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#### Awaclus

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##### RNG vs decisions
« on: October 24, 2018, 10:59:00 am »
+2

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 »
+4

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 »
+2

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 »
+1

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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#### 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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##### Re: RNG vs decisions
« Reply #16 on: July 27, 2019, 06:00:40 pm »
0

This is also a fun thought experiment on other games
For sure. One I've enjoyed doing: instead of doing the random thing, the player doing the random thing chooses the outcome (e.g. orders the cards that would be shuffled, selects the dice roll). Who wins?

For some Magic: The Gathering decks used at the world championship throughout 1998 to 2004 (or rather pairwise matches), this becomes solvable. I imagine player 1 wins at Machi Koro due to first player advantage, they can do everything player 2 can do but earlier. It's not fully coherent who decides the ordering of a shared deck. If the player receiving the card chooses which card to get, then player 1 wins Love Letter: take Princess and then Baron, play Baron.

Who replenishes the market when plants are scrapped in Power Grid? Who selects the plantation tiles in Puerto Rico? Who does the random setup in Terra Mystica? This notion is not always completely coherent, but when it is coherent it often generates interesting puzzles.

For example, the RNG player wins Monopoly.
Is there a simple argument for why this is the case? Like, the deck of random events is cashflow net negative and RNG can make decider hit it arbitrarily often while not landing on any property? Or going through a revolving door in and out of jail?

Let's do a few more.

Love Letter: if RNG plays first, RNG holds double Baron vs. Priest. If RNG player plays second, decider holds double priest, then faces double Baron. RNG wins!

Machi Koro: RNG rolls a 2 on their own turn and a 6 on decider's turn. Decider never lets RNG buy any building with their ever increasing fortune, and decider never goes above \$3 and so cannot buy any victory buildings. Stalemate!

Star Realms: RNG cannot prevent decider from playing Viper 50 times, decider makes RNG never play anything. Decider wins.

Terra Mystica: Decider builds Temple, takes... IIRC FAV11, the one that gives 2 VP for building a dwelling. Then decider digs somewhere and builds a dwelling. RNG player never scores any VP. Decider out-earns RNG on the cult tracks, scoring 8 vs. 0-or-2 on each track. This easily outweighs the end-of-game resource conversion on the part of the RNG player. Have each player select the same bonus tiles in a cyclic pattern, and have the RNG player wast their power on e.g. digs or an unspent priest if they need to take an action to make decider quit first.

Puerto Rico: Decider buys a plant matching their crop type, if applicable, and mayors once, then the role selections are Craftsman, Captain, Trader for decider and Prospector, Builder, Settler for the RNG player. The VP earnings are 2 vs. 1 every Captain, for an easy win.

Twilight Struggle: load your opponent up with enough Defcon suicide cards at once I guess? OTOH, I guess a decider can always keep the defcon high enough to survive, then just take control Europe and score it for the win.

Food Chain Magnate: decider hires a Waitress on turn 1 and sends her to work every later turn; the RNG player never hires anyone. Decider wins.

Power Grid: Decider buys plant 4, RNG buys plant 5. Nobody buys any fuel or builds any houses for an awfully large number of turns, then decider buys two coal, builds 21 houses and powers one house for an easy win.

Power Grid: The Stock Companies (v1 and/or v2): decider invests in a company and runs it half-decently, it appreciates in value. In v2, it eventually builds 15 houses and powers one of them. The RNG player never invests. Play variant 3 as base Power Grid.

Splendor: decider has the RNG player only ever pick tokens, decider buys everything in deck 1, then deck 2, then deck 3, except decider will have already hit 15 VP before deck 3 is exhausted. To accomplish this, decider somehow accumulate two chips of every color to buy a card in deck 1, and has RNG player release any necessary chips by going over 10 (focused on other colors).

Coup: RNG always claims roles they don't have and decider calls the bluff. Or, RNG takes one coin per turn and decider steals two per turn. RNG is never forced to shoot decider, decider is forced to shoot RNG.

1846: decider buys any private company, then everybody passes on the second private to make the first private pay out, and repeats doing this until the bank breaks. Decider wins. (This works in several 18xx titles, but not all of them have randomness or 2p variants. I'm using the 2p rules from https://boardgamegeek.com/thread/1616729/draft-2-player-1846-rules-game-designer). It's noteworthy that you can win before your first "real" turn.

Warpgate: RNG player always spends their action cards drawing more cards. Decider draws cards on turns 1-3, and plays a movement card on turn 4, going towards a group of trade planets. Then, every later round, draw cards on turns 1-3, then play the trade action if you have it on turn 4. If you don't have it, move some ship not in the set of trade planets, and play the trade action on the next turn (they'll be the last two cards of the deck). Eventually get all the trade tokens. They're enough points to defeat an objective that's satisfied by default.
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#### Awaclus

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##### Re: RNG vs decisions
« Reply #17 on: July 29, 2019, 09:30:43 am »
+1

Is there a simple argument for why this is the case? Like, the deck of random events is cashflow net negative and RNG can make decider hit it arbitrarily often while not landing on any property? Or going through a revolving door in and out of jail?

RNG can force an infinite loop of both players going in and out of jail (by only ever rolling 10) with decision always having to pay the fine and RNG getting out for free. In the meantime, decision can force RNG to donate all of his money to decision, but this doesn't matter because decision can't ever force RNG to owe any money to anyone. RNG, on the other hand, can slowly drain decision's money by having him pay the fine and eventually, decision will owe the fine to the bank but can't afford to pay it.
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