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[Chronos]

Whenever there's a Sea Hag, I generally find myself in the mid to late game position of having a Sea Hag after all of the curses are gone and wondering what to do with it.  Assuming there's no net positive reactions (Horse Traders), when  is it, if ever, a good idea to keep playing my Sea Hag, just to get rid of the top card of my opponent's deck?

My initial intuition is that this is a good idea if there are one or two key cards (like BM + Smithy), but then I think maybe the low probability of hitting the card you want is outweighed by the high probability of hitting a card you don't want to.

Thoughts?

[rrenaud]

If the opp's deck has started to green more than it's gotten stronger, play the the sea hag.  There is often a mid game period where people should stop playing them but don't.  But I am sure the effect is pretty minor.

[Deadlock39]

Cycling a card is going to have the same chance of discarding their good card as it has of getting them one card closer to it.    There isn't a way to strategically try to hit their good cards (unless you try to play mind games with Ghost Ship, but that is crazy).

The effect this does have is that it will push them to a reshuffle faster.  As such, you would not want to play it during the portion of the game when they are building their deck up and reshuffling gets them better cards, but you would want to play it at the end of the game when they are buying victory cards and their deck is getting weaker.

That said, I suspect the effect is so small that for a single game, it would be dwarfed by the random chance of hurting/helping your opponent by discarding good/bad cards.

[WanderingWinder]

The benefits of playing it are so small that the win percentage you're losing by 'never play sea hag' as opposed to playing optimally is negligible. The exceptions are if you somehow know what's on top of their deck, and you know it's good, or if you need to play an action, like for peddler.

[Epoch]

Any time after they've started greening.  That is the point where you start to expect that "the next shuffle is lower quality than the present shuffle," so cycling your opponent's deck starts to work for you.  Otherwise...  you're broadly going to expect that "the next shuffle is higher quality than the present shuffle," so cycling your opponent's deck works against you.

The only reason I can think of why a good quality opponent might have a mid-game "next shuffle is worse than present shuffle" is if, like, last turn you slammed them with four Curses or something, as your deck, I don't know, KC-Sea Hagged?  Or something?  So you've put a bunch of Curses in their discard pile, and now you'd like them to get those into their draw pile.  Seems like an ultra-rare case.

[HiveMindEmulator]

Cycling a card is going to have the same chance of discarding their good card as it has of getting them one card closer to it.    There isn't a way to strategically try to hit their good cards (unless you try to play mind games with Ghost Ship, but that is crazy).

But the effect of pushing one card closer is much smaller than the effect of skipping it. If, for instance, you're playing a no-trash sea hag game, and the only way you can really afford a province is on a tactician turn, playing sea hag while their tactician is not in the discard is probably a good idea. Even if hitting another card is N times more likely, skipping the tactician is probably more than N times as impactful. One mistake people make when analyzing stuff like this is assume everything is linear, but it's not.

So I guess if there is one powerful card you can track, you should probably do it. However, in most cases, it won't be like this. The general rule should be "play it when their discard pile is worse than their deck", which occurs in the case described above or, more often, simply when they start greening.

[WrathOfGlod]

"But the effect of pushing one card closer is much smaller than the effect of skipping it. If, for instance, you're playing a no-trash sea hag game, and the only way you can really afford a province is on a tactician turn, playing sea hag while their tactician is not in the discard is probably a good idea. Even if hitting another card is N times more likely, skipping the tactician is probably more than N times as impactful. One mistake people make when analyzing stuff like this is assume everything is linear, but it's not."

If you don't track cards then playing a sea hag has the effect of making their N card deck play as if it was N-1 cards this means that they will marginally draw their good cards faster on average. My general heuristic is to play it after they have had a bad turn and not after a good turn (unless they shuffled at the end) as a primitive way of guessing the relative quality of the rest of the deck.

[popsofctown]

Is it an unfair comparison to say that with perfect card counting, Sea Hag is half as good as the offense of spy when they are half through their deck, 3/4 as good if they are 3/4 of the way through their deck, and every bit as good as offensive spy if they are n-1 cards through their deck?

[Kuildeous]

If your opponent has Counting House, has no more cards in his draw deck, and has not played his Counting House since his last reshuffle, then it is in your best interest to play the Sea Hag and force a reshuffle so he has no Coppers to draw with his Copper House. This happened to me one game.*

This could apply at any time that you are sure that the opponent has a good card coming up. Maybe you'll be thwarted because he already has that card in his hand. In general, I'd say that when he greens, the Sea Hag could potentially still be useful, but only barely.

* This did not happen to me ever; I just wanted to sound cool.

[popsofctown]

If he would buy a useful card that turn, it doesn't matter if he has Counting House or not, trigger the reshuffle.

[fp]

A general rule of thumb for Sea Hag is...

Don't play it if players are still developing.
Play it is players are buying victory points.

The reasoning is quite simple. If you play Sea Hag as players are still buying new valuable cards, then you are shuffling those cards into their deck ever so sooner. So you shouldn't play it if those cards are helpful and inversely, if those cards aren't helpful, you should play Sea Hag to shuffle then in sooner.

fp

[rod-]

If he would buy a useful card that turn, it doesn't matter if he has Counting House or not, trigger the reshuffle.

The reshuffle trigger is still my least-favorite aspect of dominion.  I really wish that estates were modified to work as pseudo-pearldivers (without the +1card) so you could have at least SOME control over which cards wind up disappearing from a shuffle.

[rinkworks]

But the effect of pushing one card closer is much smaller than the effect of skipping it. If, for instance, you're playing a no-trash sea hag game, and the only way you can really afford a province is on a tactician turn, playing sea hag while their tactician is not in the discard is probably a good idea. Even if hitting another card is N times more likely, skipping the tactician is probably more than N times as impactful. One mistake people make when analyzing stuff like this is assume everything is linear, but it's not.

Your first sentence is true but misleading, because it doesn't take into account the likelihood of each of those things happening.   It's like lottery odds.  Say you get 9,999 people to put $1 in a lottery.  One name is drawn, and winner takes all.  Should you pay $1 to join the pool (bumping the pot up to 10,000)?  Well, the effect of winning would put you up $9,999 instead of down $1.  But you've only got a 1 in 10,000 chance to win, while you've got a 9,999 out of 10,000 chance of losing.  It's a zero-sum game.

Arbitrarily cycling an opponent's deck is like that.  You stand a remote chance of hurting him badly, and a great chance of helping him slightly.  It averages out to even, UNLESS you know something about the quality of his draw pile vs. the quality of his discard pile.  Only then might arbitrarily cycling have a net positive or net negative effect.

A few ways you can have some clues about an opponent's draw and discard piles:

* What cards he's purchased since the last shuffle.  Greens or no?
* Have Rabble attacks hurt the quality of his deck?
* Did the purchase of an Inn improve his draw pile?
* Have you been counting his cards and noticing what he's discarded since the last shuffle?  (Your Tactician example falls into this category; I'm not disputing that.)

Lacking any information like this, though, cycling arbitrarily will break-even over the long run.

[rrenaud]

There is also one more subtle point that is probably not worth mentioning.

Even if the expected value is 0, sometimes you want high variance (your opp is better than you, or you are behind), and sometimes you wanted low variance (you have a skill advantage, or you are ahead).

[Sigmaril]

Easy! Play a Duchess first. If he choose to keep his card, go ahead and Hag him

[DStu]

Arbitrarily cycling an opponent's deck is like that.  You stand a remote chance of hurting him badly, and a great chance of helping him slightly.  It averages out to even, UNLESS you know something about the quality of his draw pile vs. the quality of his discard pile.  Only then might arbitrarily cycling have a net positive or net negative effect.

And one other thing, which is rather academical than practical: While the average quality of the deck might stay the same, what matters in the end is the winchance. And the winchance is probably not linear in the quality, so the winchance might as well change by playing the Hag (even when the winchance was 50:50 before).
What makes all this academical is that it's not really easy to find out in which sense the winrate is not linear, and if it helps to play the Hag or hurts. But usually you would expect it to be some kind of increasing and convex in the value (or superlinear). This would indeed mean that playing the Hag would help him if you put in variance while letting the mean stay the same, as E[f(X)] >= f(E[X]), [ where E[f(X)] is the average winchance you get when when you add randomness, and, as E[X] is what he gets when you do nothing (because you don't change the mean value), f(E[X]) is what he gets when you don't play the hag.]

So I think the other two points are much more relevant, a) Information and b) Variance.

[HiveMindEmulator]

Arbitrarily cycling an opponent's deck is like that.  You stand a remote chance of hurting him badly, and a great chance of helping him slightly.  It averages out to even, UNLESS you know something about the quality of his draw pile vs. the quality of his discard pile.  Only then might arbitrarily cycling have a net positive or net negative effect.

And one other thing, which is rather academical than practical: While the average quality of the deck might stay the same, what matters in the end is the winchance. And the winchance is probably not linear in the quality, so the winchance might as well change by playing the Hag (even when the winchance was 50:50 before).
What makes all this academical is that it's not really easy to find out in which sense the winrate is not linear, and if it helps to play the Hag or hurts. But usually you would expect it to be some kind of increasing and convex in the value (or superlinear). This would indeed mean that playing the Hag would help him if you put in variance while letting the mean stay the same, as E[f(X)] >= f(E[X]), [ where E[f(X)] is the average winchance you get when when you add randomness, and, as E[X] is what he gets when you do nothing (because you don't change the mean value), f(E[X]) is what he gets when you don't play the hag.]

So I think the other two points are much more relevant, a) Information and b) Variance.

I don't quite follow what you defined as X in this example. Maybe you can be more specific?

@rinkworks: You don't really have "a great chance of helping him slightly". You either help him a lot by letting him play that key card one extra time, or you have no real effect on the game outcome. Moving a card up one place in the draw pile will often leave it on the same draw. In an extremely simplified (and thus quite inaccurate) example, say all the matters is the number of times they play card X, which is in the draw pile (or hand). Let D be the size of the draw pile + hand, and let N be the final size of the deck. Your probability of hurting him is the probability of skipping card X, which is 1/D. And your probability of helping him, by giving him one extra play of the card is (D-1)/D*1/N, since the chance of the extra card drawn being card X is 1/N. Since (D-1)/N < 1, the chance of helping is strictly less than the chance of hurting.

Anyway, the effect does not really make a meaningful difference most of the time, so it's probably not worth thinking that hard about it.

[DStu]

I don't quite follow what you defined as X in this example. Maybe you can be more specific?

So let X be quality of your opponents hand. When your (uninformed) play a Sea Hag, you don't change the average quality of his hand E[X], but add some randomness. Now the value (in terms of how it will help his winchance) of his hand f(X) will not be linear in the quality. A first guess would that the value increases faster than linear (which is not really true always, because without +buy getting more than $8 doesn't help you at all, but below $8 it should be true...), an is thus convex.  When you now average a convex function, (Jensen's inequality, link above) the average is  larger than the function applied to the average. The latter is what you would get when you would not play the Sea Hag.

qft:

Anyway, the effect does not really make a meaningful difference most of the time, so it's probably not worth thinking that hard about it.

[HiveMindEmulator]

^Win probability is not a function of your opponent's hand. It depends on the *sequence* of hands. So you have to look in a much higher-dimensional space, where I don't think the surface will necessarily be convex.

[rinkworks]

@rinkworks: You don't really have "a great chance of helping him slightly". You either help him a lot by letting him play that key card one extra time, or you have no real effect on the game outcome. Moving a card up one place in the draw pile will often leave it on the same draw. In an extremely simplified (and thus quite inaccurate) example, say all the matters is the number of times they play card X, which is in the draw pile (or hand). Let D be the size of the draw pile + hand, and let N be the final size of the deck. Your probability of hurting him is the probability of skipping card X, which is 1/D. And your probability of helping him, by giving him one extra play of the card is (D-1)/D*1/N, since the chance of the extra card drawn being card X is 1/N. Since (D-1)/N < 1, the chance of helping is strictly less than the chance of hurting.

Considering that a game might well last only 12-20 turns, it's quite inaccurate to think of "how many times can he play X?" as the determining heuristic.  How soon can he play X is also important, mainly because a delay of a single turn might mean that the game ends before he can play that X again.  Additionally, if X is an attack card, obviously the earlier you play the attack, the more potent it is, as most attacks have rippling effects beyond the turn they're played.  Also, in the event of any kind of race to deplete a pile, the one who can buy whatever they are up soonest wins the race.

I agree that the most likely outcome of moving X up one in the deck is that it stays in the same hand.  In the absence of drawing cards, there's an 80% chance of that being the outcome.  But of the remaining 20%, you're N times more likely to move X up 1 turn than of moving it back ~N turns.  Mathematically, that's a wash.  Not a net benefit.

Someone correctly pointed out that the variance is more important than the average outcome, so if you're behind, you ought to take the risk, and if you're ahead, you should avoid it.  But if you're even, it's a complete and total wash unless you have additional knowledge you can utilize in making the decision.