Two Tier Scoring

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swordsman3003
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Two Tier Scoring

#1 Post by swordsman3003 » Sun Dec 29, 2019 11:42 pm

A proposal for a new zero-sum scoring system.

Not written by me, but I am hosting this proposal as a guest post.

I am however really interested in this idea.
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Re: Two Tier Scoring

#2 Post by Claesar » Mon Dec 30, 2019 7:28 am

It looks very interesting and I like the premise.

One thing I would like to clarify after reading the article:

Lead/Rank-based scoring is used almost exclusively in Face-to-Face, because of the relatively fast end (typically predetermined at something between 1906 to 1911). As a result elimination is hard and most games end in a 5-way draw. DSS would be a horrible choice in these games because there's a need to differ between players for the final ranking.

Two-tier scoring would also not be great at F2F for this same reason. C-diplo and Carnage are the most common ones there.
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Re: Two Tier Scoring

#3 Post by Mercy » Mon Dec 30, 2019 8:31 am

Wow.

That blog post is something I could have written myself, if it were not for the fact that I usually don't spend time writing long and elaborate posts like that. In fact, long ago (shortly after the introduction of SoS) I have written a shorter version of something like this myself on the old forum, where I suggested a new scoring system that has quite some similarities with this one. I am not exaggerating when I say that literally every point under "Part 2: Why A New System?" and "Part 4: Why Use the Two-Tier Scoring System?" directly applies to the scoring system I had suggested, and my reasoning behind suggesting it, as well.

It shouldn't some as a surprise that I agree with everything jay65536 says here, and I have nothing to add. I will now discuss the scoring system I proposed years ago as well as some details in the scoring system of jay65536.


My scoring system

The scoring system I had suggested years ago was based on the observation that there are 'two' types of draws:
1) A draw where every remaining player agrees to draw;
2) A draw where there is a solo threat and a stalemate line is formed. The game is in a deadlock and is drawn out of necessity.
If the draw is of type (1), the pot should be rewarded according to DSS scoring. If the draw is of type (2), the board leader should get half the pot and the rest of the pot should be evenly split between the other remaining players.

The challenge then is making a scoring system that properly differentiates between the two. One way of doing it is just making two draw buttons. Say, make a button 'Draw' that splits the pot evenly and a button 'Deadlock' that gives more to the board leader. I could see some problems in the implementation of that, though, since it is a requirement that everybody needs to understand the difference between the two buttons. Another possibility is to say that the draw is of type (2) if and only if the board leader is above a certain number of centers, e.g. 14.


Commentary on jay65536s Two-Tier Scoring system

I think this scoring system is better than both DSS and SoS, for the reasons you mentioned in "Part 2: Why A New System?" and "Part 4: Why Use the Two-Tier Scoring System?". I also like that it handles the situation where two players get close to a solo better than my scoring system does (if it had a way of handling that properly, I cannot remember).

I would suggest to change the following
jay65536s wrote:Points are split equally among draw participants in the following cases: (a) any 2-way draw (90 each); (b) any 3-way draw in which no one has more than 12 centers (60 each); (c) any draw in which no one has more than 6 centers.
into the following:
Points are split equally among draw participants if no one has more than 12 centers or in case of a 2-way draw.
I suggest this change for the following reasons.
1) My formulation is simpler. Players want to be able to easily calculate their score in different hypothetical situations, or at least get a feel for it. Therefore, all other things being equal, I think my version is better.
2) The reason you want to reward the board leader with extra points in some cases is because you don't want to disincentivice players to go for a solo attempt. However, if the board leader has no more than 12 centers, that would be a pretty weak solo attempt, isn't it? Hence that reason does not apply. So if no one is in 'solo-danger' territory, why not split the pot evenly?

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.


Other commentary

As a remark, I have noticed that both DSS and SoS use the following formula to calculate your share of the pot in case of a draw. Your share of the pot is equal to the following:
(Your number of centers)^n / [SUMOVERALLPLAYERS (That players number of centers)^n]
In SoS, n = 2. In DSS, n = 0. Interestingly, we are actually only using extremes here. Why does no one use n = 1, which rewards point proportional to supply center count? We could hypothetically even use e.g. n = 1/2 (Sum of Roots scoring).
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Re: Two Tier Scoring

#4 Post by Octavious » Mon Dec 30, 2019 11:22 am

So the intent of this is to reward people for failed solo attempts?

Surely the only reward required for attempting a solo is the possibility of a solo? That's reward enough for any red blooded Diplomacy player. The only real benefit of this seems to be to those people who persistently fail at their solo attempts, due to a combination of poor luck and judgment, and I would argue that there is little to be gained by encouraging those people to carry on as they are, and a lot to be gained by encouraging them to change how they try and solo.
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Re: Two Tier Scoring

#5 Post by Mercy » Mon Dec 30, 2019 12:13 pm

I'd say the intent is to not disincentivice players to try to solo. If you try to get a solo, but ultimately fail, wouldn't you say that you don't deserve a lower score than if you hadn't tried anything, but just whittled down the draw?
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Re: Two Tier Scoring

#6 Post by Octavious » Mon Dec 30, 2019 12:33 pm

Mercy wrote:
Mon Dec 30, 2019 12:13 pm
I'd say the intent is to not disincentivice players to try to solo. If you try to get a solo, but ultimately fail, wouldn't you say that you don't deserve a lower score than if you hadn't tried anything, but just whittled down the draw?
No, I would not. Diplomacy is not a game that should give you a pat on the head and a well done sticker for trying hard.
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Re: Two Tier Scoring

#7 Post by Mercy » Mon Dec 30, 2019 12:53 pm

Octavious wrote:
Mon Dec 30, 2019 12:33 pm
No, I would not. Diplomacy is not a game that should give you a pat on the head and a well done sticker for trying hard.
Yet that is what we do regardless by awarding players points if they get in a draw. The question is how we should distribute the points.
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Re: Two Tier Scoring

#8 Post by Octavious » Mon Dec 30, 2019 1:10 pm

Mercy wrote:
Mon Dec 30, 2019 12:53 pm
Octavious wrote:
Mon Dec 30, 2019 12:33 pm
No, I would not. Diplomacy is not a game that should give you a pat on the head and a well done sticker for trying hard.
Yet that is what we do regardless by awarding players points if they get in a draw. The question is how we should distribute the points.
A small sized nation that secures a draw has achieved its end game aims and deserves some recognition. You believe that a large nation that spends the end game failing to solo deserves more?

A draw is a draw and should be rewarded as such. The bonus a larger nation receives is the opportunity to try for a solo. The disadvantage a smaller nation endures is the risk of elimination. If the fail to take their opportunities or succeed in avoiding elimination they deserve the same reward for the same final result.
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Re: Two Tier Scoring

#9 Post by Mercy » Mon Dec 30, 2019 1:26 pm

Octavious wrote:
Mon Dec 30, 2019 1:10 pm
A small sized nation that secures a draw has achieved its end game aims and deserves some recognition. You believe that a large nation that spends the end game failing to solo deserves more?

A draw is a draw and should be rewarded as such. The bonus a larger nation receives is the opportunity to try for a solo. The disadvantage a smaller nation endures is the risk of elimination. If the fail to take their opportunities or succeed in avoiding elimination they deserve the same reward for the same final result.
So you want to pat people on the head who manage to get into a draw as a small nation? I am all for that. I presume that we both dislike SoS scoring, then. I am not sure if you have read the entire blog post of swordsman, but it explains that under DSS scoring, players have an incentive to mop up the small players, sometimes even if it reduces their own chance of getting a solo. Do you think that is fun? This new scoring system addresses this issue.
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Re: Two Tier Scoring

#10 Post by Octavious » Mon Dec 30, 2019 1:43 pm

SoS is an abomination that only has any redeeming qualities in a face to face events where people don't have enough time to finish a game properly. Why it was ever introduced to online Diplomacy I have no idea.

As for there being a small point incentive for mopping up small players, I quite like this. It makes the gesture of not mopping up said players more meaningful. And there should be an incentive to avoid accepting ridiculously large draws
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Re: Two Tier Scoring

#11 Post by Restitution » Mon Dec 30, 2019 8:45 pm

Mercy wrote:
Mon Dec 30, 2019 8:31 am
Stuff
I really like your changes! The original idea was cool but it was too complicated, simplifying it so there is only one conditional makes it easier to understand and actually use strategically.

If there is a ton of interest in this, I might be down to implement it (though I have so many other things I'm working on, lol).
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Re: Two Tier Scoring

#12 Post by jay65536 » Mon Dec 30, 2019 9:11 pm

I don't have time to reply to everything here (I will eventually), but I'm honestly confused why people think my system is complicated to compute.

Maybe it is just my mathematician brain? I wrote up the scoring math so that one formula would cover ties as well as sole board leadership. But when there is no tie, it's actually REALLY simple: the sole board leader gets the top score (and it's just 6 points per center between 12, which is 60, and 17, which is 90), and the rest of the points are equally split among draw participants. Mercy's formula and mine are pretty much the same level of complexity to understand.

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Re: Two Tier Scoring

#13 Post by teccles » Mon Dec 30, 2019 9:25 pm

jay65536 wrote:
Mon Dec 30, 2019 9:11 pm
I don't have time to reply to everything here (I will eventually), but I'm honestly confused why people think my system is complicated to compute.

Maybe it is just my mathematician brain? I wrote up the scoring math so that one formula would cover ties as well as sole board leadership. But when there is no tie, it's actually REALLY simple: the sole board leader gets the top score (and it's just 6 points per center between 12, which is 60, and 17, which is 90), and the rest of the points are equally split among draw participants. Mercy's formula and mine are pretty much the same level of complexity to understand.
Speaking for myself (as per my comment on the blog): while I like the idea a lot, your formula has a whole lot of moving parts:
- Exceptions for three special cases.
- 11 parameters which specify the top score (yes, some of them increase by 6 each SC. But some of them don't)
- A slightly arbitrary-feeling condition of a power being one SC behind being "top tier" (why not 0? why not 2?)
I feel like I'd definitely have to go over the details every time I want to work out what my score is going to be.

Mercy's last formulation is simple enough that it can be remembered easily, and only has one magic number in it (a 12 SC cut off to start getting points per SC).
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Re: Two Tier Scoring

#14 Post by Restitution » Mon Dec 30, 2019 9:59 pm

teccles wrote:
Mon Dec 30, 2019 9:25 pm
I feel like I'd definitely have to go over the details every time I want to work out what my score is going to be.
Yep. Ideally one should be able to, at a glance, determine how much score you will have if you take x, y, z centers or someone else takes those centers.

This is one of my biggest problems with SoS, needing to break out the calculator to figure out what the score will be if x, y, z happens. DSS is very good because it's very, very easy to see what the score impact will be as a result of certain changes.

" (c) any draw in which no one has more than 6 centers."

Specifically, this condition is silly, to me.

If the intention is to give a clear solo candidate a big bonus, a 12-SC cutoff is a great, simple way to determine that. "Oh no, Austria is close to 12 SCs, we gotta cut him off"!

Not, "Oh no, Austria is at 12 SCs and we are about to eliminate the 4th player, triggering tiered scoring!" or "Oh no, we are in a multi-way draw scenario and Austria is close to 7 SCs!"

It also creates a weird incentive where, if the top player is between 7 and 12 SCs, they have an incentive to stall the game out and not play aggressively if that would cause a 4wd to turn into a 3wd, and actually remove tiered scoring.

And then there's this:

"In any draw not covered by (2), we start by getting a “top score” for the board—a number based solely on the top center count on the board. Then we get a “bottom score” by subtracting the top score from 180 and dividing by the number of draw participants minus 1. All draw participants who trail the center leader by more than 1 center receive the bottom score. All draw participants whose center counts are within 1 of the top center count split the remaining points (i.e. the top score plus however many bottom scores it takes to match the number of players)."

Which I find, basically, incomprehensible (and I have a bachelor's in CS). I don't see this kind of information being actionable in a real game, it's even more complicated than SoS.

There should be two conditions: either there's a clear leader who is a tier ahead, or there isn't.

12-SCs is nice and elegant; 66% of the way to 18 gives you a nice breakpoint.
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Re: Two Tier Scoring

#15 Post by Restitution » Mon Dec 30, 2019 10:15 pm

Mercy wrote:
Mon Dec 30, 2019 8:31 am
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.
Actually, on further thought, this might not be good and seems to carry some perverse incentives.

Let's say there is the following split:

13 - 10 - 11 SCs among remaining 3 players

The 13-SC player gets 38% of the pot, so the remaining 62% of the pot is split 2-ways, 31% each. That is not a very substantial difference between the leader and the others!

Further, let's say the 10-SC power donates 2 SCs to the 11-SC power:

13 - 13 - 8

Then each of the two leaders, again, get 38%, and the remaining player gets 24%. So, in this case, the leader did not actually lose anything even though another player broke into his tier.

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Re: Two Tier Scoring

#16 Post by Restitution » Mon Dec 30, 2019 10:41 pm

There's also this issue; in a 16-16-2 draw, the 2-sc player only gets 6% of the pot. In a 3wd scenario where 2 players are over the limit, everyone has the same incentive; just grab as many dots as possible. In a 3wd with 2 players over-limit, the game is just ppsc.

And the 2-sc player will leave with less than he bought in with; he bought in with 14% and leaves with 6% despite outsurviving 4 players and maneuvering himself into a narrow 3wd. At this point, if I were him, I would just extort SCs out of the bigger guys in exchange for not handing over the solo. Is this desirable?

The nice thing about DSS is that it prevents grinding out endgames which have foregone conclusions, but this system means that people are going to haggle and finnagle over every single SC in the endgame. This would be fine if there were negotiated draw offers (such as, I offer a draw where player x gets 60%, player y gets 30% and player z gets 10%) but lacking such a feature everyone will be incentivized to grind out dead games.

On the other hand, this is totally fine and tbh pretty awesome with games with turn limits.

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Re: Two Tier Scoring

#17 Post by RoganJosh » Mon Dec 30, 2019 11:57 pm

Let me add a small computation to get some meat on the bone.

Let's say we're playing a DSS game with a pot size of 420. Now let's pretend that you have the choice of either going for a solo, risking a 4wd, or securing a 3wd. For simplicity, we assume there are no other possible outcomes. Which is not completely unrealistic. Yes, this is the example with Germany from the article.

Let's say that you estimate the probability of soloing to be p. Question: how large should p be for the expected return to be greater if you go for the solo as compared to securing the 3wd? Please, take a second to guess the answer before you continue.

This is a simple one variable linear equation. Solo gives 420 points with probability p and a 4wd gives 105 points with probability 1-p. A 3wd gives 140 points.
So, you're solving the equation 420p + 105(1-p) = 140. The solution is p = 1/9.

So - in the example in the article - if Germany thinks there is even a 1 in 9 chance of soloing, then he should go for the solo.

The author wrote that there was an easy argument to be made that it was a mistake for Germany to go for the solo. I think that is wrong. Hindsight - knowing the outcome - maybe. But, really, no. Even in DSS you should more or less always go for the solo.

There are two main takeaways.

The first takeaway is that no one bothers enough to make these kinds of computations. Or - in other words - even DSS is too complicated for people to understand what incentives it gives. It was mentioned before that the proposed new system is too complicated. It really is. People will only see "top" or "bottom". And - yes - you will get plenty of nihilism.

The second takeaway is that no one bothers enough to make these kinds of computations. People will play according to conceived incentives, and not actual incentives. So if you want to sell a new scoring system, to change players behaviors, don't look for fairness in the equations. Look for easy to understand slogans.
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Re: Two Tier Scoring

#18 Post by ben9990 » Tue Dec 31, 2019 4:24 am

I dont like the idea that a two center difference (in the blog post version) could be dispositive - a 10, 11, 13 center board should not place the 13 center player in a separate tier

That said, SOS functionally does this, so maybe its not a big deal.

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Re: Two Tier Scoring

#19 Post by FlaviusAetius » Tue Dec 31, 2019 6:22 am

I dont like the idea of suicide players ngl
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Re: Two Tier Scoring

#20 Post by Mercy » Tue Dec 31, 2019 9:25 am

Restitution wrote:
Mon Dec 30, 2019 8:45 pm
I really like your changes! [...]
Thanks!
jay65536 wrote:
Mon Dec 30, 2019 9:11 pm
I don't have time to reply to everything here (I will eventually), but I'm honestly confused why people think my system is complicated to compute.

Maybe it is just my mathematician brain? [...] Mercy's formula and mine are pretty much the same level of complexity to understand.
I doubt it is your mathematician brain; I am majoring in mathematics and I like elegancy. I do think my version is more elegant because it accomplishes the same goals but is, in my view, easier to understand (and apparently people agree with me in that assessment).
teccles wrote:
Mon Dec 30, 2019 9:25 pm
stuff
Agree.
Restitution wrote:
Mon Dec 30, 2019 9:59 pm
also stuff
I also agree, except that to me, nothing is literally incomprehensible; it's just more complicated. Also this part:
teccles wrote:
Mon Dec 30, 2019 9:25 pm
12-SCs is nice and elegant; 66% of the way to 18 gives you a nice breakpoint.
I am actually thinking that this number may be increased a bit. If you have 12 SC's, you can even still be eliminated in most cases.
Restitution wrote:
Mon Dec 30, 2019 10:15 pm
Mercy wrote:
Mon Dec 30, 2019 8:31 am
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.
Actually, on further thought, this might not be good and seems to carry some perverse incentives.

Let's say there is the following split:

13 - 10 - 11 SCs among remaining 3 players

The 13-SC player gets 38% of the pot, so the remaining 62% of the pot is split 2-ways, 31% each. That is not a very substantial difference between the leader and the others!
I don't see that as a problem as I don't perceive the difference between 13 and 11 SCs as that big.
Restitution wrote:
Mon Dec 30, 2019 10:15 pm
Further, let's say the 10-SC power donates 2 SCs to the 11-SC power:

13 - 13 - 8

Then each of the two leaders, again, get 38%, and the remaining player gets 24%. So, in this case, the leader did not actually lose anything even though another player broke into his tier.
Mmm yeah, true. I still don't see that as a big problem though.

If you do think this is a problem, the scoring system I suggested earlier in which the board leader gets 1/2 of the pot solves this, though. More on this below.
Restitution wrote:
Mon Dec 30, 2019 10:41 pm
There's also this issue; in a 16-16-2 draw, the 2-sc player only gets 6% of the pot. In a 3wd scenario where 2 players are over the limit, everyone has the same incentive; just grab as many dots as possible. In a 3wd with 2 players over-limit, the game is just ppsc.

And the 2-sc player will leave with less than he bought in with; he bought in with 14% and leaves with 6% despite outsurviving 4 players and maneuvering himself into a narrow 3wd. At this point, if I were him, I would just extort SCs out of the bigger guys in exchange for not handing over the solo. Is this desirable?
I thought about that myself too and I do not think this is desirable. More on this below.
Restitution wrote:
Mon Dec 30, 2019 10:41 pm
The nice thing about DSS is that it prevents grinding out endgames which have foregone conclusions, but this system means that people are going to haggle and finnagle over every single SC in the endgame. This would be fine if there were negotiated draw offers (such as, I offer a draw where player x gets 60%, player y gets 30% and player z gets 10%) but lacking such a feature everyone will be incentivized to grind out dead games.
That is not how DSS works. In DSS, a solo threat may deliberately move back to give other powers the opportunity to eliminate a 1-2 SC power, when the game should have been drawn already. That is one of the reasons why I think we can improve on DSS.

The scoring system I suggested in my first post gives 1/2 of the pot to the board leader if the board leader has more than a certain number of SCs (say at least 14) and divides the rest of the pot between the other players. Then if you are in a 3-way draw, and you are not the board leader, you will always receive 1/4 of the pot, and the board leader has no incentive to let you lose SCs.

This scoring system would not have the issues you mentioned above, though we should think about what we do when there are multiple board leaders. I think I have found a solution.

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. In your example where the pot is split 13-13-8, or 16-16-2, the power with the least number of SCs would receive 1/4 of the pot (just as if there was only 1 board leader who got half the pot and the other half was divided between him and the other non-board leader), and the other two players each receive 3/8 of the pot.
RoganJosh wrote:
Mon Dec 30, 2019 11:57 pm
Let me add a small computation to get some meat on the bone.

Let's say we're playing a DSS game with a pot size of 420. Now let's pretend that you have the choice of either going for a solo, risking a 4wd, or securing a 3wd. For simplicity, we assume there are no other possible outcomes. Which is not completely unrealistic. Yes, this is the example with Germany from the article.

Let's say that you estimate the probability of soloing to be p. Question: how large should p be for the expected return to be greater if you go for the solo as compared to securing the 3wd? Please, take a second to guess the answer before you continue.

This is a simple one variable linear equation. Solo gives 420 points with probability p and a 4wd gives 105 points with probability 1-p. A 3wd gives 140 points.
So, you're solving the equation 420p + 105(1-p) = 140. The solution is p = 1/9.

So - in the example in the article - if Germany thinks there is even a 1 in 9 chance of soloing, then he should go for the solo.

The author wrote that there was an easy argument to be made that it was a mistake for Germany to go for the solo. I think that is wrong. Hindsight - knowing the outcome - maybe. But, really, no. Even in DSS you should more or less always go for the solo.

There are two main takeaways.

The first takeaway is that no one bothers enough to make these kinds of computations. Or - in other words - even DSS is too complicated for people to understand what incentives it gives. It was mentioned before that the proposed new system is too complicated. It really is. People will only see "top" or "bottom". And - yes - you will get plenty of nihilism.

The second takeaway is that no one bothers enough to make these kinds of computations. People will play according to conceived incentives, and not actual incentives. So if you want to sell a new scoring system, to change players behaviors, don't look for fairness in the equations. Look for easy to understand slogans.
I disagree with both of your takeaways. I bother enough to make these kinds of computations, and so do you, it seems.

And yes, Germany should go for the solo if his chance to solo is greater than 1/9. But he should refrain from going for the solo if his chance to solo is smaller than 1/9. That's bad.

Don't assume people are stupid, either. Especially in press games players can explain to each other why it is in their incentive to do certain things based on the scoring system.

I agree with the upside of having a scoring system that can be understood easily, but that should not stop us from making one that gives players the right incentives, if properly understood. There has to be a balance between the two.
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