ra is an example, but he uses it elsewhere. He's a math PhD who often writes in his books about how to break down these problems, and I wonder if he approached it the same way or reached the same conclusion.

https://boardgamegeek.com/boardgame/12/ra

There's another dice puzzle which can also be named "go first" dice.

There exist three dice where A beats B, B beats C and C beats A, statistically speaking. So if you consistently throw A and B, A has a higher chance of rolling a greater number than B, etc.

Kind of rock-paper-scissors, so whoever "goes first" and picks the die will (statistically) lose against a smart opponent who chooses second.

I used to play at low level but officially sanctioned Magic: The Gathering tournaments at a local shop, and there was no official policy for deciding who would play first, My understanding is that this is even the case at higher-level prize tournaments, and I think generally only high profile matches are live streamed (e.g. ones with well-known players or the final few rounds after most people are eliminated), and I read an article from a professional player a few years back about how sometimes people try to get away with things when the traditional "high roll" method is used (i.e. each player rolls their own die and the highest roll goes first, with repeats for ties), like bringing extremely large dice that essentially don't roll so much as just flop once, making it easy to toss in a way to force a given result. I'd also seen people use "evens and odds", where one person calls even or odd and the other rolls a die, with the choice of whether to go first going to the person who called if they were right and to the roller if not, but this still gives the person rolling some influence if they're unscrupulous. The article proposed that having one player call evens or odds but having _both_ players roll would eliminate any ability for cheating (assuming both players roll at once), since even being able to choose your result exactly would not be able to affect the chance of the sum being even or odd without knowing what your opponent would roll.

Some notes

- Permutation-fairness is likely too strong of a condition at anything other than serious competitive tournaments. When playing a game casually, usually the group first sits down in a circle and then chooses someone to go first, taking turns clockwise from there; then if the method for choosing the first player is first-player-fair then it is also place-fair.

- Although having to sometimes resolve ties is annoying, there is one major benefit of choosing turn order by rolls of the same dice: You are guaranteed that any non-fairness in the dice do not affect turn order determination.

My wife and I play a game with dice called 100.

You roll two dice, and you have to decide which permutation to keep, and it gets added to your score. So, if I roll [3,1] then I can add 13 or 31 to my score. The aim is to take turns and get closest to 100 without going over, and you can pass your turn if you think you're close enough.

It can also be played with more dice, ie 1000, 10000, but it gets easier the more dice there are.

Am I missing something here? Site just redirects to https://phpwebhosting.com/not-configured.html?target_host=tr...

I too love the effort put into solving this problem. It's a totally different mindset from my own and clearly they get a lot of satisfaction from it. I myself would be satisfied with understanding that it is possible but non-trivial to do this, and immediately propose a trivial alternative like choosing from a deck of playing cards. Society needs both kinds of people! (and everyone in between)

I really enjoyed reading this but the way I’ve always done this is number the players and roll and single die.

Can I ask a dumb question? I understand the permutation argument for why using 5 six-sided dice can't possibly be fair, but looking at the serpentine example, I'm having trouble intuiting why. At a glance, removing any one of those dice looks pretty fair.

I don't understand why sites like this are not HTTPS, yet.

I'm not giving you a security exception to read your blog.

Let's Encrypt is free.