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[AdamH]

DISCLAIMER: I'm well aware that this discussion is largely theoretical in nature. I won't attempt to inject any application of the findings here into actual Dominion strategy -- that's for the reader to you. The purpose of this is simply to calculate an expected draw for your Scrying Pools given a certain deck composition.

Sometimes there's Scrying Pool on a board and there's no trashing. Sometimes it's the only non-terminal source or the best source of draw, enough that if you're going to try and increase your hand size, that's what you're going for. Is it worth it? Say it with me...

Depends on the kingdom.

But how do you tell? I, for one, am no stranger to getting crushed by people who go for Scrying Pool when I've decided it wasn't worth it.

I talked about this in that third link ("crushed") above, but sometimes it's helpful for me to try and calculate one's expected draw with Scrying Pool based on action density, and when I visualize that number in the context of the specific deck I'd try to build, sometimes I can see Scrying Pool's value more clearly. Unfortunately for me this usually happens after the game is over so I'm working on that.

So there are a couple of numbers that I want to talk about computing and maybe I'm close to getting this right. First off, I want to only look at the case where there's no trashing. Let's be generous, though, and say that the only non-Action card you have to buy is a single Potion and everything else is an Action card and not worry about greening, so you have 11 non-Action cards in your deck. I want to know how many Action cards (X) you have to have in your deck to make Scrying Pool's expected draw be E cards.

A simple calculation to make would look like:

Code: [Select]

Expected Draw = 1+(X/11)

and this is what I've been using to get close to this number, but I'm thinking maybe this isn't accurate enough (it doesn't even attempt to take the Spy-effect into account, among some other little things). I want to find either something that's closer and still easy to compute or just find some benchmarks for "E=2 when X=this and E=3 when X=that, etc."

Depending on your deck, you may also want to consider the case that discarding a non-Action card is just as good as drawing it because of the cycling or the nature or your payload (Let's say you're playing Highway/Market Square/Scrying Pool with no other support whatsoever; it doesn't matter if you're drawing your Coppers or not, you just want to find and play as many of your action cards as possible so Scrying Pool gets a boost here).

Here are some of my attempts at refining this calculation:

1. Replace "X" with "X-1" because you can't draw the Scrying Pool you play.
2. Split the formula into two separate cases:

Code: [Select]

E = P(top card was an action)*(Expected draw in that case) + P(top card wasn't an action and you discarded it)*(Expected draw in that case[+1])

-- and you can insert a "+1" where I pointed that out if you're counting the cycling advantage from the card you discarded.

Let me try and come up with expressions for these things

Code: [Select]

P(top card was an action) = (X-1)/(X+10)
P(top card was not an action) = 11/(X+10)

Expected Draw given we saw an action = 2 + (X-2)/11
Expected Draw given we saw a non-action) = 1 + (X-1)/10

The only thing I can think of that this doesn't take into account is the fact that your deck composition changes with each Action card you draw, making it slightly less likely each time that you hit another Action. I can't think of a concise formula for this and I'm terrible with summations so I don't have that formula right away (so far my attempts at coming up with this have just looked wrong to me so I just need to think about this a little more).

This stuff is messy but at least I could plug it into Excel or something and get some numbers and charts that could potentially be useful. I also know there are quite a lot of math people around here that would at least find this intelectually stimulating if nothing else. Help is appreciated, particularly in telling me where I'm wrong.


EDIT 1


Without the spy effect taken into account, but with deck changing taken into account, I believe I'm pretty close with the following formula.

Code: [Select]

Let E = expected draw from playing your Scrying Pool (sans Spy effect)
    X = the number of Action cards in your deck

E(X) = 1 + (X-1)/(X+10) + [(X-1)/(X+10)]*[(X-2)/(X+9)] + [(X-1)/(X+10)]*[(X-2)/(X+9)]*[(X-3)/(X+8)] + ... + [(X-1)/(X+10)]*[(X-2)/(X+9)]*...*[(X-n)/(X+11-n)] + ... + 0 when n eventually reaches X

E(X) = 1 + sum(i=1 up to X-1 (inclusive),
               prod(j=0 up to i-1 (inclusive),
                    (X-1-j)/(X+10-j)
               )
           )

So if that's correct, then we have

Code: [Select]

Let Ea(X) = Expected draw given you saw an Action card
    Eb(X) = Expected draw given you discarded a non-Action card

Ea(X) = 2 + sum(i=1 up to X-2 (inclusive),
                prod(j=0 up to i-1 (inclusive),
                     (X-1-j)/(X+10-j)
                )
            )

Eb(X) = 1 + sum(i=1 up to X-1 (inclusive),
                prod(j=0 up to i-1 (inclusive),
                     (X-1-j)/(X+9-j)
                )
            )

So if we take from above

Code: [Select]

Let Pa(X) = probability that the top card was an action
    Pb(X) = probability that the top card was not an action

Pa(X) = (X-1)/(X+10)
Pb(X) = 11/(X+10)

Then we get

Code: [Select]

E(X) = Ea(X)*Pa(X) + Eb(X)*Pb(X)

I think this is correct, yeah? I'll plug this into Excel when I get a chance and try to make some pretty pictures.


EDIT 2


Using this handy website I'm able to run simple Python code from all kinds of computers; I've written a little bit of Python to compute these formulas. Here's the code, for those interested.

Code: [Select]

#!/usr/local/bin/python2.7

def Ea(X, NA):
  sum = 2.0
  for i in range(1, X-1):
    prod = 1.0
    for j in range(i):
      prod *= (float(X)-2.0-float(j))/(float(X)+NA-2.0-float(j))
    sum += prod
  return sum
     
def Eb(X, NA):
  sum = 1.0
  for i in range(1, X):
    prod = 1.0
    for j in range(i):
      prod *= (float(X)-1.0-float(j))/(float(X)+NA-2.0-float(j))
    sum += prod
  return sum
   
def Pa(X, NA):
  return (float(X)-1.0)/(float(X)+float(NA)-1.0)
 
def Pb(X, NA):
  return (float(NA))/(float(X)+float(NA)-1.0)
 
def EaSimple(X, NA):
  return 2.0 + (float(X)-2.0)/(float(NA))
 
def EbSimple(X, NA):
  return 1.0 + (float(X)-1.0)/(float(NA-1))
 
def Esilly(X, NA):
  return 1.0 + (float(X))/(float(NA))
   
for X in range(2, 31):
  #for NA in range(6, 21):
  for NA in range(11, 12):
    EofX = Ea(X, NA)*Pa(X, NA) + Eb(X, NA)*Pb(X, NA)
    EsimpleofX = EaSimple(X, NA)*Pa(X, NA) + EbSimple(X, NA)*Pb(X, NA)
    print "%2d" % X,
    print "%2d" % NA,
    print "%2.4f" % Esilly(X, NA),
    print "%2.4f" % EsimpleofX,
    print "%2.4f" % EofX

I imported the printed results into Excel and messed around with them until it seemed like it was as visually appealing as I could easily get it. The following are the results from this:



The interesting result here is the difference between the blue and green lines.

So my back-of-the-envelope calculation of E(X) = 1 + (X/11) gives a pessimistic viewpoint of the draw you can expect; the actual expected draw is usually about a half a card more than this calculation gives, or you can get away with having 3-4 less Action cards in your deck than this calculation would imply in order to get the expected draw you're after.

This next picture attempts to show the effect of a different number of non-action cards in your deck on expected draw.



I really don't think any less than 11 non-actions is all *that* relevant to actual Dominion, since if you're trashing your cards and drawing with Scrying Pool the name of the game is minimizing that number and you actually have some power to do so. I wanted to make a 3-D graph with two independent variables but all the Excel options for that look terrible, so I decided to just color-code it. Do with this data what you want, but it seems to me that you probably need to add one extra Action card per non-Action card to your deck in order to make sure your Scrying Pools are expected to draw the same amount. Keep in mind that now you have two extra cards to draw.

[JW]

Sometimes there's Scrying Pool on a board and there's no trashing. Sometimes it's the only non-terminal source or the best source of draw, enough that if you're going to try and increase your hand size, that's what you're going for. Is it worth it? Say it with me...

Seems like you have the correct basic approach.

Edit: Call the average cards drawn by Scrying Pool in a deck of X actions (including Pool) and C non-actions, assuming that you discard non-actions with the self-spy effect, S(X,C).

S(X,C)=  (X^2 + 3*X*C + C^2 - 2*C -1)/ [ (X+C-1) * (C+1)]

See my post below for details.

The self-spy effect is a big part of the card's power. Consider a simple case where the self-spy effect is not trivial but can be included by hand. Assume that your hand consists of 1 Scrying Pool and nothing else. You have 2 other actions (besides the Scrying Pool) and 1 non-action ("Potion") in your deck.

Then the average number of cards that you draw is 2.667: you draw all three cards if the Potion is on top or on bottom (2/3), but only draw two cards if the Potion is second. Without considering the self-spy effect, you average 2 cards (1/3 chance each of 1, 2, and 3 cards).

Here's my calculation, which includes the self-spy effect:
6 actions including Scrying Pool, 11 non-actions: 1.7292 average cards drawn
7 actions including Scrying Pool, 11 non-actions: 1.8529 average cards drawn
8 actions including Scrying Pool, 11 non-actions: 1.9722 average cards drawn
9 actions including Scrying Pool, 11 non-actions: 2.0877 average cards drawn
10 actions including Scrying Pool, 11 non-actions: 2.2 average cards drawn
11 actions including Scrying Pool, 11 non-actions: 2.3095 average cards drawn
12 actions including Scrying Pool, 11 non-actions: 2.4167 average cards drawn
13 actions including Scrying Pool, 11 non-actions: 2.5217 average cards drawn
14 actions including Scrying Pool, 11 non-actions: 2.625 average cards drawn

I realize this would be easier to absorb as a picture, may add that later.

One small point that won't matter for calculations with a lot of stop cards but will matter with few stop cards is that you are typically going to have a 5-card hand that includes Scrying Pool, so if you assume that you already have Scrying Pool + 4 other cards in hand, then the maximum that you can draw off Pool is limited by the number of cards left in your deck.
 

[SCSN]

Sometimes there's Scrying Pool on a board and there's no trashing. Sometimes it's the only non-terminal source or the best source of draw, enough that if you're going to try and increase your hand size, that's what you're going for. Is it worth it? Say it with me...

It is.

[Titandrake]

Pool is a weird card, but in practice every time I think I can skip Pool I lose. I don't think it's as bad as skipping Witch, but it's pretty close.

[amalloy]

Pool is a weird card, but in practice every time I think I can skip Pool I lose. I don't think it's as bad as skipping Witch, but it's pretty close.

Of course Pool is usually a big deal, but exceptions do arise. I skipped it in a league match with ADK, opting to focus on Torturer/FV instead, and in two different casual games to instead try silver-flooding.

Now, obviously I am not the best player in the world, and neither were my opponents in any of those games. Particularly in the Trader/Scheme game I'm not sure I was as far ahead as he thought when he resigned. But there are definitely boards in which the time invested to get Scrying Pool just won't pay off.

[AdamH]

EDIT: THESE NUMBERS ARE NOT CORRECT: SEE OP FOR MOST UP-TO-DATE NUMBERS


Update on numbers:

I wrote some Python code to compute the numbers and I'm going to import it into Excel to make it into pretty pictures. My source code is below (I just did an online portal to execute the Python code since I'm working from a different computer than where my Python is set up)

Code: [Select]

#!/usr/local/bin/python2.7

def Ea(X):
  sum = 2.0
  for i in range(1, X-1):
    prod = 1.0
    for j in range(1, i):
      prod *= (float(X)-1-float(j))/(float(X)+10-float(j))
    sum += prod
  return sum
     
def Eb(X):
  sum = 1.0
  for i in range(1, X):
    prod = 1.0
    for j in range(i):
      prod *= (float(X)-1-float(j))/(float(X)+9-float(j))
    sum += prod
  return sum
   
def Pa(X):
  return (float(X)-1.0)/(float(X)+10.0)
 
def Pb(X):
  return (11.0)/(float(X)+10.0)
 
def EaSimple(X):
  return 2.0 + (float(X)-2.0)/(11.0)
 
def EbSimple(X):
  return 1.0 + (float(X)-1.0)/(10.0)
 
def Esilly(X):
  return 1.0 + (float(X))/(11.0)
   
for X in range(2, 51):
  EofX = Ea(X)*Pa(X) + Eb(X)*Pb(X)
  EsimpleofX = EaSimple(X)*Pa(X) + EbSimple(X)*Pb(X)
  print "%2d" % X,
  print "%2.4f" % EofX,
  print "%2.4f" % EsimpleofX,
  print "%2.4f" % Esilly(X)

It computes all of the intermediate values so I can compare them.

An interesting result (one that makes me question the correctness of my code, actually) is that Eb(X)*Pb(X) was always equal to one. This result seems like it shouldn't be but I can't think of a good reason why and the numbers I get all pass at least a simple sanity check. Did I do something wrong? Is there some reason why this should be the case?

Here's the output of the program:

Code: [Select]

2 1.1667 1.1750 1.1818
 3 1.4615 1.3371 1.2727
 4 1.6758 1.4890 1.3636
 5 1.8659 1.6327 1.4545
 6 2.0414 1.7699 1.5455
 7 2.2058 1.9016 1.6364
 8 2.3611 2.0288 1.7273
 9 2.5088 2.1522 1.8182
10 2.6500 2.2723 1.9091
11 2.7857 2.3896 2.0000
12 2.9167 2.5045 2.0909
13 3.0435 2.6174 2.1818
14 3.1667 2.7284 2.2727
15 3.2867 2.8378 2.3636
16 3.4038 2.9458 2.4545
17 3.5185 3.0525 2.5455
18 3.6310 3.1581 2.6364
19 3.7414 3.2627 2.7273
20 3.8500 3.3664 2.8182
21 3.9570 3.4692 2.9091
22 4.0625 3.5713 3.0000
23 4.1667 3.6727 3.0909
24 4.2696 3.7735 3.1818
25 4.3714 3.8738 3.2727
26 4.4722 3.9735 3.3636
27 4.5721 4.0727 3.4545
28 4.6711 4.1715 3.5455
29 4.7692 4.2699 3.6364
30 4.8667 4.3680 3.7273
31 4.9634 4.4656 3.8182
32 5.0595 4.5630 3.9091
33 5.1550 4.6600 4.0000
34 5.2500 4.7568 4.0909
35 5.3444 4.8533 4.1818
36 5.4384 4.9496 4.2727
37 5.5319 5.0456 4.3636
38 5.6250 5.1415 4.4545
39 5.7177 5.2371 4.5455
40 5.8100 5.3325 4.6364
41 5.9020 5.4278 4.7273
42 5.9936 5.5229 4.8182
43 6.0849 5.6178 4.9091
44 6.1759 5.7126 5.0000
45 6.2667 5.8073 5.0909
46 6.3571 5.9018 5.1818
47 6.4474 5.9962 5.2727
48 6.5374 6.0904 5.3636
49 6.6271 6.1846 5.4545
50 6.7167 6.2786 5.5455

Next up: pretty pictures and results

[AdamH]

EDIT: THESE RESULTS ARE NOT CORRECT: SEE OP FOR MOST UP-TO-DATE RESULTS AND STUFF

RESULTS



Something alarming I see is that I thought taking deck composition into account after drawing actions would actually decrease the expected draw but it seems to have increased it. Not sure if I messed up numbers, but other than that this passes a sanity check.

So the calculation I had been doing gave me an expected draw of a whole card less than what one could actually expect to draw.

"In order to expect to draw X cards, you need at least Y actions in your deck" -- some notable milestones:

2 cards, 6 actions
3 cards, 13 actions
4 cards, 22 actions
5 cards, 32 actions

[Titandrake]

For the Eb(X)*Pb(X) = 1 thing, I have a semi-explanation.

Suppose that your deck is infinite in size, so the odds of revealing a non action card is always the same.

Eb(X) = expected number of draws when you discarded non-action, meaning expected number of draws with no information about top card.

Pb(X) = probability of revealing non-action. (The chance for the top card is the same as the chance for the entire deck, with the assumption made earlier)

Eb(X) = P(draw at least 1 card) + P(draw at least 2 cards) + P(draw at least 3 cards) + ...
(If you draw n cards, it'll get counted in n of the probabilities above. I just discovered this trick last weekend so I'm abusing it wherever I see it.)

So,

Eb(X) = 1 + (1 - Pb(X)) + (1 - Pb(X))^2 + (1 - Pb(X))^3 + ...
= 1 / (1 - (1-Pb(X)) = 1 / Pb(X)

And Eb(X) * Pb(X) = 1.

There might be some trickiness with a deck that actually changes in action proportion, but I think there's a theorem where sifting doesn't affect probability of drawing key cards that I'll cite here. I don't know if it applies, and actually forgot how that theorem goes, but there you go, a semi-explanation.

[AdamH]

Suppose that your deck is infinite in size

This makes that particular issue very clear to me. Thank you. +1

[AdamH]

Sometimes there's Scrying Pool on a board and there's no trashing. Sometimes it's the only non-terminal source or the best source of draw, enough that if you're going to try and increase your hand size, that's what you're going for. Is it worth it? Say it with me...

It is.

SCSN, you know I love you man <3 , but in a 1-card Kingdom with just Scrying Pool you can't tell me I should go for it. OK well you could but I wouldn't believe you. There has to be a line somewhere and I've noticed that finding that line is a weakness in my game. Calculating this expected draw has been really enlightening when looking back on these games so I want to do it.

Yeah people get lots of respect for snarky comments around here, but the fact that a comment that dismisses what I'm trying to talk about gets more respect than what I'm saying leads me to believe that maybe lots of people out there just blindly follow your hyperbole in this case.

If they should then I want to see evidence to back it up. Otherwise I'll just have to be frustrated about that dynamic and try to figure it out for myself; and then one day I'll be a better player than you (and all those people who +1ed you and didn't look at my math) because I put in the work.

<3

[Mic Qsenoch]

This math can not help you play better Dominion at all, I'm sorry.

[DG]

I think you may be better served looking at how the drawing of the scrying pools relates to the non-action cards. The big problem with this maths though, compared even to ventures, is that is built on shifting sand. Adding even one vp card to a scrying pool deck will change the maths. Even weak drawing cards such as mystics can change the viability of scrying pools for drawing.

[AdamH]

Yeah I'm aware of that. The point of this is that my intuition on how much I can expect to draw with a Scrying Pool is off (too low). I did some crude math and it got higher and it's still too low. Now every time I do more precise math it gets better, to the point where it's much higher than I ever expected.

How does it combo with other cards? Well of course math can't tell you that you have to play the game. But when this particular fundamental is off then I'm not getting anywhere.

And when decks green of course that changes. Yeah I can do math on that too and see how it affects things and if that's surprising.

[silverspawn]

This math can not help you play better Dominion at all, I'm sorry.

? why not

[-Stef-]

...
Yeah people get lots of respect for snarky comments around here, but the fact that a comment that dismisses what I'm trying to talk about gets more respect than what I'm saying leads me to believe that maybe lots of people out there just blindly follow your hyperbole in this case.
...

Sorry to read your frustration here (& welcome to the internet).

It's not plain 'upvoting snarky comments' though. You really do over-complicate a question here. Sure you can try to revert to math when your intuition about Scrying Pool fails you, but in this case the required math is very hard to write down. I'd say you're still miles and miles away from getting the math right, whereas a simple rule like "just buy it unless you can explain a rubber duck why you shouldn't" is actually very helpful. At least it is to me. If that rule somehow fails you, I suggest you try SCSN's rule for a while ("just buy it").

[Mic Qsenoch]

This math can not help you play better Dominion at all, I'm sorry.

? why not

Because it focuses on one aspect of SP in an extremely specific case. It says next to nothing about when/how to go about playing an SP deck. The numbers it produces are restricted to a case that doesn't exist in real games of Dominion. How many cards SP can draw isn't even that important, in a real game we want to know roughly how many turns it will take to get the engine going, how many turns will the game last, what can I do once I'm drawing lots of cards etc. The other kingdom cards affect these important questions.

Adam says he has improved his intuition through this exercise, but how? Has he played any more SP games and actually seen the real amount of card draw or effectiveness of the card?

It's also a sophisticated approach to addressing a question of limited strategic importance, which just further enforces the wrong assumption that good Dominion strategy is obtained through analysis and calculation. The analysis and calculation that is popular on this board doesn't help that much with playing Dominion, it's just something to do for fun. It's cool to do things like this, but not helpful for getting better at the game.

Adam wants "evidence" that SCSN's views are correct, he already has it, SCSN is better than him at Dominion. You can't get better evidence that someone's ideas about Dominion are good than them winning lots of Dominion games.

[Davio]

Are we assuming 1 SP and cantrip actions?
Or just multiple SP's and no other actions?

A big part of SP is that it can draw other SPs.

If you want accurate simulations, you should look at multiple SPs and multiple payload or cantrip actions.

[AdamH]

Stef/MQ, thanks for taking the time to write something thoughtful. That's sort of the whole point of this.

I can plug a bunch of numbers into Excel and come up with a pretty picture. Is it helpful? You say probably not but to be honest, if my numbers are correct then I have a very useful conclusion: Take the standard X/11 expected draw, add one to it and you're really close to what the real number is. Knowing that I should have added one to my number in my league match against soulnet would probably have made me go for the Pools.

If I know why I'm going for the card and what benefit I can expect to get from it, then it helps me understand how to build my deck around it appropriately. You're not going to convince me that doing this math has no value because I've already found value in it.

It was SCSN himself who said "If you keep experimenting, keep practicing and keep learning, one year from now you can be better than any player is today." It's also well-known that he's had lots of success with just playing a lot of games with something he thinks he's bad at until he's not bad at it anymore.

Unfortunately, as a result of that second discussion I found out that playing lots of games and trying to get a feel for it works much better for him than it does for me. I played so many games with Familiar and I don't feel like I really got that much better. I felt like I didn't have any direction or purpose.

Don't get me wrong, I plan to play games like this, but I actually want to get something out of it so I'm trying to have some direction when I do this. It's totally OK that SCSN's advice needs a little bit of tweaking before I can make it click in my head; in fact I'd be really surprised if I'm the only person out there that needs a little bit of adjustment.

Am I getting too frustrated that people enjoy snark in the middle of a conversation that I'm taking seriously? Probably. Am I wrong when I say people +1ing a hyperbolic statement means they think my discussion isn't useful and I should just blindly do what SCSN says? Probably. But that is what it looks like sometimes and I've been told many times that it's funny when I get mad.

-----------------------

Davio, Labs can draw other Labs too -- yes this analysis alone won't tell you what you need to draw your deck. I'm just trying to think of SP as "+1 Action, +E(X) Cards", like a Lab variant, to help me better understand its value in a given kingdom -- the rest of the work with understanding how useful it is is something I'm already OK at.

There are other useful numbers to be found here, though. That's for sure.

[AdamH]

[...] the wrong assumption that good Dominion strategy is obtained through analysis and calculation. The analysis and calculation that is popular on this board doesn't help that much with playing Dominion, it's just something to do for fun. It's cool to do things like this, but not helpful for getting better at the game.

Allow me to stone-cold disagree with this statement though. Calculation isn't everything but without it my "intuition" means absolutely nothing. Maybe it doesn't help you but it helps me to be able to compare SP to other non-terminal draw cards using numbers. I know what Lab does and I want to know at what point SP becomes better than Lab -- when does it become the same as having two Labs?

[silverspawn]

...
Yeah people get lots of respect for snarky comments around here, but the fact that a comment that dismisses what I'm trying to talk about gets more respect than what I'm saying leads me to believe that maybe lots of people out there just blindly follow your hyperbole in this case.
...

Sorry to read your frustration here (& welcome to the internet).

It's not plain 'upvoting snarky comments' though. You really do over-complicate a question here. Sure you can try to revert to math when your intuition about Scrying Pool fails you, but in this case the required math is very hard to write down. I'd say you're still miles and miles away from getting the math right, whereas a simple rule like "just buy it unless you can explain a rubber duck why you shouldn't" is actually very helpful. At least it is to me. If that rule somehow fails you, I suggest you try SCSN's rule for a while ("just buy it").

It if was mountebank - fine, but you only go SP on roughly half of all boards, which is why I also thought the comment was stupid. I think it's once again a case where we give the games with scrying pool more weight, because they are more interesting, and take a lot longer to play. But they don't form a vast majority of all games. Maybe - probably - a majority of time playing, but not a majority of games.

Because it focuses on one aspect of SP in an extremely specific case. It says next to nothing about when/how to go about playing an SP deck. The numbers it produces are restricted to a case that doesn't exist in real games of Dominion. How many cards SP can draw isn't even that important, in a real game we want to know roughly how many turns it will take to get the engine going, how many turns will the game last, what can I do once I'm drawing lots of cards etc. The other kingdom cards affect these important questions.

Adam says he has improved his intuition through this exercise, but how? Has he played any more SP games and actually seen the real amount of card draw or effectiveness of the card?

It's also a sophisticated approach to addressing a question of limited strategic importance, which just further enforces the wrong assumption that good Dominion strategy is obtained through analysis and calculation. The analysis and calculation that is popular on this board doesn't help that much with playing Dominion, it's just something to do for fun. It's cool to do things like this, but not helpful for getting better at the game.

Adam wants "evidence" that SCSN's views are correct, he already has it, SCSN is better than him at Dominion. You can't get better evidence that someone's ideas about Dominion are good than them winning lots of Dominion games.

I see. I don't necessarily agree, but it's a good reason.

[silverspawn]

Adam wants "evidence" that SCSN's views are correct, he already has it, SCSN is better than him at Dominion. You can't get better evidence that someone's ideas about Dominion are good than them winning lots of Dominion games.

That's not a proof though. Even if we assume that skill is directly proportional to rating, which isn't true, then skill is not one dimensional. I think it's reasonable to say that skill is roughly identical to "how well can you play an average board", and that demonstrates the problem: every board is different. A lower skill player can always be better at a specific board, or even with a specific card, than a higher level player. I'm not suggesting that SCSN isn't better with SP, he probably is, but I don't like "he has a higher rating, therefore he's right" as a black box argument.

[Mic Qsenoch]

Adam wants "evidence" that SCSN's views are correct, he already has it, SCSN is better than him at Dominion. You can't get better evidence that someone's ideas about Dominion are good than them winning lots of Dominion games.

That's not a proof though. Even if we assume that skill is directly proportional to rating, which isn't true, then skill is not one dimensional. I think it's reasonable to say that skill is roughly identical to "how well can you play an average board", and that demonstrates the problem: every board is different. A lower skill player can always be better at a specific board, or even with a specific card, than a higher level player. I'm not suggesting that SCSN isn't better with SP, he probably is, but I don't like "he has a higher rating, therefore he's right" as a black box argument.

The only proof is playing games of Dominion, words/reasons/arguments are bullshit. So I rely heavily on rating in evaluating strategy advice, simply as a heuristic. I don't have a way of rigorously checking the strategy advice (I can't play enough games). The higher rated player is more likely to be right. It's a probabilistic thing, hey look at that, Dominion has some probabilities too.

[...] the wrong assumption that good Dominion strategy is obtained through analysis and calculation. The analysis and calculation that is popular on this board doesn't help that much with playing Dominion, it's just something to do for fun. It's cool to do things like this, but not helpful for getting better at the game.

Allow me to stone-cold disagree with this statement though. Calculation isn't everything but without it my "intuition" means absolutely nothing. Maybe it doesn't help you but it helps me to be able to compare SP to other non-terminal draw cards using numbers. I know what Lab does and I want to know at what point SP becomes better than Lab -- when does it become the same as having two Labs?

The calculation has no useful information, the number of expected cards drawn doesn't answer even the simple question you propose "when better than Lab?". Cards drawn with SP are more likely to be actions, so there's no straightforward way to compare the two. And this question is barely useful!

Your intuition means everything, your calculation means nothing because it doesn't apply in actual games of Dominion that can be played. Unless you change your calculation to include way more effects, which you won't be able to do as there are just too many. And then you have to update the calculation on each new board! It's hopeless. Almost anyone's hunches are a million times more valuable for playing Dominion than a neutered calculation about SP decks with all junk cards remaining and anonymous action cards without play effects. Certainly yours are!

Here's a comment Adam you can take it or leave it. This is based on your posts and watching your videos. You think too much, you overvalue "understanding", you want things to have reasons so your brain can feel comfortable about the decisions you make in the game.

You agonize over decisions whenever two options seem close. This is the absolute worst time to agonize! It wastes brain power. If two options are close in your mind, flip a coin and pick one. The time to stop and think is when you check the board for interactions, whenever you don't have any ideas at all, when the play order of your actions matters a lot, or when you need to check piles for auto-wins. Don't think the rest of the game. Don't even try to justify what you're doing, just do what immediately seems best.

Dominion games aren't won by playing "optimally". They are won by making reasonable choices and being opportunistic whenever the shuffles go your way, and by not missing chances for wins *right now* (or next turn, or next two turns).

You'll forgive me if I don't write anything else in this discussion, I've already word vomited enough for one day.

[silverspawn]

The only proof is playing games of Dominion, words/reasons/arguments are bullshit. So I rely heavily on rating in evaluating strategy advice, simply as a heuristic. I don't have a way of rigorously checking the strategy advice (I can't play enough games). The higher rated player is more likely to be right. It's a probabilistic thing, hey look at that, Dominion has some probabilities too.

completely agree, I just wanted to make a point that it's not a proof.

[WalrusMcFishSr]

Not meaning any offense to anyone (I find this type of heady discussion quite interesting), but this thread gave me a bad feeling of deja vu as soon as I saw it...

[AdamH]

MQ, you're not understanding how this is helpful because you're just being dismissive. You think I think too much and analyze and understand too much, I consider that a hallmark of my strategy and the reason why I've slowly and steadily improved as a player over the past several years. Maybe it doesn't work for you, and that's OK; I'm not asking you to read this topic or understand it if you don't want to; but I can't tell you how many people have told me that watching me explain things and understand them and break them down on my videos has made things clearer to them and made them a better player. I'm not alone in the way I think about the game and there's value in it. Maybe not for you, but for me and other people out there.

You can't convince me that what I do is wrong. You can't convince me that the way I play or the way I think about the game is wrong. Why would you ever want to do that? Why are you saying that I should think a certain way when you aren't me?! Just because I'm not you or SCSN or in the A division of the Dominion League doesn't mean I can't get there unless I do everything just like you. You aren't perfect. The moment you decide you can't learn more about Dominion is the same moment you'll stop getting better, and I promise you if you stop getting better at Dominion it's only a matter of time before I'm better than you.

Look, I have a lot to learn and that's clear. There's even good advice in your post. Unfortunately what I'm seeing is that I've posted something that's helping me understand a part of the game. You reply and say it's useless because understanding the game is useless, you should just play it. If you've got something helpful to say, can't you just say the helpful thing? If you think this is useless then just don't post in the thread. If you think that talking about Dominion is useless and you should just play more to get better, then why do you even post on the forums? If you think I'm saying something that's incorrect then tell me that, but at least be constructive instead of dismissive.

Or is that too much to ask of people on these forums? Jeez.

[SCSN]

Am I getting too frustrated that people enjoy snark in the middle of a conversation that I'm taking seriously? Probably. Am I wrong when I say people +1ing a hyperbolic statement means they think my discussion isn't useful and I should just blindly do what SCSN says?

I think you just mistook for snark my genuine though economically expressed thoughts on the matter. I understand the perspective that takes offense at my comment, but in the end I said in two words what Stef and Mic explained in multiple paragraphs.

I do think this whole approach is counter-productive mental masturbation, regardless of your efforts to convince yourself of the contrary. The very interesting book Hare Brain, Tortoise Mind: How Intelligence Increases When You Think Less has a chapter called "Premature Articulation: How Thinking Gets in the Way of Learning", which captures neatly what you're doing here. If you're a toddler that still has some trouble walking the solution is not to teach him physics, anatomy and physiology, but just to let him stumble and fall and stand up again. So my advice to you is to buy Scrying Pool whenever you see it except when there's a really strong rush strategy, stop deliberately thinking about whether you made the right call and just let your brain calibrate itself over time. I know this is not what you want to hear, but following it will be infinitely more helpful than writing 20 mathematical papers.

It if was mountebank - fine, but you only go SP on roughly half of all boards

That's just way off. If I had to ballpark it I go for SP ~92% of the time and judging by the times I've regretted it I estimate its optimal purchase frequency to be around 88%. I expect most people to be baffled by this, so here's another one: for Golem I think it's around 70%.

[silverspawn]

That's just way off. If I had to ballpark it I go for SP ~92% of the time and judging by the times I've regretted it I estimate its optimal purchase frequency to be around 88%. I expect most people to be baffled by this, so here's another one: for Golem I think it's around 70%.

I just don't think that's true. Like, I don't think you're playing it wrong, but I have a hard time believing that you buy it 92% of your games. are there stats for that? it's possible that I don't buy it enough I guess, but I don't buy it that much less than my average opponent.

[Mic Qsenoch]

I'm not against this kind of analysis, I have no problem with these posts. I just want to make it clear to anyone reading that it won't help anyone be better at Dominion. It's good practice for calculating probabilities, and fun to think about how deck composition and the various parts of SP affect the probabilities.

I try to convince you you're wrong because I want to see you get better, I'm just a nice guy like that.

Also, I'm not writing replies just for you, I'm writing so that other people also don't follow your approach because I think it sucks. I'm writing so people don't waste their time thinking about these things if they want to get better at the game. You are welcome to say your approach works once you reach the top of the board, until then you are just blowing smoke.

I participate in these forums to learn about tournaments, to read game reports/help threads (the absolute best possible resource for learning Dominion, aside from the game itself), to hear about streams/videos, and to tell people they're wrong when I think they're full of crap. It's fun times.

[AdamH]

Thanks for your concern. I never stated I was doing anything but calculating an expected draw for SP given deck composition; as one of many tools used to evaluate a kingdom in certain circumstances. If you want me to make that clear in the OP in big bold letters because you think I'm saying something that I never actually said, fine. I'll get to it.

If you think the way I get better at Dominion sucks, fine. Don't get better at Dominion that way. I'm me, on the other hand, and since I'm not you I'll continue doing what has been working for me for years now. I mean you'd come across like less of a jerk if you said "that's not going to work for everyone" or "that doesn't work for me" instead of "it sucks" -- especially since I'm steadily getting better at Dominion by doing this and it's consistently working. Just because I'm not at the top of the leaderboard doesn't mean everything I say is wrong. Just because it's not exactly what you think (and you're at the top of the leaderboard) doesn't mean everything I say is wrong and it doesn't mean everything you say is right.

Thanks for the rest of the advice you gave me too. There's good stuff in there and I didn't ignore it. It's not good advice by virtue of the fact that you said it and you're at the top of the leaderboard, but rather because I can admit there are weaknesses in my game (like the one this topic is designed to help!) and some of the things you mentioned are among them. I'm working on it.

Now if it's OK with you all, I'm going to finish drawing my conclusions from this math when I get a chance, since nobody has pointed out any huge errors in my calculations. If you still think it's worthless, then go ahead and let this thread die once I'm done with it (it will only be a day or two, I'm sure you can handle it). Please try and be constructive.

[silverspawn]

If he's right and your way of doing things isn't effective for you, telling you so seems pretty productive. whether or not it's reasonable for him to claim such a thing is of course up for debate.

[2.71828.....]

I just assume Adam is doing the math because it is fun, and this is what nerds do for fun.  Right?

[2.71828.....]

but on the "learning to play Dominion better" side of the house, I think this article (which I think it might be better under articles not General discussion?) is a great learning tool for some, while others will be fine without it.  People learn different ways.  I think Adam's Jack article where he really analyzed Jack was extremely beneficial to his play, and I think this will be similarly beneficial.  Now, not everyone learned from that [the jack] article, and not everyone will learn from this one.  Which is fine. However, if analyzing a card to the nth degree is what really helps solidify how to use that card, then this has accomplished its purpose.  And because he is a great guy share it for the rest of us who may learn something from it.

[JW]

An interesting result (one that makes me question the correctness of my code, actually) is that Eb(X)*Pb(X) was always equal to one. This result seems like it shouldn't be but I can't think of a good reason why and the numbers I get all pass at least a simple sanity check. Did I do something wrong? Is there some reason why this should be the case?

Here's the output of the program:

Code: [Select]

2 1.1667 1.1750 1.1818
 3 1.4615 1.3371 1.2727
 4 1.6758 1.4890 1.3636
 5 1.8659 1.6327 1.4545
 6 2.0414 1.7699 1.5455
 7 2.2058 1.9016 1.6364
 8 2.3611 2.0288 1.7273
 9 2.5088 2.1522 1.8182
10 2.6500 2.2723 1.9091
11 2.7857 2.3896 2.0000
12 2.9167 2.5045 2.0909
13 3.0435 2.6174 2.1818
14 3.1667 2.7284 2.2727
15 3.2867 2.8378 2.3636
16 3.4038 2.9458 2.4545
17 3.5185 3.0525 2.5455
18 3.6310 3.1581 2.6364
19 3.7414 3.2627 2.7273
20 3.8500 3.3664 2.8182
21 3.9570 3.4692 2.9091
22 4.0625 3.5713 3.0000
23 4.1667 3.6727 3.0909
24 4.2696 3.7735 3.1818
25 4.3714 3.8738 3.2727
26 4.4722 3.9735 3.3636
27 4.5721 4.0727 3.4545
28 4.6711 4.1715 3.5455
29 4.7692 4.2699 3.6364
30 4.8667 4.3680 3.7273
31 4.9634 4.4656 3.8182
32 5.0595 4.5630 3.9091
33 5.1550 4.6600 4.0000
34 5.2500 4.7568 4.0909
35 5.3444 4.8533 4.1818
36 5.4384 4.9496 4.2727
37 5.5319 5.0456 4.3636
38 5.6250 5.1415 4.4545
39 5.7177 5.2371 4.5455
40 5.8100 5.3325 4.6364
41 5.9020 5.4278 4.7273
42 5.9936 5.5229 4.8182
43 6.0849 5.6178 4.9091
44 6.1759 5.7126 5.0000
45 6.2667 5.8073 5.0909
46 6.3571 5.9018 5.1818
47 6.4474 5.9962 5.2727
48 6.5374 6.0904 5.3636
49 6.6271 6.1846 5.4545
50 6.7167 6.2786 5.5455

Next up: pretty pictures and results

I get different results compared to your first column. We agree on the case with 2 actions (including Pool) at 1.1667. However, I get 1.3205 for the 3 actions case compared to your 1.4615 and all my subsequent numbers are lower too. 

3 actions isn't that hard to do by hand. To get 3 cards you need on top either (Action, Action) or (StopCard, Action, Action). Action, action has probability 2/13 * 1/12= 1/78. (Stop, Action, Action) has probability 11/13 * 2/12 * 1/11= 1/78 as well. So 3 cards has probability 1/39= 2.56%.

To get 1 card you need on top (Stop Card, Stop Card). That happens 11/13 * 10/12= 55/78 = 70.51%. The rest of the time we know you get 2 cards, 26.92%. Weight these and they average 1.3205.

I similarly get that the weighted total for the "non-action on top" case is 1 exactly, which suggests that we differ in the "action on top" case.

Here's my calculation, which includes the self-spy effect:
6 actions including Scrying Pool, 11 non-actions: 1.7292 average cards drawn
7 actions including Scrying Pool, 11 non-actions: 1.8529 average cards drawn
8 actions including Scrying Pool, 11 non-actions: 1.9722 average cards drawn
9 actions including Scrying Pool, 11 non-actions: 2.0877 average cards drawn
10 actions including Scrying Pool, 11 non-actions: 2.2 average cards drawn
11 actions including Scrying Pool, 11 non-actions: 2.3095 average cards drawn
12 actions including Scrying Pool, 11 non-actions: 2.4167 average cards drawn
13 actions including Scrying Pool, 11 non-actions: 2.5217 average cards drawn
14 actions including Scrying Pool, 11 non-actions: 2.625 average cards drawn

[theright555J]

As I see it, this whole thread really boils down to something like this:

Adam: "I think I undervalue SP, possibly because I underestimate the drawing power in lack of trashing. Let me do some math to assess this...huh, looks like the draw is better than I thought"

Ivory Tower: "SP is just really good, just use it"

Adam: "But the math, as well as my thought process, helps me play better!"

Ivory Tower: "No it doesn't"


I have learned a great deal from Adam and WW's videos as well as from their thought processes. Adam has a lot of guts to put stuff like this out there when he's NOT #1 on the leaderboard. It's really quite discouraging to see top-tier players actually disparaging a thought process!!

As my daughter's viola teacher says, practice doesn't make perfect, practice makes easy. Unfortunately, if one is doing the skill wrong in the first place, then it's the bad habits that become easy! Therefore, any time an expert backhandedly dismisses an idea or concern from someone less experienced and just says "practice more" or "you just got unlucky" or " it will come in time" it brings my piss to a boil.

[jsh357]

I dunno.  I feel like the more you put next to each other, the worse it looks.  I have no idea what YOU people are debating; to me it's clear you don't want ten of these things.  Maybe if you draw a bunch of cool looking cards like Highway and Salvager it's a legitimate use of time.  The Witch kind of resembles a kid's drawing, but it's kind of cool how her hair rises magically while gazing into the pool.  Still, the less you look at Scrying Pool the better it holds up.

On the other hand, the Potion is quite fashionable, which adds a few style points that help to compensate. 



Wait a minute... ARE THOSE SCROLLS IN THE CABINET BACK THERE?

[JW]

For the Eb(X)*Pb(X) = 1 thing, I have a semi-explanation.

Suppose that your deck is infinite in size, so the odds of revealing a non action card is always the same.
...
There might be some trickiness with a deck that actually changes in action proportion, but I think there's a theorem where sifting doesn't affect probability of drawing key cards that I'll cite here. I don't know if it applies, and actually forgot how that theorem goes, but there you go, a semi-explanation.

There's a proof by induction here . 

Call F(X,C) the number of cards you draw on average with Scrying Pool in a deck that started with X actions (including Scrying Pool) and C stop-cards, given that you’ve put one stop card on the bottom of the deck (and the Pool was in your hand). Note that the chance that the top card is a non-action is C/(X+C-1).

If the current top card is an action you’re done, drawing only one card. While if the current top card is an action, you average 1 card + the amount you’d get when the problem repeats as if your deck started with X-1 actions instead of X actions.

So, F(X,C) satisfies the recursive relationship: F(X,C)= (C-1)/(X+C-2) + [1+ F(X-1,C)]*(X-1)/(X+C-2) =  1 + (X-1)* F(X-1,C) / (X+C-2)

Conjecture that F(X,C) = (X+C-1)/C.

Trivially, F(1,C) = 1, because if Scrying Pool is your only action it draws one card. So the formula works in that case.

Assume that F(k,C)= (k+C-1)/C. Then using the recursive relationship, it’s just a bit of algebra to show that F(k+1,C) = (k+C)/C.

Since this formula for F(X,C) works for X=1, and it works for X=k+1 if it works for X=k, it’s true for any positive integer X (for any C).

[DG]

Mathemetician - how many scrying pools do we need?
Engineer - keep adding scrying pools until they draw the deck.

Scrying pools generally follow the advice for basic engine building but often amplify the effects. There's not much point giving basic advice though. Scrying pools support extreme decks but they're impossible to categorize. There aren't really any rules of thumb for scrying pools since the kingdoms do really make a difference. The maths is really hard and isn't actually going to be a better asset than practicing engine building.

[SCSN]

That's just way off. If I had to ballpark it I go for SP ~92% of the time and judging by the times I've regretted it I estimate its optimal purchase frequency to be around 88%. I expect most people to be baffled by this, so here's another one: for Golem I think it's around 70%.

I just don't think that's true. Like, I don't think you're playing it wrong, but I have a hard time believing that you buy it 92% of your games. are there stats for that? it's possible that I don't buy it enough I guess, but I don't buy it that much less than my average opponent.

I asked AI if he can get these numbers easily and will let you know what he responds, but I went ahead and looked at my last 20 pro games with SP on board: I bought it in 19 of them.

I'm not at all surprised that you and your average opponent underbuy it a lot because almost everyone does, including players as good as AI himself and WW (based on my experience playing him I'm pretty sure he goes for it <50% of the time).

As my daughter's viola teacher says, practice doesn't make perfect, practice makes easy. Unfortunately, if one is doing the skill wrong in the first place, then it's the bad habits that become easy! Therefore, any time an expert backhandedly dismisses an idea or concern from someone less experienced and just says "practice more" or "you just got unlucky" or " it will come in time" it brings my piss to a boil.

You're confusing practice and repetition. Of course when you repeat the same damn thing over and over again you can't expect any different results, that would be insane. Practice, on the other hand, consists of varying what you're doing and sticking with what works. In the words of Niels Bohr: "An expert is a person who has found out by his own painful experience all the mistakes that one can make in a very narrow field."

That's why my advice to AdamH is not to simply play a shitload of Scrying Pool games, but to specifically go for them on every single board and to make them work no matter what, without allowing him to talk himself out of it. That way his brain will learn through sheer experience the very rare circumstances under which it should be skipped.

And it's quite ironic that you accuse the most vocal proponent of "when your feet are stuck in the mud, it's time to get your hands dirty" of occupying an ivory tower. Instead of trying to convince people of what I think, I'm always encouraging them to go out and see for themselves.

[JW]

For the Eb(X)*Pb(X) = 1 thing, I have a semi-explanation.

There's a proof by induction here. 

There's a similar proof by induction of the result for the case where your top card is an action (or you can skip it based on the results of the other case, as Titandrake points out below).

With those two results, you can calculate the average cards drawn by Scrying Pool in a deck of X actions (including Pool) and C non-actions, assuming that you discard non-actions with the self-spy effect. Call this S(X,C).

S(X,C)=  (X^2 + 3*X*C + C^2 - 2*C -1)/ [ (X+C-1) * (C+1)]

That formula gets you, among others, the results I've posted earlier. Useful sanity checks: With X=1, this equals 1 (Pool is your only action). With X=3 and Y=1, this equals 2.667, for which I showed the direct derivation above. 

Note that this formula has assumed that Scrying Pool is the only card in your hand, and that you can potentially draw all of your other cards. That will slightly overstate the number of cards you draw on average in decks with fewer than 5 stop cards, but of course Scrying Pool is great in those circumstances so a formula telling you the exact number is not necessary.

[CG19]

I'm going to weight this thread more towards the math side here. TL;DR: More math.

Without using code, you can get the E(X) (aka average draw) with some easy math. Following a very similar problem on this site:
http://math.stackexchange.com/questions/75968/expectation-of-number-of-trials-before-success-in-an-urn-problem-without-replace

We can simplify the summation down to 1+a/(p+1) where a = actions in deck and p = non-actions in deck (this is without the spy effect). Now, we can add the spying-effect by saying that p/t times (where t = a+p is the total cards in deck), the spy-effect finds a non-action card and a/t times it doesn't. So we get to p/t*(1+a/p) + a/t*(1+a/(p+1)). Using the same set of cards as before:

6 actions not including Scrying Pool, 11 non-actions: 1.53 average cards drawn (1.50 without spy effect)
7 actions not including Scrying Pool, 11 non-actions: 1.62 average cards drawn (1.58 without spy effect)
8 actions not including Scrying Pool, 11 non-actions: 1.70 average cards drawn (1.67 without spy effect)
9 actions not including Scrying Pool, 11 non-actions: 1.79 average cards drawn (1.75 without spy effect)
10 actions not including Scrying Pool, 11 non-actions: 1.87 average cards drawn (1.83 without spy effect)
11 actions not including Scrying Pool, 11 non-actions: 1.96 average cards drawn (1.92 without spy effect)
12 actions not including Scrying Pool, 11 non-actions: 2.04 average cards drawn (2.00 without spy effect)
13 actions not including Scrying Pool, 11 non-actions: 2.13 average cards drawn (2.08 without spy effect)
14 actions not including Scrying Pool, 11 non-actions: 2.21 average cards drawn (2.17 without spy effect)

What I don't take into account is current discard and deck size. If you only have 1 card in your deck that is a non-action and you discard it, the spy-effect does not help because it shuffles the non-action right back in. The equation gets more complicated with every caveat, but I think the first equation without the spy effect is simply enough to use when you need it

[shark_bait]

here's another one: for Golem I think it's around 70%.

Can we all just take a moment of silence for Council Room.

[Titandrake]

For the Eb(X)*Pb(X) = 1 thing, I have a semi-explanation.

Suppose that your deck is infinite in size, so the odds of revealing a non action card is always the same.
...
There might be some trickiness with a deck that actually changes in action proportion, but I think there's a theorem where sifting doesn't affect probability of drawing key cards that I'll cite here. I don't know if it applies, and actually forgot how that theorem goes, but there you go, a semi-explanation.

There's a proof by induction here. 

Call F(X,C) the number of cards you draw on average with Scrying Pool in a deck that started with X actions (including Scrying Pool) and C stop-cards, given that you’ve put one stop card on the bottom of the deck (and the Pool was in your hand). Note that the chance that the top card is a non-action is C/(X+C-1).

If the current top card is an action you’re done, drawing only one card. While if the current top card is an action, you average 1 card + the amount you’d get when the problem repeats as if your deck started with X-1 actions instead of X actions.

So, F(X,C) satisfies the recursive relationship: F(X,C)= (C-1)/(X+C-2) + [1+ F(X-1,C)]*(X-1)/(X+C-2) =  1 + (X-1)* F(X-1,C) / (X+C-2)

Conjecture that F(X,C) = (X+C-1)/C.

Trivially, F(1,C) = 1, because if Scrying Pool is your only action it draws one card. So the formula works in that case.

Assume that F(k,C)= (k+C-1)/C. Then using the recursive relationship, it’s just a bit of algebra to show that F(k+1,C) = (k+C)/C.

Since this formula for F(X,C) works for X=1, and it works for X=k+1 if it works for X=k, it’s true for any positive integer X (for any C).

Technically, I think this only holds when C >= 2. Your recursive definition for F(X, C) relies on assuming the non-action card you discarded will not be revealed again by this pool, and you need at least 2 stop cards total in your deck to guarantee this. But yes, that works.

Edit: For the case where you keep an action on top, I don't think the induction is that complicated, since I think you can skip induction entirely. If you label that case A(X,C), you can cheat it by saying

A(X,C) = 1 + F(X-1,C+1).

You are guaranteed to draw 1 card. Then, you have a deck of X-1 actions including Pool, and C stop cards, but you don't know that the last card of your deck is a stop card. So, add a virtual stop card to the bottom of your deck (which is why you get the C+1.)

I think this gives you the same result, but your S(X,C) aggregates both results and I don't have time to combine the two right now.

[JW]

I'm going to weight this thread more towards the math side here. TL;DR: More math.

Without using code, you can get the E(X) (aka average draw) with some easy math. Following a very similar problem on this site:
http://math.stackexchange.com/questions/75968/expectation-of-number-of-trials-before-success-in-an-urn-problem-without-replace

We can simplify the summation down to 1+a/(p+1) where a = actions in deck and p = non-actions in deck (this is without the spy effect). Now, we can add the spying-effect by saying that p/t times (where t = a+p is the total cards in deck), the spy-effect finds a non-action card and a/t times it doesn't. So we get to p/t*(1+a/p) + a/t*(1+a/(p+1)). Using the same set of cards as before:

6 actions not including Scrying Pool, 11 non-actions: 1.53 average cards drawn (1.50 without spy effect)

The bolded part a/t*(1+a/(p+1)) isn't correct because you have an action on top of your deck, which the formula isn't taking into account. See my post directly above yours.

[JW]

There's a proof by induction here. 

Call F(X,C) the number of cards you draw on average with Scrying Pool in a deck that started with X actions (including Scrying Pool) and C stop-cards, given that you’ve put one stop card on the bottom of the deck (and the Pool was in your hand). Note that the chance that the top card is a non-action is C/(X+C-1).

If the current top card is an action you’re done, drawing only one card. While if the current top card is an action, you average 1 card + the amount you’d get when the problem repeats as if your deck started with X-1 actions instead of X actions.

So, F(X,C) satisfies the recursive relationship: F(X,C)= (C-1)/(X+C-2) + [1+ F(X-1,C)]*(X-1)/(X+C-2) =  1 + (X-1)* F(X-1,C) / (X+C-2)

Conjecture that F(X,C) = (X+C-1)/C.

Trivially, F(1,C) = 1, because if Scrying Pool is your only action it draws one card. So the formula works in that case.

Assume that F(k,C)= (k+C-1)/C. Then using the recursive relationship, it’s just a bit of algebra to show that F(k+1,C) = (k+C)/C.

Since this formula for F(X,C) works for X=1, and it works for X=k+1 if it works for X=k, it’s true for any positive integer X (for any C).

Technically, I think this only holds when C >= 2. Your recursive definition for F(X, C) relies on assuming the non-action card you discarded will not be revealed again by this pool, and you need at least 2 stop cards total in your deck to guarantee this. But yes, that works.

I think it still works for C=1. C=1 is a trivial case, since you put the one non-action on the bottom and draw your entire deck, F(X,1)= X. And indeed the formula works there, and this result holds since the chance the top card is a non-action is 1/X (the Scrying Pool is in your hand).

[Titandrake]

There's a proof by induction here. 

Call F(X,C) the number of cards you draw on average with Scrying Pool in a deck that started with X actions (including Scrying Pool) and C stop-cards, given that you’ve put one stop card on the bottom of the deck (and the Pool was in your hand). Note that the chance that the top card is a non-action is C/(X+C-1).

If the current top card is an action you’re done, drawing only one card. While if the current top card is an action, you average 1 card + the amount you’d get when the problem repeats as if your deck started with X-1 actions instead of X actions.

So, F(X,C) satisfies the recursive relationship: F(X,C)= (C-1)/(X+C-2) + [1+ F(X-1,C)]*(X-1)/(X+C-2) =  1 + (X-1)* F(X-1,C) / (X+C-2)

Conjecture that F(X,C) = (X+C-1)/C.

Trivially, F(1,C) = 1, because if Scrying Pool is your only action it draws one card. So the formula works in that case.

Assume that F(k,C)= (k+C-1)/C. Then using the recursive relationship, it’s just a bit of algebra to show that F(k+1,C) = (k+C)/C.

Since this formula for F(X,C) works for X=1, and it works for X=k+1 if it works for X=k, it’s true for any positive integer X (for any C).

Technically, I think this only holds when C >= 2. Your recursive definition for F(X, C) relies on assuming the non-action card you discarded will not be revealed again by this pool, and you need at least 2 stop cards total in your deck to guarantee this. But yes, that works.

I think it still works for C=1. C=1 is a trivial case, since you put the one non-action on the bottom and draw your entire deck, F(X,1)= X. And indeed the formula works there, and this result holds since the chance the top card is a non-action is 1/X (the Scrying Pool is in your hand).

Oh, right I was thinking that F(X,1) should be X+1, but you already have one card in hand.

[CG19]

I'm going to weight this thread more towards the math side here. TL;DR: More math.

Without using code, you can get the E(X) (aka average draw) with some easy math. Following a very similar problem on this site:
http://math.stackexchange.com/questions/75968/expectation-of-number-of-trials-before-success-in-an-urn-problem-without-replace

We can simplify the summation down to 1+a/(p+1) where a = actions in deck and p = non-actions in deck (this is without the spy effect). Now, we can add the spying-effect by saying that p/t times (where t = a+p is the total cards in deck), the spy-effect finds a non-action card and a/t times it doesn't. So we get to p/t*(1+a/p) + a/t*(1+a/(p+1)). Using the same set of cards as before:

6 actions not including Scrying Pool, 11 non-actions: 1.53 average cards drawn (1.50 without spy effect)

The bolded part a/t*(1+a/(p+1)) isn't correct because you have an action on top of your deck, which the formula isn't taking into account. See my post directly above yours.

Correct! So I guess it should be a/t*(2+a/(p+1)). I'll recheck the expected values, but it should bump up the spy-effect pretty well. I thought the difference looked rather low.

[AdamH]

JW and I exchanged a couple of PMs and we got my code to be what looks to be correct. It's certainly not calculated the same way you all are doing it now, so I'll be interested to see if you end up with the same numbers. I'm going to edit my previous posts to take out wrong things and put in right things with pretty pictures and add in a lovely disclaimer and maybe some conclusions.

[silverspawn]

I'm not at all surprised that you and your average opponent underbuy it a lot because almost everyone does, including players as good as AI himself and WW (based on my experience playing him I'm pretty sure he goes for it <50% of the time)

oh, well that's good to know.

[markusin]

It if was mountebank - fine, but you only go SP on roughly half of all boards

That's just way off. If I had to ballpark it I go for SP ~92% of the time and judging by the times I've regretted it I estimate its optimal purchase frequency to be around 88%. I expect most people to be baffled by this, so here's another one: for Golem I think it's around 70%.

I think 88% might be a good estimate in the end. Scrying Pool is just so good so often. I used to lose to it all the time, but then I started using it myself and started winning a lot when my opponent would ignore it. When I do win by ignoring it, say to go for a rush or something, the game still ends up being pretty close.

Golem? Meh, I feel it breaks even a lot of the time, as in your decks performs about as well as it would have without the Golem. Whatever, we have another thread for Golem (also started by Adam).

[Witherweaver]

Scrying Pool Math:

Scrying Pool = <3

QED.

[silverspawn]

Scrying Pool Math:

Scrying Pool = <3

QED.

that's awful and wrong in every single way

[Witherweaver]

Scrying Pool Math:

Scrying Pool = <3

QED.

that's awful and wrong in every single way

The Horse says: Doctorate denied.

[JW]

We can simplify the summation down to 1+a/(p+1) where a = actions in deck and p = non-actions in deck (this is without the spy effect). Now, we can add the spying-effect by saying that p/t times (where t = a+p is the total cards in deck), the spy-effect finds a non-action card and a/t times it doesn't. So we get to p/t*(1+a/p) + a/t*(1+a/(p+1)). Using the same set of cards as before:

The bolded part a/t*(1+a/(p+1)) isn't correct because you have an action on top of your deck, which the formula isn't taking into account. See my post directly above yours.

Correct! So I guess it should be a/t*(2+a/(p+1)). I'll recheck the expected values, but it should bump up the spy-effect pretty well. I thought the difference looked rather low.

That still doesn't look quite right, because if you have that action on top of your deck then going forward you have one fewer action in your deck. So presumably it should be (a-1) instead of the bolded a.

In your notation, my result is: 1 + [(A+2P+1)*A] / [(P+1)(A+P)].

[jomini]

when speaking about expert opinion, we have to remember that dominion is not just pick a set of cards and go; it is also a huge amount of decisions about what order to make buys. SCSN is likely better at picking a set of cards, but it is possible that the bigger issue is the ability to optimally play the game once a set of cards are chosen.

For instance say SCSN takes SP 92% of the time and wins those matches 80% of the time. Does that mean it is the dominant strategy 92% of the time? Not without a lot more priors. For instance, if SCSN wins 75% of his games anyways, there is extremely little information to be gained by his win rate. Further, it may well be that SCSN plays SP better than some alternative cards. Say we could exhaustively have a computer search all possible end game states for each decision SCSN makes after he picks SP and the rest of his strategy. He makes the optimal move 98% of the time. If, with other cards, he makes the optimal move only 95% of the time on average, then cards that intrinsically (e.g. you computed every possible game) are 3.16% better or less will still result in SP being the better call for SCSN, but worse for everyone else.

There are, broadly, two big skills in dominion. Strategic - picking a strategy from the cards in the kingdom and tactical - making correct decisions, particularly with timing, while playing the strategy.

Elite players win too many games simply off raw tactical skill to accurately gauge card strength off their win rate. It is only when they are playing equally skilled opponents that card strength has a larger impact on win rate that player tactical skill. SCSN could almost certainly beat his average opponent most of the time even if he intentionally picked the second best strategy on the board. Flawlessly playing the 2nd best strategy will very often beat marginally flawed play of the best strategy.

This is why Big Money is so strong until you really get to know the game. Your ability to screw up decisions (when should I start greening, should I take two components at $7 or a gold when I will eventually need both, do I reveal this moat against a Margrave or do I draw for 3 good cards, etc.) is relatively low. So even though the strategy, if played perfectly, is weaker than another, also played perfectly; a casual player can come close to perfection (say 80%) with BM strats but only makes 33% correct plays with a complicated engine build. The engine has to be more than twice as strong to come out ahead.

Disambiguation is really hard. I see way, way too much variance in the annual card rankings to say that player skill is sufficient to determine card strength.

That being said, SCSN is totally correct about the best course of action to get better. Improving your tactical play happens when you put yourself into difficult challenges and you get very good at getting the maximum amount out of a strat. Picking individual cards or classes of cards and always playing them will make you much better at playing them optimally.

[AdamH]

That being said, SCSN is totally correct about the best course of action to get better. Improving your tactical play happens when you put yourself into difficult challenges and you get very good at getting the maximum amount out of a strat. Picking individual cards or classes of cards and always playing them will make you much better at playing them optimally.

To be clear, I fully intend on playing lots of games like this, but playing games and getting a gut feeling and not doing anything else isn't enough for me. I want to have some idea of why things are happening and some of the theory behind it or else it's not going to click for me.

All of the other things they're saying I should do are great advice. But should I do them instead of theoretical stuff? No. Absolutely not. And if the implication is that I don't have the "brain power" to do both of them and "gut" is the more important thing to focus on if I have to pick one; well my response is that I don't think I should have to choose. If the best players in the world can only do one of those things then it's not going to be long before someone else can do both and be better. I'm trying to be that person.

[Titandrake]

Meanwhile I'm sitting here trying to get my math/probability calculation skills to go from 2/10 to something more like 3/10...

Personally, I don't get much value out of hard mathematical analysis like this - I'm noticing that I come up with some probabilities to figure out likely outcomes, then rely on my gut in the end.

If there was a Council Room, I'd be using that a lot, because the data helps give qualitative results - "I buy this much less than everyone else, am I underestimating it?" Whereas this feels more like the article on Treasure Map, "how likely is it to have these collide by this turn if I buy this many": not that useful and yet not that useless.