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Author Topic: Infinite stalemate game  (Read 4717 times)

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pubby

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Infinite stalemate game
« on: January 23, 2015, 04:19:21 am »
+5

The only way to end a Dominion game is by running out of piles. Buying/gaining cards from a pile is optional, and so it is possible for games to last an infinite number of turns.

Do any scenarios exist where the only optimal play for both players is to prolong the game? Think of a situation where a player will lose next turn if they attempt to buy/gain from any pile.

Here's a silly example I thought of:
Both players have only 1 card in their deck: chapel. The chapel pile is empty and both copper and curses are down to 1 card each. The copper pile has an embargo token on it. Attempting to buy a copper will lose you the game. The optimal move is to pass your turn every turn.

Can you think of any other scenarios? Bonus points for a scenario that could realistically happen.
« Last Edit: January 23, 2015, 05:37:48 am by pubby »
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faust

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Re: Unwinnable infinite game
« Reply #1 on: January 23, 2015, 04:51:34 am »
+4

Basically any Golden Deck (Monument or Bishop/Fortress) with both players being able to generate the same amount of VP tokens every turn, 2 supply piles empty and 1 down to 2 copies, no +buys and any additional card clogging up your deck works.

Things get more difficult without VP tokens. Here's something: No +buy, both of you have 2 Highways, and can play KC/KC/Smugglers/Smugglers/Smugglers each turn. Two piles are empty. If you gain any kingdom card or VP card, your opponent can empty that pile AND buy another VP card. Money doesn't help your deck. You do nothing.
« Last Edit: January 23, 2015, 04:53:19 am by faust »
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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.

qmech

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Re: Unwinnable infinite game
« Reply #2 on: January 23, 2015, 05:01:25 am »
0

Hard mode: make sure the stalemate is reachable through best play on both sides.
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faust

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Re: Unwinnable infinite game
« Reply #3 on: January 23, 2015, 05:33:35 am »
+3

Hard mode: make sure the stalemate is reachable through best play on both sides.

I actually think the Smugglers scenario satisfies this. On a board as described above, it is probably the best strategy. If both players reach a state where they can play their deck, and player 1 drains a second pile via Smugglers, then the described scenario is reached.
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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.

RTT

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Re: Infinite stalemate game
« Reply #4 on: January 23, 2015, 09:22:54 am »
0

not with knights though but those dont have to be in the kingdom.
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AdamH

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Re: Infinite stalemate game
« Reply #5 on: January 23, 2015, 09:55:59 am »
+1

I've heard of Pirate Ship games where stalemates are theoretically possible. You can't build up your deck without buying Copper but if you buy Copper your opponent can just hit it with their Pirate Ships and get a token, which allows them to build up their deck.

Optimal play gets you there? That's weird... I don't know if that can be done.
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RTT

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Re: Infinite stalemate game
« Reply #6 on: January 23, 2015, 10:42:08 am »
+1

well sometimes bishop fortress is the optimal play. i had that once where me and my oponent had a golden deck that trashed 5 or 4 fortresses a turn and at the end of his turn each player was leading in points. It would have never ended if he wasnt deciding to try to win another way (remodeling to provinces) but that was not the correct move as i continued trashin for points and won.

http://gokosalvager.com/static/logprettifier.html?20140304/log.513bdc80e4b0da6e3de056bb.1393955690770.txt
« Last Edit: January 23, 2015, 10:43:25 am by RTT »
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Awaclus

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Re: Infinite stalemate game
« Reply #7 on: January 23, 2015, 12:35:40 pm »
+3

Happened in this game.

http://gokosalvager.com/static/logprettifier.html?20131009/log.51102b6ee4b06719e45eef9d.1381323260969.txt

We used the forum dice roll feature to determine the winner. Yed lost the roll and resigned.

Dingan

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Re: Infinite stalemate game
« Reply #8 on: January 23, 2015, 02:09:37 pm »
0

well sometimes bishop fortress is the optimal play. i had that once where me and my oponent had a golden deck that trashed 5 or 4 fortresses a turn and at the end of his turn each player was leading in points. It would have never ended if he wasnt deciding to try to win another way (remodeling to provinces) but that was not the correct move as i continued trashin for points and won.

http://gokosalvager.com/static/logprettifier.html?20140304/log.513bdc80e4b0da6e3de056bb.1393955690770.txt

I know no log = didn't happen, but I had something very similar to this happen to me once.  The game got to a point where I had 5 Fortresses and 4 Bishops, while opponent had 5 Fortresses and 5 Bishops (I can't remember if we had anything else, maybe a couple other villages or something, not sure).  I was several points ahead, but knew I was donezo (unless he drew all 5 Bishops in his hand a couple times, or something), so I went Remodel to turn Fortresses into Duchies for the 3 pile, but it was too late.
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liopoil

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Re: Infinite stalemate game
« Reply #9 on: January 23, 2015, 05:09:57 pm »
+2

Happened in this game.

http://gokosalvager.com/static/logprettifier.html?20131009/log.51102b6ee4b06719e45eef9d.1381323260969.txt

We used the forum dice roll feature to determine the winner. Yed lost the roll and resigned.
Yed had a forced win. He can first empty the curses by buying one and trashing one every turn. Then there are 4 estates left. He buys a copper, awaclus gets 2 estates tops with possession, then yed gets an estate, trashing a scrying pool or something. Now awaclus buying the last estate will still leave him a point behind, and the game 3-piles on estates, scrying pools, and curses.
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liopoil

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Re: Infinite stalemate game
« Reply #10 on: January 23, 2015, 05:10:55 pm »
0

well sometimes bishop fortress is the optimal play. i had that once where me and my oponent had a golden deck that trashed 5 or 4 fortresses a turn and at the end of his turn each player was leading in points. It would have never ended if he wasnt deciding to try to win another way (remodeling to provinces) but that was not the correct move as i continued trashin for points and won.

http://gokosalvager.com/static/logprettifier.html?20140304/log.513bdc80e4b0da6e3de056bb.1393955690770.txt

I know no log = didn't happen, but I had something very similar to this happen to me once.  The game got to a point where I had 5 Fortresses and 4 Bishops, while opponent had 5 Fortresses and 5 Bishops (I can't remember if we had anything else, maybe a couple other villages or something, not sure).  I was several points ahead, but knew I was donezo (unless he drew all 5 Bishops in his hand a couple times, or something), so I went Remodel to turn Fortresses into Duchies for the 3 pile, but it was too late.
...buy the last bishop? Or was it in the trash?
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werothegreat

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Re: Infinite stalemate game
« Reply #11 on: January 24, 2015, 08:12:17 am »
0

If both players have 5 Lighthouses and are able to reliably play at least one every turn, then Ambassadoring the same card you buy every turn would suffice.  Not sure why you'd want to do that, though.
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Holger

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Re: Infinite stalemate game
« Reply #12 on: January 24, 2015, 08:32:23 am »
0

Happened in this game.

http://gokosalvager.com/static/logprettifier.html?20131009/log.51102b6ee4b06719e45eef9d.1381323260969.txt

We used the forum dice roll feature to determine the winner. Yed lost the roll and resigned.
Yed had a forced win. He can first empty the curses by buying one and trashing one every turn. Then there are 4 estates left. He buys a copper, awaclus gets 2 estates tops with possession, then yed gets an estate, trashing a scrying pool or something. Now awaclus buying the last estate will still leave him a point behind, and the game 3-piles on estates, scrying pools, and curses.

It wasn't quite forced, yed might have drawn Copper, poss. and all green cards on his next regular turn, allowing awaclus to get all four estates.
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amalloy

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Re: Infinite stalemate game
« Reply #13 on: January 24, 2015, 04:36:31 pm »
+3

well sometimes bishop fortress is the optimal play. i had that once where me and my oponent had a golden deck that trashed 5 or 4 fortresses a turn and at the end of his turn each player was leading in points. It would have never ended if he wasnt deciding to try to win another way (remodeling to provinces) but that was not the correct move as i continued trashin for points and won.

http://gokosalvager.com/static/logprettifier.html?20140304/log.513bdc80e4b0da6e3de056bb.1393955690770.txt

I know no log = didn't happen, but I had something very similar to this happen to me once.  The game got to a point where I had 5 Fortresses and 4 Bishops, while opponent had 5 Fortresses and 5 Bishops (I can't remember if we had anything else, maybe a couple other villages or something, not sure).  I was several points ahead, but knew I was donezo (unless he drew all 5 Bishops in his hand a couple times, or something), so I went Remodel to turn Fortresses into Duchies for the 3 pile, but it was too late.

4 Bishops is much better than 5 in a Bishop+Fortress deck. His 5th Bishop will, on most turns, be playable on nothing, netting him one extra point. But eventually something terrible will happen to him, making up for all those extra points. The most drastic scenario is of course drawing a starting hand of 5 Bishops, as you said, which costs him 12 points, but lesser versions of this will crop up quite often. Perhaps he draws Fortress and four Bishops, into a fifth Bishop: that's just as bad. Or he draws a second Fortress, trashes it, and then draws the fifth Bishop: that costs him 9 points. Basically he needs to rely on his fifth Bishop being drawn after his fourth Fortress (or anyway his fourth village, if he has other villages). That's far from guaranteed, and he will gradually fall behind despite his better best-case turn.
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Dingan

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Re: Infinite stalemate game
« Reply #14 on: January 26, 2015, 07:10:09 am »
0

well sometimes bishop fortress is the optimal play. i had that once where me and my oponent had a golden deck that trashed 5 or 4 fortresses a turn and at the end of his turn each player was leading in points. It would have never ended if he wasnt deciding to try to win another way (remodeling to provinces) but that was not the correct move as i continued trashin for points and won.

http://gokosalvager.com/static/logprettifier.html?20140304/log.513bdc80e4b0da6e3de056bb.1393955690770.txt

I know no log = didn't happen, but I had something very similar to this happen to me once.  The game got to a point where I had 5 Fortresses and 4 Bishops, while opponent had 5 Fortresses and 5 Bishops (I can't remember if we had anything else, maybe a couple other villages or something, not sure).  I was several points ahead, but knew I was donezo (unless he drew all 5 Bishops in his hand a couple times, or something), so I went Remodel to turn Fortresses into Duchies for the 3 pile, but it was too late.
...buy the last bishop? Or was it in the trash?

Yes I had trashed it somewhere along the way for some reason.
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jomini

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Re: Infinite stalemate game
« Reply #15 on: January 27, 2015, 12:46:24 pm »
+1



4 Bishops is much better than 5 in a Bishop+Fortress deck. His 5th Bishop will, on most turns, be playable on nothing, netting him one extra point. But eventually something terrible will happen to him, making up for all those extra points. The most drastic scenario is of course drawing a starting hand of 5 Bishops, as you said, which costs him 12 points, but lesser versions of this will crop up quite often. Perhaps he draws Fortress and four Bishops, into a fifth Bishop: that's just as bad. Or he draws a second Fortress, trashes it, and then draws the fifth Bishop: that costs him 9 points. Basically he needs to rely on his fifth Bishop being drawn after his fourth Fortress (or anyway his fourth village, if he has other villages). That's far from guaranteed, and he will gradually fall behind despite his better best-case turn.
Actually, I think this is slightly wrong. The odds that you will get a Bish on the bottom of the deck are .5. In that case you score 13 VP. Of the 50% of the other cases, you will have a Fortress followed by a Bish (moving from bottom up) 5/9 - so .28 overall. In this case you will score just 9 VP. In like manner we can fill out the odds and the payouts and then find a weighted expectation value. For 4 Bish we have 12 VP/turn as our expectation value, for 5 Fort/5 Bish we have 10.0 (negligably higher than 10 VP per turn). The expectation value is actually slightly higher as when we wiff (e.g. we end up with Bish x3 in hand), we leave 3 Forts on the bottom to draw at 100% odds in the top of the next hand (shuffle). Basically when you hit 13 VP, you reset, when you don't you have some compensatory skewing in the next hand to make 13 VP more likely.

So all of this is just the numbers behind your stuff above. However, nothing says we can't opportunistically trash Bish with Bish the first time we hit it. In this case, our expectation before we trash Bish with Bish is slightly more than 11.5.  Still bad right? Well yes and no. If the other guy can eventually 3 pile the board, you might want to take a gamble if you are behind 1 or 2 VP. Sure you most likely will fall further behind, but you have a non-zero chance of getting ahead if you go 5 Bish (and then trash Bish with Bish when you whiff).




This gets better still if we have additional villages, one of the problems is that we care, a lot, about where the last Fort is. If you had a deck of 5 Bish/5 Fort/10 Farming villages, you really only care about drawing 4 Villages (for enough +action to play Bishop) and 1 Fort. In that setup we most likely will have plenty of actions to play all the Bishops even if the bottom of the deck is Fort x4. This makes your odds of getting 13 VP go up (non-trivial calculation). Unless I'm getting the statistics wrong here, optimal play should be to go for 5 Bish with excess villages, and trash Bish -> Bish when it first shows up, preferably with a bunch of excess villages if possible. This should be your best option when you are behind (it likely won't win, but you have a possibility of scoring >12 VP which you don't if you just mirror). Particularly if you are something like 2 VP behind and the other guy can pile out Bish (buy final Bish on his final turn), Fort, and another cantrip; you should roll the dice on Bish #5 first and then trash the sucker the first time he collides with Bish #4.

If you are slightly behind against Bish X4/Fort X5, then you likely are better off going for Bish X5 than trying to quickly snatch up some Provinces.
« Last Edit: January 27, 2015, 12:48:50 pm by jomini »
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Dingan

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Re: Infinite stalemate game
« Reply #16 on: February 09, 2015, 02:15:20 pm »
0

Don't know if it was already mentioned, but couldn't a stalemate situation arise where:
  • 2-player game
  • No Colonies
  • Opponents are tied in VP
  • There is 1 Estate, 1 Duchy, 1 Province, and 10 curses left
  • There are no sources of alt-VP
  • There is no way to trash cards from your deck
  • Embargos have been TR'd/KC'd to the point where there are 2 Embargo tokens on the Estate pile, 4 on the Duchy, 7 on the Province, and at least 1 on all other supply piles
« Last Edit: February 09, 2015, 02:16:40 pm by Dingan »
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amalloy

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Re: Infinite stalemate game
« Reply #17 on: February 09, 2015, 03:05:19 pm »
+1

Don't know if it was already mentioned, but couldn't a stalemate situation arise where:
  • 2-player game
  • No Colonies
  • Opponents are tied in VP
  • There is 1 Estate, 1 Duchy, 1 Province, and 10 curses left
  • There are no sources of alt-VP
  • There is no way to trash cards from your deck
  • Embargos have been TR'd/KC'd to the point where there are 2 Embargo tokens on the Estate pile, 4 on the Duchy, 7 on the Province, and at least 1 on all other supply piles

Certainly this is possible. Note though that there are more conditions needed. For example (and this is probably still incomplete):

  • Nobody has, or can get, a Watchtower
  • Border Village, HoP, Feast, other non-buy gainers are not in the supply
  • Cursing attacks such as Witch and IGG are not available.
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Dingan

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Re: Infinite stalemate game
« Reply #18 on: February 19, 2015, 07:56:04 am »
0

Another scenario:

Both players have nothing in their deck, but have Prince'd a Stonemason (or some other mandatory trasher).  The only thing they can ever afford is a copper or curse.  They can empty these piles, but there is no other empty pile, so they cannot empty a third pile.  There is no source of alt-VP, so they are tied in VP (assuming they have an equal amount of curses).
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Davio

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Re: Infinite stalemate game
« Reply #19 on: February 19, 2015, 11:22:04 am »
0

Ruins?
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qmech

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Re: Infinite stalemate game
« Reply #20 on: February 20, 2015, 03:32:13 am »
0

Ruins are not in the game unless there are Looters.
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Dingan

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Re: Infinite stalemate game
« Reply #21 on: February 20, 2015, 07:36:13 am »
0

Ruins?

This:

The only thing they can ever afford is a copper or curse ... there is no other empty pile

implies this is a game without ruins.
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faust

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Re: Infinite stalemate game
« Reply #22 on: February 20, 2015, 11:16:37 am »
0

I thought about a different version of this puzzle: A finite stalemate game.

Can anyone think of a situation where it is optimal for both players to not gain anything for n turns, then gain something the (n+1)st turn? I'm thinking pin game, but I think the pinned player wants to buy stuff there to help drain piles...
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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.

TheOthin

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Re: Infinite stalemate game
« Reply #23 on: February 20, 2015, 12:36:03 pm »
0

I thought about a different version of this puzzle: A finite stalemate game.

Can anyone think of a situation where it is optimal for both players to not gain anything for n turns, then gain something the (n+1)st turn? I'm thinking pin game, but I think the pinned player wants to buy stuff there to help drain piles...

Something would have to be happening over the course of that time. Maybe something with Knights?
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ephesos

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Re: Infinite stalemate game
« Reply #24 on: February 20, 2015, 12:43:15 pm »
0

Someone who's pinned by KC/KC/Goons/Masquerade with Copper and Curses gone. As the pinning player, I wait N turns until their whole deck is gone, then I buy up a third pile to end the game.
Wrote a bot to do this once, but I could never figure out how to count the opponent's deck, so I just made N 100. Seemed to work fine.
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