[proposition] town:scum winrate calculator

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Post Post #50 (ISO) » Wed Jun 15, 2016 3:35 pm

Post by callforjudgement »

EV calculations for Open setups with power roles normally work like this: town decide in advance whether a power role claim will save the person being lynched, and which power roles will counterclaim (and what to do if you get a counterclaim). Then you break the game into probabilities depending on who gets forced to claim first (on the understanding that if the claim hasn't been agreed in advance to be something that saves the player making it, the player just gets lynched).

The thing is, even an apparently simple setup can be really complex in terms of EV calculation:

Town Vigilante
2 Vanilla Townies
Mafia Roleblocker (can both kill and act)
"must lynch", i.e. no-lynching is not allowed

Instead of running through this myself, I think it might be interesting to throw some of the questions it leads to into this thread. Here's the first: suppose the vig is the first player to be run up, and claims to avoid being lynched (it's easy to prove that the vig will clam if they're run up, as town would just lose otherwise). Should scum counterclaim?

(For what it's worth, this setup is simple enough that it's possible to account for every possibility, but there's still something of a combinatorial explosion…)
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Post Post #51 (ISO) » Wed Jun 15, 2016 3:45 pm

Post by zMuffinMan »

i am aware definitions (assumptions) need to be made, and although it may have been missed in the long-winded rambling, the point i was trying to make is that those assumptions don't necessarily reflect reality (e.g. even if you say this *should* happen, because it's ideal, it assumes that enough people play in an ideal way that it really is going to be the case a majority of the time)

to put it more simply, if, say, 70% of people who roll cop don't play the way you say they should play as a cop, why is that even an assumption you're making?

like, yes, if you know all the variables in play, and make assumptions, you can say, "with these assumptions, EV = X..." but so what if that has little to no bearing on how a game plays out in reality?

ftr, i've thought about writing a program similar to what you're suggesting, making certain assumptions about certain roles and how they will play, but i realised it'd be all for moot if my assumptions are wrong AND it requires me to think of every possible interaction and every possible factor in a game that can affect different actions and decisions AND while EV is important, the amount of swing in a setup is equally (if not more) important unless the setup is being played a lot of times (and even then, meh)
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Post Post #52 (ISO) » Wed Jun 15, 2016 3:55 pm

Post by zMuffinMan »

In post 50, callforjudgement wrote:Should scum counterclaim?
my initial thought was "no", because realistically scum should only counter-claim if it increases their odds of winning, which it wouldn't unless i'm misunderstanding something

but upon reflecting, potentially "yes", because of the above. lol
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Post Post #53 (ISO) » Wed Jun 15, 2016 4:04 pm

Post by BROseidon »

you should just make assumptions that optimize pv and assume that behavior outside of that is a faction shooting itself in the foot.
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Post Post #54 (ISO) » Wed Jun 15, 2016 4:19 pm

Post by zMuffinMan »

i dunno. you should probably just assume the very worst of players and make assumptions based on that, because outside of how common it is for a player to shoot its own faction in the foot, as you put it... as an example, ideal play for a cop who has a guilty n1 is to not claim and not make it super-obvious they're a cop, while getting their guilty lynched. how often does that even happen? i think that's a rare enough occurrence that making an assumption like that doesn't serve much purpose, despite it being optimal
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Post Post #55 (ISO) » Wed Jun 15, 2016 4:49 pm

Post by Smithereens »

Oh yes for sure muffin. Nothing ever goes according to plan. However, when balancing a game, we make the assumption that both town and mafia are playing at equal levels in order to determine if one side is objectively more advantaged than the other. It doesn't really matter if complexity is added in after, so long as we know that the playing field is even. Thus we can determine optimal play and draw formulas etc even if everyone is retarded.

These outcomes and equations don't describe actual games. They describe games that depict how town and scum would play if they were playing perfectly. You can't really study the balance of a game if you add in all those confounders. But even still, many of the complexities associated with roles can still be studied and reduced to mere numbers.

Take this example: In a game of professional baseball, officials will methodically ensure that the bats used by each team meet certain standards so that the game is fair. That's basically what we're doing here.
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Post Post #56 (ISO) » Wed Jun 15, 2016 6:06 pm

Post by callforjudgement »

In post 50, callforjudgement wrote:Town Vigilante
2 Vanilla Townies
Mafia Roleblocker (can both kill and act)
"must lynch", i.e. no-lynching is not allowed

Instead of running through this myself, I think it might be interesting to throw some of the questions it leads to into this thread. Here's the first: suppose the vig is the first player to be run up, and claims to avoid being lynched (it's easy to prove that the vig will clam if they're run up, as town would just lose otherwise). Should scum counterclaim?
Taking a shot at analyising the setup myself:

First, let's consider the counterclaim situation from town's point of view. If scum are run up and claim vig, do town counterclaim during the day, or with a bullet at night (only counterclaiming if run up)?

If town don't counterclaim during the day, then what happens depends on the next player to be run up. There's a 1/3 chance it's the vig, who claims, and then town have to lynch the correct player to win (EV here depends on whether the strategy involves lynching the first claimant, the second claimant, or both). There's a 2/3 chance it's a VT, who gets lynched, and then town has a 1/2 chance of winning at night (depending on whether scum blocks the vig or the other VT). Thus, depending on their strategy, town win rate here is somewhere between 1/3 and 2/3 (specifically, it's (1+
x
)/3, where
x
is the probability with which town lynch the first player who claims vig after being run up).

If town do counterclaim, then their win rate depends on whether they lynch the first (run up) or second (counterclaim) vig claim. Let's call the probability with which they lynch the first claim
y
.

Town's win rate when they pick the scum first is thus
a
*(1+
x
)/3 + (1-
a
)*
y
, where
a
is the probability with which the vig stays silent after a scum fakeclaim.

Now suppose town run up the vig first, who claims.

If scum don't counterclaim, then what happens again depends on the next player to be run up. There's a 1/3 chance it's the scum, who claims, and then town have to lynch the correct player to win. There's a 2/3 chance it's a VT, who gets lynched, and town just loses when scum block+kill the vig. Thus, town win rate is somewhere between 0 and 1/3; specifically, it's (1-
x
)/3. If scum counterclaim, then the town win rate is just 1-
y
. Thus, the town win rate in this situation is
b
*(1-
x
)/3 + (1-
b
)*(1-
y
), where
b
is the probability with which scum stays silent after a town vigclaim.

Scum have control of
b
here; town have control of
a
,
x
, and
y
. Town are trying to maximise
E
=
a
*(1+
x
)/3 + (1-
a
)*
y
+
b
*(1-
x
)/3 + (1-
b
)*(1-
y
), where
E
is the win rate in vig-run-up plus the win rate in scum-run-up (clearly claiming strategy is irrelevant in the VT-run-up case). For any given
a
,
x
, and
y
, scum will choose
b
to minimize the function in question. The minimum will either be at an endpoint of the range (
b
= 0 or
b
= 1), or at a critical point of
E
(i.e.
a
,
x
, and
y
have values such that changing
b
has no impact on the result); d
E
/d
b
is (1-
x
)/3-(1-
y
), or
y
-2/3-
x
/3, so the critical points for
b
happen when
y
-2/3-
x
/3 = 0, i.e.
y
*3 =
x
+2. We can apply the same reasoning to
a
,
x
, and
y
. In each case, the variable will be minimum, maximum, or at a critical point. This means that all four of these conditions hold:

a
= 0 or
a
= 1 or
y
*3 =
x
+1
b
= 0 or
b
= 1 or
y
*3 =
x
+2
x
= 0 or
x
= 1 or
a
=
b

y
= 0 or
y
= 1 or
a
=
b


Clearly, we can't have both
y
*3 =
x
+1 and
y
*3 =
x
+2. So either
a
or
b
(possibly both) is at an endpoint of the range. If
a
isn't 0 or 1, then we discover that either
x
or
y
isn't at an endpoint (there's no way to do
y
*3 =
x
+1 with integers), which means that
a
=
b
, and thus that
b
isn't 0 or 1 either, a contradiction. So we know that, at least,
a
must be 0 or 1; either the town vig never counterclaims or always counterclaims, there's never any benefit from randomizing. We can consider the two cases separately.

a
= 0 (town always counterclaims):

Here, we have
E
=
y
+
b
*(1-
x
)/3 + (1-
b
)*(1-
y
). It's fairly clear that in this situation,
x
will be 0 (which can be observed either mathematically, or by observing that if town always counterclaims, a vig claim under pressure must come from scum, as a genuine vig would have counterclaimed earlier). This gives
E
=
y
+
b
/3 + (1-
b
)*(1-
y
). Increasing
y
can only help town here, no matter what the value of
b
; town will institute a policy of always lynching the original claimer in the case of a counterclaim. So
E
= 1 +
b
/3. Scum will thus always counterclaim (setting
b
to 0), safe in the knowledge that they'll get the vig lynched. Our final
E
value is 1, and the scenario generally is one in which claims under pressure are ignored, meaning town always wins if they run up scum and never wins if they run up the vig.

a
= 1 (town never counterclaims):

In this case, we have
E
= (1+
x
)/3 +
b
*(1-
x
)/3 + (1-
b
)*(1-
y
).
y
is going to be minimized here; town never counterclaims, and thus a counterclaimer is always going to be lynched. Thus this simplifies to
E
= (1+
x
)/3 +
b
*(1-
x
)/3 + (1-
b
). Then we discover (unsurprisingly) that scum will never counterclaim (i.e. b = 1), as doing so would just help town lynch them, simplifying further to
E
= (1+
x
)/3 + (1-
x
)/3. Finaly,
x
turns out to be irrelevant (because nobody counterclaims, it's equally likely for the scum to be run up before the vig, or vice versa). The final
E
value is 2/3, worse than town could achieve in the other scenario.


So we've solved the setup: whenever there's a claim, it just gets counterclaimed and the original claimant gets lynched. In other words, there's no actual point to claiming in the first place. For anyone wondering about the EV, it's relatively easy to calculate due to the lack of claims and counterclaims: 1/4 from lynching the scum D1, plus 1/2 (VT lynch) * 1/2 (vig not blocked) * 1/2 (vig shoots scum) from vigging the scum N1, = 3/8 total overall.

I'm actually a little disappointed here; I was hoping for a mixed strategy (and tried to construct the example to obtain one, but I hadn't fully run through the analysis at the time). In fact, as soon as we wrote out the list of critical point conditions and observed that two of them contradicted each other, we could have skipped much of the analysis; this situation ruled out a mixed strategy, and the pure strategies can be mostly determined via common sense, with the only non-obvious factor being that town will never lynch a counterclaimer. (This is because town
always
counterclaim, and thus it gives them an advantage if scum decide not to counterclaim, while not hurting if scum always counterclaim because the situation is symmetrical.) Hopefully the logic above at least shows how mixed strategies are calculated, though, and thus what sort of work would have to go into this sort of tool (you couldn't really brute-force anything but the simplest setups as there's an exponential or even tower-of-exponentials explosion).
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Post Post #57 (ISO) » Wed Jun 15, 2016 7:23 pm

Post by Smithereens »

+1
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Post Post #58 (ISO) » Wed Jun 15, 2016 7:45 pm

Post by zMuffinMan »

In post 55, Smithereens wrote:how town and scum would play if they were playing perfectly
i would be surprised if this has ever happened in the history of mafia

but that still misses the point

we're not talking about playing optimally here, necessarily. for example, it's optimal for a cop not to claim if they can push their lynch without making it obvious they're a cop, but it's silly to assume that the player who rolls cop both thinks this and is able to pull it off. you're suggesting we assume a cop will claim if they have a guilty, which also isn't the case in a lot of games. little things like this might seem silly, but they determine the difference between scum team shooting at random and cop getting a different result and scum team killing cop so no more results

*shrug*
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Post Post #59 (ISO) » Wed Jun 15, 2016 8:46 pm

Post by callforjudgement »

Apparently, there are people who run Open games offsite in which a Cop needs multiple guilties in order to balance the game. As such, Cops don't claim their guilties immediately in the site meta in question, and VTs take steps to try to keep the cops hidden (fake-breadcrumbing results and the like).

The main reason cops claim immediately on mafiascum.net is that site meta tends to avoid powerful single roles, and as such a definite guilty is typically more valuable than any use you'd get out of your unrevealed role in future. (And even when it isn't, scum might incorrectly assume you're 1-shot and not kill you for fear of a doctor.)
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Post Post #60 (ISO) » Thu Jun 16, 2016 1:10 am

Post by Smithereens »

In post 58, zMuffinMan wrote:
In post 55, Smithereens wrote:how town and scum would play if they were playing perfectly
i would be surprised if this has ever happened in the history of mafia

but that still misses the point

we're not talking about playing optimally here, necessarily. for example, it's optimal for a cop not to claim if they can push their lynch without making it obvious they're a cop, but it's silly to assume that the player who rolls cop both thinks this and is able to pull it off. you're suggesting we assume a cop will claim if they have a guilty, which also isn't the case in a lot of games. little things like this might seem silly, but they determine the difference between scum team shooting at random and cop getting a different result and scum team killing cop so no more results

*shrug*
To address this situation specifically, ideally a cop does not need to out to push a lynch on a scum, that's perfect play by town. Perfect play by mafia would be that they identify that the push is caused by information not normally accessible to town, ergo investigative role. So a cop would out themselves even if they don't out themselves if both sides play perfectly.

We assume they play perfectly due to long term probability. Mistakes made by one side are cancelled out by mistakes made by the other side. I'm not sure what the original point was anyway.
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Post Post #61 (ISO) » Thu Jun 16, 2016 1:40 am

Post by zMuffinMan »

In post 60, Smithereens wrote:Perfect play by mafia would be that they identify that the push is caused by information not normally accessible to town, ergo investigative role.
uh... this isn't necessarily the case if it's done right, which is the whole point. i'm not talking about coming out at the start of a day and going "hey guys, i have a hunch X is scum..."

you can work your way around to it, write a proper case, etc, and regardless of whether they're "perfect" scum, unless there's literally no holes in their play, you can push a lynch on them in a standard way without outing yourself as a cop.

i'm harping on about this example in particular, but that's because cops are the most basic of basic roles in a setup and how to treat them in terms of working out EV isn't even obvious (at least imo)
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Post Post #62 (ISO) » Thu Jun 16, 2016 2:33 am

Post by BROseidon »

In post 54, zMuffinMan wrote:i dunno. you should probably just assume the very worst of players and make assumptions based on that, because outside of how common it is for a player to shoot its own faction in the foot, as you put it... as an example, ideal play for a cop who has a guilty n1 is to not claim and not make it super-obvious they're a cop, while getting their guilty lynched. how often does that even happen? i think that's a rare enough occurrence that making an assumption like that doesn't serve much purpose, despite it being optimal
Assume reasonable behavior?

Cop claims guilty as soon as it happens is pretty safe to assume, for instance
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Post Post #63 (ISO) » Thu Jun 16, 2016 2:34 am

Post by BROseidon »

Like, muffin, you're basically complaining that you can't balance setups because one side might outplay the other, which is a little silly
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Post Post #64 (ISO) » Thu Jun 16, 2016 2:36 am

Post by zMuffinMan »

no, i'm saying the differences in opinion on how you "should" play a role make it borderline impossible to reasonably predict actions

for example, i personally would never claim a guilty as a cop unless i felt it absolutely necessary to get my lynch or it was at the point of the game where it didn't matter, and the odds that i would feel it necessary to do that to push a lynch are pretty low (this may differ for others, so *shrug*)
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Post Post #65 (ISO) » Thu Jun 16, 2016 3:32 pm

Post by Smithereens »

"the differences in opinion on how you "should" play a role make it borderline impossible to reasonably predict actions."
We can objectively determine the best possible outcome with long term probability.

A cop has outed himself and there is a doc in the game. Should the doc doc the cop?
Scenario 1 (the existence of the doc is unknown): Obviously yes.
Scenario 2 (the existence of the doc is known): in this wifom the maf suspect that the doc is going to protect the cop, so they should target another player. The doc knows this, so he docs a different player instead of the cop. The mafia consider this possibility, so they want to NK the cop instead etc etc.
In this situation there are two Nk outcomes:
1) The NK is against the cop (50%)
2) The NK is not against the cop (50%)
The doc knows that mafia are equally likely to choose either option. With a player pool of say, 8 townies, one of which is the cop and another the doc, there is a 50% chance that the cop will be the NK target and a 7.1% chance that any given other player will be NK target. in a game where the doc chooses to protect another player, the NK is expected to be prevented 7.1% of the time. Protecting the cop should prevent the NK 50% of the time. Therefore the doc knows that his best bet is to protect the cop. Knowing this, the mafia's best bet is to shoot someone else. So the probability of a NK against a random other player becomes 14.3% in a game of 8. So long as this percentage remains lower than 50%, the doctor's optimal course of action is to always protect the cop. Likewise the mafia should always shoot someone else, and that's how we define optimal play in an infinite regression.
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Post Post #66 (ISO) » Thu Jun 16, 2016 4:00 pm

Post by zMuffinMan »

that logic is flawed but that's not really important

with regard to figuring out optimal play in the long run, not claiming cop and pushing your lynch is going to produce higher EV for town than claiming a guilty, no? so theoretically, if we're saying "base it on optimal play", we should be assuming that any time a cop gets a guilty, he gets the guilty lynched the next day without outing himself, right?

im harping on about this one example because a cop is the most basic of basic roles when it comes to decision-making, since results are binary, and i don't think it's even that easy to determine how a program should treat it when determining balance. with roles where there are a lot more options in how to play them, this becomes even more of an issue (especially if you don't even know how to optimally play a role - eg if you're assuming a role will do X if Y happens but it's many times better to do Z, then all your calculations are going to be off...)

like I've said from the beginning, the issue isn't writing the program, it's determining what rules to use. and what rules you use can significantly affect the odds...
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Post Post #67 (ISO) » Fri Jun 17, 2016 2:10 am

Post by Smithereens »

Optimal play for the cop would to get a guilty lynched and then not die, but optimal play for mafia would be to kill a cop that lynches a mafia. Therefore if both play optimally, cop will lynch a guilty but mafia will NK a cop that does that. Cop's are not the most basic roles. Their results are knowledge, which cannot be quantified or measured. A doctor does not produce 'psychological' effects such as the gaining of knowledge, so a doc is simpler than a cop. The rules we use for cop need to assume that there is only 1 best possible outcome. This is a reasonable assumption to make so working from there is really not a big deal.
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Post Post #68 (ISO) » Fri Jun 17, 2016 3:03 am

Post by callforjudgement »

If the Doctor blocks a kill, that's worth some proportion of an inno on the targeted player (depending on how much the Doctor believes they were responsible for the protect; after all, they can't know it wasn't another role). Cop results are much more black and white. As such, it's possible that Doctor's an even more complex role; in addition to defending your innos in thread (something a Cop also wants to do), you have to work out how reliable they are. (Not to mention that a Doctor benefits more than a Cop does from predicting the kill, which is another fairly complex skill.)
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Post Post #69 (ISO) » Fri Jun 17, 2016 3:48 am

Post by zMuffinMan »

In post 67, Smithereens wrote:Optimal play for the cop would to get a guilty lynched and then not die, but optimal play for mafia would be to kill a cop that lynches a mafia.
... look, you're argument seems to be that scum will know who the cop is anyway if he doesn't out himself to get a lynch on scum to happen, but that's not the case, because people push lynches hard all the time and they can't all be cops... so realistically optimal play for a cop is to not claim and push their lynch and there's no optimal play for scum here, unless they somehow
know
the push is coming as the result of a guilty... like you keep coming back to "well scum will know," but know, they won't... sure, they might shoot someone who's on the wagon of their lynched scum buddy (which, now that i think about it, is another factor you'd have to weigh into any program you make to simulate real games), but they're not going to know they were pushed because of a guilty unless the cop isn't playing optimally...

but this single example isn't so important, so i don't really care enough to keep arguing this. cop
is
a simple role because there's no nuances in when/where to claim, what your result might mean, etc that you have with others roles... and that isn't even a simple black-and-white thing. the more complex a game gets, the more factors that come into play, the harder it is to write any sort of rule (or "definition") for standard behaviour. and i'm not even talking about one role (although there are a lot of complexities with a single role) - i'm talking about how a role interacts with other roles. e.g. what do you do when a tracker gets redirected to a roleblocker who blocked the person who ended up getting killed that night? who claims? who does scum decide to kill as a result of it?

sure, it's theoretically
possible
to work out a set of rules for an enormous amount of potential interactions (if you had the time and will to do it...) but you'd need to know that you've covered everything (or at least enough that anything you haven't covered won't greatly affect the EV) and you'd need to know that these rules are actually
good rules
(because, at the very least, your idea about what constitutes a good rule seems vastly different to mine, so who's the arbiter in the end?)
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Post Post #70 (ISO) » Sun Jun 19, 2016 3:00 am

Post by mole »

For mountainous I'd use a spreadsheet: https://docs.google.com/a/mimhoff.com/s ... sp=sharing

And cop is even more complicated than that though.

A cop could also have a string of innocent results which improves the chance you catch a mafia for a couple of days, until the cop and all the known innocents are dead. But to track that you'd have to track how many innocents the cop knows about, and whether they have since been killed. Even before you decide when your cop will tell everyone this info.
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Post Post #71 (ISO) » Sun Jun 19, 2016 3:28 am

Post by Smithereens »

In post 70, mole wrote:For mountainous I'd use a spreadsheet: https://docs.google.com/a/mimhoff.com/s ... sp=sharing

And cop is even more complicated than that though.

A cop could also have a string of innocent results which improves the chance you catch a mafia for a couple of days, until the cop and all the known innocents are dead. But to track that you'd have to track how many innocents the cop knows about, and whether they have since been killed. Even before you decide when your cop will tell everyone this info.
Nice :D

Is there anything you mean for us to see in the spreadsheet? cuz it seems we don't have permissions. Or maybe that's me O.o
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Post Post #72 (ISO) » Sun Jun 19, 2016 4:24 am

Post by mole »

Try it now.

It gives you the same results as the other tools in this thread, but now it's a spreadsheet!
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Post Post #73 (ISO) » Tue Aug 30, 2016 11:03 am

Post by TGGC »

I did the tool mentioned up there (http://games-net.de/hosted/tggc/trash/m ... 0&do=Go%21)
Everything is double checked and should work fine for any combination of values. If you have questions, problems or find an error, just let me know.
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Post Post #74 (ISO) » Thu Apr 27, 2017 12:51 pm

Post by Ircher »

Feel like this thread ought to be moved to open setup discussion simply cuz it contains a very useful link.
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