@aardvarkpepper Dang. To be frank, your story is a lot more philosophical in nature than I actually feel is necessary to worry about. But that’s because I should have explained something:
—This whole tournament idea is actually for a playtest. Aside from having a fun, quick-moving tournament, I’m trying to gather statistics from a large pool of players to see if there is any bias involved. The “bias” I’m talking about relates to my earlier vague reference that “turn order is important.”
In each five player game, there are five “colors,” each with their own situation on the board. My hope is that any innate advantages or disadvantages of being one color vs another color are outweighed by the much larger influence of the players themselves. After the tournament is over, my idea is to scramble the players and start over, as many times as possible to weed out any issues with the game’s balance. If there is any position in the turn order which is noticeably doing worse or better, it should eventually become evident.
The reason I’ve been putting “fair” in quotes is largely due to how ambiguous being “fair” is, given so many variables and human choices, strategies, and real-life motives. However, my objective is to make a balanced game rather than an objectively “fair” tourney. In order to gather at least somewhat accurate data, I am trying to lesson individual player influence to help make the bracket “fair,” at least on paper.
This whole “21 participant, 5-player” model that I’m so stubbornly adhering to was created by asking myself, “What’s the smallest number of games a participant could play and still get a chance at being each color (position in the turn order) and also facing every other participant?” Well, the answer is that each participant would have to play at least 5 games obviously, and since there would have to be 4 opponents in each of those games, there would be 20 opponents total plus the first guy (= 21).
So 21 is the smallest magic number for 5 player games. And it offers the unique distinction of only requiring opponents to see each other once (which, in my opinion, helps to lesson individual player influence over the data I desire). The next magic number for 5 player games is 42. Likewise, 7 is the first magic number for 3 player games, and 14, 21, etc. are the next ones.
The only reason to use larger “magic numbers” is to compensate for what @CWO-Marc has noted: even with all these stipulations, there are still many inconsistencies, as a 3 or 4 or 5 player game has groups of players than could potentially be switched around and manipulated to change results. But…as we’ve all come to conclude, it is also easy to just swap one player with another to get another valid matchup. So if I scramble the players after each tourney, I’ll get closer and closer to an accurate answer.
My hope is that the game is already “fair enough” to the point that skill level, luck, and matchups play a much larger role than the color (or rather, the place in the turn order) that each participant plays. As I’m sure some of you are probably thinking, there are easier ways to playtest with acceptable accuracy than to go to all this trouble building something around a “magic” number that was created through preset rules. But being the curious, stubborn person that I am, I want to solve it this way; it’s fascinating; it’s unique; and although it’s imperfect, it will work for me if it’s indeed possible.
The 3 player variant that you did some awesome number crunching with @aardvarkpepper is a very good frame with which to build an intense 7 player tournament with perhaps 21, or even 105 games if desired. The 4 player variant, which I’ve found several solutions for, was so enjoyable in playtesting, that we’ve already reused it several times. I must take the time to examine more in detail, aardvark, the way you solved the 3P version, as it might work better than everything I’ve tried so far.
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.
@cwo-marc You hit the nail right on the head. Fairness is definitely a quality that is largely based on opinion and perspective. Luckily, I don’t have anyone actually complaining to me that the game is unfair—I think most of the players are much more focused on tricking and influencing each other enough to claim a win—which is exactly what I’d expect (and maybe even encourage) in a free-for-all with 3, 4, or 5 people.
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.
As you may have noticed, I’ve been almost maintaining a bit of secrecy about what this game is, and that’s because it’s actually something unique we came up with. This is just me being hopeful and daydreaming, but I think the game may have market value, even if only in a digital format. There have been flaws, and we’ve been tweaking the setup try to fix that. I think the setup is at a decent standard now, but hosting these tournaments will give me a good idea of whether or not it needs more work.
I must say, I’m amazed at the depth of thought you and aardvark have explained in relation to what actually goes on in the minds of players in large competitions. I have been focusing so much on the math and given little thought to the amount of psychology involved in tactics and motivations. It’s something I’ll definitely try to consider more in any future tournaments where outside factors and real-life pressure can play a heavy role.
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.
Success!
I hope I’m not crying wolf (I will double check later), but I believe I found a solve!
Thanks so much for your help gentlemen! Your ideas gave me an idea for a fun way to do this:
A more generic table can be created by making the top row “player 1 through player 21” and substituting likewise wherever the roundel or chip repeats. Each column represents a game, with position in the column determining turn order.
@cwo-marc For the turn order, I would like to go back to the Axis and Allies analogy. In A&A depending on your place in the turn order, the nation that you’re playing changes. And that’s a big deal. Not because you’re going first, third, or last, but because of drastic differences between nations. With our game, it’s a similar scenario. Red is the last player that moves. But the fact that red goes last isn’t so much the issue. It’s the fact that being red gives the player a certain predetermined setup that no other color has. And if necessary, we could change that setup. Turn order, in and of itself, doesn’t play much role. Rather it’s who you’re playing with and what color you are. But indirectly, turn order does determine your color AND who moves before and after you, so it actually does play a big role in this setting.
Think of it like this. In Pacific 1940 the are five people. Let’s say I’m third in the turn order, so I’m China. If every game we play, China gets totally destroyed, even when Japan loses. We might want to consider giving China some more forces to help counter that. Now Axis and Allies is WAY more complicated and team-based, so that would indeed be a very simplistic way of doing this. But I hope that gives a better idea why turn order is important, because it isn’t just about when you get to play; it also determines what nation/color/or whatever that you get.
Now, to put another idea in A&A terms, let’s talk about “Bob” again. Let’s say that he was playing as China. And he complains that another guy, “Joe,” the UK player, never sent him any help through Burma, because maybe Joe is an adventurous guy trying to fight Japan in Java. So if Bob keeps getting Joe on his team, he might feel like he’s at a disadvantage, and more importantly, it might skew my data to make it look Bob is always losing and obscure the real problems, such as maybe China is too weak.
These are imperfect correlations, but I think it might explain a little better what I’m trying to say.
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:
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.
I think you need to decide whether you’re trying to use statistics and “science” to justify a position, or research a position.
As I’m sure some of you are probably thinking, there are easier ways to playtest with acceptable accuracy than to go to all this trouble building something around a “magic” number that was created through preset rules. But being the curious, stubborn person that I am, I want to solve it this way; it’s fascinating; it’s unique; and although it’s imperfect, it will work for me if it’s indeed possible.
Given that quote, I think you already know what I’m talking about. So why say something you already know? Because I’m saying, as a friend would, just a reminder about perspective. I’m not saying it’s wrong to do what you want to do how you want to do it. But is it really about detecting balance issues in the game, or is it about exploring a mathematical process you find interesting? (In reading through comments after starting this reply, I noticed CWO Marc already brought this up, pardon the repetition).
Is it about getting a product out the door, or about things you may find along the way? If the latter, then by all means, most certainly look into the round robin tournament algorithm. If the former, well.
As you may have noticed, I’ve been almost maintaining a bit of secrecy about what this game is, and that’s because it’s actually something unique we came up with. This is just me being hopeful and daydreaming, but I think the game may have market value, even if only in a digital format.
I wish you every success, but from what I’ve seen, my understanding is you can secure rights to things related to physical processes, but you cannot protect ideas. So you can copyright artwork and such, but you cannot protect, say, the idea that a certain number of dice are rolled and having a number or less indicates a “hit”. You can secure rights to the “Tap” symbol in Magic the Gathering but you can’t prevent other games from rotating cards to indicate conditions. And so forth. I am not a specialist in copyright or trademark, or a lawyer, nor do I have much familiarity with international law, but you may find what I’m saying to be the case.
Am I saying to not maintain secrecy? No. Maintain secrecy by all means. I think were you to open the idea to others, you would find few others that would be able to contribute meaningfully, and a lot of unasked for unproductive advice that even when accurate (which it often wouldn’t be) simply wouldn’t apply to your particular circumstance. When you have the product in release state, you’ll have the advantage of competitors needing to take time to reverse engineer the systems (so to speak) so even if they copy it, they may not be as successful. (Though be aware someone can take what you’ve done, slap on a few distinct features to try to make it more marketable and/or profitable, then overtake your market share. There are literally companies that do only this.)
All right, so the project is secret, but still, a few comments.
As I understand it, Elo/Glicko something like respective player ratings are plugged into a formula, and you get an output that you can interpret to get the expectation a particular player wins or loses. After a match is completed, a player’s rating rises or falls, which is how the system self-corrects.
You probably already know that the, is it “k” value changes depending on the situation? Particularly in I think chess federations FIDE and USCF (are those the names?), you have a high k value while playing your first I think it’s 25 matches, so your rating can correct more quickly. After you finish your provisional games you’re assigned a rating, and instead of using the provisional k value which is high, you then use a new k value for all future games, which reduces the ratings gain/loss you would otherwise get. Then after you reach - I don’t know, Master or something, your k-value decreases yet again. And I think Glicko maybe introduces something like a consideration for real time elapsed, like if you don’t play for a long time, you may get rusty, but your fundamentals are still strong so you can catch up, but maybe you don’t get the rust off because you’re not active . . . colloquially speaking.
So obviously it’s going to be a little more complicated because instead of using only two ratings, you’d ideally use multiple player ratings.
Well, that much is pretty predictable I guess, but I think the tricky part is assigning instead of a constant k-value (or even a k-value that varies under different conditions), you assign a non-straight-line curve to player actions.
Suppose I say you have a 10% chance to win a game, and that you typically play a “conservative” game. If you know your chances are 10%, then aren’t you more likely to play differently than your usual manner? This may increase your chances or decrease them, but functionally you’re using an entirely different playstyle.
So you see what I mean about using a non-straight-line curve, right?
Anyways, there’s a lot of hurdy-gurdy mysticism mumbo-jumbo in the mathematics because who really knows if the curve you’re using is correct? Just as I think initially for Elo systems they used a standard distribution for prediction but it was found the normal distribution was more accurate, or was it vice versa? I forget. And I won’t even get into the specific value of k; anyways you use data received to correct the formula . . . er . . . okay, hopefully you understand what I’m getting at.
Which is, instead of having a readily understandable system that perhaps breaks apart under examination (as I mentioned when talking about cabals of players, differing motivations, and changing playstyles) - instead of saying “well, if you take this PARTICULAR case then the whole testing regimen falls apart” - instead of having a more readily understandable process, instead you use some god-awful mutation of Glicko which is itself a mutation of Elo, which has all sorts of complicated variables and nonlinear this and that, and if investors dare to try to understand it, you just look them in the eye, nod, and say you understand their concerns and can they perhaps point out which specific part of the equation should be changed? You say major federations using ratings formula found it necessary to change formulas over time, you show how you’ve built in self-correcting measures so feedback from data self-corrects the specified variables in the formula, and I’ll bet you dollars to donuts they’ll suddenly find something very interesting just outside the window.
Mind, you do need some serious mathematics and methodology in case someone takes your report back to some serious analysts, but your system is even somewhat reasonable, then by using formulas instead of an extensive and rigid paradigm, you have more freedom.
Of course, you then say that the formulas need something like 5,000 inputs (or whatever) to arrive at a baseline, then say you expect the formulas to change depending on data, then you’re covered and everyone’s covered . . .
Right. It’s not that you are TRYING to be obscure (wink wink). It just works out that way (wink wink).
Say you have a 5 player game that ends up with 4 losers. If it’s a board game played live, you know the actual people, maybe you’re all watching a movie on TV or whatever, sure, fine, whatever, if someone loses again and again they don’t care, they’re there for the pizza and wings and football game. But digital? They’re not there for the pizza, wings, football game, if digital you’re taking them away from that. So as manipulative as it is, it’s even more important for players to feel their in-game actions matter, that they had fun, that they had a chance to not just win, but win on their terms, and the game needs to be designed that way.
But you have to can’t get carried away with comebacks, you have to make sure the game ends before too long. Don’t get me wrong, there’s players that want the fourteen-hour-experience, and if you really want to cater to that, then by all means, totally ignore what I’m saying here (know your audience and cater to them!) But if you’re going for a casual market, then players don’t want to get into a game that offers every opponent a comeback, then the game just gets drawn out.
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In closing - all right, so your process is secret, and you will do things your way. Sure. But I want you to realize cabal formation in multiplayer groups is natural and expected, that player motivation is going to be different to simply trying to “win” some abstract “game” by the rules in the box. You must not think of things in terms of one player “bullying” another. Nor should you assume every player is going to play it fair, above board, and try to win only by themselves and for themselves. It should be completely natural and expected in your predictive model that when there are two friends in a game, one of which is bad at the game and knows s/he is bad, that that friend will sacrifice themselves to assure the other’s success. It should be completely natural and expected that a player that is losing badly will try to eliminate a winning player, not because they are spiteful, it’s very natural for players to want to feel that they are personally significant and that their decisions matter; by changing the outcome of the game they feel they accomplish just that.
more philosophical in nature than I actually feel is necessary
If you already have your core clientele and marketing strategy all worked out, of course it is more philosophical than necessary.
But since you know I have no way of knowing you’ve done that based on what you’d written to that point, it’s a thinly veiled hint that I’m wasting my time - and perhaps yours, if I may be so bold.
It is indeed a valuable life lesson to remember that no matter how much time one may spend on something, if it is not exactly what someone else wanted, such effort may at best go unremarked, or even earn a reprimand.
I thank you for your reminder to value others’ time, and to spend one’s own time wisely. It really is good to be reminded of these things time to time.
being the curious, stubborn person that I am
Me too, you know? Me too. I hope you will view what I have written in this light, and pardon me for not having developed and presented an algorithm for you to get the special round robin tournament generator you wanted. (Edit - having read this after it was written, I feel that may have come off as having intended disrespect. That was not the intent; as I commented earlier I do think it’s just an application of combination and permutation mathematics; I could write a computer program to generate and test sequences myself if I had some more programming knowledge; I don’t imagine it would take more than half a day if I were familiar with the commands I’d need. (Though if I’m wrong about the core mathematics (which would become evident in the check step after data generation) it would take a lot longer.)) But you can see if it really isn’t a big project (which I think it might be), no disrespect was intended, if I could have done, I think I might have done it for a bit of diversion even. Well, I still have a ways to go with programming so eh.
I saw from a post in this thread you seem to have already found one solution; congratulations and I wish you every success with your project! I would imagine, that solution being found, that your objectives with this thread are met?
@cwo-marc I think I’ve been unintentionally making confusion by mumbling several things together —
playtesting the game itself
hosting a tournament
and figuring out the mathematical relationship I wanted
For the tournament, players would not get to determine their color or position in the turn order because I’m going to base it off of the chart I showed earlier. However, after playtesting, the final game should be just like Axis and Allies and most other games, in which players pick or agree on who plays what. So the game isn’t going to decide that for the players. I’m just arbitrarily choosing to do so to try to make a tournament for playtesting. My goal is to eventually have a game where all colors and places in the turn order are equally appealing.
Anyway, I don’t want to take any more of your time, but before you leave, I want to thank you for all the helpful questions and information you’ve shared! I knew I’d get some excellent advice from this forum.
@aardvarkpepper Now that you and Marc have talked through so many different aspects that I wasn’t even considering, I must admit, doing the tournament this way wasn’t necessary, and it’s far from being fool-proof. It’ll work, but as you guys pointed out, it doesn’t account for some variables or human choice.
But, my motivation (aside from having fun with everybody that’ll play) is definitely to research how biased the game might be. I’m concerned that there might be significant advantages or disadvantages to certain colors, and if there are, I hope to spot them and make adjustments. I’m thinking that the setup is already pretty well balanced, but I want to put everything to the test, because we’re not sure yet.
I really haven’t given much thought to any of the legal or marketing aspects of actually selling a game, so I’m glad you brought that up as well for me to seriously think about. And make no mistake, I appreciate every argument you’ve given. The mathematical relationships you discussed are part of what made solving this so easy. Your advice is always welcome and helpful.
If I ever think about hosting another tournament or convincing someone to accept the wacky house rules and games we come up with, I’ll be sure to read this thread again. But right now it looks like I’m ready to organize a tournament! So until we meet again, take care gentleman.
