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[theory]

rrenaud has updated the Best and Worst openings page: http://councilroom.com/openings

I, for one, am shocked at how well Tournament/Ambassador does.  In retrospect, it makes sense, since Tournament's additional buying power is going to be pretty important in a drawn-out Ambassador game, and the fact that it keeps cycling your deck means you can keep up in the Ambassador race.  Still, I'm surprised it does better than Caravan/Ambassasdor.

I am also consistently surprised by how well Ambassador/Ambassador does, since I have always believed that Silver/Ambassador with a second Ambassador later is superior to Ambassador/Ambassador -> Silver.  Perhaps the stats for Silver/Ambassador are tainted with too many games of people falling behind in the Ambassador race because they didn't respond to Ambassador/Ambassador with a second Ambassador.

[Jimmy Jimmy]

Re ambassador/ambassador: Yeah that surprised me too. I would have thought silver/ambassador would be considerably superior because, as you set out in your March article on basic opening probabilities, there's a 30.3% chance of your two ambassadors colliding.

The stats are probably furthered contaminated by players who ignore ambassador altogether as an opening card - to their peril.

[shark_bait]

I for one, am a fan of the ambassador / silver opening.  By opening with ambassador / ambassador, you risk the having the actions conflict.  Additionally, you severely limit your ability to get to 5 gold in turns 3 or 4.  By getting that one silver, you have a good chance of being able to purchase a powerful 5-cost action.  In almost all cases, I agree that you should pick up another ambassador to keep up with the passing of estates and coppers.  The following game I recently played illustrates the benefits of ambassador / silver while showing potential pitfall of the double ambassador opening.  My opening allows me to get an early tactician followed by kings court.  The game was practically over after my first tactician hand and my opponent resigned soon after.  Here's the the log if you're interested.

http://councilroom.com/game?game_id=game-20110412-173225-6dd7b6d9.html

[Mean Mr Mustard]

3   Mountebank / Lighthouse   5/2

Lighthouse:  surprised by this but it is undervalued or overvalued quite a bit depending on the board.
 
5   Caravan / Ambassador   4/3

The caravan in the top five seems out of place to me.  I am learning to severely dislike this card and usually avoid it.

12   Trading Post / Lighthouse   5/2

Oddball combo.  No synergy I see here.  Looking further, Tradepost +any cantrip is ranked very strong.  /Pearl Diver?  I think I would rather TP/-

14   Monument / Chapel   4/2

I am really surprised.  I love both of these cards but they are both terminal and have a tendency to clash during the important trashing turns.  I would have thought old fashioned silver/chapel is better. 

16   Witch   5/-

Jeez.  This game is broken.

23   Laboratory / Chapel   5/2

Old favorite from BSW.

32   Young Witch / Ambassador   4/3

Ugh.  Try this or Hag / Amb against a top player and get creamed.

33   Potion / Ambassador   4/3

Jeez.  This game is broken.

Level -6  -14.447 ± 9.374      Watchtower / Embargo   3/2

Two potentially great cards.  When used under the right circumstances either can really shine.

[adf]

I'm surprised Wishing Well/Ambassador doesn't rate better. If you hit your Wells perfectly, they're Labs, and even if they hit 60-70% of the time (which should be doable over the aggregate of the first two or three reshuffles) I think they're better than Caravans because they're faster and they don't get tripped up by reshuffles.

Mountebank/Lighthouse seems obvious because you get a non-terminal that provides solid coin and protects against the most devastating attack in the game.

I've started moving away from Ambassador/Ambassador, but I used to take it all the time. The 70% chance of no clash in reshuffle 1 is worth the 30% chance of a clash. And in the case of a clash you're still returning two cards.

[timchen]

Caravan/ambassador:
Caravan is a strong card. It is pretty much a lab when you cannot draw your entire deck. I am only appreciating how strong it is not until rather recently. It works well with ambassador too.

Monument/Chapel:
I actually have thought of this one. It is actually not that hard: in the first few times when you draw chapel with silver, the correct thing to do is to trash everything else. It therefore does not make any difference (unless there is some savvy $2 cards to buy) if you replace the silver by the monument. On the other hand, in a chapel-province game, the few extra VP tokens can make a huge difference.

On the technical side, I have some trouble understanding what the ratings mean and how they are calculated. For one thing, if I open the same thing with say, LV 5 with my opponent, but I am at LV 40 and he is LV 0, now we become LV 45 and LV 5, which persumably would have a different win rate I think. (Or does the win rate depend only on the difference of the level of the players? If so it seems a rather strong assumption for the levels to follow a normal distribution.) Even for asymmetric openings, it seems very impossible for every kind of player match up to shift for the same amount for the different openings. It would be easier for me to understand if there is just an explanation of how it is calculated.

[variance]

What up Theory, great site, TY.

I love amb/amb, the times that you get both ambs in your hand on turn 3 or 4; you have 2 copper or 2 estates anyway....thoughts?

[rrenaud]

On the technical side, I have some trouble understanding what the ratings mean and how they are calculated. For one thing, if I open the same thing with say, LV 5 with my opponent, but I am at LV 40 and he is LV 0, now we become LV 45 and LV 5, which persumably would have a different win rate I think. (Or does the win rate depend only on the difference of the level of the players? If so it seems a rather strong assumption for the levels to follow a normal distribution.) Even for asymmetric openings, it seems very impossible for every kind of player match up to shift for the same amount for the different openings. It would be easier for me to understand if there is just an explanation of how it is calculated.

TrueSkill makes the normal distribution assumption.  So holding variance constant, if you add a fixed level to both you and your opponents skills, the model says there is no difference.  The hard part is to understand TrueSkill.  http://www.moserware.com/2010/03/computing-your-skill.html . 

The openings data also makes the assumption that you and your opening form a team and that you add your skills together with equal weight on the player skill and opening skill.  I'd guess the player matters much more than the opening, but optimizing this for the right criteria for multiplayer games is hard, http://stats.stackexchange.com/questions/10067/maximum-of-two-or-more-gaussian-distributions-with-known-and-possibly-different).  Likewise the opponents team does the same thing, then you model a little noise from the outcome of the game, and then your combined normal distributions fight.

[Stoc]

I've always preferred Ambassador / Silver, and with some of the other top players here posting the same thing I'm a surprised how big the gap is. Top 10 vs. 61st is a big gap.

I'm trying to think of a $5 card that would necessitate going Ambassador / Silver, though, and I can't come up with one.

[WanderingWinder]

Let me preface this by saying that Ambassador is my least favourite card.
I always go ambassador/silver. I most usually pick up a second ambassador on my second reshuffle, especially if my opponent has an ambassador.
However, there are some cases where you can get around picking up an ambassador at all, and I pretty much always go for them (whether I should or not).

[painted_cow]

Ye, like many of you stated, i also like Ambassador/Silver a lot more than Amb/Amb. You get the second Ambassador in turn 3 or 4 like always.

1st you have no chance of colliding 2 action cards when its most crucial.
2nd and even more important, you already have a silver (oh really...). If you open with Amb/Amb you sometimes have trouble to get to 3 coin.

Imho Amb/Amb is only better if there are really cheap villages for 2 or 3 coin.

[J.Co.]

I am also consistently surprised by how well Ambassador/Ambassador does, since I have always believed that Silver/Ambassador with a second Ambassador later is superior to Ambassador/Ambassador -> Silver.  Perhaps the stats for Silver/Ambassador are tainted with too many games of people falling behind in the Ambassador race because they didn't respond to Ambassador/Ambassador with a second Ambassador.

I dunno, I think having the silver won't be as beneficial, since you're often trying to get rid of coppers. By doing that, you've got less money to spend, so getting $5 cost cards won't be easy, with or without a silver.

On the other hand, while colliding ambassadors is a risk, having two of them can mean up to four cards out of your deck and two into someone else's deck by the second shuffle. Even if they use Ambassador on you (just once until your second shuffle), they're already behind. I'd imagine whoever gets his/her deck streamlined (as much as it can be in an ambassador game, anyway) faster has the better chance of staying ahead. Getting a second Ambassador later is a buy that won't see an effect until it may be too late.

It's kind of like your famed Chapel theory. Even if you have four coppers, there should be little hesitation in trashing all four of them. Same with Ambassador. You should be trying to get rid of as much as you can as quickly as possible and slowing down your opponent as much as possible before getting your deck into gear.

[theory]

That does sound persuasive, but it seems like you'd run into major buying power issues if you can only keep playing Ambassador.

I guess it depends on whether the $5's help Ambassador.  Laboratory or Hunting Party, I can see Silver / Ambassador >> Ambassador/Ambassador.  But guided once sold me on how Ambassador / Silver / Ambassador beats Ambassador / Ambassador ... he'll have to show up here and explain why

[Death to Sea Hags]

There's a lot of two-terminal openings - many more than most players would consider playing.  I suspect, then, that in many of these cases, its just feast or famine: if you don't collide, you win.  If you collide as P1, you've basically lost your P1 advantage.  If you collide as player 2, you lose.

What I find more interesting is the different synergies.  TP-Haven beats TP-Lighthouse?  But Attack-Lighthouse beats Attack-Haven.  Mountebank is better with Haven, Witch isn't? 

Also, trashing is key - all level 6+ openings include trashing!

[adf]

TP-Lighthouse doesn't guarantee that there's an attack in the set. Attack-Lighthouse does.

[WanderingWinder]

Also, trashing is key - all level 6+ openings include trashing!

Mountebank/Lighthouse begs to differ. Other than that, if you can call Ambassador trashing (not really a stretch), then you're right. Also your point is well taken.
Also, why is Witch/Lighthouse so low?

[chwhite]

Also, trashing is key - all level 6+ openings include trashing!

Mountebank/Lighthouse begs to differ. Other than that, if you can call Ambassador trashing (not really a stretch), then you're right. Also your point is well taken.
Also, why is Witch/Lighthouse so low?

Well, I'd expect it to be worse than Mountebank/Lighthouse because you run the risk of drawing the Lighthouse dead from the Witch.  In general, if you actually buy anything with your $2, Mountebank will be superior for that reason.

[DG]

Tournament/ambassador will be exceptionally strong. You'll get plenty of uses of the tournament as +1 card/+1 coin as ambassador games are usually very slow to develop. The deck will cycle quickly allowing you to play the ambassador more often. You can still get value from an ambassador hand after with playing an ambassador and two cards with it, negating a common problem for hands with ambassadors. There are no complications with actions clashing. Very few strategies are  excluded by this opening.

A successful ambassador deck is likely to be small so that when you buy a province it will hit the tournaments almost immediately. The prizes will allow you to expand your deck quickly with quality, killing off an opponent who might still be competitive if you expand your deck badly.

[WanderingWinder]

Tournament/ambassador will be exceptionally strong. You'll get plenty of uses of the tournament as +1 card/+1 coin as ambassador games are usually very slow to develop. The deck will cycle quickly allowing you to play the ambassador more often. You can still get value from an ambassador hand after with playing an ambassador and two cards with it, negating a common problem for hands with ambassadors. There are no complications with actions clashing. Very few strategies are  excluded by this opening.

A successful ambassador deck is likely to be small so that when you buy a province it will hit the tournaments almost immediately. The prizes will allow you to expand your deck quickly with quality, killing off an opponent who might still be competitive if you expand your deck badly.

And for some reason I wasn't thinking about this half an hour ago and theory crushed me. Really really badly. Very strong opening.

[variance]

I guess pulling an amb on both 3rd and 4th turn, though rare (anyone know the %?), is super strong.  Why else is amb/amb ranked so high on the best/worst list?

[DStu]

I guess pulling an amb on both 3rd and 4th turn, though rare (anyone know the %?)

Having the first ambassador in the first 10 is 10/12=5/6, the second one in the five cards of the other hand is 5/11 (11 and not 12 cause one of the "bad" places is already occupied by the other ambassodor).
This gives you a chance of 25/66 which is roughly 38%.

[Amaranth]

32   Young Witch / Ambassador   4/3

Ugh.  Try this or Hag / Amb against a top player and get creamed.

It's not one that would have occured to me but I guess part of it might be that the Ambassador makes it harder for opponents to draw their banes.

There's also the fact that unlike other +card termminals, you don't care if you draw Ambassador off the Witch.

[Reyk]

Isn't it strange that Envoy with some terminals (Amb, Chapel, Masquerade) is better than Silver/Envoy? Ok, they are all trashers but still ... Would you ever open Envoy/Masquerade?

[boloni]

Isn't it strange that Envoy with some terminals (Amb, Chapel, Masquerade) is better than Silver/Envoy? Ok, they are all trashers but still ...

I would say it's because ambassador, chapel and masquerade are much better opening cards than envoy.

Would you ever open Envoy/Masquerade?

I wouldn't. Instead I would open Masquerade/Silver which is ranked better than Envoy/Masquerade. Ambassador/Silver and Chapel/Silver are also better than Ambassador/Envoy or Chapel/Envoy.

[david707]

I guess pulling an amb on both 3rd and 4th turn, though rare (anyone know the %?)

Having the first ambassador in the first 10 is 10/12=5/6, the second one in the five cards of the other hand is 5/11 (11 and not 12 cause one of the "bad" places is already occupied by the other ambassodor).
This gives you a chance of 25/66 which is roughly 38%.

That's correct, although it's worth mentioning that the chances they come up in the same hand is about 30%, leaving about 32% that an one ambassador is in hand 3 or 4 and the other is in hand 5.

[Death to Sea Hags]

32   Young Witch / Ambassador   4/3

Ugh.  Try this or Hag / Amb against a top player and get creamed.

It's not one that would have occured to me but I guess part of it might be that the Ambassador makes it harder for opponents to draw their banes.

There's also the fact that unlike other +card termminals, you don't care if you draw Ambassador off the Witch.

This must be a case where the increased win-rate from non-collision swamps the greater loss-rate of risk of collision.

YMMV.  If you like high-risk strats, this is for you.  If your play is geared towards minimizing risk and variance, then this is not for you.

[rrenaud]

Ambassador cancels/reflects cursing attacks.  I get my deck small.  You give me a curse.  You know where that cuse is going to end up?  In your deck.

[Amaranth]

Ambassador cancels/reflects cursing attacks.  I get my deck small.  You give me a curse.  You know where that cuse is going to end up?  In your deck.

I think that in general the deck with Young Witch will be drawing and playing both the Ambassador and the Young Witch more frequently than its opponent, unless the other player opened with a better card drawer than Young Witch. So their deck is likely to become small first.

[rrenaud]

You go amb/YW.  I'll go amb/amb.  My deck is going to shrink faster than yours.

[painted_cow]

I always love it when players get some Cursing cards when I have Ambassador deck. It saves me the pain to buy to Curse myself to hand it over :-)

[Amaranth]

You go amb/YW.  I'll go amb/amb.  My deck is going to shrink faster than yours.

That's certainly possible, but I imagine amb/YW will have an easier time buying good cards. Anyway, I got a chance to try it out (versus Ambassador/Caravan) and went on to lose, though the fact that neither of them showed up until turn 5 didn't help matters.

[drg]

Ambassador cancels/reflects cursing attacks.  I get my deck small.  You give me a curse.  You know where that cuse is going to end up?  In your deck.

I think that in general the deck with Young Witch will be drawing and playing both the Ambassador and the Young Witch more frequently than its opponent, unless the other player opened with a better card drawer than Young Witch. So their deck is likely to become small first.

Giving out curses to someone who's going heavy on ambassadors usually isn't a very good plan.

You go amb/YW.  I'll go amb/amb.  My deck is going to shrink faster than yours.

That's certainly possible, but I imagine amb/YW will have an easier time buying good cards. Anyway, I got a chance to try it out (versus Ambassador/Caravan) and went on to lose, though the fact that neither of them showed up until turn 5 didn't help matters.

Getting both your opening cards on T5 vs a competent player is usually a loss, especially when ambassador is in play, regardless of small differences in what you buy other than the ambassador.  I drew both ambassadors on T5 once after my opp played his on T3 and T4... that game was not fun.

[Geronimoo]

I simulated this (players buy only treasure and green cards after the opening):

Young Witch/Ambassador (57%) - Ambassador/Ambassador (40%)


Compare it to this:

Sea Hag/Ambassador (35%) - Ambassador/Ambassador (62%)
 

[rrenaud]

I am especially distrustful of big money simulations for heavy trashing type decks.  Heavy trashing -> small deck -> easy combos.

[Death to Sea Hags]

Getting both your opening cards on T5 vs a competent player is usually a loss, especially when ambassador is in play, regardless of small differences in what you buy other than the ambassador.  I drew both ambassadors on T5 once after my opp played his on T3 and T4... that game was not fun.

I'd be interested to know what the win percentages are on comebacks from this.  Except against very poor play, I find that if I end up with a t5 collision and the other player doesn't, I am going to lose, period.

Especially as player 2.... 


But I might be wrong! Is there really much hope of a turn around?  Maybe strategy needs to change drastically - no more actions, only Big Money will save the day.

What has to happen subsequent to this for me to win?  There might be a further break-point in the game that has such a high impact on win rates, but I'm not sure what it would be.

[Geronimoo]

I am especially distrustful of big money simulations for heavy trashing type decks.  Heavy trashing -> small deck -> easy combos.

You're right of course. I added these openings to a simulator bot who's going for a Market/Peddler engine (with a few Worker's Villages and Smithies). This results in a reversal of win%:

Young Witch/Ambassador (40%) - Ambassador/Ambassador (58%)

The Witch player always ends up with decks bloated with Curses, Coppers and Estates.

[Toskk]

Sorry for a slight thread necro, but on the topic of best/worst openings and Ambassador.. I was curious as to why Ambassador / Ambassador ranks in soooo high? The best/worst opening data has essentially always suggested that anytime Ambassador is present, regardless of what you might pair it with you're better off going Ambassador / Ambassador (the only exceptions being Tournament and Caravan). This does not seem intuitive to me, what with the extra early chance of terminal collisions. Further, going over to the win rate by card accumulation, while a 1 Ambassador lead produces a significant increase in win chances, a 2 Ambassador lead produces a significant decrease in win chances. Taken together, this data doesn't really add up.. am I just missing something, or is Ambassador / Ambassador somehow overrated as an opening?

[fencingmonkey]

I have the same aversion to turn-3 collision, but this thread is making me reconsider it. Remember: if your worst-case scenario is drawing amb/amb/C/C/E turn 3, you are still drawing amb.

As discussed above, there's roughly a 33% chance of a collision with 2 terminals. If that happens, you shrink your deck by 2 and increase your opponent's by 1. If not, you shrink by 4 and increase them by 2. In other words, amb/amb gives you an advantage of (1/3*3 + 2/3*6) = 5 cards. That's incredibly powerful.

Or, if you think better in words than in math (how did you end up here?) the payoff for best-case amb/amb is SO good that it's worth the non-trivial chance of getting an early collision.

[guided]

One of the big qualitative leaps I made as a Dominion player was in realizing that terminal collisions are not the end of the world. Putting extra terminals in your deck and having them not collide is so strong that going out of your way to avoid even the possibility of collision means you will have weaker draws on average than if you just eat the collisions that do happen with high terminal density. By extension, when building a village-based engine from the ground up you should usually weight your buys significantly toward the terminals rather than the villages when you have a choice, filling in the villages later as needed.

Now, for the specific example of an Ambassador/Ambassador opening, I prefer Ambassador/Silver followed by another Ambassador at turn 3 or 4. Why? Because the double-Ambassador opening will very likely have enough collisions in the first few shuffles (given a steadily decreasing deck size) to completely wipe out the advantage of sometimes getting an Ambassador on both turn 3 and turn 4. Put another way, if I open A/S/A while you open A/A, you may hit twice before turn 5 but it's likely we will both get the same number of hits by turn 10 (or whatever) since you will tend to have more collisions. Meanwhile I start up a Silver which may allow me to buy another helpful card early. Note this reasoning is quite specific to Ambassador!

[DG]

Ambassador/ambassador is strong if there are village style cards in the kingdom. Ambassador/silver is stronger if there are some laboratory style cards in the kingdom. Ambassadors always depend upon the continuation since there is always a continuation after the ambassadors have resized the decks. Talking about them in a general sense isn't very helpful.

[Fangz]

I dunno, though. Amb/amb tends to snowball - a heavily ambassadored player will have difficulty playing his ambassador often, and so fall even further behind. Even in the case of a collision it isn't that disastrous: chances are you aren't going to buy anything too awesome in your ambassadoring turn anyway. The risk of collision is equal to the risk of drawing ambassador/silver/?/?/? and a full 70% of the time that means only a $4 hand (and the remainder of the time, that means no trashing on your first turn). You can judge that the reward outweights the risk. It all depends on what other cards are available, obviously - e.g. if there's a hunting party then go for it. If there's fishing village I'd go amb/amb every time.

Personally I've played both ways and I have no idea which is stronger.

[guided]

Even in the case of a collision it isn't that disastrous:

It's not disastrous. It's just that if you have one more collision than your opponent over the first few shuffles, you've lost your tempo advantage. And it's quite likely you will have one more collision. The tempo generally evens out before you ever get to worrying about continuations. The Silver opening is slightly more likely to get an early strong card, while the Ambassador opening is slightly more likely to trim faster. In my experience Silver has the more likely upside in most cases.

The difference between these two openings is not large in any case.

[HiveMindEmulator]

Sorry for a slight thread necro, but on the topic of best/worst openings and Ambassador.. I was curious as to why Ambassador / Ambassador ranks in soooo high? The best/worst opening data has essentially always suggested that anytime Ambassador is present, regardless of what you might pair it with you're better off going Ambassador / Ambassador (the only exceptions being Tournament and Caravan). This does not seem intuitive to me, what with the extra early chance of terminal collisions. Further, going over to the win rate by card accumulation, while a 1 Ambassador lead produces a significant increase in win chances, a 2 Ambassador lead produces a significant decrease in win chances. Taken together, this data doesn't really add up.. am I just missing something, or is Ambassador / Ambassador somehow overrated as an opening?

A few notes about how these numbers work out:
1. Just because one opening is higher than another doesn't mean you should always choose it. The best/worst openings are conditioned on one player actually using that opening. So just because ambassador/ambassador is so high doesn't mean you should always get it.
2. The win rate by difference in number is about the *difference* in number, not the total number. Going ambassador/ambassador doesn't mean you have a 2 ambassador lead. If your opponent gets 1 ambassador, it's a 1 ambassador lead. And if they don't get ambassador as an opening, but get 1 later, it's still a difference of 1. That data is hard to use for thinking about openings.

Regarding amb/amb vs amb/silver, I agree with DG that it depends on what's on the board. For me, there has to be a reason to buy the silver. In a vacuum, if there were no other cards in the kingdom, I'd prefer double ambassador, because you want 2 eventually and the silver doesn't do a whole lot the first time through the deck anyway. As fangz said, drawing silver with ambassador is just as bad as drawing ambassador with ambassador a lot of the time anyway. The real difference occurs when you draw them apart. A second ambassador gives more deck size advantage, while silver can let you get to $5. If there is a $5 worth shooting for early that can help during the ambassador phase of the game, like a lab or treasury or upgrade, I'll go for the amb/silver opening. Otherwise I prefer to stay on top of the ambassadors and just buy the silver on turn 3 or 4.

[guided]

A second ambassador gives more deck size advantage

I'm trying to point out the thing that most people seem to miss here, which is that even if you don't collide at turn 3 or 4, you're likely to get an extra collision or two compared to an A/S/A opponent in the next few turns, which gives up the tempo advantage. You only realize the tempo upside if you don't get an extra collision early, which is unlikely. The Silver opening's upside is also unlikely, so again, there's not much space between the two openings.

FWIW at one point I did extensive playtesting of A/S/A vs. A/A when neither deck was allowed to buy any other Kingdom cards beyond 2 Ambassadors, and A/S/A had a clear (if not huge) edge. Other Kingdom cards can of course change the equation.

[HiveMindEmulator]

^But like you said, you're not really concerned with collisions. What's more important is how many times you get to play ambassador (and how many cards you return per play). Even if you get an extra collision on turn 6 or something, you got in an extra play by getting the second ambassador sooner. After 4 turns, if you've played 2 ambassadors and he's played one, you're ahead. Yeah you're more likely to get a collisions the next time through the deck, because your deck is smaller, since you're ahead on ambassadoring... Amb/silver does not get ahead because it avoids collisions and shrinks the deck better; it gets ahead because it has more buying power.

[Fangz]

Plus if you get such a thin deck that ambassadors colliding become inevitable, you could just ambassador your ambassador.

FWIW at one point I did extensive playtesting of A/S/A vs. A/A when neither deck was allowed to buy any other Kingdom cards beyond 2 Ambassadors, and A/S/A had a clear (if not huge) edge. Other Kingdom cards can of course change the equation.

Well, that may be so, but BM+ambassador situations might not reflect on the 'average game', whatever that is...

[guided]

^But like you said, you're not really concerned with collisions. What's more important is how many times you get to play ambassador (and how many cards you return per play).

An extra collision will tend to equalize the number of times you get to play Ambassador, if it happens within the first few shuffles. That's my whole point. A/S/A "catches up" if it has fewer collisions. When A/A realizes its turn 3/4 upside of playing one both turns, it becomes highly likely that A/S/A will catch up with fewer collisions. A/A only realizes its true upside if it gets through at least a couple more shuffles maintaining a lead in Ambassador plays, which isn't likely.

I'm not saying something radical here! It's just a small thing that people seem to overlook. They think "ah-ha! I played them at turn 3 and turn 4 so I must be way ahead now!" Because not quite: collisions in the next few turns do matter.

[rod-]

The collisions after the first two plays are not all that important:  If you've outplayed ambassador 2:1 in the first reshuffle, your deck has reached 10 cards while your opponent's is 14 cards.  Your ambassadors are obviously more likely to collide than theirs, but not by that great of a margin.  1 collision means your deck is 12 cards while your opponent's has reached 14, but you were 1 turn(a full 5 cards) ahead on the 3rd reshuffle and with a 12 card deck are about 60% likely to be playing another ambassador this turn to go back to 11 and push them to 15.  A second collision at this point doesn't really impact you because your deck is still very nearly 1 turn faster per shuffle, and when you reach 0 collisions for a shuffle you've cleared down to ~8-10 cards and will push everything that gets ambassadored to you back faster than it gets sent.

[Empathy]

Question: Should the cards you pass around with ambassador not matter?

Because it seems to me that hitting a double estate with one ambassador can somewhat cancel out the effect of two single-estate ambassadors. Likewise double ambassador has a good chance of hitting an ambassador/copper hand even without a collision (so does amb/silver, but it doesn't matter as much). I have no idea if that would actually help A/S/A versus AA. You could pollute with coppers instead (guaranteeing deck size advantage) but then you probably lose pretty badly on money-tempo.

I like rod's method of analysis, but I do think the choices you make on ambassador, and the event "double estate" might have a bigger impact than the collision event.

[guided]

The collisions after the first two plays are not all that important:  If you've outplayed ambassador 2:1 in the first reshuffle, your deck has reached 10 cards while your opponent's is 14 cards.  Your ambassadors are obviously more likely to collide than theirs, but not by that great of a margin.  1 collision means your deck is 12 cards while your opponent's has reached 14, but you were 1 turn(a full 5 cards) ahead on the 3rd reshuffle and with a 12 card deck are about 60% likely to be playing another ambassador this turn to go back to 11 and push them to 15.  A second collision at this point doesn't really impact you because your deck is still very nearly 1 turn faster per shuffle, and when you reach 0 collisions for a shuffle you've cleared down to ~8-10 cards and will push everything that gets ambassadored to you back faster than it gets sent.

You're assuming that every Ambassador sends away 2 cards regardless of deck size, and (even more critically) flat-out assuming you don't collide at turn 3 or 4. Yes, it becomes more likely that you'll hit the upside when you don't collide at the first opportunity, but the point is that even when you don't collide at 3/4 your chances of coming out ahead are not as high as you might think.

The BM+2xAmbassador example is designed to illustrate that no special mid-price (particularly $5) cards are required for A/S/A to realize a small advantage in at least one particular case.

I am rapidly losing my motivation to argue this rather unimportant point (the two openings are extremely close in strength in the general case) just for the sake of convincing others to re-examine their assumptions. I am not trying to tell you A/S/A is a vastly stronger opening in any case at all, only that those who think it's obviously inferior to A/A in most circumstances should reconsider and exercise some critical thinking past turn 4. While I feel rod- has made logical errors in drawing the conclusions he does, it is at least heartening to see somebody engaging with the problem in a thoughtful way.

[rod-]

Ambassador was my worst card for a long time, although it seems i've made some headway on it in recent times, to where it's almost average.  My win rate without is still a good deal higher than my win rate with, though.  Some ambassador games are entirely determined on the rate at which you pare your deck down, some are determined more by the amount of money you amass while you're trimming, some games you can just ignore it and still win.  Clearly, you are going to be making approximations when thinking about openings in a vacuum. 

The thing i've noticed about ambassador wars is that (if the game is in fact one of the 1st type) you can lose them VERY early.  Being as few as 4 cards behind is insurmountable.  That's why amb/amb is such a strong opening - you're giving yourself the opportunity to get 4 cards ahead.  It won't happen every time, but if will happen. 

It's important to remember that not every ambassador game is type 1 - most are closer to type 2.  But there are boards where it's entirely critical.

[HiveMindEmulator]

An extra collision will tend to equalize the number of times you get to play Ambassador, if it happens within the first few shuffles. That's my whole point. A/S/A "catches up" if it has fewer collisions. When A/A realizes its turn 3/4 upside of playing one both turns, it becomes highly likely that A/S/A will catch up with fewer collisions. A/A only realizes its true upside if it gets through at least a couple more shuffles maintaining a lead in Ambassador plays, which isn't likely.

OK, I see your point. In the case of no collision on turn 3/4, it may "catch up", particularly in the BM case when nothing other than ambassadoring can happen until you can hit $6 (though in that case silver/silver is best anyway). A/A will not typically get an insurmountable lead, but I don't A/S can ever really get ahead unless you use the silver to buy something good. And in the case of collision, it's often even anyway, since even though you both played one ambassador, the silver probably didn't do any good (unless you bought something for $2 or you got ASCEE or ASCCC). It is definitely a small difference either way, except in the case where there is a useful $5, where getting the silver at the outset is huge.

Because it seems to me that hitting a double estate with one ambassador can somewhat cancel out the effect of two single-estate ambassadors. Likewise double ambassador has a good chance of hitting an ambassador/copper hand even without a collision (so does amb/silver, but it doesn't matter as much). I have no idea if that would actually help A/S/A versus AA. You could pollute with coppers instead (guaranteeing deck size advantage) but then you probably lose pretty badly on money-tempo.

I'm pretty sure you should almost always be going for deck size, favoring removing 2 coppers over 1 estate. The buying power of coppers is not that important beyond getting a couple silvers, and removing coppers increases your chances of getting double estate in subsequent turns

[guided]

you're giving yourself the opportunity to get 4 cards ahead.

...and possibly give up this lead anyway. Stop for a moment, and simply bring the possibility into your mind of getting a tempo/deck size advantage and then subsequently losing it. This possibility exists. And it is extra-linearly likely with A/A since the deck has shrunk and collisions quickly become more likely. It is not anything like "gg" at the moment that you get Ambassador at both turn 3 and turn 4.

A/S/A is behind in deck size by about 1.5 cards on average at the 2nd shuffle and can catch up by having one fewer collision before the 3rd. For A/A, collision at 3/4, the 2nd Ambassador falling to turn 5 (much more likely than a single Ambassador falling to turn 5), or collision at 5/6 is enough to nullify the hoped-for advantage. Of course in having the advantage nullified you may end up "catching up" to the Silver opening by buying a Silver (or whatever) when A/S/A buys nothing. Neither opening has an kind of huge advantage here.

A/A is strong. A/S/A is strong. They're both strong. I posit based on extensive experience (and playtesting of simple toy cases) that A/S/A is perhaps slightly stronger on a healthy percentage of boards. I am not asking you to strike down A/A and abandon it as some kind of terrible second-class opening for nooblings.


late edit: Corrected "one card" to "1.5 cards" - I neglected the gained card in both directions.

[guided]

I don't A/S can ever really get ahead unless you use the silver to buy something good.

This is exactly right. A/S/A gets ahead by getting something good with the Silver early on. A/A gets ahead by avoiding enough collisions to build a definitive deck size advantage. What people all too often fail to realize is that A/A's upside is only on a par with A/S/A's upside in likelihood; it is not vastly more likely.

[Toskk]

This has been a good discussion, although I'm still not entirely convinced, as my big worry isn't T3/T4, but T5/T6. To me, the worst-case scenario of A/A just seems a heck of a lot worst than the worst-case scenario of A/S/A:

A/A Player T3: Draws A/A/C/C/C -> returns two coppers, gives opponent 1 copper. No buy.
A/S/A Player T3: Draws A/C/C/C/E -> returns two coppers, gives opponent 1 copper. No buy.
A/A Player T4: Draws E/E/E/C/C -> No buy.
A/S/A Player T4: Draws S/C/C/E/E -> Buy Ambassador.
A/A Player T5: Draws A/A/E/C/C -> returns two coppers, gives opponent 1 copper. No buy.
A/S/A Player T5: Draws A/A/C/C/E -> returns two coppers, gives opponent 1 copper. No buy.

A/A Deck: 2x Ambassador, 5x Copper, 3x Estate
A/S/A Deck: 2x Ambassador, 5x Copper, 1x Silver, 3x Estate

As you can see, worst-case the A/A player hasn't bought a single card between turns 3-5, while the A/S/A player was able to pick up their second Ambassador and is a Silver ahead. Sure, the odds of this worst-case scenario aren't high, but I really don't see the A/A player catching up at this point.. they're a full turn behind the A/S/A player by T6. The A/A player *may* outpace the A/S/A player with better luck on collisions, but is it really worth the risk of possibly capitulating at T5?

[rrenaud]

What you really want is the average case.  Double terminal openings tend to have high variance.

A heuristic guide as to when to A/A and when to go A/S would be nice.

I'd guess that expensive, non-terminal actions Hunting Party and Lab tilt towards A/S. 

Cheap village like cards, native village, village, shanty town, hamlet probably tilt toward A/A.

[DStu]

What you really want is the average case.  Double terminal openings tend to have high variance.

A heuristic guide as to when to A/A and when to go A/S would be nice.

I'd guess that expensive, non-terminal actions Hunting Party and Lab tilt towards A/S. 

Cheap village like cards, native village, village, shanty town, hamlet probably tilt toward A/A.

I'm not so sure about this, but also only guessing. From what I've read here, I would think that also important $3 would more likely want you take the Silver, as you in this case can pick them up while ambassadoring estates, improving your cylce/deck while trashing. That would be consistent with guideds BM-experiments, as in this case the important $3 is silver.

[guided]

I find it at least plausible that cheap Villages (and in general boards that support village-based engines) argue more for A/A, but I have no specific evidence to offer in one direction or the other.

[Fangz]

The one thing to add that we haven't addressed is 3+ player games. I feel like going Amb/Amb is stronger then, since with multiple players ambassadoring, your deck can grow very very quickly indeed, you need that early ambassador pair to keep things under control.

[guided]

I'm less inclined to rush Ambassadors with 3 or more players. My limited experience is that Ambassador is much less overpowering in multiplayer. Games are shorter (particularly with Ambassador helping to empty the Estates) and if other people are using it you can't necessarily get the definitive tennis-match victory you are looking for in 2p. It's definitely a card I would tend to buy at 3p; I'm just much less inclined to get 2 of them and aggressively trim down to a tiny deck.

[Fangz]

Really? My experience with ambassador 3 players tends to be that it always reaches a ballooning situation where possibly one player (me if I'm lucky) remains in control of his deck, whereas the remaining players find themselves with massively ballooning decks that they can't even find their ambassadors in.