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### AuthorTopic: [Discussion] DominionStrategy Qualifier for 2012 US National Championships  (Read 39821 times)

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#### WanderingWinder

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #100 on: June 25, 2012, 11:44:07 am »
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Advancing in the tournament is not about expected value, it's about the chance of being near the max, and the chance that players win is not 1/N.  I am not sure if simple models will really help you or mislead you.

Of course, still the best answer is don't have 3p games.  Can we come up with a system using alternates or previously qualified players filling up the brackets?
I don't like substitutes. By definition they can't win; do I still expect they are playing to win? Using qualified players may solve this problem, but then the field is probably stronger than it should be.
Wait, why can't substitutes win?

#### zxcvbn2

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #101 on: June 25, 2012, 11:45:36 am »
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Advancing in the tournament is not about expected value, it's about the chance of being near the max, and the chance that players win is not 1/N.  I am not sure if simple models will really help you or mislead you.

Of course, still the best answer is don't have 3p games.  Can we come up with a system using alternates or previously qualified players filling up the brackets?
I don't like substitutes. By definition they can't win; do I still expect they are playing to win? Using qualified players may solve this problem, but then the field is probably stronger than it should be.
Wait, why can't substitutes win?

You can't qualify for the semis twice. You're just in or out.
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#### WanderingWinder

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #102 on: June 25, 2012, 11:46:03 am »
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Quote
Advancing in the tournament is not about expected value, it's about the chance of being near the max, and the chance that players win is not 1/N.  I am not sure if simple models will really help you or mislead you.

That's true, but there's already so much variance in 3 and 4 player games. By the time you put together values for skill, turn order, shuffle luck, etc., in four games it's practically a roll of the dice anyway, at least in such a small sample size (which I realize we don't have time for a larger one).
But I don't think the answer to large variance is to throw our hands up and say 'ah, it's a crapshoot anyway, screw it'. I think we should try to do the best we can, given the limitations we're given. And there's much MORE variance in 4p than 3p. But I mean, if you think that it's such a roll of the dice anyway, then why have a tournament at all? Now, obviously there's a lot of luck, but I still think there's SOME appreciable amount of skill....

#### WanderingWinder

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #103 on: June 25, 2012, 11:46:43 am »
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Advancing in the tournament is not about expected value, it's about the chance of being near the max, and the chance that players win is not 1/N.  I am not sure if simple models will really help you or mislead you.

Of course, still the best answer is don't have 3p games.  Can we come up with a system using alternates or previously qualified players filling up the brackets?
I don't like substitutes. By definition they can't win; do I still expect they are playing to win? Using qualified players may solve this problem, but then the field is probably stronger than it should be.
Wait, why can't substitutes win?

You can't qualify for the semis twice. You're just in or out.
Right, but then he says that using pre-qualified players SOLVES this problem, so I assumed he DIDN'T mean those when he was talking about substitutes.

#### rrenaud

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #104 on: June 25, 2012, 11:57:35 am »
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One thing about hard pods that we can do is try to keep them as balanced as possible using the iso level.

The first round algo was rank by iso level, assuming that we have a perfectly sized player pool, split into N/4 groups where N = num entrants.  Each of the N/4 groups gets exactly one top rated player, and the other 3 players are random.

I could make those other 3 player pools have as close to the average rating as possible.

That's right guys, I'll be solving (or approximating) NP-complete problems for you.  http://en.wikipedia.org/wiki/Partition_problem

Of course, this assumes iso level is the right way to balance groups, which is just the first kind of reasonable thing I could think of.
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#### timchen

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #105 on: June 25, 2012, 11:59:01 am »
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Quote
Right, but then he says that using pre-qualified players SOLVES this problem, so I assumed he DIDN'T mean those when he was talking about substitutes.
Two problems here. Either the substitute player may not play to win (if we just find some random player) or the player is too strong (if we use the qualified players).
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#### zxcvbn2

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #106 on: June 25, 2012, 11:59:33 am »
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Quote
Advancing in the tournament is not about expected value, it's about the chance of being near the max, and the chance that players win is not 1/N.  I am not sure if simple models will really help you or mislead you.

That's true, but there's already so much variance in 3 and 4 player games. By the time you put together values for skill, turn order, shuffle luck, etc., in four games it's practically a roll of the dice anyway, at least in such a small sample size (which I realize we don't have time for a larger one).
But I don't think the answer to large variance is to throw our hands up and say 'ah, it's a crapshoot anyway, screw it'. I think we should try to do the best we can, given the limitations we're given. And there's much MORE variance in 4p than 3p. But I mean, if you think that it's such a roll of the dice anyway, then why have a tournament at all? Now, obviously there's a lot of luck, but I still think there's SOME appreciable amount of skill....

I agree with you. I don't think that there's any system that will give us meaningful results in four games' time. I think a Swiss would probably come closest. I totally agree with you. There's so much more variance in 4p than 3p. But the alternative is either turning people away, having players play who have no chance of winning, or... ?

Obviously I like my system more, that rewards 4p more than theory's. I get the feeling nobody else here does, though because it has a lower ceiling for 3p games. I also really like rotating opponents. Theory's basically said that's off the table. Basically I've come to that point of saying, "Screw it, it's a crapshoot, anyway."

I'm sure if we do a multi-player tourney again, we'll be able to do some major rehauling of the format. For now, I just think we have to deal with the unbalanced nature of 3 player games vs. 4 player games.
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#### timchen

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #107 on: June 25, 2012, 12:18:06 pm »
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I think you miscalculated. with 20 points (6-4-2-0) as a threshold, there are actually 15 cases (6 of them being 2 first places and 2 second places) so the chance for 4p is 15/256. 15/256 and 5/81 are close enough to me.

This difference is caused by the quantization of the points. it does not give definite advantage to 3p or 4p however (depending on the threshold.)

Just to respond to someone says he's absolutely correct: in 4p the cases are
1-1-1-1 (1) 24 pts
1-1-1-2 (4) 22 pts
1-1-1-3 (4) 20 pts
1-1-2-2 (6) 20 pts

so this is 15/256.

in 3p
the cases are
1-1-1-1 (1) 24 pts
1-1-1-2 (4) 21 pts

this is 5/81.

Sorry if this sounds a bit harsh, but I usually find claim of correctness based on who you are kinda laughable.
Yeah, but this is still a problematic way of looking at it. There are this many ways of getting qualified, of course, but it is simply easier to get these cases form the 3p.
Importantly, it's not about getting to a point threshhold, it's about beating the other guys. The 3p are much more likely to produce a point total that is in the top X, as opposed to the 4p.
No it is not simpler for 3p, as the simple calculation shows. If you think otherwise, it's better to show via at least some qualitative arguments instead of just saying it...

As I said, using the points, getting 4th place/3rd place in a 4/3p game is as serious as getting 1st. So if the players are roughly equally skilled the above calculation is valid. You can generalize the calculation for a good player (say I just randomly choose that he can get #1 50% and #2 50% in a 4p game; in a 3p game to have the same expectation value of points he then should get #1 2/3 of the time and #2 1/3 of the time. Putting the numbers in then (suppose the threshold is 20 points again)

4p: 1/16+1/16*4+1/16*6=11/16
3p: 16/81+32/81=16/27.

Actually in this case 4p is better!

To qualitatively understand this, note that #2 in 3p game is worth a bit less than #2 in a 4p game. To get above the same threshold you thus need to push harder for #1 in a 3p game which may or may not be easier than trying to score #2 in a 4p game. The situation can change if you play with the percentages and the threshold, but I think the difference of the chances are too small to matter. To put this way. In the tournament using this scoring system, I am perfectly willing to concede my chance of playing in a 3p game.
« Last Edit: June 25, 2012, 12:27:51 pm by timchen »
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#### zxcvbn2

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #108 on: June 25, 2012, 12:22:57 pm »
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I think you miscalculated. with 20 points (6-4-2-0) as a threshold, there are actually 15 cases (6 of them being 2 first places and 2 second places) so the chance for 4p is 15/256. 15/256 and 5/81 are close enough to me.

This difference is caused by the quantization of the points. it does not give definite advantage to 3p or 4p however (depending on the threshold.)

Just to respond to someone says he's absolutely correct: in 4p the cases are
1-1-1-1 (1) 24 pts
1-1-1-2 (4) 22 pts
1-1-1-3 (4) 20 pts
1-1-2-2 (6) 20 pts

so this is 15/256.

in 3p
the cases are
1-1-1-1 (1) 24 pts
1-1-1-2 (4) 21 pts

this is 5/81.

Sorry if this sounds a bit harsh, but I usually find claim of correctness based on who you are kinda laughable.
Yeah, but this is still a problematic way of looking at it. There are this many ways of getting qualified, of course, but it is simply easier to get these cases form the 3p.
Importantly, it's not about getting to a point threshhold, it's about beating the other guys. The 3p are much more likely to produce a point total that is in the top X, as opposed to the 4p.
No it is not simpler for 3p, as the simple calculation shows. If you think otherwise, it's better to show via at least some qualitative arguments instead of just saying it...

As I said, using the points, getting 4th place/3rd place in a 4/3p game is as serious as getting 1st. So if the players are roughly equally skilled the above calculation is valid. You can generalize the calculation for a good player (say I just randomly choose that he can get #1 50% and #2 50% in a 4p game; in a 3p game to have the same expectation value of points he then should get #1 2/3 of the time and #2 1/3 of the time. Putting the numbers in then (suppose the threshold is 20 points again)

4p: 1/16+1/16*4+1/16*6=11/16
3p: 16/81+32/81=16/27.

Actually in this case 4p is better!

I *believe* in this case he's referring to something that is more difficult to describe analytically, which is that a better player is more likely to win a 3p game than a 4p game because the game itself is more swingy. In other words, suppose each was a weighted die, one with three sides and the other with four, but the 3-sided die  was more unevenly weighted than the 4-sided.

Councilroom may or may not have statistics to support this. I doubt it, because there are so few games among high-level players played with 3 or 4 players. I will say, it seems correct. This doesn't mean it is.
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#### WanderingWinder

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #109 on: June 25, 2012, 12:30:00 pm »
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I think you miscalculated. with 20 points (6-4-2-0) as a threshold, there are actually 15 cases (6 of them being 2 first places and 2 second places) so the chance for 4p is 15/256. 15/256 and 5/81 are close enough to me.

This difference is caused by the quantization of the points. it does not give definite advantage to 3p or 4p however (depending on the threshold.)

Just to respond to someone says he's absolutely correct: in 4p the cases are
1-1-1-1 (1) 24 pts
1-1-1-2 (4) 22 pts
1-1-1-3 (4) 20 pts
1-1-2-2 (6) 20 pts

so this is 15/256.

in 3p
the cases are
1-1-1-1 (1) 24 pts
1-1-1-2 (4) 21 pts

this is 5/81.

Sorry if this sounds a bit harsh, but I usually find claim of correctness based on who you are kinda laughable.
Yeah, but this is still a problematic way of looking at it. There are this many ways of getting qualified, of course, but it is simply easier to get these cases form the 3p.
Importantly, it's not about getting to a point threshhold, it's about beating the other guys. The 3p are much more likely to produce a point total that is in the top X, as opposed to the 4p.
No it is not simpler for 3p, as the simple calculation shows. If you think otherwise, it's better to show via at least some qualitative arguments instead of just saying it...

As I said, using the points, getting 4th place/3rd place in a 4/3p game is as serious as getting 1st. So if the players are roughly equally skilled the above calculation is valid. You can generalize the calculation for a good player (say I just randomly choose that he can get #1 50% and #2 50% in a 4p game; in a 3p game to have the same expectation value of points he then should get #1 2/3 of the time and #2 1/3 of the time. Putting the numbers in then (suppose the threshold is 20 points again)

4p: 1/16+1/16*4+1/16*6=11/16
3p: 16/81+32/81=16/27.

Actually in this case 4p is better!
I do give a qualitative reason. I mean, it's actually a numbers reason that I didn't provide numbers for.  The thing is that you need to score in the top 4 out of all the players. It's not about reaching any particular threshhold. If it were 'everyone who scores at least X points advances' then, if X were 20, your calculations would be valid. But there is no fixed value for X. My argument is that it is more likely for things to work out so that the qualifying value, whatever that turns out to be, will be met by a 3p than a 4p. But these things are not independent. How well 3p guys score, how well 4p guys score, has a direct impact on how many you NEED to score to get through. Of course it's also dependent on how many sign up, which makes it too much math to do, I think. If you can somehow explain to me that 20 points is the absolute threshold where below it, you can't qual, and above it, you can, then I will accept your calculations. But that's not what's happening. My entire point is that the threshold is NOT valid.

But ok, here's another reason. There's the inherent difference between 3p and 4p games. Of course, this means that some players may be better suited to one or the other, which is a big real issue, but let's even ignore that for the moment. Let's even use your example - 2/3 win, 1/3 2nd in 3p should be much easier to achieve than 50% win and 50% 2nd in 4-player. Just much much easier. This is down to the nature of the very games themselves - there's more randomness inherent to the 4p. This is the hugest source of inequity.

Of course, I don't have particular evidence to support this, and I don't think the data really exists either way. But I'm quite confident in my assessment of that, and I think most people will agree.

#### timchen

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #110 on: June 25, 2012, 12:50:53 pm »
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Do you mean that in terms of points, a good player will have a higher expectation value of score in a 3p game than in a 4p game using this point system? If that is the case then sure, playing in a 3p game is better. I have no reason to believe this is the case however.

If the expected score is the same but the variance is different, you can play with the numbers. But it seems to me the variance tends to be larger in a 4p game for a good player, simply because he has more places to be. If this is so then indeed 3p games are better for good players, as if his expected score is higher than the threshold he would like the variance to be as small as possible. But as I showed in a typical case above, at least we can say the difference is not that significant.

@WW:
for the threshold, I have no particular reason to choose 20 points, except it seems to get a similar outcome to our first day qualifiers. Now you can actually calculate for every possible threshold (say, from 15 to 24), and you will see that the advantage oscillates between 3p and 4p. I don't see how a fixed number entrant criteria can fix the threshold to be favorable to 3p in this case.

Quote
But ok, here's another reason. There's the inherent difference between 3p and 4p games. Of course, this means that some players may be better suited to one or the other, which is a big real issue, but let's even ignore that for the moment. Let's even use your example - 2/3 win, 1/3 2nd in 3p should be much easier to achieve than 50% win and 50% 2nd in 4-player. Just much much easier. This is down to the nature of the very games themselves - there's more randomness inherent to the 4p. This is the hugest source of inequity.
Yeah, by this you just mean the expected score will be higher for the same player in 3p. I dunno. I don't think so, but maybe you are right.
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#### Rabid

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #111 on: June 25, 2012, 12:55:00 pm »
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In the current format surely game 4 of each match only usually matters to 1 possibly 2 of the players playing it?
This opens up a huge potential for collusion.
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#### WanderingWinder

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #112 on: June 25, 2012, 01:01:20 pm »
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Do you mean that in terms of points, a good player will have a higher expectation value of score in a 3p game than in a 4p game using this point system? If that is the case then sure, playing in a 3p game is better. I have no reason to believe this is the case however.

If the expected score is the same but the variance is different, you can play with the numbers. But it seems to me the variance tends to be larger in a 4p game for a good player, simply because he has more places to be. If this is so then indeed 3p games are better for good players, as if his expected score is higher than the threshold he would like the variance to be as small as possible. But as I showed in a typical case above, at least we can say the difference is not that significant.

@WW:
for the threshold, I have no particular reason to choose 20 points, except it seems to get a similar outcome to our first day qualifiers. Now you can actually calculate for every possible threshold (say, from 15 to 24), and you will see that the advantage oscillates between 3p and 4p. I don't see how a fixed number entrant criteria can fix the threshold to be favorable to 3p in this case.
Sure, you can do it for every possible threshhold, but that still misses my point. My point is that the threshold is correlated to the other things, i.e. I think it's more likely for the threshold to be in a place that favours a 3p  than a 4p pod. The main point I'm trying to make in this section is that you aren't taking into account that the threshold is moving based on how people do, which is an important factor.
I missed where we were discussing a fixed number of participants.
I do actually think that the rotating number of participants is going to be a big thing too, and I seriously doubt that 20 points is going to be a very stable point for qualification. But this has more to do with how many people I expect to sign up, and a little with the inherent randomness of the game, than anything else.

Quote
Quote
But ok, here's another reason. There's the inherent difference between 3p and 4p games. Of course, this means that some players may be better suited to one or the other, which is a big real issue, but let's even ignore that for the moment. Let's even use your example - 2/3 win, 1/3 2nd in 3p should be much easier to achieve than 50% win and 50% 2nd in 4-player. Just much much easier. This is down to the nature of the very games themselves - there's more randomness inherent to the 4p. This is the hugest source of inequity.
Yeah, by this you just mean the expected score will be higher for the same player in 3p. I dunno. I don't think so, but maybe you are right.
Okay, I guess we just disagree here.

#### timchen

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #113 on: June 25, 2012, 01:38:20 pm »
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Quote
Sure, you can do it for every possible threshhold, but that still misses my point. My point is that the threshold is correlated to the other things, i.e. I think it's more likely for the threshold to be in a place that favours a 3p  than a 4p pod. The main point I'm trying to make in this section is that you aren't taking into account that the threshold is moving based on how people do, which is an important factor.
My argument is that in any case, the threshold should change in a continuous manner. That is, if the competition is fierce, the threshold moves down, if there are clearly better players, the threshold moves up. And since I think the threshold favors either 3p or 4p in a oscillatory manner (the difference being small anyway), I don't see how the dependency on various things can somehow always fix the threshold at some places that favors 3p. Like say, at 22 it favors 3p (just say that randomly, not that it is correct), now the threshold can change depend on a few things as said, it can move to 21 or 23 lets say. But at 21 and 23 it may very well favor 4p. and at 20 or 24 it can again favor 3p. You know what I mean.
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#### WanderingWinder

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #114 on: June 25, 2012, 02:04:44 pm »
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Quote
Sure, you can do it for every possible threshhold, but that still misses my point. My point is that the threshold is correlated to the other things, i.e. I think it's more likely for the threshold to be in a place that favours a 3p  than a 4p pod. The main point I'm trying to make in this section is that you aren't taking into account that the threshold is moving based on how people do, which is an important factor.
My argument is that in any case, the threshold should change in a continuous manner. That is, if the competition is fierce, the threshold moves down, if there are clearly better players, the threshold moves up. And since I think the threshold favors either 3p or 4p in a oscillatory manner (the difference being small anyway), I don't see how the dependency on various things can somehow always fix the threshold at some places that favors 3p. Like say, at 22 it favors 3p (just say that randomly, not that it is correct), now the threshold can change depend on a few things as said, it can move to 21 or 23 lets say. But at 21 and 23 it may very well favor 4p. and at 20 or 24 it can again favor 3p. You know what I mean.
Right. But I don't think that it's oscillatory, and I'm not at all convinced that the difference is small. At any rate, I don't feel like taking the time to calculate all that stuff out right now.

#### metzgerism

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #115 on: June 25, 2012, 02:17:05 pm »
+1

In the current format surely game 4 of each match only usually matters to 1 possibly 2 of the players playing it?
This opens up a huge potential for collusion.
I agree with this, with the current format, and I believe it's a bigger problem than points vs. wins and actually spins my opinions in the reverse a bit.

Let's say one player has won games 1-3.
Is there any reason to play game 4? Probably not.

Let's say one player has 2 wins, and another has 1.
Is there any reason for the other two players to play game 4? Probably not.

Let's say the first 3 games have been split, one apiece to three players.
Is there any reason for the 0-3 player to play game 4? Probably not.

We've already got the evidence that this is likely the case - you need at least 2 wins to advance, and even then it's probably a play-in of some sort. I prefer WTA obviously, and since that's likely what the brackets and championships will use, you've got to have it. Regardless, a point system will mitigate this, but absolutely cannot eliminate it. I mentioned my preference for a best-of-X/first-to-3-wins series - a first-to-2-wins may actually be the best option, with additional games played in 3p to find 2nd place in a pod.

That said, it's a little too late - we're just going to have to run with it as-is.
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#### metzgerism

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #116 on: June 25, 2012, 02:18:26 pm »
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Since we're changing it up for each qualifying round, maybe we can do the next round as a "first-to-2" or "first-to-3" wins scenario? The kingmaking problems become a little more clear that way, however.
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#### WanderingWinder

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #117 on: June 25, 2012, 02:21:30 pm »
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Since we're changing it up for each qualifying round, maybe we can do the next round as a "first-to-2" or "first-to-3" wins scenario? The kingmaking problems become a little more clear that way, however.
But you can't do that, because you have more pods than qualifiers.

#### knaacku

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #118 on: June 25, 2012, 03:18:29 pm »
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Here is the game 2 log for Fading, TheBlueFool, and knaacku that was missing prior.  We did have the order of 2nd and 3rd wrong, but I do not think that changes anything. (1st knaacku, 2nd TheBlueFool, 3rd Fading)

http://councilroom.com/game?game_id=game-20120624-120138-896ed538.html
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#### metzgerism

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #119 on: June 25, 2012, 04:39:58 pm »
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Since we're changing it up for each qualifying round, maybe we can do the next round as a "first-to-2" or "first-to-3" wins scenario? The kingmaking problems become a little more clear that way, however.
But you can't do that, because you have more pods than qualifiers.
Right, and I gave it a little thought and decided it's probably not worth it.

In a 3p "best 2/4," you're going to have an obvious problem: game 2 will, with ideal political play, never be won by the winner of game 1. Neither of the other players can allow it and survive. Game 3 returns back to a veritable free-for-all, with the exception that both previous winners would rather hand game 3 to the trailing player and finish it in game 4. Effectively, a best-of-4 makes for marginalized, politicized games 2 and 3. In 4p "best 2/5," the same issues arise in the same way for games 2, 3, and 4 (game 3 being a little odd this way). Even in best 3/9, the same problems exist, only with more chances to use and/or ignore them.

I do believe that the issue is with which format best represents a multiplayer game. One method, probably the cleanest, is to have single or double (or triple) elimination in one-off games - unfortunately, we've long established that one-offs are not even closely representative of Dominion skill, and having a loser's bracket as the mitigating factor doesn't really solve that. Another possibility is swiss, which is good because all players have similar stakes in all games - the main downside being that the length of the tournament needs to be long enough to be anything more than single-elimination in practice.

I'm not beyond saying that multiplayer Dominion just makes for bad tournaments...upon serious analysis, it all feels like a crapshoot.
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#### Polk5440

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #120 on: June 25, 2012, 04:41:56 pm »
+1

The problem we have isn't that we have a lack of tournament experience. We know how to set up two-player tournaments (or at least have a general idea about what's fair and what's not)....The trouble is that we're dealing with a type of tournament that we've largely avoided. We were forced to play 3-4 player matches, as to maintain a similar format to nationals/ world championships. This is something that we're pretty unfamiliar with, and we had a very short time to put it together.

That's a good point.

However, the two main problems, comparing across pods and how to deal with seating advantage, are present no matter the number of players, so it still may be helpful to think about how to structure tournaments from scratch with only two players even if you are interested in 4 player versions. For instance, what I proposed deals with these two problems for 2 players. The problems are just compounded with 4 players.

For three/four players, in order to eliminate comparing across pods one could either
1) Add an extra round to have playoffs among winners of pods (so we don't need to determine whether a winner of a three player game is the same as a four player game).
or
2) Have only four pods and more people and 4-player only matches within each pod round robin style. The problem with more than 2 players per game is how to pick a winner out of a pod.
a) One way this can be done is to play a "reasonable" number of matches and choose the Condorcet winner, if present, and sudden-death playoff among Condorcet ties otherwise.
b) Another way to do this is to play a "reasonable" number of matches (of 4 players per match) allocate a "point" to each winner in the match-up. To earn the point for the match, play first to three wins to earn the point (rotating seating appropriately). This preserves the inventive to win a little better than allocated points per place and playing a fixed number of games.

Really, it seems like most of the problems boil down to: In order to have better tournaments, more games need to be played. This is especially true with 4 players.
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#### WanderingWinder

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #121 on: June 25, 2012, 05:35:22 pm »
0

Since we're changing it up for each qualifying round, maybe we can do the next round as a "first-to-2" or "first-to-3" wins scenario? The kingmaking problems become a little more clear that way, however.
But you can't do that, because you have more pods than qualifiers.
Right, and I gave it a little thought and decided it's probably not worth it.

In a 3p "best 2/4," you're going to have an obvious problem: game 2 will, with ideal political play, never be won by the winner of game 1. Neither of the other players can allow it and survive. Game 3 returns back to a veritable free-for-all, with the exception that both previous winners would rather hand game 3 to the trailing player and finish it in game 4. Effectively, a best-of-4 makes for marginalized, politicized games 2 and 3. In 4p "best 2/5," the same issues arise in the same way for games 2, 3, and 4 (game 3 being a little odd this way). Even in best 3/9, the same problems exist, only with more chances to use and/or ignore them.

I do believe that the issue is with which format best represents a multiplayer game. One method, probably the cleanest, is to have single or double (or triple) elimination in one-off games - unfortunately, we've long established that one-offs are not even closely representative of Dominion skill, and having a loser's bracket as the mitigating factor doesn't really solve that. Another possibility is swiss, which is good because all players have similar stakes in all games - the main downside being that the length of the tournament needs to be long enough to be anything more than single-elimination in practice.

I'm not beyond saying that multiplayer Dominion just makes for bad tournaments...upon serious analysis, it all feels like a crapshoot.

I think you're over-estimating the effect of politicization. I mean, the semi-targeted attacks (does different things depending on player - spy, to some extent thief, and especially swindler), are the biggest thing here, I think, but even there, by helping opponent A to hurt opponent B, you're hurting yourself pretty badly.

#### WanderingWinder

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #122 on: June 25, 2012, 05:50:10 pm »
+1

Ok. I've woken up more now, and I have a way of phrasing my nebulous-sounding concerns better (I hope!), and I've got some math. What timchen is looking at is exactly what to do to figure out the probability of any one player reaching a certain threshold of points. Incidentally, here's the breakout of that, from 15 points up:
4p chance              Points      3p chance
1/256  ~= 00.39%   23-24      1/81  ~= 01.23%
5/256  ~= 01.95%   22              1/81  ~= 01.23%
5/256  ~= 01.95%   21              5/81  ~= 06.17%
15/256 ~= 05.86%   19-20      5/81  ~= 06.17%
35/256 ~= 13.67%   17-18      15/81 ~= 18.52%
66/256 ~= 25.78%   16              15/81 ~= 18.52%
66/256 ~= 25.78%   15               31/81 ~= 38.27%

As we can see, 19-20 is by far the closest, and the rest have pretty substantial gaps.

However, the important thing comes here: what will the threshold actually be? To figure out what you'd expect this to be, you'd need to know what the maximum score from any given pod will be, and then how many pods there are. And it's going to be different for 3p pods and 4p pods, so you actually need to know how many of each type you have.
Now, if we make a few assumptions, we can actually break this down. For now, let's assume that every game is independent of every other (bad because of player skill and seating order), and that every person is equally likely to get every place in every game.
Note that we can't just use what timchen started and I expanded on above, because the different players in a pod are NOT independant of each other. i.e., if one guy gets 1st 4 times, we know that nobody else got first, so those possibilities get sliced out, if somebody gets 1st twice, nobody else can get 1st more than twice, if two guys get first twice, nobody else can get first, etc.
Anyway, in a 3p pod, there are 6 distinct orderings that any game can finish in. 4 games makes 1296 orderings, but we get one factor of 6 back because at the end, if we switch all the As for Bs, or all the Bs for Cs, etc., this doesn't make a substantive difference (i.e. ABC ABC ABC ABC == BCA BCA BCA BCA == BAC BAC BAC BAC == CAB CAB CAB CAB, etc.).
So we have 216 possibilities. Now, I'm not going to sit and count them all out right now, because I don't have time. But just take the top case, 24 points. To get 24 points, someone had to score perfect. The other guys could have (B gets 2nd 4 times, C gets 3rd 4 times; B gets 2nd thrice; B gets 2nd twice, etc.).

...aaaaand I realize my counting brain has gone offline here. Someone else who is skilled in the mathematical arts want to progress this?

#### timchen

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #123 on: June 25, 2012, 07:33:20 pm »
0

... lol. Was looking forward to the following.

Also the delta between 4p and 3p advancing chances is indeed oscillatory as a function of threshold.

Coincidentally, this suggest that setting a fixed cut at 19-20 is probably a good idea (instead of fixing only 4 people can proceed.)
« Last Edit: June 25, 2012, 07:37:23 pm by timchen »
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#### WanderingWinder

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##### Re: Re: DominionStrategy Qualifying Day Results
« Reply #124 on: June 25, 2012, 10:34:24 pm »
0

Okay. Think I'm counting right again now? 3p pod means 216 order-invariant possible result lines, with 8 of these giving perfect scores, for a ~3.7% chance. 4p pod means 13824 lines with 216 of them giving perfect scores, for a rate of only 1.5625%. As the number of possibilities grows, though, it becomes increasingly difficult to count them, because there's LOTS of different cases, especially for the middle ones like 18 points. But the point is, a 3p pod is more likely to produce players at these higher point threshholds, even if any particular player may not be so much favoured. Which is important insofar as the 3p pods increase the threshold more quickly than 4p pods do.
Of course, this is just intuitive also. If I add an extra player to the pod, it is more likely that the top player in that pod will score worse, even with the added points injected. Certainly you see this at the top end, which is what is really important.
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