Two Tier Scoring

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jay65536
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Re: Two Tier Scoring

#21 Post by jay65536 » Tue Dec 31, 2019 12:53 pm

Again, longer reply later, but I almost laughed out loud when I saw this:
If there is an n-way draw and the board leader has more than the 'magic' number of SCs, and you are not the board leader, the way I have formulated my scoring system, you always receive 1/(2(n-1)) of the pot (you can check). If there is an n-way draw with a shared board lead above the 'magic' number of SC's, you can simply follows the same formula: so the non-board leader receive 1/(2(n-1)) of the pot, and the rest of the pot is divided among the board leaders.
Up to a change in how the board leader's score is determined, this is exactly the same mechanism as my system. (You can check.) More complicated, indeed.

Mercy
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Re: Two Tier Scoring

#22 Post by Mercy » Tue Dec 31, 2019 2:34 pm

Yes, as I said in my first post here, your scoring system is quite similar to the scoring system I proposed years ago on the old forum. It is considerably simpler, though; however I admit that the way I wrote about it in the part of my post you quote is too difficult to understand; it should be possible to write it down in a better way.

Mercy
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Re: Two Tier Scoring

#23 Post by Mercy » Tue Dec 31, 2019 2:42 pm

Just for clarity, I no longer fully stand behind the following, because of the reasons that Restitution pointed out.
Mercy wrote:
Mon Dec 30, 2019 8:31 am
Interestingly, if my above formulation is being used, it is only a little extra tweak to formulate Two-Tier Scoring as follows:
A solo is worth the full pot. In case of a draw, any player with more than 12 supply centers receives a proportion of the pot equal to the proportion of supply centers he or she controls. The rest of the pot is distributed evenly among the other surviving players.
It's simple, clean, and, I think, has the same upsides as the version of Two-Tier Scoring proposed in the blog.
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Mercy
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Re: Two Tier Scoring

#24 Post by Mercy » Tue Dec 31, 2019 3:30 pm

I think the following is a more elegant way of writing down 'my' scoring system and how it treats all the special cases.
If a player wins the game, that player is awarded the full pot.

If the game ends in a draw, and no player has at least 14 supply center, the pot is split evenly between the surviving players.

If the game ends in a draw, and at least one player has at least 14 supply centers, then the player with the most supply centers is awarded half of the pot*. The rest of the pot is split evenly among the other surviving players.

*If two players have at least 14 supply centers and they are tied in supply center count, each of them is awarded a quarter of the pot plus half of what any other player is awarded.
The scoring system jay65536 proposes is very similar to this. The main differences are the following (but please correct me if I am wrong):
- The award for topping the board is not half the pot, but proportional to the number of centers you control, if your number of centers is above a certain threshold (which depends on the size of the draw).
- There is also an award for 'almost' topping the board, namely if you are within 1 center of the board topper.

Summary of my thoughts on it:

I find it an interesting idea to make the award for a board top depend on the number of centers you control. I think the upside is that it takes away some 'jumps'. In my proposed system, if you get 14 centers, you suddenly jump a lot in reward. In jay65536s system that is more gradual. The downside I see it that it makes the scoring system a tad more complicated.

I am against making the threshold dependent on draw size; I think it is fine to make it 12 centers in all cases instead of making it 6 centers in case of a big draw. I have argued before why I think so, as well as Restitution (I agree with him on this).

Specially awarding 'almost' board toppers sounds interesting to me as well. It would cause a little bit more competition among the strongest players. Ultimately I don't think it is worth making the system more complicated, though.

jay65536
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Re: Two Tier Scoring

#25 Post by jay65536 » Tue Dec 31, 2019 4:17 pm

OK, I think I have time to write a real reply now.

The "TL;DR" of this post is basically, I read all your responses. Some of the issues that have been raised, I'm not worried about, for reasons I'll explain below. Some seem legit, but are issues that I am hoping to be able to work out in play-testing, not in theoretical debate.

So, let me start with the complaint that the system is too hard to understand. Based on what I have read, I am still convinced that my error here is not in making a system that is too hard, but in writing up the system in a way that obscures what it's really doing. (I think the evidence for that is that Mercy's proposed alternative is to use my scoring mechanism, just with a different proposal for what the top scores should be. And I think that the issues with the top scores are things that should probably be hammered out in play-testing, rather than just debated.)

Here's an alternate explanation for how my system really works. The idea is that in any draw where there is one sole board leader who leads by more than 1 center, that one player receives the top score, and all other draw participants divide the remaining points.

So...that seems pretty easy, right?

The reason why it LOOKS more difficult is because there are two conditions the system has to account for. One is ties at the top, and the other is draws that do not include all survivors.

The way my system deals with ties is exactly the same as Mercy's proposed addition to his (her?) own system above. If you're not the board leader, we compute what your score would have been if there had been only one board leader. Then all the non-board leaders get that score, and the board leaders split the rest. (This is the language about assigning the "bottom score" to all powers that finish off the lead.)

Everything else that makes the system look complicated is actually just adjusting to create one set of rules that covers non-DIAS draws. For example, let's say a 2way draw is voted while the top 2 powers are tied at 15. We want those players to still get 90 points each, right? So there needs to be a special case carved out in the rules for that. That's why (2)(a) exists. Notice that in a DIAS game, the only possible way to achieve a 2way draw is a 17/17 split, and then that rule would be unnecessary because the top score for 17 centers is already 90. So (2)(a) is not an exceptional case, but a special case.

Another example would be a 3way draw that is voted early, before anyone has broken into double-digit centers. Imagine, for example, that the center counts are something like 9/7/7/4/4/3, and the bottom 3 powers are voting themselves out. A "normal" application of my scoring mechanism would have the 9-center power getting 57 points and the 7-center powers each getting 123/2 = 61.5. But having the top power earn less is an aberration, so we need a special case to cover this. That's why (2)(b) exists. Notice that in a DIAS game, it is not possible to have a 3way draw without at least one power having at least 12 centers (since 34/3 is between 11 and 12), so this rule would never be necessary in a DIAS game.

Finally, (2)(c) exists because 7 centers is literally the smallest center count it is possible to have while leading by more than 1 center. So below that, I don't care to have my system distinguish between draw participants. (Also, how often is a game going to end that way??) I suppose it would be possible to try to compute reasonable top scores for lower center counts, and eliminating rule (2)(c) in the process, but it seems like it hardly matters at that point. This is totally something that I would be willing to revise after play-testing, though.

So, yeah, if that makes my system seem simpler, then I guess I messed up in the way I originally chose to describe it. But it really is a simple system--the top score is fixed for each center count, and everything flows from there. And furthermore, a lot of what is being described as "too much" just totally falls away when you simply make draws include all survivors.

The second complaint that I feel like I should address is Claesar's comment that the system wouldn't be good for FtF tournaments. Claesar, I'm guessing you play exclusively in Europe? In North America, a lot of what you said about FtF tournaments simply isn't true. North American tournament games are actually played with no fixed end year, unlike European tournaments. (Most tournaments have "timed" final rounds, by which is meant, the end time of the game is a random time--not year--selected from a range.) By your argument, in theory, that should mean that it would be OK to have draw-based scoring in a NA tournament. And indeed, some tournaments in NA do still use systems that are primarily draw-based. (The Dixiecon system, for example, is primarily draw-based with a secondary center count component. It also differentiates losses by rank and center count, as a tertiary component.)

However, even in North America, Carnage and SoS have almost entirely uprooted draw-based scoring. So the idea that draw-based scoring would be used, except for the existence of timed rounds, is contradicted by experience. There is another reason why draw-based scoring is not used in FtF tournaments.

That reason is a segue into the last point, which addresses the array of comments about how the system is no good because it removes the importance of draw-whittling. Removing the importance of draw-whittling is the point.

One of my motivations for trying to invent a new system is because I wanted to see if I could make a system that preserved good parts of every system I like, while cutting out the bad parts. (When it comes to scoring systems, I'm a pluralist: there are not many widely used systems that I would say I categorically dislike.) Draw-whittling is a bad part of draw-based scoring. There is no theoretical argument for why this is the case; it is the case because it is the documented experience of an entire generation of tournament players. And it is for this reason, not the necessity of adapting to timed games, that explains why draw-based scoring has declined in use in North America. Experience says that the games simply become more fun when the top powers are able to take a large draw without their score being affected.

So, while giving a good reward to solo bids is one of the things my system does, having a top score for all board-leading center counts has the separate effect of undercutting the need for all of the top powers to collaborate to engage in draw-whittling. This is intentional.

That's why Mercy's proposed "simplification" is actually a sizable alteration, in my view, and I see it as a shift away from one of the goals of the system. If we just say all draws are equally split until someone has 13+ centers, that means that until we reach the point where someone grows large enough, draw-whittling is still a major motivator of play. My hope--and once again, this is something that I'd like to actually play-test--is that until the game reaches a state where someone has >12 centers, the motivation to play on would be for someone close to the top rank to try to grab it, or else for the board leader to extend their center lead instead of trying to whittle the draw down. I expect (and again, I can't know this without play-testing) that there would not be a whole lot of 60/40/40/40 draws that actually get voted up, but the negotiations over how to resolve the game would change.

Now, if you actually like draw-whittling, then you can just ignore my system; it's not for you, and that's OK with me.
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Restitution
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Re: Two Tier Scoring

#26 Post by Restitution » Tue Dec 31, 2019 5:47 pm

jay65536 wrote:
Tue Dec 31, 2019 4:17 pm
stuff
For the sake of my sanity, can you reformulate the system assuming that DIAS is true (which it is in webdip)?

Also: I think it is tautologically true that if you want to reward small-SC powers making it to the draw, you *necessarily* also reward draw whittling, simply by virtue of scoring being zero-sum. There is necessarily some compromise. I think you are actually underselling your system in this post; it provides incentives for top players to yeet their scores, middling players to: make it to top tier, suppress the top tiers, slim the draw, and small players to: stay in the draw, suppress top tiers.

Having competing incentives is good, you're underselling that draw slimming is still a part of your system.

Why is PPSC no longer supported on webdip? That seems like the most elegant system to me.
Last edited by Restitution on Tue Dec 31, 2019 5:52 pm, edited 1 time in total.

Claesar
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Re: Two Tier Scoring

#27 Post by Claesar » Tue Dec 31, 2019 5:48 pm

I've only played CarnageCon outside of Europe. Unsurprising, it featured Carnage scoring. It's the epitome of rank-based scoring. Technically most rounds were unlimited, but the other players were unhappy when I vetoed a Draw at 10PM (I had a shared board top). They were relieved when I accepted at midnight-ish. We were the last board playing

The same organisation will host Worlds next year.

teccles
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Re: Two Tier Scoring

#28 Post by teccles » Tue Dec 31, 2019 5:52 pm

Jay: apologies if I’ve missed this, but could you spell out a little more why your top scores are what they are? I do think one reason your system feels complicated is that these values don’t have an obvious explanation.

Restitution
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Re: Two Tier Scoring

#29 Post by Restitution » Tue Dec 31, 2019 5:53 pm

teccles wrote:
Tue Dec 31, 2019 5:52 pm
Jay: apologies if I’ve missed this, but could you spell out a little more why your top scores are what they are? I do think one reason your system feels complicated is that these values don’t have an obvious explanation.
I agree, there's no clear reason why each SC count has a given score.

This is why I like Mercy's system, where players above 12 just get an amount proportional to their SC count. Very easy to calculate.

jay65536
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Re: Two Tier Scoring

#30 Post by jay65536 » Tue Dec 31, 2019 7:12 pm

The lack of a "multi-quote" feature on this forum is getting very pronounced for me, as I try to reply to these questions.
For the sake of my sanity, can you reformulate the system assuming that DIAS is true (which it is in webdip)?
Sure. I'll also separate what happens when there is not a tie at the top, from when there is, since that may also help clear things up.

1. Each game is worth 180 points.

2. All solos get 180 points. Any draw in which no one has more than 6 centers is equally split.

3. In any draw in which one player leads all other players in center count by more than one center, that player receives the top score, which is a function of that player's center count. All other draw participants split all remaining points.

4. In any draw in which multiple players are within one center of the top center count (and that top center count is at least 7), the draw participants who are not within 1 center receive the same score they would have received in (3), and the other draw participants split the remaining points.
[C]ould you spell out a little more why your top scores are what they are? I do think one reason your system feels complicated is that these values don’t have an obvious explanation.
I buried the reason inside the original article and didn't talk about it much, so it's understandable that you missed it. Let me explain how I got them.

Obviously in a fixed-sum system where losses receive no points, a 17/17 2way draw has to be 90 points (out of 180). That is why the top score for 17 centers is 90: because that way, if you hit 17 centers, you have no scoring incentive to knock people out to make a 2way draw.

Similarly, a 3way draw with equal center counts ought to be worth 60 points. But because there are 34 centers, the closest it can possibly be is 12/11/11 or else 12/12/10. That is why the top score for 12 centers is 60 points: because that way, if you hit 12 centers, you have no scoring incentive to knock people out to make a 3way draw. (It's also one reason why I want being within one center of the top to qualify for the top tier: because in a 12/11/11 scenario, it seems extremely unfair that the 12-center power should get extra points when this is mathematically the closest 3way draw possible.)

The top scores for 13-16 centers are what they are because I wanted to make each center between 12 and 17 worth the same amount of points. 30 divided by 5 equals 6, so that's why it goes 90/84/78/72/66/60.

The top scores for all lower center counts are what they are for a similar reason: to prevent someone from gaining points without gaining centers. For example, why is the top score for 10 centers 60? Because if it were any less, someone leading the board on 10 centers would have a scoring incentive to set up a 12/12/10 3way draw with them still being on 10--that is, they have an incentive to give up their center lead to whittle the draw down. That's the scenario I don't want.

Other center counts work on the same principle, just with different numbers. For example, 7 centers: I want a 7-center board lead to be worth at least as much as a 7-center small draw. The most that can be worth is 54 (when it's 14/13/7), so that's why that top score is what it is.

When spelled out this way, of course, you might ask why I can't just extend this down to cover all center counts down to 4 (it would go 54, 51, 51 for 6, 5, and 4, respectively) and then get rid of the rule that draws are split equally in everyone-under-7 draws. I guess I could; it just didn't seem that important since that's really a fringe case. (And I guess if I'm being really honest, it's also a holdover from an earlier version of the system that I abandoned.)

You might also ask why the top scores have to be exactly equal to the 3way draw score, as opposed to larger. That's because to do so would devalue a solo. I don't want someone who finishes with three 12-center 3ways to be ahead of a solo and 2 eliminations just because one or more of them was a 12/12/10.

And Restitution, I agree with you that the conflicting endgame incentives are a selling point of my system.
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Restitution
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Re: Two Tier Scoring

#31 Post by Restitution » Wed Jan 01, 2020 1:08 am

jay65536 wrote:
Tue Dec 31, 2019 7:12 pm
stuff
Am I right in thinking that a leading player's score should be equal to their score in the worst-case 3wd? Or should it be equal to their score in the best-case 3wd?

jay65536
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Re: Two Tier Scoring

#32 Post by jay65536 » Wed Jan 01, 2020 2:36 am

It's the best-case small draw. That means a 2way if you have 17 and a 3way otherwise. The worst-case 3way draw is when you allow someone to reach 17 and you have 15 or fewer--it's always 45 points regardless of your own center count.

tr1285
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Re: Two Tier Scoring

#33 Post by tr1285 » Wed Jan 01, 2020 4:51 pm

I think I would like this system as an improvement over the current options. One thing I'm not sure about is if you should disincentivize whittling down the draw so much as to allow the top player the same score regardless of draw size.

After crunching some numbers I think my own preferred system might be a hybrid between the Calhamer Points and a simple points-per-center approach.

For a pot size of 60 + 34n:
1. A Solo takes all the points of course
2. In any draw, 60 points are split equally among the participants, plus each player gets n points per center.

For n=5 (my choice):

Draw / Points / Percentages
14/10/10 = 90, 70, 70 = 39.1%, 30.4%, 30.4%
15/14/5 = 95, 90, 45 = 41.3%, 39.1%, 19.6%
16/16/2 = 100, 100, 30 = 43.5%, 43.5%, 13%
12/9/7/6 = 75, 60, 50, 45 = 32.6%, 26.1%, 21.7%, 19.6%
11/11/6/6 = 70, 70, 45, 45 = 30.4%, 30.4%, 19.6%, 19.6%
8/8/7/6/5 = 52, 52, 47, 42, 37 = 22.6%, 22.6%, 20.4%, 18.3%, 16.1%

teccles
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Re: Two Tier Scoring

#34 Post by teccles » Wed Jan 01, 2020 7:55 pm

Jay: Thanks for the explanation on the rationale for top scores, that makes perfect sense. I wonder whether, in practise, you don't need to worry so much about people draw-whittling. For example, your 7 centre top score is based on a rather absurd scenario, where a board leader on 7 centres has the power to ensure a 7/13/14. So it might be fine to change the top score to something simpler (with SCs/34 being a natural option), despite the theoretical issues with that.
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jay65536
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Re: Two Tier Scoring

#35 Post by jay65536 » Wed Jan 01, 2020 8:24 pm

teccles wrote:
Wed Jan 01, 2020 7:55 pm
Jay: Thanks for the explanation on the rationale for top scores, that makes perfect sense. I wonder whether, in practise, you don't need to worry so much about people draw-whittling. For example, your 7 centre top score is based on a rather absurd scenario, where a board leader on 7 centres has the power to ensure a 7/13/14. So it might be fine to change the top score to something simpler (with SCs/34 being a natural option), despite the theoretical issues with that.
I mean, it's not just the assumption that it could happen in the same game; it's the straight fact that I don't want a 7-center 3way to be worth more than a 7-center board top. In the set of all X-center finishes, I want board tops to be the highest possible score for all X.
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Restitution
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Re: Two Tier Scoring

#36 Post by Restitution » Tue Jan 07, 2020 2:52 am

I think I have come up with a better two-tier system. Though, maybe it was proposed earlier and I am just rephrasing:

Given that a 17-SC finish should give exactly 50% of the pot, and given that no leader should earn less than 33.333% of the pot (because this is what they would make in a 3WD in DSS, and the leader should not care about cutting members of the draw):

The reward for a leader should scale between 33.333% and 50%, depending on their performance.

33.3333% * 34 SCs = 11.333, rounds to 12.

So, here is a system: anybody at or above 12 SCs earns a share of the pot exactly equal to their portion of the SC count. (This is because 12 is exactly the point at which a player would make more points through PPSC than through a 3-way DSS).

The remaining pot is split among all other players, equally.

I think it achieves everything we have been looking for: The 1 or 2 leaders are totally disincentivised from cutting small powers, and only look for growth - they rationally just grab as many SCs as possible (which means, basically, they put themselves on track for a solo). The middle of the pack want to a) cut down the leader's lead, b) prevent the leaders from soloing, c) get into a leadership position, d) cut out small powers (all depending on their assessment of the situation). And the small powers are encouraged to try to get a wide draw without a big leader.

It also prevents more than 2 leaders from existing, which is possible with some of the other systems proposed.

Furthermore, it's very easy to explain. Some people get DSS, some people get PPSC, and there's no complicated calculation about who is and isn't a leader: Either you have more than 33% of the SCs or you don't.

Does anybody see a downside with this system?
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Restitution
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Re: Two Tier Scoring

#37 Post by Restitution » Tue Jan 07, 2020 3:04 am

jay65536 wrote:
Wed Jan 01, 2020 8:24 pm
I mean, it's not just the assumption that it could happen in the same game; it's the straight fact that I don't want a 7-center 3way to be worth more than a 7-center board top. In the set of all X-center finishes, I want board tops to be the highest possible score for all X.
Okay, so, here is an amendment/reformulation of the system I just proposed that deals with leaders in games without anybody >= 12 SCs.

In any game, the top two players *always* earn PPSC, regardless of their SC total. If >2 players are tied for top, then there is no top. If there is a top but >1 players are tied for second, there is only one top player.

This means, if the leader is at 7 and a draw occurs now, he will earn precisely as much as he would earn if the game went to a 3WD conclusion and he was still at exactly 7 SCs - because any 3WD is effectively PPSC for everyone.

Then, all remaining players split the remaining pot by DSS.

EDIT: to clarify, by PPSC I mean ((player's SCs) / (all SCs)), not ((player's SCs) / (all owned SCs)). In any edge cases where a leader would end the game with more points through DSS than through PPSC, they are given DSS instead.

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Re: Two Tier Scoring

#38 Post by jay65536 » Wed Jan 08, 2020 1:51 pm

So, I don't know how best to structure my reply to emphasize the main point I want to make, so I guess I'll just lead with that.

I don't think anyone here is going to convince anyone else of whose theoretical system is better. The point is supposed to be to see if people want to try play-testing. In that sense, I suppose my idea is stalled, since I don't think it can be further improved by debate--I just want to see what happens in games where people are playing to the system, and if adjustments are necessary I can make them off of that.

OK, so with that out of the way...Restitution, I think your proposed system is significantly more different from my system than you think it is. I wouldn't presume to say which is "better" because, as I said above, neither has been tested, but I can tell that they are different. (Although I will say that in developing my system, I considered a version that's somewhat similar to your proposal, so that's why I know they're different.)

The way your system works, if I understand correctly, is that for any fixed pot size (let's say 180 for comparison's sake), the top 2 powers get points proportional to their center counts, and the remaining points are divided equally among all players, with the caveat that if 3+ players are tied for the lead, the draw is equally split, and if 2+ players are tied for second, there is only one board topper who gets center points.

So, here are just 2 ways I notice your system is different:

1) It removes the value of being the board leader, and actually takes away a lot of the incentive for the top 2 powers to compete with each other. For example, in an 11/11/6/6 scenario, your system has the top two powers both earning ~58.24 points (out of 180) and the bottom two powers each earning ~31.76 points (out of 180). At the same time, because the top 2 powers' point totals will simply depend on their center counts, there is no real need for them to fight; while they don't care about draw-whittling, they also care less about competing. The 11-center powers don't care where they get their next centers from. However, the bottom 2 powers do care about draw-whittling.

In my system, the same 11/11/6/6 would play differently. Now each 11-center power would earn 50 points, and each 6-center power would earn 40. Notice that the reward for the small powers is higher in my system; that's intentional, and it's because I'm penalizing the two tied powers at the top for neither of them creating "separation" and actually topping the board. If instead it were 12/10/6/6, the 6-center powers' scores would be unchanged, and the top power would now "take" 10 points from second place, making it 60/40/40/40. That is a much higher point swing than it is in your system, and again, that is intentional. I want that board-topping reward to be there, and I want the second-place power to have motivation to fight the first-place power.

2) In endgames with multiple large powers, the scores for small powers who survive into the draw are quite small compared to my system. This is particularly pronounced when there are more than 3 powers in the endgame. For example, let's say we have a 16/12/6 3way draw. Again on a 180-point scale, your system breaks down ~84.71/~63.53/~31.76. The 16-center power is rewarded similarly to my system, but the two smaller powers are certainly not! My system would break down 84/48/48, basically awarding points to the 16-center solo threat and making no distinction between the two powers who are combining to stop the solo. In so doing, it ups the reward for the 6-center power a bit (from the rough equivalent of an equal 6way to the rough equivalent of an equal 4way) and penalizes both smaller powers for allowing someone else to reach 16. This is intentional.

But what happens if it's 16/12/4/2? Now in your system, the top two scores are unchanged, and the already scant reward for the small powers is split in half. The 4- and 2-center powers are each now earning less than 10% of the total points. My system would break this down 84/32/32/32--basically equally penalizing all 3 bottom powers for neither topping the board nor eliminating enough opponents. (And 32 points is at least close to 20% of the points.)

There are others, but this is long enough. The point is just to show that the two systems can be quite different in games that aren't roughly equal 3ways.

Claesar
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Re: Two Tier Scoring

#39 Post by Claesar » Wed Jan 08, 2020 2:05 pm

Jay's point #1 would matter a lot for F2F. That's an important perk in tournaments.

Puscherbilbo
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Re: Two Tier Scoring

#40 Post by Puscherbilbo » Wed Jan 08, 2020 10:42 pm

Did not yet find the time to read all your posts. so apologies if the system has been suggested already.
How about splitting scores 50/50 between SoS (or other ranked distributions) and DSS.
Allowing for a more meaningful reward for small nations while still rewarding to some degree the bigger powers.

The point that should be avoided at all costs is players willfully allowing solos to lvl the playingfield in a multiroundcompetetion.
Also i would like to avoid that medium powers automatically settle for draw and there is no reward or incentive other than to eliminate players.
So a system should basically reward both eliminations and centers

However i would disagree that 2 draws with 17 centers are worse than 1 solo and 1 bust. There should be a higher jump than from any other centerincrease and significantly so.

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