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Awaclus

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RNG vs decisions
« on: October 24, 2018, 10:59:00 am »
+1

2-player Dominion, one player gets to make all decisions for both players and the other gets to choose how each deck is shuffled. Obviously in most kingdoms, the decision player can just have the RNG player buy out all the Curses and never buy anything else and then it just becomes a matter of time until the decision player can force good enough draws to win the game, but this might not be the case in every game.

1) Can you think of a kingdom where the RNG player can force a stalemate?
2) Can you think of a kingdom where the RNG player can force a win?
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faust

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Re: RNG vs decisions
« Reply #1 on: October 24, 2018, 02:31:07 pm »
0

I think the first question to ask is: is there a board where, if player A never does anything (and controls the shuffles), player B cannot win?

For this, some calculations. So  the idea is to have a useless kingdom (maybe all Villages or something) so you are limited to basic Treasures. But we add Bandit Camp! So basic Treasures are worth negative VP.

So we start with 7 Coppers. Player B only ever gets the highest-value Treasure they could possibly buy. As long as the average coin per hand is less than or equal to 5, it will be possible to arrange the shuffle in a way to only give $5 or less hands, forcing to buy another Silver. After adding s Silvers, your average $ per turn is 5*(7 + 2s)/(10 + s). This is larger than 5 if s > 3. Thus, adding 4 Silvers allows you to get at least 1 Gold every 5 shuffles.

Now start adding Golds until your money density is high enough for Provinces. Current money total is 15, and total cards in deck are 14. We now want that 5*(15 + 3g)/(14 + g) > 7. This is the case when g > 23/8. We need to add 3 Golds and then are guaranteed to get Provinces. Of course adding a Province lowers money density, so we have to get more Golds to make up for it. How many? We want a Gold-per-Province ratio such that the average money density remains at least 7, i.e. 5*3g/(p + g) >= 7, or g/p >= 7/8. So to add 8 Provinces, we're adding 7 more Golds.

Now, we added 4 Silvers and 10 Golds total. With Bandit Fort, that's -28 VP. We have a total of 32 cards, with Wall that's another -17 VP, so -45 VP total, but all the Provinces are worth 48 VP, which is still a net profit of 3VP.

Possible ways to improve to really get a stalemate situation:
- add Heirlooms. With Cursed Gold, you'd effectively have 1 Copper less to Start with. But it also means the kingdom includes Pooka, i.e. trashing, which probably implies that we can be more efficient for Wall points - the shuffle might be able to force "useless Pooka", i.e. a Pooka always paired with 3 Estates and Cursed Gold, but that means that the rest of the draws will be better, so is probably not viable. Lucky Coin might work, but Fool is hard to account for.
- Wolf's Den? For our strategy it doesn't really do anything, but maybe there's a way to get some negative VP out of this? In conjunction with Heirlooms possibly?
- other kingdom cards that influence the setup, but I can't think of any that would make a difference.

Anyway since the "easy" case is already very hard to do, I strongly believe that the answer to 1) and 2) is no.
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Mic Qsenoch

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Re: RNG vs decisions
« Reply #2 on: October 24, 2018, 02:58:48 pm »
+3

I don't understand at all, decision player can obviously always win a 3-pile ending of Curses/Copper/Estates. And of course a million other ways too.
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GendoIkari

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Re: RNG vs decisions
« Reply #3 on: October 24, 2018, 04:35:33 pm »
+1

I don't understand at all, decision player can obviously always win a 3-pile ending of Curses/Copper/Estates. And of course a million other ways too.

I'm inclined to agree... the only things that exist that I can see making a potential impact on this are Landmarks, Shelters, and Heirlooms.

Events cannot even possibly matter, because the controlling player can always just choose to completely ignore events.

Landmarks such as Wall and Bandit Fort cannot matter because controlling player can force other player to purchase all the Coppers if they want.

Heirlooms cannot make a difference that I can see; controlling player can get around Cursed Gold by just waiting until other player has bought all 10 Curses; etc.
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Awaclus

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Re: RNG vs decisions
« Reply #4 on: October 25, 2018, 12:49:05 pm »
0

I arrived to the same conclusion. This is also a fun thought experiment on other games; for example, the RNG player wins Monopoly.
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trivialknot

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Re: RNG vs decisions
« Reply #5 on: October 25, 2018, 01:25:50 pm »
0

A variant: what if the RNG player, instead of being controlled by the Decision player, simply does nothing all game?

In the base set, I estimate you need 4 silvers before you can afford gold, and then 10 gold to be able to afford the 8th province.  With Wall and Bandit Fort, that's 64+3-28-17=22 VP.  So that's still winning.
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trivialknot

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Re: RNG vs decisions
« Reply #6 on: October 25, 2018, 01:33:13 pm »
0

Thinking about it more, those estimates aren't quite right.  With only four silver, the RNG player can prevent you from ever hitting $6.  But 22VP is a large enough lead that I'm convinced of the result even if the calculation is a bit off.
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GendoIkari

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Re: RNG vs decisions
« Reply #7 on: October 25, 2018, 02:56:33 pm »
0

A variant: what if the RNG player, instead of being controlled by the Decision player, simply does nothing all game?

In the base set, I estimate you need 4 silvers before you can afford gold, and then 10 gold to be able to afford the 8th province.  With Wall and Bandit Fort, that's 64+3-28-17=22 VP.  So that's still winning.

Provinces give 6, not 8. So 48+3-28-17 only gives 6VP; which is only room for 2 more Silver or Gold than your 4/10 estimate.

I think the way to win here is actually to just get a single Estate or Duchy; and then find things you can 3-pile. You can definitely pile out Copper; which I think gives you enough money density to pile out 2 other things.
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GendoIkari

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Re: RNG vs decisions
« Reply #8 on: October 25, 2018, 03:00:23 pm »
+1

It just occurred to me that the answer to part 2 in the OP is a very trivial "no". Assuming the decision player doesn't choose to end the game in a loss; there's no possible thing that exists that would ever allow RNG player to ever end the game. Stalemate would be the best he can hope for.
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faust

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Re: RNG vs decisions
« Reply #9 on: October 25, 2018, 03:48:09 pm »
+1

A variant: what if the RNG player, instead of being controlled by the Decision player, simply does nothing all game?

In the base set, I estimate you need 4 silvers before you can afford gold, and then 10 gold to be able to afford the 8th province.  With Wall and Bandit Fort, that's 64+3-28-17=22 VP.  So that's still winning.
So... literally the same thing I already posted?
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trivialknot

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Re: RNG vs decisions
« Reply #10 on: October 25, 2018, 08:11:07 pm »
0

Sorry faust, I can't read. Or count the number of VP in the province pile for that matter. Where's the foot in mouth emoji?
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trivialknot

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Re: RNG vs decisions
« Reply #11 on: October 25, 2018, 08:41:01 pm »
+1

I think the first question to ask is: is there a board where, if player A never does anything (and controls the shuffles), player B cannot win?

For this, some calculations. So  the idea is to have a useless kingdom (maybe all Villages or something) so you are limited to basic Treasures. But we add Bandit Camp! So basic Treasures are worth negative VP.

So we start with 7 Coppers. Player B only ever gets the highest-value Treasure they could possibly buy. As long as the average coin per hand is less than or equal to 5, it will be possible to arrange the shuffle in a way to only give $5 or less hands, forcing to buy another Silver. After adding s Silvers, your average $ per turn is 5*(7 + 2s)/(10 + s). This is larger than 5 if s > 3. Thus, adding 4 Silvers allows you to get at least 1 Gold every 5 shuffles.

Now start adding Golds until your money density is high enough for Provinces. Current money total is 15, and total cards in deck are 14. We now want that 5*(15 + 3g)/(14 + g) > 7. This is the case when g > 23/8. We need to add 3 Golds and then are guaranteed to get Provinces. Of course adding a Province lowers money density, so we have to get more Golds to make up for it. How many? We want a Gold-per-Province ratio such that the average money density remains at least 7, i.e. 5*3g/(p + g) >= 7, or g/p >= 7/8. So to add 8 Provinces, we're adding 7 more Golds.

Now, we added 4 Silvers and 10 Golds total. With Bandit Fort, that's -28 VP. We have a total of 32 cards, with Wall that's another -17 VP, so -45 VP total, but all the Provinces are worth 48 VP, which is still a net profit of 3VP.
Although I gave basically the same solution, I realized a problem with it: shuffle skipping.  With 4 silver, the RNG player can carefully order cards in such a way that you always hit $5.

T1: SCCCE
T2: SSCEE
T3: SCCC (shuffle) E
T4: SSCEE
T5: SCCC (shuffle) E
etc.

So you need 5 silver, and then you need 11 gold to be able to pick up the 8th province.  If you only have 10, then you could have the following shuffles:
T1: GGCPP
T2: GGCPP
T3: GGCPP
T4: GGCPE
T5: SSSCE
T6: SSCCE
T7: GG (shuffle) CPP

So the total VP is 48 (provinces) + 3 (estates) - 32 (bandit fort) - 19 (wall) = 0 VP.  Which is losing.

I think you still win by buying a couple duchies though.
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Kirian

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Re: RNG vs decisions
« Reply #12 on: October 26, 2018, 12:34:10 am »
0

Apropos of exactly nothing in this thread except minor mistakes among cards:

Bandit does not combo with Bandit Fort, but Bandit Camp combos with either.
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faust

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Re: RNG vs decisions
« Reply #13 on: October 26, 2018, 03:22:50 am »
0

I think the first question to ask is: is there a board where, if player A never does anything (and controls the shuffles), player B cannot win?

For this, some calculations. So  the idea is to have a useless kingdom (maybe all Villages or something) so you are limited to basic Treasures. But we add Bandit Camp! So basic Treasures are worth negative VP.

So we start with 7 Coppers. Player B only ever gets the highest-value Treasure they could possibly buy. As long as the average coin per hand is less than or equal to 5, it will be possible to arrange the shuffle in a way to only give $5 or less hands, forcing to buy another Silver. After adding s Silvers, your average $ per turn is 5*(7 + 2s)/(10 + s). This is larger than 5 if s > 3. Thus, adding 4 Silvers allows you to get at least 1 Gold every 5 shuffles.

Now start adding Golds until your money density is high enough for Provinces. Current money total is 15, and total cards in deck are 14. We now want that 5*(15 + 3g)/(14 + g) > 7. This is the case when g > 23/8. We need to add 3 Golds and then are guaranteed to get Provinces. Of course adding a Province lowers money density, so we have to get more Golds to make up for it. How many? We want a Gold-per-Province ratio such that the average money density remains at least 7, i.e. 5*3g/(p + g) >= 7, or g/p >= 7/8. So to add 8 Provinces, we're adding 7 more Golds.

Now, we added 4 Silvers and 10 Golds total. With Bandit Fort, that's -28 VP. We have a total of 32 cards, with Wall that's another -17 VP, so -45 VP total, but all the Provinces are worth 48 VP, which is still a net profit of 3VP.
Although I gave basically the same solution, I realized a problem with it: shuffle skipping.  With 4 silver, the RNG player can carefully order cards in such a way that you always hit $5.

T1: SCCCE
T2: SSCEE
T3: SCCC (shuffle) E
T4: SSCEE
T5: SCCC (shuffle) E
etc.

So you need 5 silver, and then you need 11 gold to be able to pick up the 8th province.  If you only have 10, then you could have the following shuffles:
T1: GGCPP
T2: GGCPP
T3: GGCPP
T4: GGCPE
T5: SSSCE
T6: SSCCE
T7: GG (shuffle) CPP

So the total VP is 48 (provinces) + 3 (estates) - 32 (bandit fort) - 19 (wall) = 0 VP.  Which is losing.

I think you still win by buying a couple duchies though.
Good point. I'm not convinced Duchies help though. Each Duchy is worth 2 VP effectively, and you're going to have to add a Gold (-3 VP) for almost every Duchy that you get in order to ensure that you can still empty the Provinces afterwards.
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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.

ghostofmars

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Re: RNG vs decisions
« Reply #14 on: October 26, 2018, 06:22:38 am »
0

What if we add two constraints on the decision player: When deciding for the RNG player, he must play all treasures and buy the most expensive card he can afford. Now it should be possible for the RNG player to win, right?
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faust

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Re: RNG vs decisions
« Reply #15 on: October 26, 2018, 08:38:53 am »
0

What if we add two constraints on the decision player: When deciding for the RNG player, he must play all treasures and buy the most expensive card he can afford. Now it should be possible for the RNG player to win, right?
Certainly; all you have to do is craft a board where BM is optimal and no kingdom cards cost $3, $6 or $8. Bonus points if you can avoid cards costing $4 so that RNG can open double Silver.
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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.
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