@charlesdegaulle said in I Need Help!  Special Round Robin Tournament Algorithm:
It’s not uncommon for an experienced veteran to get “bullied” by even just 1 silly opponent who goes out of his way to destroy the threat of a skilled player early on by focusing only on him (much to the harm of both players).
I’ll tell you a hypothetical story, just sit back and relax for a bit.
Let’s say Player 7 is a nice person by some estimations, they like spending time with their friends, many of whom like playing this board “game” (note the quotation marks). Now let’s say that Player 7 feels that they’re mostly wasting their time with this “game” that they’re not very good at, and doesn’t feel that they are being respected. So to make a lasting impression on others, Player 7 decides to “take down” whoever is thought as the “strongest” player in whatever “game” they play in. That way player 7 earns respect, maybe even admiration on some level. Some others might call this “bullying” but really, Player 7 is just playing their own game, which isn’t necessarily “the game” with all these nice pat rules that the others think they’re playing. Understand the distinction between game (life goals) and “game” (board game), this distinction comes up later.
Let’s say every player in the group is more skilled at the game the lower their number is, and that this is generally recognized by the group. So 6 can expect to beat 7, all other things being equal, 2 can expect to beat 5 handily, etc.
Let’s say Player 1, instead of just being skilled at the “game” with all the formal rules, also is good at the GAME and by that I mean player psychology, making friends, just those sorts of things. And they understand of course players 6 and 7 are going to feel marginalized and disrespected, that there’s a pecking order. See?
So now we get some very interesting results.
In Game One with players 1, 2, and 3, in Game Two with players 1, 4, and 5, and Game Three with players 1, 6, and 7, the same thing happens. When appropriate, 1 treats opponents with respect and consideration, so conspiring against 1 is reduced. After all, if other players immediately conspire against 1, it’s basically admitting they’re not on 1’s level, so 1 wins anyways in a sense. And if it sometimes suits 1 to be dominant to players that like to be dominated (oh my), then let’s say that happens too.
But that isn’t the interesting part, we’re assuming that all things being equal 1 was favored to win anyways, so if 1 wins, no big deal right?
But that’s not the end of it. Though player 1 wins all their games, what of the other games that other players play? If 1 wants to destroy their competition in the game sense (not the “game” by its nice clean rules, but the GAME, like life shenanigans), then what?
We know that player 2 already lost the game in which 2 matched against 1. But player 2 is also playing with 5 and 6 in game four, and with 4 and 7 in game five. And because 1 is masterful at the game, 1 says “well, you know, I like 2, but really, 2 needs to be taken down a notch”. So player 5 and 6 conspire so player 5 wins against player 2 in game four, and players 4 and 7 conspire so player 4 wins against player 2 in game five.
But wait, you say, that’s crazy? Of course it’s crazy. But because 2 is losing all their games, 2 becomes the new 7. That’s just how it works, even though it’s not how it’s SUPPOSED to work, we KNOW that 2 is better at the “game”, but that certainly isn’t what the game results will show. But the outcome, as weird as it is, still suits everyone except number 2, so you can easily see how it happens with a little encouragement.
But it’s all more or less okay because the best player is still the best, and if one player’s not where they ought to be in the ranking that’s just how it goes, oh well? But that’s not even the twist ending. The twist in our little story is NOBODY is where they ought to be, actually the SEVENTH best at the “game” is player 1, it’s only that the seventh player is actually a master of the GAME OF LIFE (not the “board game”). And all the little complications in the OP were entirely besides the point to someone that understood how to work the system.
Instead of meekly playing along and accepting the rules of the board “game” and the supposed rating system, the least skilled at the “game”, the REAL player 7, has such a strong grip on the GAME (in terms of getting people do what one wants them to do in real life) that they are, without dispute so far as the other players are concerned, player 1! Is that really what was desired?
I’m not saying it plays out dramatically like in a movie. You can do a lot with a slight wrinkle in the nose, some subtle hand gestures, and a friendly smile, maybe a little shrug to help things along. It doesn’t even have to be conscious manipulation, it’s really just a very natural part of social interaction that one influences others in these subtle ways.
So the real question is not whether it happens, it’s how much it happens, and whether the OP system counters it, (which it doesn’t, right?)
What is the “mathematical solution”? If you have a nice clean simple answer, there’s going to be factors you’re not accounting for mathematically, and that means the system will be subject to manipulation via manipulation of the variables that ought to be accounted for but aren’t. It is not required that someone understand or even have intent to manipulate a system, people naturally puzzle out what works, just like you don’t need real intent or understanding of calculus to nearautomatically catch a ball when it’s thrown to you.
I’d imagine if you wanted a real mathematical address to player skill in the “game”, the mathematical solution would have to be very dirty indeed. Like you’d have to use a modified Glicko with additional multipliers based on turn order, and accounting for multiple players with rating having different motivations for game outcomes. I’m not even sure you could call such a system Elo / Glicko any more; if Glicko is really “different” enough to Elo that it deserves a name change, surely a system with so many added variables and considerations would have its own name.
But as I see it, CWO Marc’s first post in the thread  it really returns to, what is the purpose of the system? If you want a mathematical answer you need to define the mathematical question. So what is it that’s wanted, exactly?

A system to accurately assess player strength (which would require a load of data crunching, there’s just no other way to get the corrective numbers for a particular population)

A system to notso accurately assess player strength, but which requires less work

A system that doesn’t even pretend to try to assess player strength at all, but could still be of interest to players.
https://www.youtube.com/watch?v=THUGHEJjjGc
Is Bill Murray REALLY assessing mathematically a horsesized duck versus a hundred ducksized horses? Clearly not. It’s schmoozing, but 800,000+ views; apparently some people found it interesting.

A device to get players interested in playing more games

A device to convince players of the legitimacy of game outcomes. I’ll point out again the OP system just doesn’t mathematically account for “bullying”.
Or something else, what? Once you can clearly define the issue, and I mean clearly and honestly and openly in all ways, even if those end up maybe looking a little shameful, only then can you really define mathematical solutions.
@charlesdegaulle said in I Need Help!  Special Round Robin Tournament Algorithm:
The “21” model is the smallest logical number (for a 5player game) that is “fair” in my group. The extra stipulations are there to give everyone an even chance without too much luck being involved.
What are the specific criteria for “fair”? Consider, why does the one designing the model not account for and predict a lower ranked player hitting a higher ranked player, offplaying it as “bullying”, even though apparently this is enough of a trend to be noticeable? Could it be that the predictive model need be reexamined and reconsidered from the ground up?
If you’re looking for a mathematical solution, you need to not look at how you want things to be, but how things really are.
But the OP is about a “special round robin tournament algorithm”? All right, then, cautions given so let’s look at that then.

“It’s a five player game in which turn order matters” / 2) Every person plays five games. This is really just one thing mathematically. Or procedurally. Whatever. An nplayer game in which every player plays n games, isn’t that the parameter?

“Each player will only see every other player once” which is another thing,

21 players = 21 games. Well, we’ll look at this. I don’t think this should be taken for granted.
Okay, so let’s look at the 3 player game (abbreviated hereafter as 3PG). What if have only 3 players? Fine, that works out, just one game. But what if we have 4 players? Well then there’s a problem. You can have a 3PG game with players 1, 2, and 3, but for there to be a game with player 4, then you need either 1 and 2, or 1 and 3, or 2 and 3 to play a game together again. There’s no way around it, you simply can’t have a roundrobin for 3PG with 4 players with the specified conditions.
But you define the parameters by specifying the player count and the value for n in nPG? All right, if you say so, but you do see where that’s a bit odd. Rather than saying here are the number of players you have and let’s figure out a system, you stick on so many parameters and requirements it’s quite the pincushion. I really think if you’re designing a system you would do best to consider a system that doesn’t have so many requirements for participants. Of course I’m not familiar with your situation, for all I know you have total control over the number of participants, so eh.
Apparently it works for 7 players for 3PG, or 13 players for 4PG, but what is that says 21 will be the magic number for 5PG? Really, you just take nPG, then take (n(n1))+1? If it were that simple, then shouldn’t it be easy to figure out your brackets?
Let’s take a cute little example. Let’s say that we use nCr mathematics to resolve 7 players in 3PG. First let’s say that we’re using nCr instead of nPr in the first place because numbers can just switch name tags; it doesn’t REALLY matter if you’re doing 1,2,3 vs 1,3,2 if “2” and "3’ can just switch name tags. Then we say how do we figure out how many games a player plays? 6C2 = 15, 5C1 = 5, divide to get 3. Wow, what a great answer, so simple, and completely wrong for all I know, but eh.
If this combination thing is so great then what happens if you go from 7C3 = 35 to 6C2 = 15? Oh, let’s not worry about THAT, it looks weird but TRUST me (wink wink). 6C2 = 15, 5C1 = 5, divide to get 3, it’s magic! don’t think about anything else! frantic hand gestures!
12C3 = 220, 11C2 = 55, amazing! Divide and you get 4! Magic!
20C4 = 4845, 19C3 = 969, divide and you get 5!
Hey well (shrug) it’s not like I’m really thinking things out, just kinda throwing some nCr stuff at the wall and seeing what sticks, you know? But if that IS actually all you have to do, then you just set up a program to generate the nCrs and to handle the “division” part so you get a nice clean output. I mean, how hard could THAT possibly be?
Look yeah, let’s just take the 7 player 3PG. Here, you have 6C2.
2,3
2,4
2,5
2,6
2,7
3,4
3,5
3,6
3,7
4,5
4,6
4,7
5,6
5,7
6.7
Then you have 5C1
3
4
5
6
7
And how do you get from there to 1,2,3 / 4,1,5 / 6,7,1?
First, you define the ordered permutations by defining the place of “1” then fill in the rest. So we’re arbitrarily taking 1,2,3, what next? Well the thing about 5C1 is it doesn’t actually represent 3, 4, 5, 6, and 7, does it? Of course not, because there 3, 4, 5, 6, and 7 are abstractions that can “name tag switch” with 2.
So actually what you’re doing procedurally is you defined the location of 1, then you took the first sequence in 6C2 which is “2,3”, defining the permutation “1,2,3”. Then since you want to eliminate any repetition, you’re *eliminating all remaining elements in 6C2 that have the “2” or “3” elements (“1” is already eliminated).
So from 6C2, you’re selecting “2,3” and eliminating “2,4”, “2,5”, “2,6”, “2,7”, “3,4”, “3,5”, 3,6", and “3,7”. But then why do you divide 15 by 5 to get 3?
Hm, how to explain it. It just works out that way mathematically. There’s fifteen elements in 6C2, five elements in 5C1. The elements in 5C1 are mutable until they’re defined by their selection in 6C2. Every element in 5C1 recurs in 6C2 five times; once defined the element is removed from both lists.
So when “2,3” is defined, then 5+4 elements are removed (one of those being the permuatation “1,2,3”). When “4,5” is defined, 3+2 elements are removed (one of those elements being the permutation “4,1,5”). When “6,7” is defined, 1+0 elements are removed (the permutation “6,7,1”, for a total of 15 removed elements.
It’s less about “dividing” and more about that’s just how it works out, just like how you get the sum of the integers 1 through n by (n(n+1))/2.
I could be totally wrong about how it all works, but that’s how it seems to me.
(Edit  when I write, for example, “divide and you get 4!”, I mean divide and you get four, exclamation mark for emphasis. Not 4! in the mathematical sense (=24))