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Author Topic: The Mathematics of Venture Chaining AND The Sociology of Internet Forums  (Read 28544 times)

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Kirian

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #75 on: August 15, 2013, 10:25:46 am »
+3



Awww, it's like an ABC after-school special.

You young'uns may have to go look that up.  *grumble mumble get off my lawn*
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DStu

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Re: The Mathematics of Venture Chaining
« Reply #76 on: August 15, 2013, 11:57:59 am »
0

Man you're right. This is not the most strategically helpful article in the world. It's not as if when I'm playing Dominion I'm (consciously) performing advanced combinatorics in my head. Mostly I thought it was an interesting mathematical result, and I wanted to share with you guys.
I knew I shouldn't had commented in this thread because my defense dor dondon would come out more as an attack on you than an actual defense.  I did it anyway, sorry for this.

/leave thread

edit:
Quote

OK, seems like it would be worth to read page 3 after all...
« Last Edit: August 15, 2013, 11:59:25 am by DStu »
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Wingnut

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #77 on: August 15, 2013, 02:16:24 pm »
+10

What I learned from this thread:

Many of us are jerks sometimes
We can all hash our differences out
The result is usually some memes
And I still can't understand any basic mathematical equations or anything of the sort
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meandering mercury

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Re: The Mathematics of Venture Chaining
« Reply #78 on: August 15, 2013, 02:42:45 pm »
+4

This is now off-topic because it's on-topic, but the original "authoritative one-liner":

I think it is far more useful to view Venture as a Treasure card that augments the value of another Treasure card in your deck by $1.

... is misleading when it comes to chaining and is the point of the OP. If you break down the equation and add in the + d, then it's:

value of venture = $1 + $d + $v/(1+N)

$1: this is the $1 you get from playing the venture
$d: this is the value of an average, non-venture Treasure card. So far, this is dondon's point.
$v/(1+N): this is (approximately) the ratio of venture to non-venture treasures in your deck and is the value of chaining. There's a +1 in the denominator so it's not quite the ratio.

In other words, if you have as many Ventures as non-Venture treasure cards (plus one), your Venture will get you, on average, $2 + the average Treasure card. If your Venture:other treasure ratio is 2:1, then it's about $3 + the average Treasure card. Thinking of ventures as being "1 + ratio of venture to non-venture" might be informative if you're trying to decide between e.g. buying a gold or buying a venture in the mid-game.

OK, back to your regularly scheduled programming ...
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dondon151

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Re: The Mathematics of Venture Chaining
« Reply #79 on: August 15, 2013, 06:55:46 pm »
0

This is now off-topic because it's on-topic, but the original "authoritative one-liner":

... is misleading when it comes to chaining and is the point of the OP. If you break down the equation and add in the + d, then it's:

You're definitely correct mathematically. However, Venture can augment the value of another Venture by $1, and each Venture only adds $1 by itself. I think, though, that my original point wasn't supposed to be concerned with the average "value" of Venture at all (but of course it sounds a little ambiguous).
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Polk5440

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Re: The Mathematics of Venture Chaining
« Reply #80 on: August 15, 2013, 07:12:03 pm »
0

Well, I guess to be really clear,
value of venture = $1 + $d + $v/(1+N)
is the expected coin of PLAYING a Venture from your hand given you have v other Ventures in your deck and discard pile, N non-venture treasures in your deck and discard pile, and know nothing about what's in your deck and discard pile other than the constant average value of non-venture treasures, d.
« Last Edit: August 15, 2013, 07:13:05 pm by Polk5440 »
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WanderingWinder

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #81 on: August 15, 2013, 07:19:25 pm »
0

You can go through the average value of playing a venture (if you want to - and I did this a couple years ago - it's sorta nifty and you want to use induction; induction is definitely your friend here), you can. But generally, in terms of gameplay, what is the decision you are making? Well, venture vs gold is a pretty significant one - basically take the gold for a long time. And venture chaining is important.

And then you get to weird edge cases of 'but what about what I have in my hand and/or running out of treasures' and 'but what about special treasures' that just make the whole exercise less fun. So, the other side degenerate case where all or virtually all your treasures are ventures is also not hard to understand, and the in-between stuff usually doesn't make much difference. So it's sort of a cool math problem, but as for Dominion decision-making, dondon's point is generally right, and getting more precise on this would take time probably better spent elsewhere.

Anyway, if you want to do it, induction my friend.

dondon151

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #82 on: August 15, 2013, 07:24:19 pm »
0

Well, venture vs gold is a pretty significant one - basically take the gold for a long time.

Wait, really? I mean, it depends on whether there is Copper trashing and several other factors, but I usually go by taking only 1 Gold over Ventures.
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Watno

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #83 on: August 15, 2013, 07:30:10 pm »
+2

Well, when optimizing average value, you want venture over gold once you have more gold than copper.
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dondon151

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #84 on: August 15, 2013, 10:05:35 pm »
0

Well, when optimizing average value, you want venture over gold once you have more gold than copper.

I feel like this is not necessarily true. Already having Ventures in your deck increases the average value of further Ventures, plus you have an effectively smaller deck. If you simulate BMU with Colonies, a bot that buys only Ventures at $6 absolutely crushes a bot that doesn't buy any Ventures (~73-21), even though according to you, one would pretty much never want a Venture at all in such a deck. I can't claim that the bots were optimized, but it shouldn't matter for the purpose of this argument.

Venture is relatively worthless in Province games, though, where hitting $8 is easy enough, unless you can couple it with +buy. The problem is that it also skips the vast majority of cards that give +buy.

EDIT: Scratch that last part. BMU Venture-only beats normal BMU by a fair margin as well (~51-40). Clearly average value isn't the metric that you want to optimize when buying your first Venture.
« Last Edit: August 15, 2013, 10:12:47 pm by dondon151 »
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WanderingWinder

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #85 on: August 15, 2013, 10:14:20 pm »
0

Well, when optimizing average value, you want venture over gold once you have more gold than copper.

I feel like this is not necessarily true. Already having Ventures in your deck increases the average value of further Ventures, plus you have an effectively smaller deck. If you simulate BMU with Colonies, a bot that buys only Ventures at $6 absolutely crushes a bot that doesn't buy any Ventures, even though according to you, one would pretty much never want a Venture at all in such a deck. I can't claim that the bots were optimized, but it shouldn't matter for the purpose of this argument.

Venture is relatively worthless in Province games, though, where hitting $8 is easy enough, unless you can couple it with +buy. The problem is that it also skips the vast majority of cards that give +buy.

For your first venture, only looking at expected value (i.e. average), then golds +platina/3 = coppers is the break even point. If you already have ventures, that shifts toward needing fewer golds than otherwise, as there is the chance of hitting the other ventures, etc. The important point with the colony games is that this is also only taking into account the right now (and not factoring in cycling, though cycling is less important). So, you can anticipate at some point later on that you will get platinum, so while venture may not be the best for my next shuffle, it might be for the rest of the game. Nevertheless, I think you still tend to take a gold or two there to help you hit your platina. Not that you are playing for many treasures in your colony-gaining deck anyway...

PSGarak

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #86 on: August 15, 2013, 10:18:26 pm »
0

Thing is, Venture self-synergizes. It's value is superlinear in the number of Ventures you have (the second Venture is worth more than the first), which makes it non-convex and therefore not amenable to strictly local optimization. Maybe right now buying a single Venture is not as good as buying a single Gold, but buying 4 Ventures is better than buying 1 Gold & 3 Ventures. So you need to incorporate the value added to future Venture purchases into the current purchase decision.

But touching on something else Dondon said, I rate Venture on a given board more by the ability to trash copper than the ability to gain other coin. Not only does it add average value, it also puts in a better value floor.
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WanderingWinder

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #87 on: August 15, 2013, 10:28:23 pm »
0

Thing is, Venture self-synergizes. It's value is superlinear in the number of Ventures you have (the second Venture is worth more than the first), which makes it non-convex and therefore not amenable to strictly local optimization. Maybe right now buying a single Venture is not as good as buying a single Gold, but buying 4 Ventures is better than buying 1 Gold & 3 Ventures. So you need to incorporate the value added to future Venture purchases into the current purchase decision.

But touching on something else Dondon said, I rate Venture on a given board more by the ability to trash copper than the ability to gain other coin. Not only does it add average value, it also puts in a better value floor.
Well, 'it will be better later' is a concern indeed, BUT if the period of the game which is better for me coming up is longer than the later period where going venture is better, it's not worth it, not to mention that better deck now helps you get better deck later, generally, and you need to make sure the game goes long enough you can meaningfully get to that magic venture number. So, this comes in sometimes, but not very often - the greedy algorithm is not such a bad approximation, usually, of what you want to do.


Of course, there are edge cases.

dondon151

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #88 on: August 15, 2013, 11:12:16 pm »
0

So, this comes in sometimes, but not very often - the greedy algorithm is not such a bad approximation, usually, of what you want to do.

If BM-Venture beats BMU, doesn't this suggest the exact opposite?
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eHalcyon

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #89 on: August 16, 2013, 12:10:51 am »
0

So, this comes in sometimes, but not very often - the greedy algorithm is not such a bad approximation, usually, of what you want to do.

If BM-Venture beats BMU, doesn't this suggest the exact opposite?

Not necessarily.  What are the buy rules for BM-Venture?  In this setup, Venture is always better than Silver (with some reshuffle control edge cases I guess).  Even if BM-Venture always buys Venture over Gold, that could just be an indication that even though Gold > Venture, it makes up for it because Venture >> Silver.

What you actually need to test is BM-Venture without Gold vs. BM-Venture with Gold (probably with different versions that eventually favour Venture over Gold after some threshold).
« Last Edit: August 16, 2013, 12:12:00 am by eHalcyon »
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dondon151

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #90 on: August 16, 2013, 12:11:54 am »
0

BM-Venture has the same exact buy rules as BMU, only replace Gold with Venture. So it buys zero Golds.

EDIT: Oh, I see what you mean by the Venture vs. Silver thing.
« Last Edit: August 16, 2013, 12:19:37 am by dondon151 »
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eHalcyon

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #91 on: August 16, 2013, 12:12:57 am »
0

BM-Venture has the same exact buy rules as BMU, only replace Gold with Venture. So it buys zero Golds.

Yeah, so that doesn't necessarily say that you should prefer Venture over Gold.
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meandering mercury

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Re: The Mathematics of Venture Chaining
« Reply #92 on: August 16, 2013, 02:40:06 am »
+3

You're definitely correct mathematically. However, Venture can augment the value of another Venture by $1, and each Venture only adds $1 by itself. I think, though, that my original point wasn't supposed to be concerned with the average "value" of Venture at all (but of course it sounds a little ambiguous).

Well, venture vs gold is a pretty significant one - basically take the gold for a long time.

Wait, really? I mean, it depends on whether there is Copper trashing and several other factors, but I usually go by taking only 1 Gold over Ventures.

I feel like this is not necessarily true. Already having Ventures in your deck increases the average value of further Ventures, plus you have an effectively smaller deck. If you simulate BMU with Colonies, a bot that buys only Ventures at $6 absolutely crushes a bot that doesn't buy any Ventures (~73-21), even though according to you, one would pretty much never want a Venture at all in such a deck. I can't claim that the bots were optimized, but it shouldn't matter for the purpose of this argument.

Venture is relatively worthless in Province games, though, where hitting $8 is easy enough, unless you can couple it with +buy. The problem is that it also skips the vast majority of cards that give +buy.

EDIT: Scratch that last part. BMU Venture-only beats normal BMU by a fair margin as well (~51-40). Clearly average value isn't the metric that you want to optimize when buying your first Venture.

See, this is the sort of thing that the math helps you address concretely. The idea of Venture adding $1 to the average Treasure is extremely useful, but what exactly is the value of the Venture chain? Your intuition breaks down, and then your experience tells you the wrong thing.

This is why the math is helpful: you should *definitely* prefer Gold to Venture.

The value of the venture is

$1 + $d + $v/(1+N)

Let's say you go with your proposed 1 gold, then ventures, strategy. Let's say, generously, that you have 7 silvers in your deck. Then d = 1.4.

Now you start buying ventures because even though short-term it's bad, you believe that long-term you'll get nice pretty chains. How many ventures do you have to get before your venture (chain) does better than the gold? In order to get the total value above $3, you need v/(1+N) > 0.6, and with (7 coppers + 7 silvers + 1 gold) = 15 cards, you need to get 9 Ventures. You could get fewer silvers, but then d goes down. You could get more silvers to improve d, but then N goes up.

Yes, reshuffling and cycling helps a bit, but the more important effect is that, as WW points out, the Gold provides more value now. The Venture provides more value later ... but the math says, does it really? The Venture provides more value when you get 9 Ventures; are you really going to get 9 Ventures in the game?

Certainly you can drown in equations and there is no substitute to real, in-game experience. But the real experience is misleading because *your* experience mainly tells you, Venture is good, buy Venture, remember those awesome Venture chains; you don't realize that without trashing, the Venture chains you remembered fondly will just fizzle compared to gold until you get, like, all 10 Ventures, at which point the game is long over.

A corollary to these results is that in a Colony game without trashing, the reason why Ventures do better than Gold is not because of Venture chains. In order to get an extra $1 out of Venture chains, you need to buy more Ventures than are available in the stack. The reason why Ventures do better than Gold is primarily because of the chance that they'll run into a lucky platinum. (plus, as pointed out, one bot can buy venture at $5, and the other can't)
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GeoLib

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #93 on: August 16, 2013, 03:24:08 am »
+3

From simulation.
BMU that buys opportunistic ventures on 5 (above duchy if prov<=6 and silver) = BMU+V
BMU that buys venture over gold = BMV
BMU that buys X golds then venture over gold = BMV+XG

In a province game:
BMU+V beats BMU 63/29
BMU+V beats BMV 58% to 32%
BMU+V beats BMV+1G 52/38
BMU+V beats BMV+2G 47/42
BMU+V beats BMV+3G 46/43
BMU+V and BMV+4G essentially tie with 45/45 as do BMU+V and BMV+5G although the edge tips from BMU+V to BMV+5G in the fractions of a percent
In an "ultimate" simulation BMV+6-8G beats BMU+V 45/44 and then it tips back to BMU+V against BMV+9G. This seems like basically noise. I guess that one game where the bot buys 10 golds rather than a venture because it keeps hitting 6 is too much. Regardless, it seems pretty clear that in this set-up you want a lot of golds.

With colonies:
BMU+V crushes BMU 75/22
but BMV beats BMU+V 50/46
BMV+1G beats BMU+V 50/45
BMV+2G beats BMU+V 51/44
BMV+3G beats BMU+V 50/46

So it looks like for BMU colony games you want 2 golds before going for venture
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DG

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #94 on: August 16, 2013, 06:57:03 am »
0

Did you optimize the coin threshhold for buying victory cards in the BMU style bots? The default big money bots buy provinces with 18 coins in the deck so if we buy ventures (2 coins) instead of gold (3 coins) we need more turns buying coins before buying provinces.
« Last Edit: August 16, 2013, 02:44:23 pm by DG »
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timchen

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #95 on: August 16, 2013, 02:18:22 pm »
+1

Basically one can think of the formula this way:

Think about the remaining deck containing v ventures and N other treasure. The N treasures divide v ventures into N+1 clusters. the average number of ventures in each cluster is therefore v/(N+1). The ventures you draw by playing a venture is just the first cluster.

One corollary of thinking this way is that if you have 2 ventures in hand, the "extra" expectation value of the two ventures via venture chaining is just 2v/(N+1). Of course v is smaller in this case as you have 1 more venture in hand.

One reason I found dondon's comment dismissive is that the comment that venture is just a random treasure with +$1 is basically just what's printed on the card. Sure it is phrased in a different way so it appears people might have to think about it, but it actually contains less information than the original post. From this short comment one cannot infer directly how effective is venture chaining. And as we see dondon actually overestimates the effectiveness of that.

So, while being terse is not guilty in itself, why don't people think about the things in OP that he doesn't know first? (instead of, you know, posting some comment berating the original post without really knowing its content.)
« Last Edit: August 16, 2013, 02:19:25 pm by timchen »
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GeoLib

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #96 on: August 16, 2013, 04:19:34 pm »
0

Did you optimize the coin threshhold for buying victory cards in the BMU style bots? The default big money bots buy provinces with 18 coins in the deck so if we buy ventures (2 coins) instead of gold (3 coins) we need more turns buying coins before buying provinces.

Oh good call. I didn't. I just took BMU and added a venture buy. Does anyone know the value the simulator counts venture as. My guess would be $1, not $2 because that's how much it adds to the total treasure value (when all treasures are played).
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dondon151

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #97 on: August 16, 2013, 04:29:14 pm »
0

And as we see dondon actually overestimates the effectiveness of that.

Uh, not true. Look up several posts. My conclusion (that Venture is relatively worthless in Province games) mostly agrees with GeoLib's simulation.

And then I erroneously revised it based on a bad simulation, but I blame that on me making a bad simulation, not on overestimating Venture.
« Last Edit: August 16, 2013, 04:54:11 pm by dondon151 »
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GeoLib

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #98 on: August 16, 2013, 04:43:41 pm »
0

Is there a better way to optimize the coin threshold other than just trying a whole bunch of them?
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WanderingWinder

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Re: The Mathematics of Venture Chaining AND The Sociology of Internet Forums
« Reply #99 on: August 16, 2013, 05:00:00 pm »
0

why don't people think about the things in OP that he doesn't know first?

Probably because it's impossible to know what someone else does or does not know.
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