Dominion Strategy Forum

Please login or register.

Login with username, password and session length
Pages: 1 ... 9 10 [11]  All

Author Topic: How high can you go  (Read 12913 times)

0 Members and 1 Guest are viewing this topic.

ephesos

  • Explorer
  • *****
  • Offline Offline
  • Posts: 305
  • Shuffle iT Username: Ephesos
  • Respect: +261
    • View Profile
Re: How high can you go
« Reply #250 on: June 26, 2016, 04:54:35 pm »
0

I don't know if people are still looking now, but last night I thought of some things for "the other strategy", with KC-Golem-Black Market

1. It's hard to play both tactician and buy Save. That was now part of the plan but doesn't really work.
2. Your setup turn can be improved a lot by doing KC-Golem into 3x {tactician, draw silver}. You get to play 3 tacticians that way.
3. In fact, if you start with KC-KC-Golem into tact, and trash overgrown estate during setup, you can activate 6 tacticians.

-> that means starting hand goes up to 35 from just 12. I don't really have a good feel for the consequences though. I think it's still not enough to beat the 1E727 unless something else shows up.
Unfortunately I think this leads to an unbounded solution:

1. Tactician
2. 10 golems into king's court golem chain and play alternating tacticians and black markets (or any way to gain cards in hand)
3. keep repeating to put ever more tacticians in play leading to an arbitrarily large hand size (over the course of many turns)

This works because tactician never discards golems. You can get the golems all in play before resolving a tactician by always resolving the king's courts first.

So basically the KC golem plan is useless.

I was thinking about this before, it's not unbounded because you have to have a Golem in hand at each step in the Golem chain, in order to play it with KC. But when you play a Tactician, you discard all your Golems, and Black Market doesn't let you get a Golem in hand, only Silver. So you can only get another Golem with another source of limited draw i.e. Overgrown Estate.
Logged

ephesos

  • Explorer
  • *****
  • Offline Offline
  • Posts: 305
  • Shuffle iT Username: Ephesos
  • Respect: +261
    • View Profile
Re: How high can you go
« Reply #251 on: June 26, 2016, 05:20:42 pm »
0

Why is the 2/3-1/3 ratio optimal? HoPs also make it more complicated.
I tried to do out the math on it, you can say something like:
Start with M cards, M constant, let X be the number of Silver and M-X the number of Banks.
Then you have 2X+(M(M+1)-X(X+1))/2 money, 2X^2+X*((M(M+1)-X(X+1))/2) gains (or -(X^3)/2+(3/2)X^2+(M^2*X)/2 when you eliminate the 1 for being too small)
Derivative of that is -3/2*X^2+3X+M^2/2, set it equal to 0, and you get X~=M/sqrt(3), or .58 M

Hmmm, when I did that before, it came out to 2/3. Looks like I flipped a sign before :P

Anyway, you can do the same math with HoP Rocks, just have M-(X/2) be the number of Banks and account for an extra X HoP treasures when figuring the Bank value. Did it out and got something like X=(8+2sqrt(31))/15 M (where the number of HoPs is X/2), or 63% HoP, 37% Banks
« Last Edit: June 26, 2016, 05:33:25 pm by ephesos »
Logged

-Stef-

  • 2012 & 2016 DS Champion
  • *
  • Online Online
  • Posts: 1521
  • Respect: +3962
    • View Profile
Re: How high can you go
« Reply #252 on: June 26, 2016, 06:37:52 pm »
0

When using both the banks and the horns I end up around 1E930.
Logged
Join the Dominion League!

AdrianHealey

  • Mountebank
  • *****
  • Offline Offline
  • Posts: 2244
  • Respect: +762
    • View Profile
Re: How high can you go
« Reply #253 on: June 26, 2016, 07:00:51 pm »
+2

I don't get the assignment: if all piles are infinite, how is the answer not always: 'infinite amount of gains'?
Logged

scott_pilgrim

  • Jester
  • *****
  • Offline Offline
  • Posts: 963
  • Respect: +1869
    • View Profile
Re: How high can you go
« Reply #254 on: June 26, 2016, 07:10:54 pm »
+2

I don't get the assignment: if all piles are infinite, how is the answer not always: 'infinite amount of gains'?

You want to find the highest number of gains you can get in a single turn, such that it is not possible to get more gains than that.  I don't think there's anything in Dominion that lets you get infinity gains in one turn.  There are a ton of kingdoms that have unbounded amounts of gains you can get.  For example, a Village Smithy kingdom has no solution, because for any number I try to say is the maximum number of gains, you can get more gains than that.  The goal is to set up a kingdom which has a bounded, but still huge, number of gains, in one turn.  It reminds me of the busy beaver problem.

Here's a simple example of a kingdom with a bounded number of gains in one turn:

Kingdom is just Squire. Play 2 for +2 Actions, then 3 for +2 buys. So X=7, Y=8.

Taking an early (and temporary) lead!
Logged

Watno

  • Margrave
  • *****
  • Offline Offline
  • Posts: 2579
  • Shuffle iT Username: Watno
  • Respect: +2682
    • View Profile
Re: How high can you go
« Reply #255 on: June 26, 2016, 07:25:46 pm »
+3

I don't think there's anything in Dominion that lets you get infinity gains in one turn. 
You just need a single card for that: Rats.
Also, if there's just Village and Smithy in the kingdom, you can't gain more than one card.
Logged

eHalcyon

  • Adventurer
  • ******
  • Offline Offline
  • Posts: 8689
  • Respect: +9078
    • View Profile
Re: How high can you go
« Reply #256 on: June 26, 2016, 09:22:44 pm »
0

Also, if there's just Village and Smithy in the kingdom, you can't gain more than one card.

Yeah, this tripped me up earlier with my Baked kingdom.  The goal is to get a large but bounded number of gains.  An unbounded number of coins and/or cards (which easily translates into coins) is actually fine unless there is a way to translate those into an unbounded number of gains.  Village-Smithy is unbounded cards/coins, but not gains since there's no source of +Buy or other gaining. 

But the most efficient ways to get tons of gains are such that they are only bounded if coins/cards are also bounded, and the solutions so far have been able to achieve huge hands that are bounded by keeping +actions very limited: Crossroads, Necropolis, Golem (which can't play itself).
Logged

liopoil

  • Margrave
  • *****
  • Offline Offline
  • Posts: 2577
  • Respect: +2447
    • View Profile
Re: How high can you go
« Reply #257 on: June 26, 2016, 11:08:11 pm »
0

Why is the 2/3-1/3 ratio optimal? HoPs also make it more complicated.
I tried to do out the math on it, you can say something like:
Start with M cards, M constant, let X be the number of Silver and M-X the number of Banks.
Then you have 2X+(M(M+1)-X(X+1))/2 money, 2X^2+X*((M(M+1)-X(X+1))/2) gains (or -(X^3)/2+(3/2)X^2+(M^2*X)/2 when you eliminate the 1 for being too small)
Derivative of that is -3/2*X^2+3X+M^2/2, set it equal to 0, and you get X~=M/sqrt(3), or .58 M

Hmmm, when I did that before, it came out to 2/3. Looks like I flipped a sign before :P

Anyway, you can do the same math with HoP Rocks, just have M-(X/2) be the number of Banks and account for an extra X HoP treasures when figuring the Bank value. Did it out and got something like X=(8+2sqrt(31))/15 M (where the number of HoPs is X/2), or 63% HoP, 37% Banks
When I did it I got that it was definitely 2/5ths horns, assuming that you have a negligible number of treasures in play beforehand. Reasoning:

You have t treasures to play, and choose h of them to be HoP. Turning a bank into a HoP results in losing 3*h coins and gaining 2*(t - h) coins. When is this a fair deal? When 3h = 2t - 2h, or h = 2/5 * t.

If you have x treasures in play beforehand (particularly for the first black market), then it comes out to h = (2t - x)/5.

EDIT: Okay, so I got 10^1192, but I'm pretty sure I calculated it wrong.

EDIT: Now 10^448, so still doing something wrong...

EDIT: Now 10^673, which, while reasonable and better than anything I had before, is clearly unoptimized and possibly miscalculated still.
« Last Edit: June 26, 2016, 11:32:13 pm by liopoil »
Logged

ephesos

  • Explorer
  • *****
  • Offline Offline
  • Posts: 305
  • Shuffle iT Username: Ephesos
  • Respect: +261
    • View Profile
Re: How high can you go
« Reply #258 on: June 26, 2016, 11:53:24 pm »
0


When I did it I got that it was definitely 2/5ths horns, assuming that you have a negligible number of treasures in play beforehand. Reasoning:

You have t treasures to play, and choose h of them to be HoP. Turning a bank into a HoP results in losing 3*h coins and gaining 2*(t - h) coins. When is this a fair deal? When 3h = 2t - 2h, or h = 2/5 * t.

If you have x treasures in play beforehand (particularly for the first black market), then it comes out to h = (2t - x)/5.

Don't you optimize gains instead of coins? i.e. losing a HoP loses you twice the number of Raids you are buying in gains, but gaining the Bank helps you buy more Raids.
Logged

faust

  • Torturer
  • *****
  • Offline Offline
  • Posts: 1567
  • Shuffle iT Username: faust
  • Respect: +1964
    • View Profile
Re: How high can you go
« Reply #259 on: June 27, 2016, 03:31:27 am »
0

I don't think there's anything in Dominion that lets you get infinity gains in one turn. 
You just need a single card for that: Rats.
Also, if there's just Village and Smithy in the kingdom, you can't gain more than one card.

Rats doesn't give infinity gains, only one. As stated in the sentence directly after the one you quoted,

There are a ton of kingdoms that have unbounded amounts of gains you can get.
Logged
Since the number of points is within a constant factor of the number of city quarters, in the long run we can get (4 - ε) ↑↑ n points in n turns for any ε > 0.

liopoil

  • Margrave
  • *****
  • Offline Offline
  • Posts: 2577
  • Respect: +2447
    • View Profile
Re: How high can you go
« Reply #260 on: June 27, 2016, 06:02:01 am »
0


When I did it I got that it was definitely 2/5ths horns, assuming that you have a negligible number of treasures in play beforehand. Reasoning:

You have t treasures to play, and choose h of them to be HoP. Turning a bank into a HoP results in losing 3*h coins and gaining 2*(t - h) coins. When is this a fair deal? When 3h = 2t - 2h, or h = 2/5 * t.

If you have x treasures in play beforehand (particularly for the first black market), then it comes out to h = (2t - x)/5.

Don't you optimize gains instead of coins? i.e. losing a HoP loses you twice the number of Raids you are buying in gains, but gaining the Bank helps you buy more Raids.
Considering raid is only remotely relevant on the last black market play, and you'll want to max out coins then anyway because one coin turns into two silvers
Logged

AdrianHealey

  • Mountebank
  • *****
  • Offline Offline
  • Posts: 2244
  • Respect: +762
    • View Profile
Re: How high can you go
« Reply #261 on: June 27, 2016, 06:16:48 am »
0

I don't get the assignment: if all piles are infinite, how is the answer not always: 'infinite amount of gains'?

You want to find the highest number of gains you can get in a single turn, such that it is not possible to get more gains than that.  I don't think there's anything in Dominion that lets you get infinity gains in one turn.  There are a ton of kingdoms that have unbounded amounts of gains you can get.  For example, a Village Smithy kingdom has no solution, because for any number I try to say is the maximum number of gains, you can get more gains than that.  The goal is to set up a kingdom which has a bounded, but still huge, number of gains, in one turn.  It reminds me of the busy beaver problem.

Here's a simple example of a kingdom with a bounded number of gains in one turn:

Kingdom is just Squire. Play 2 for +2 Actions, then 3 for +2 buys. So X=7, Y=8.

Taking an early (and temporary) lead!

So the trick is: creating a kingdom that even with infinite amount of supply piles still is only limited in the amount of cards you can have in your hand and, thus, the amount of gains you can have?

Is that the idea?

So smithy/champion/squire is not an option, because, well, that will have infinite gains. (Smithy eternally and than squire eternally.)
Logged

liopoil

  • Margrave
  • *****
  • Offline Offline
  • Posts: 2577
  • Respect: +2447
    • View Profile
Re: How high can you go
« Reply #262 on: June 27, 2016, 06:57:42 am »
0

Yes. Our solutions work because it's impossible to get, say, 10^827382927 coins in a single turn even though it is possible to get 10^600+. And converting coins to gains is the only way to gain a very large number of cards while staying bounded.
Logged

Watno

  • Margrave
  • *****
  • Offline Offline
  • Posts: 2579
  • Shuffle iT Username: Watno
  • Respect: +2682
    • View Profile
Re: How high can you go
« Reply #263 on: June 27, 2016, 07:51:30 am »
0

I don't think there's anything in Dominion that lets you get infinity gains in one turn. 
You just need a single card for that: Rats.
Also, if there's just Village and Smithy in the kingdom, you can't gain more than one card.

Rats doesn't give infinity gains, only one. As stated in the sentence directly after the one you quoted,

There are a ton of kingdoms that have unbounded amounts of gains you can get.
I thought what Scott wanted to say that he tought on any given turn, the amount of gains you have would still be finite (but you can get arbitrarily high finite numbers here). However, with Rats there is a single turn (with a finite amount of turns taken before) where you can gain any number of cards you wnat (with that number being picked after the turn).
Logged

Schneau

  • Saboteur
  • *****
  • Offline Offline
  • Posts: 1167
  • Shuffle iT Username: Schneau
  • Respect: +1424
    • View Profile
    • Rainwave
Re: How high can you go
« Reply #264 on: May 12, 2017, 12:00:32 am »
+2

I just discovered this thread and spent the last hour reading it. It is amazing.

Why did this thread stop? Is 1E930 really the max? I was hoping it would end over 1E1000.
Logged

liopoil

  • Margrave
  • *****
  • Offline Offline
  • Posts: 2577
  • Respect: +2447
    • View Profile
Re: How high can you go
« Reply #265 on: May 12, 2017, 03:45:37 pm »
+1

I just discovered this thread and spent the last hour reading it. It is amazing.

Why did this thread stop? Is 1E930 really the max? I was hoping it would end over 1E1000.
Well, I don't think I ever got above 10^673 and it would take a lot of work for me to try to replicate what stef did... so it kinda died off.

EDIT: this was before empires and the seconds editions though I think... so maybe it's worth coming back to.
« Last Edit: May 12, 2017, 03:54:59 pm by liopoil »
Logged

Schneau

  • Saboteur
  • *****
  • Offline Offline
  • Posts: 1167
  • Shuffle iT Username: Schneau
  • Respect: +1424
    • View Profile
    • Rainwave
Re: How high can you go
« Reply #266 on: May 12, 2017, 07:57:38 pm »
+1

I just discovered this thread and spent the last hour reading it. It is amazing.

Why did this thread stop? Is 1E930 really the max? I was hoping it would end over 1E1000.
Well, I don't think I ever got above 10^673 and it would take a lot of work for me to try to replicate what stef did... so it kinda died off.

EDIT: this was before empires and the seconds editions though I think... so maybe it's worth coming back to.

It included Empires, since the best solution included Rocks, which is Empires. Looks like it was before second editions.
Logged

Ewokonfire

  • Pawn
  • **
  • Offline Offline
  • Posts: 1
  • Respect: +2
    • View Profile
Re: How high can you go
« Reply #267 on: May 29, 2017, 09:05:03 am »
+2

Sorry to necropost, but I just read through this and I can't work out if I'm stupid, or everybody who posted in this thread is (I suspect the former).  Once you can convert money to gains, surely Catapult + Pathfinding leads to a trivial infinite combo.  You put your +1 card on Catapult, then all your Rocks give you +1 card (as they're from the same pile), which draws you another Rocks, so you can play an unbounded number of Rocks in a single turn, and gain unbounded money.  What am I missing here?
Logged

-Stef-

  • 2012 & 2016 DS Champion
  • *
  • Online Online
  • Posts: 1521
  • Respect: +3962
    • View Profile
Re: How high can you go
« Reply #268 on: May 29, 2017, 09:14:27 am »
0

Sorry to necropost, but I just read through this and I can't work out if I'm stupid, or everybody who posted in this thread is (I suspect the former).  Once you can convert money to gains, surely Catapult + Pathfinding leads to a trivial infinite combo.  You put your +1 card on Catapult, then all your Rocks give you +1 card (as they're from the same pile), which draws you another Rocks, so you can play an unbounded number of Rocks in a single turn, and gain unbounded money.  What am I missing here?

You're not wrong, we just missed that. Removing pathfinding from the solution won't reduce it by much but it was certainly invalid in its current form.
Logged
Join the Dominion League!
Pages: 1 ... 9 10 [11]  All
 

Page created in 0.108 seconds with 22 queries.