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@thedesertfox

As I recall, one of Larry’s introductions encourages people to meet with WWII vets and hear their stories, since those who are still with us are now elderly. I once had the privilege, many years ago, of having a long talk in Ottawa with a retired WWII naval officer who had commanded a Royal Canadian Navy corvette (HMCS Kamloops) during the Battle of the Atlantic. A couple of years later I had a chat, in Halifax, with one of the sailors who’d served on his ship, in the engine room. It was particularly interesting to note that the enlisted men were sometimes told cover stories about the operations in which the ship was engaged, whereas the officers (or at least some of the officers) knew the full picture.

@thedesertfox said in May 31:

@barnee

Yeah, there’s plenty of ways to do it and whatever works fine, for me though I feel that taking a break from the game for at least today would be the right thing, but nobody is required or should feel entitled to do that. Just honor the fallen however you feel honors them.

And indeed, Larry Harris himself paid tribute to his dad’s WWII combat service in the introduction to some of his A&A rulebooks.

In A&A, players know in advance who will play which power, and they know in advance which power gets to play in what order. They know in advance who will play which power because, prior to the start of the game, they’ll have come to an agreement about it – either by a random draw, or by mutual agreement, or whatever. And they know in advance which power gets to play in what order because the turn order is prescribed in the rulebook:

1. Germany
2. Soviet Union
3. Japan
4. United States
5. China
6. United Kingdom
7. Italy
8. ANZAC
9. France

In your game, it sounds as if one or both of those elements have been discarded. It sounds as though the five colours don’t play in a predetermined order, and it sounds as though the players don’t get to choose which colour they play. Rather, it sounds as if these things are determined either by the game system, or by the actions of one or more players, or both. To use the analogy of a five-part train consisting of a locomotive and three passenger cars and a caboose, it sounds as if the game starts with somebody somehow being assigned to the driving the locomotive (and therefore getting to play first). The results of that person’s first-player actions then determine who gets assigned to the first passenger car (and therefore getting to play second). And so on, until the last remaining person ends up being put into the caboose. And it sounds as if one of your playtesters is complaining about being stuck in the caboose, a problem which you’re wondering could be remedied by changing the configuration of the caboose to make it more comfortable. At least that’s how I’m interpreting your (understandably) cryptic remark that turn order determines what colour/nation you get. I say “understandably cryptic” because I realize you want to keep the details secret, which is fair enough and which is why I don’t think I’ll have any further questions on the subject. Good luck with your project.

The fact that you haven’t said anything specific about how your game works actually makes it easier to discuss it from a broad theoretical perspective. The theoretical issue I’m wondering about at this point relates to what you mentioned here: “The main complaints I’ve gotten are more like, “Bob kept screwing me up because he moves right before I do.” That’s where the idea of only having to play against every opponent once came to mind, in an attempt to lessen some of the luck of the draw that’s involved.”

What I’m wondering is: to what degree are you trying to eliminate (or compensate for) factors which give one player a potential advantage over another? And with all those factors eliminated, what does the game outcome actually end up hinging on? There has to be some way for Player X to gain advantage over Player Y, because otherwise there would be no mechanism for winning the game. I’m not expecting any answers, since your game is confidential; I’m just framing these as questions for you to think about.

Even though nobody wants to play a game that’s unfair, the flip side is that wants to play a game which is so perfectly balanced and utterly fair that it’s dull and pointless. Games are inherently conflictual. Okay, an exception can be made for certain Euro-style games in which the focus is on teamwork rather than winning and losing, and perhaps that’s what you’re designing, but I’m going to assume that we’re talking about conventional adversarial games. Let’s take A&A, and the conflict which inspired it, WWII. In simplistic terms, the Axis powers start out in a position of military strength and economic weakness, while the Allies start out in a position of military weakness and economic strength. Is this situation balanced? Arguably yes, since each side has both strengths and weaknesses. Is the situation symmetrical? Definitely not, because the strengths and weaknesses are different on the two sides. Is it fair? That’s a matter of opinion. Does it make for an interesting game and an interesting historical event? Absolutely yes.

One final observation about turn order, by the way. The quintessential no-luck-involved game, chess, has a turn order: White plays first. Does this mean that chess has a deep structural flaw and is inherently unfair? I don’t think so. There was a Soviet chess grandmaster (I can’t remember his name, so I’ll call him So-and-so) who was once asked if he preferred playing White or Black. He answered, “It doesn’t matter to me. If I play White, I win because I play first. If I play Black, I win because I am So-and-so.” Blaming one’s defeat on turn order alone is simplistic, unless the game is so badly designed that turn order inherently gives one side such a clear advantage. In such a case, the bias will be obvious without having to play dozens of game to verify its existence.

Your response to aardvarkpepper provides some interesting insights into why you’re developing this matrix, and it also raises some new questions. I’m not sure I’ve fully grasped what you’re getting at, but it sounds as if you’re trying to generate a massive amount of evidence in order to make a point by the sheer brute force of quantity. What I’m most unclear about is the nature of the point you’re trying to make by doing this. To put it as a question: are you trying to test your game’s level of balance for your own benefit (possibly because its balance can’t be proved from theoretical considerations alone) or are you trying to prove its balance to someone else who needs to be convinced? If it’s the latter, I doubt that a bunch of statistical data is going to overcome their scepticism. I don’t know if you’ve ever seen the original pilot movie for the eventual TV series Voyage to the Bottom of The Sea, but there’s a scene depicting an argument at the U.N. between two scientists, each of whom thinks the other is an idiot. Scientist B produces a stack of calculations which he says proves that he’s correct, puts them on Scientist A’s desk and says he’s welcome to check the figures for himself. Scientist A mumbles for a few seconds, looking at random through the stack of papers, then puts them down in disgust and declares that there’s no point in checking these figures because “I cannot be wrong!” So much for the scientific method.

When I talk about proving fairness (or lack thereof) from theoretical considerations, I’m referring to this sort of thing, quoted from something I remembered from Wikipedia: “Robert Feinerman has shown that the game of dreidel is “unfair”, in that the first player to spin has a better expected outcome than the second player, and the second better than the third, and so on. Feinerman, Robert (1976). “An ancient unfair game”. The American Mathematical Monthly. 83 (8): 623–625.” Or to use a different model, consider the buttered toast phenomenon. If a slice of toast, buttered on one side, were flung energetically into the air a hundred times under perfectly random conditions, it would land buttered-side-down exactly (or almost exactly) 50% of the time. Then why is it that, in real life, buttered toast sliding off a plate tends to land buttered-side-down significantly more often than 50% of the time? Because in real life it involves conditions which are relatively standardized rather than random. The slice of toast typically starts out sitting on plate buttered-side-up being held somewhere between waist height and chest height by a human being roughly five to six feet tall. If the plate is accidentally tilted, the toast slides off and falls with an acceleration of 1G (9.8 m/s squared). Over the distance at which it was dropped, this typically gives the slice of toast enough time to turn over once but not twice – so it hits the floor buttered-side-down. I don’t know if all this has ever been proved empirically, by running hundreds of toast experiments (half of them randomized and half of them standardized), but even if it’s been done the numbers wouldn’t change my mind; the theoretical argument I’ve mentioned sounds convincing in and of itself. In fact, I’d find it much more interesting to read additional theoretical arguments demonstrating that the above model is wrong than to read masses of numerical data demonstrating that the above model is right.

But anyway, I agree that the matrix is, if nothing else, an interesting theoretical exercise in its own right – kind of like solving a “Seven Bridges of Konigsberg” type of problem. I don’t have the mathematical background to crack it, so unfortunately I can’t be of any help with it.

Here are a few additional thoughts, based on your “I’ve tried many, many matchups, but they all have thus far failed” comment and on aardvarkpepper’s analysis, which among other things raises the question of whether the kind of matrix you’re contemplating actually achieves the fairness it’s supposed to deliver.

To illustrate the argument I’m about to make, I’ll use the relatively simple case of the three-player matrix, which has already been solved, my version of it being:

A B C
1 2 3
4 1 5
6 7 1
2 6 4
7 5 2
3 4 7
5 3 6

The first point to consider is this. You’ll note that every player does indeed play in each position (A, B and C) exactly once, and that every player plays against every other player exactly once, as required. At first glance, everything looks perfectly symmetrical, and therefore perfectly “fair”, because the assumption being made is that “symmetry equals fairness.” It turns out, however, that there are two problems here. Problem one has already been pointed out by Aardvarkpepper: the concept that “symmetry equals fairness” is questionable. Problem two is that the matrix is only symmetrical when you view it it terms of every player playing against every other INDIVIDUAL player. It stops being symmetrical when you view it it terms of every player playing against every possible COMBINATION of players. In the above example, for instance, 1 gets to play against three combinations of players; 2&3, 4&5, and 6&7, but never gets to play against all the other possible combinations of players (such as 2&6). If you work from the assumption that “symmetry equals fairness”, the logical conclusion would be that your players would all have to play a game against all the other possible combinations of players, not just against every individual player. This would not only increase the number of games to be played, it also require you to drop the requirement that each play play each position only once.

The second point to consider is this. As noted, the above matrix does not include all the possible three-player combinations. Some of the possible combinations are “valid”, in the sense that they work within the matrix, while others are “invalid”, in the sense that they wreck the matrix. This may explain the “I’ve tried many, many matchups, but they all have thus far failed” problem you’ve been running into. Your five-player matrix experiments presumably all use the following match-ups as their starting points, since those match-ups are quick and easy to identify…

1 2 3 4 5
1 6 7 8 9
1 10 11 12 13
1 14 15 16 17
1 18 19 20 21

2 6 10 14 18
2 7 11 15 19
2 8 12 16 20
2 9 13 17 21

…but I’m wondering if those match-ups include “invalid” ones that make the rest of the matrix impossible to complete? I’m not saying that your contemplated 5-player matrix is impossible; I’m saying that working it out may require a good deal of mathematical knowledge (which I certainly don’t have). If you look at the Wikipedia articles on round-robin tournaments…

https://en.wikipedia.org/wiki/Round-robin_tournament
https://en.wikipedia.org/wiki/Tournament_(graph_theory)

…you’ll note that the scheduling algorithms for even the relatively simple case of rotating two-player match-ups are quite complicated, or at least (to borrow a phrase from Calvin and Hobbes) look complicated “to the untutored eye of the ignorant layman.” This doesn’t appear to be a problem you’ll be able to solve by continuing to “diligently working on at least finding ways that DON’T work for the 5-player, 21-participant bracket.”

Which brings me to a practical suggestion. The problem you’re working on is extremely difficult to solve if the only solution that’s acceptable to you is one that meets your requirements perfectly. If you settle for “almost perfectly,” however, it immediately becomes quite manageable. I assume from your remark “I’ve tried many, many matchups, but they all have thus far failed” that every time you try to set up a matrix, there are always one or two match-ups that don’t fit. How about simply living with them? As has ben discussed above, the premise that “symmetry equals fairness” has got a couple of conceptual problems with it, so it seems to me that having a couple of non-fitting outliers in your matrix is hardly going to be a fatal flaw. And you’ll note that the Wikipedia articles mention the concept “dummy competitors”, whose function seems to be make scheduling algorithms work. If the professionals need to use loopholes of this type, there’s no dishonour in your doing likewise.

Incidentally, I’ve just checked my table against yours (which is oriented horizontally rather than vertically)…

Game 1. Game 2. Game 3. Game 4. Game 5. Game 6 Game 7.
----1---------4---------6---------2----------7----------3---------5
----2---------1---------7---------5----------4----------6---------3
----3---------5---------1---------6----------2----------4---------7

…and I see that we got the same results in three of the cases and similar but different ones in the remaining four, so it looks as if the problem has at least two valid answers. This is is good news because this suggests that even your five-player matrix should be workable in at least one way.

Now to resume. By now we’ve generated a tentative table of three-number match-up lines which looks like this:

1 2 3
1 4 5
1 6 7
2 4 6
2 5 7
3 4 7
3 5 6

As a check, I then mentally went through the following two-number combination: one-two, one-three, one-four, etc., all the way up to three-seven, to see if each pair of numbers in the combination can be found on one and only one line of the table. Answer: yes. So – unless I’ve overlooked something, which I hope I haven’t – we now have a complete list of who gets matched with whom, which meets the required “play with everyone but only once” condition.

Now to figure of the turn order. Each player gets to play in each position (A, B and C) once. The first cluster of unedited lines is:

1 2 3
1 4 5
1 6 7

If we move the 1 by one and two spaces respectively, we get:

1 2 3
4 1 5
6 7 1

Now for these unedited lines:

2 4 6
2 5 7

The first line already has the 2 in the A position, and the 4 and the 6 don’t conflict with anything above, so it’s tentatively fine as is (more on this later). The second line needs to have the 2 in the C position. Putting the 7 in the B position would conflict with an earlier line, so our only choice is to put it in the A positition:

7 5 2

Our table thus far is:

1 2 3
4 1 5
6 7 1
2 4 6
7 5 2

We’re now left with these two unedited lines:

3 4 7
3 5 6

Here we run into a problem: the lines can’t be arranged to avoid conflicts. I therefore had a second look at that “tentatively fine” earlier line and tried switching it around. That fixed the problem, with the rest of the table fell into place. The full final result is…

1 2 3
4 1 5
6 7 1
2 6 4
7 5 2
3 4 7
5 3 6

…which can be verified by noting that each number appears once in each column:

A B C
1 2 3
4 1 5
6 7 1
2 6 4
7 5 2
3 4 7
5 3 6

I think I’ve found a way for you to work out your five-player matrix without too much trouble. I tried to see if there was a reasonably systematic / programmatic way to work out the answer for the simpler case of a three-player matrix; there is, and you should be able to scale it up. It’s a manual process that involves checking things – but it uses a process of elimination, so some parts of the process get shorter as you progress, which makes it manageable.

It also breaks the problem down into stages, so that you don’t have to try to figure out all the variables at once by an excruciating process of trial and error. Crucially, it first determines only the issue of “who plays with whom in each game”, and it leaves until a second stage the issue of who plays which turn order position.

Here’s the approach I took, after studying the photo you posted (which answered a lot of my earlier questions).

For clarity, I designated the players with numbers (1, 2, 3, etc.), and I designated the turn order positions with capital letters (A, B and C). You might want to give Roman numerals to the games.

I started with Player 1 (hereafter just 1). The game has three turn order positions, and therefore also has three players per game. 1 has to play once in each turn order position, which means three games. There are three players per game, so at each game 1 plays against two opponents. The opponents are required to all be different at every game, so this means that 1 will play against six other people, numbered from 2 to 7. Thus:

1 plays against 2, 3, 4, 5, 6, and 7

Since it’s a three-person game, this translates into:

1 plays against (2 & 3) and (4 & 5) and (6 and 7)

All three of 1’s match-us are now spoken for (though note that we haven’t yet nailed down which turn order position he plays in each pairing; that will come later), so we can turn to the next player to consider. Here’s where the process of elimination comes in. The next player to consider is 2. If we were replicating the pattern of this line…

1 plays against 2, 3, 4, 5, 6, and 7

…you’d think this that corresponding “2” line would look like this…

2 plays against 1, 3, 4, 5, 6, and 7…

but in actuality it looks like this…

2 plays against 4, 5, 6, and 7

…with two numbers (1 and 3) eliminated because 2 has already played against both of them. (The systematic check to apply at every step of this procedure is the question, “has this been used before?”)

At first glance, if we follow what we did with 1, this…

2 plays against 4, 5, 6, and 7

…translates into…

2 plays against (4 & 5) and (6 and 7)

…but if we apply the “has this been used before?” test we see that (4 & 5) and (6 and 7) were both used with 1, so we can’t use them with 2. Switching two of the digits, however, gives this…

2 plays against (4 & 6) and (5 and 7)

…which works because 4 hasn’t previously played against 6, and 5 hasn’t previously played against 7.

At this point I put together a little table to see if anything was missing. The results I have so far are:

The “1” match-ups

1 2 3

1 4 5

1 6 7

The “2” match-ups:

2 4 6

2 5 7

This leaves the “3” match-ups to work out. 3 has already been used with 1 and 2, but has not been used with 4, so I wrote the partial line:

3 4

We need another number to complete the line. 5 comes after 4, but it’s been used with 4 previously. Next comes 6, but that’s also been used with 4 previously. Next is 7, which has no previous use with 4, so the line becomes:

3 4 7

I then went back over all the previous lines to check whether there are any remaining numbers with which 3 had not yet been used. Yes indeed: 5 and 6. This generated the following line:

3 5 6

I don’t know if the forum has a posting length limit, so I’m going to break off here and continue in a second post in a few moments.

Thanks for the additional information. It helps to know that this isn’t for A&A because a single A&A game can take an awfully long time to play, never mind 21 of them. I have some follow-up questions.

1. You say that each game involves five players (that’s clear enough) and you say that “every opponent will only be seen once” over the course of the multiple games. What I’m wondering is whether “every opponent” refers to specifc individuals or whether it refers to a unique combination of individuals. To explain myself, let’s take five people and designate them as A, B, C, D and E, and let’s use person A as our reference point. In the first game, person A will face B, C, D and E. So far so good. In the second game, will person A need to face four entirely different individuals (say F, G, H, I and J)? Or does it suffice to face a different combination of individuals which can include some of the members of the previous group but not all of them – for example, B, C, D, E and F? If it’s absolutely prohibited for any of the players to ever face any of the same individuals in any of the games, regardless of what combinations they’re in, this implies a rather large number of people.

2. Is a “different opponent” defined by who the physical persons are, or is it defined by the position in which they’re playing? In other words, if person B is playing the #1 position (which I assume means the person who plays first in the turn order) in one game, and if person B is playing the #2 position in another game, is person B considered to be a different oppenent, even though it’s the same individual?

3. When you say “turn order matters”, do you mean that players are defined (within your system) in terms of what – in an A&A context – is the prescribed order in which the nations on the board get to take their turns? Or do you mean something else?

4. How many game rounds are there? (I define a round as each player in the turn order playing one turn.) Are people allowed to play multiple rounds until a game is finished?

5. Is the “you’ll never play the same opponent twice” concept an end in itself? Or does it relate to a larger concept for winning the tournament itself? To use an imperfect analogy: in the NHL playoffs, most teams never get to play against each other; they play in pairs, and each pairing eventually results in one team eliminating the other. Each winning team then meets another team which similarly eliminated another team in the first round, and so forth, with the number of teams surviving each round dropping by 50% at each round, until ultimately only two teams face off in the finals. There’s a clear goal and a clear progression. Is there some sort of similar goal in your rotational system? You seemed to indicate that one of your aims is to compensate for the fact that some players are strong and experienced, and thus can potentially bully weaker ones. Fair enough, but does this mean that the objective is simply to allow everyone to experience one game against every possible level of opponent, but that there’s no specific overall outcome regardless of who wins which games?