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[-Stef-]

Suppose we're playing Borinion, a much more boring game then Dominion. Two players, P1 and P2, take consecutive turns. P1 gets to start, and due to the unfair nature of this game it ends after 9 turns. So P1 always gets to make one extra turn. Both players start at 100 points, and every turn they choose...

option A: 60% chance at +3 points, 40% at -3 points
option B: 50% chance at +5 points, 50% at -5 points
option C: 40% chance at +10 points, 60% at -10 points

If you play this game in isolation, it's easy: always option A is best. In fact, it's the only option with a positive mathematical expectation. Simulation over one million games shows that if both players follow this simple minded strategy, P1 will win approximately 555000 vs 445000 (55.5 win% for P1).

However, this game is not about scoring as many points as you can. It's about scoring more points than your opponent. And thus, plans that might look bad at first glance end up being better after all.

Suppose we leave the game plan of P1 what it was ('always go A'), but experiment a bit with P2. If we change it to 'option A when ahead, option B when behind' the win% for P1 drops to 51%. 'option A when ahead, option C when behind' works even better; now P1 only wins 46% of the games. We can improve P2 further by delaying the risky things, and enforcing option A on his first turn (42.5% for P1) or his first two turns (42.1% for P1).

I'm starting to get convinced the optimal strategy is actually quite complex. I think it uses all 3 options, and also includes the 'turns to go' and 'actual point difference' (not just 'am I behind or ahead'). But I won't go into that - I didn't name it Borinion for nothing.

So what does this game teach us about Dominion? Most of all that the optimal strategy always involves your opponent. Even on kingdoms without any attacks, it matters a lot what he's doing and how well he's doing. If you both start out the same, but he has some shuffle luck and you don’t, it's time for crazy things. I he stumbles where you thrive, try to buy safe cards.

It also suggests it’s good to have options. If you create an option B, C or D for yourself to use later on, that choice itself is already good. Engines give you much more options than BigMoney. So in playing engines right, it's not just about 'how can I effectively build this engine in solo play'. It also requires a good feeling/understanding for taking risks. Simulators we use today don't understand it at all, and that may very well be the reason variants of BigMoney strategies do so well in simulation and so poorly in reality.

Opening two terminals isn't all that bad just because they might collide. P1 has no real reason to take this risk, but P2 is already behind at the start. It of course depends on the questions 'how good is it if they don't collide' and 'how bad is it if they do'.

One of my favorites is double Steward, because if they collide I can still get rid of 2 cards. I wasn't going to do much more on a turn with steward without collision anyway. Don't get me wrong - I'm not happy with the collision at all - but it's not a game losing disaster either. And depending on the kingdom, being able to get rid of 4 cards in the first round may very well be winning. As a rule of thumb, opening with two terminals is too soon to take risks though.

It’s not easy to define risky or safe things in general. A safe choice that happens often midgame is adding a little more +actions to your deck than the bare minimum. Another safe choice is to stop playing your engine where you could draw some more cards, just to prevent a reshuffle. Maybe you can put a good card on top for next turn?

Risky things could include adding more cards that require other cards (Baron, Remodel, Forge) or buying slightly too many terminals in general. Village + smithy is more risky than laboratory + laboratory. Swindler has the risk build-in all by himself. So do treasure map and tournament, but they’re not really an addition to a deck - they require building your entire deck around them.

In the endgame it can get quite complex because of the ending conditions of a Dominion game. The most common risky thing is buying the next-to-last province. There is a rule for not doing it, but even if it made you lose, that doesn't automatically say it was a bad thing to buy it. If your deck is not so good, and you're losing the long run for sure, try to sneak out a victory now.

To summarize: constantly figure out whether you’re ahead or behind. If you’re ahead consolidate, if you’re behind make a plan to get back. As player 2, you’re behind when you start - do something with it.

[WanderingWinder]

Your game is actually not so hard to 'solve'. You just have to work from the end. And take some time. Probably a couple hour or so and you're done. Faster if you can program the thing.
i.e. if coming into the ninth turn, P1 is...:
more than 3 points ahead, he takes option a and wins 100% of the time. (100%)
exactly 3 points ahead, he takes option a and wins 60% of the time, ties 40% of the time. (80%)
from two points ahead through two points behind, he takes option a and wins 60% of the time (60%)
three to four points behind: he takes option b and wins 50% of the time (50%)
five to nine points behind: he takes option c and wins 40% of the time (40%)
ten points behind: he takes option c and ties 40% of the time (20%)
more than ten points behind: he always loses (0%).

To find out what P2 should do on turn 8, you look at the different starting values, and what each thing gets you, except now, instead of having 100% win, 100% tie, 60s and 50s and 40s, you have to look at how much of a lead/deficit that eighth turn gets you, and then use the probabilities from the ninth turn to get your win percentage. Every extra step adds many more possibilities, so it grows pretty quickly in the number of computations. But it took me longer to type this than it did to calculate.

[WanderingWinder]

Dominion-wise, this has a nice point, that you need to adjust to your opponent, but... there really isn't any advice here as to how to do that. 'When you're ahead, consolidate.' Well, sort of. But mostly how? What do you mean by ahead? What do you mean by consolidate? You have to know what you mean by this in order to put your advice into any kind of good use. Because a lot of people will read this and think 'oh, I have more points than him right now, playing my BM deck, I need to absolutely buy green every step of the way so that I have a big lead before his engine roars into life'. Which is basically wrong - you want to build more in this situation, generally, as the money player, with a number of exceptions of course.
'When you're behind, take risks'. Well, again, what do you mean? What does behind mean? Does it mean points, deck quality, both? How do I know when I'm behind?
Finally, there's also some kind of inherent strength of the strategies. What do I mean? To use your example, if you extend it past 9 turns to like 40 turns (more like a typical game of dominion...), then there's going to be no circumstance where before, I dunno, turn 8 (I don't know when it is exactly, without calculating), you should play anything other than option A. Because it's just stronger. In Dominion terms, two strategies have to be very very close for you to take a different road just because of seat. Now, later in the game, you need to make different plays more often, because the situations change a lot more. But the idea that I should play some strategy other than the optimal one that player 1 just took, just because I'm player 2, is generally a losing proposition.

Finally finally (yeah, I know I said finally earlier...), the one really concrete thing. Double steward. It's generally a really bad opening, unless you have a $2 you really want to buy early on (lighthouse, hamlet, crossroads). In those cases, it can be pretty darn strong. Otherwise, in your best case scenario, all you're doing is trash 4 cards all reshuffle long, and buying nothing. Which is almost never better than, say, opening steward/silver (or some other 3 or 4 you want) and then picking up your second steward on the reshuffle.
Having said that, I think people are way too scared of things like terminal collision. Thing is, if you're going to have that bad luck, like the 20% of the time worst luck, you're probably going to lose on any strategy (if you're opponent is much good). So go for the risks anyway. Indeed, I think people don't take risks nearly enough, and this is true in the P1 or P2 seat.

[-Stef-]

I know I didn't include any real practical advice. I'm just trying to point out that 'best in quarantine' and 'best in an actual game' can be very different things.

--

For a set where I actually would open steward/steward as P2 I refer to a post of Jeebus (http://forum.dominionstrategy.com/index.php?topic=2469.0) where he references this game http://councilroom.com/game?game_id=game-20120430-220116-745a70e9.html

I want to draw my deck with highways/markets ASAP, and I'm not that interested in anything else. No good draw available, so getting rid of 4 cards is really good. As P1, or against an opponent going for the alchemists, fishing/steward feels better & safer.

[Obi Wan Bonogi]

Great article Stef.  This is how I think about risk every single game.  It feels the most rewarding when it leads to successful lines which in isolation would look very curious or downright terrible.  You have enough bewildered opponents that I'm sure you can attest to that. 

Because a lot of people will read this and think 'oh, I have more points than him right now, playing my BM deck, I need to absolutely buy green every step of the way so that I have a big lead before his engine roars into life'.

The amount of people who read Stef's article and somehow conjure up this nugget should not be classified as "a lot," perhaps "a few people will read this and think ..." or "one person will read this and think..."   

The article is about risk-taking in engine builds and how to think about risk-taking strategically.  I don't see how anyone could distort his points into BM green buying gibberish.

the one really concrete thing. Double steward. It's generally a really bad opening

He gives MANY concrete examples of risky options that take into account how he thinks about risk-taking.  (Baron, Forge, Swindler, +terminals, treasure map, etc) 

Don't mind WW.  This is the best article I have come across on here.

[WanderingWinder]

Great article Stef.  This is how I think about risk every single game.  It feels the most rewarding when it leads to successful lines which in isolation would look very curious or downright terrible.  You have enough bewildered opponents that I'm sure you can attest to that. 

Because a lot of people will read this and think 'oh, I have more points than him right now, playing my BM deck, I need to absolutely buy green every step of the way so that I have a big lead before his engine roars into life'.

The amount of people who read Stef's article and somehow conjure up this nugget should not be classified as "a lot," perhaps "a few people will read this and think ..." or "one person will read this and think..."   

The article is about risk-taking in engine builds and how to think about risk-taking strategically.  I don't see how anyone could distort his points into BM green buying gibberish.

Obviously hyperbole. But not really that far off the mark. And if you think people won't read this as needing to green faster... you don't know what you're talking about.

the one really concrete thing. Double steward. It's generally a really bad opening

He gives MANY concrete examples of risky options that take into account how he thinks about risk-taking.  (Baron, Forge, Swindler, +terminals, treasure map, etc) 

No, he gives many vague examples. Baron is not a concrete example, because it's a single card, which is clearly not always the risky option. You can see this about any of the other things as well, except the extra terminals, which, while a very good tip (the best thing in the article, perhaps), is definitely not a concrete example.

Don't mind WW.  This is the best article I have come across on here.

I'm not trying to say that it's a bad article. If he can't take some criticism, he really ought not to be writing. There hasn't been a single article on here that hasn't gotten some, and there are several articles better than this one, despite this being one of the better articles. The points that -Stef- is trying to make are absolutely critical, but I think they can be made more clearly.

[WanderingWinder]

I know I didn't include any real practical advice. I'm just trying to point out that 'best in quarantine' and 'best in an actual game' can be very different things.

--

For a set where I actually would open steward/steward as P2 I refer to a post of Jeebus (http://forum.dominionstrategy.com/index.php?topic=2469.0) where he references this game http://councilroom.com/game?game_id=game-20120430-220116-745a70e9.html

I want to draw my deck with highways/markets ASAP, and I'm not that interested in anything else. No good draw available, so getting rid of 4 cards is really good. As P1, or against an opponent going for the alchemists, fishing/steward feels better & safer.

How is fishing village/steward, picking up a second steward on the reshuffle, not better in either case? Seriously, this is one of the things I'm talking about: if this is the right play as second player, I really don't know how it's not the right play as first player, too. If taking the risks are worth it, it's going to be worth it either way.

[DG]

I'd agree with the philosophy of Stef's article but I'd give some caution as to how players should let it affect their play. Unless you're really on top of your game and can see entirely how a kingdom will play out, you can lose games by pursuing a riskier strategy that doesn't work. For example, if you can see a flaw in the risky strategy then sometimes that minor flaw is actually a misjudgement of a major flaw. Perhaps your opponent isn't going to play as well as you're giving them credit for. Perhaps your opponent will draw badly enough for a safe strategy to beat them, and that happens quite often. I know that some of my worst self-inflicted losses have come through overestimating an opponent's advantage and using that as an excuse to play some desperately poor (risky) Dominion.

As a game progresses though I'd certainly agree that you should evaluate your strategies on the chance of winning rather than points per turn. The PPR is the best example of that but it goes a lot further. Always prepare your deck so that it has a chance to win on best draws.

Long games tend to reduce the chance of the first player winning through the extra turn so that's the simplest thing the second player can try as metagame. This would mean avoiding game accelerators (governor, vault, council room, embassy) but using more attacks and variable vp cards.

the one really concrete thing. Double steward. It's generally a really bad opening

Not for players who realise that the village is a good card.


edit- thanks for pointing out the error

[Obi Wan Bonogi]

 
Stef used Baron as a passing example.  I don't think Stef should be required to teach everyone how to build with Baron in order to use the concept in high-level article about risk-taking. 

Seriously, this is one of the things I'm talking about: if this is the right play as second player, I really don't know how it's not the right play as first player, too. If taking the risks are worth it, it's going to be worth it either way.

You literally don't know what he is talking about.  Regardless of any specific example the whole point is precisely "how to think about when the right play is not always the right play and if the risks are worth it."

[Grujah]

I actually liked the article quite a bit.
Stuff like this always impress me, especially the sim. Though, I expect quite a different result with larger number of turns (as a lucky C wouldn't cause such a big spike).

Long games tend to reduce the chance of the second player winning through the extra turn so that's the simplest thing the second player can try as metagame. This would mean avoiding game accelerators (governor, vault, council room, embassy) but using more attacks and variable vp cards.

You mean "the first player winning through the extra turn".

[Empathy]

Liked the article.

My preferred example for the "p2 likes variance, p1 likes certainty" is the following game:

First to 43 points wins. Each player has two available strategies:

-Safe: 43 points in 20 turns, no matter what
-Risky: 10% chance of getting 43 points in 15 turns, 90% chance of self-destructing and never getting 43 points.

p1 will always play the safe strategy, and p2 the risky one.

As for the domi example:

Governor (5/2 as p1) vs Smuggler (3/4 as p2)?

I'm not sure to what extent this proves the point, or just my failure at using the simulator, but I hope it makes sense.

[O]

This is pretty true for two player dominion but even more applicable for 4-player dominion. Your odds of winning as P4 with equally skilled opponents are so slim that the stupidly luck-dependent treasure map strategy starts to look pretty good...

Actually, in four-player dominion even P1 has more incentive to pick the more luck dependent strategy, though I'm not sure its proportionately so.

[Geronimoo]

Very interesting article, Stef, but add a few example games, please.

[ARTjoMS]

Some good strategies.

Here are some examples.

Strategy: open 2 stewards, play steward  6 turns in a row, buy gold, mining village, goons etc and dont forget to gg.

Another really good strategy: get 5-2 split on trating-post governor deck.

Open silver potion - buy familiars and make them not miss shuffle.

Want more? Ask -Stef- .

[WanderingWinder]

 
Stef used Baron as a passing example.  I don't think Stef should be required to teach everyone how to build with Baron in order to use the concept in high-level article about risk-taking. 

So first it's a concrete example, now it's a passing example? I agree, it's a passing example. I don't think he needs to explain every card in this article. But I would like some examples of 'I played this risky strategy here, here's why I did it.' If it had that, it could be one of the finest articles I've read (since it's already pretty good).

Seriously, this is one of the things I'm talking about: if this is the right play as second player, I really don't know how it's not the right play as first player, too. If taking the risks are worth it, it's going to be worth it either way.

You literally don't know what he is talking about.  Regardless of any specific example the whole point is precisely "how to think about when the right play is not always the right play and if the risks are worth it."

And my whole point is that he isn't explaining how to think quite right. Because you shouldn't be taking more risks as p2 than p1. Okay, you should, but it's a really really small difference. You'll notice the difference once every hundred games, at most. Probably once every few hundred games. Does this make it un-important? In the scheme of things, yeah. I'm fine talking about it, because it's not nothing. But I think way too much emphasis is being put here on the p1 vs p2 thing. I think prima facie that the difference in positions at the start of a game p1 to p2 is being blown out of proportion here. The larger points that he's trying to make about taking risks, particularly when behind (though I think here only later in the game, where you actually are significantly behind!) are absolutely important, and sort of getting lost with this p1 vs p2 thing.

[RisingJaguar]

Disclaimer, I didn't read the full article but tried to find a game that followed Player 1: safe - Player 2: Risky (or at least unfamiliar)

http://councilroom.com/game?game_id=game-20120102-125145-aeff94ae.html
Player 1: Wharf-BM on 5/2
Player 2: Wharf -> highways. 
I'm not sure if I just had the superior strategy or just superior luck here (I do get great draws... or at least make great use of my draws? No I definitely got good draws), but the idea is that going different as second player gave me a better chance than the mirror would.  Or at least that's how I feel.  The player going as engine can make a lot of mistakes with their purchases and its 'best' to take the BM approach with player 1. 

I hope this is along the right track of the article. 

PS. It is REALLY hard to try and find games like this, looking at logs takes so much time. 

[WanderingWinder]

I'm not sure if I just had the superior strategy or just superior luck here (I do get great draws... or at least make great use of my draws? No I definitely got good draws), but the idea is that going different as second player gave me a better chance than the mirror would.  Or at least that's how I feel.  The player going as engine can make a lot of mistakes with their purchases and its 'best' to take the BM approach with player 1. 

An excellent example of what I'm disagreeing with. Pretty sure that the highway route is just better here. But my larger point is... like, if it increases your win% as player two to play the 'riskier' strategy, then it probably also increases your chances as player 1 to play the 'riskier' strategy, at least just looking from the outset. At some point in the games, you do go to playing riskier. But you don't want to plan that from the beginning. Again, 'I can't play the mirror, because that gives me a disadvantage as p2' is wrong thinking. You ALREADY HAVE the disadvantage as player two; playing a worse strategy is generally going to compound that.

[Empathy]

 
Stef used Baron as a passing example.  I don't think Stef should be required to teach everyone how to build with Baron in order to use the concept in high-level article about risk-taking. 

So first it's a concrete example, now it's a passing example? I agree, it's a passing example. I don't think he needs to explain every card in this article. But I would like some examples of 'I played this risky strategy here, here's why I did it.' If it had that, it could be one of the finest articles I've read (since it's already pretty good).

Seriously, this is one of the things I'm talking about: if this is the right play as second player, I really don't know how it's not the right play as first player, too. If taking the risks are worth it, it's going to be worth it either way.

You literally don't know what he is talking about.  Regardless of any specific example the whole point is precisely "how to think about when the right play is not always the right play and if the risks are worth it."

And my whole point is that he isn't explaining how to think quite right. Because you shouldn't be taking more risks as p2 than p1. Okay, you should, but it's a really really small difference. You'll notice the difference once every hundred games, at most. Probably once every few hundred games. Does this make it un-important? In the scheme of things, yeah. I'm fine talking about it, because it's not nothing. But I think way too much emphasis is being put here on the p1 vs p2 thing. I think prima facie that the difference in positions at the start of a game p1 to p2 is being blown out of proportion here. The larger points that he's trying to make about taking risks, particularly when behind (though I think here only later in the game, where you actually are significantly behind!) are absolutely important, and sort of getting lost with this p1 vs p2 thing.

Example:
Consider two strategies: Geronimoo's optimized tournie bot, and a tournie/baron bot I quickly scrambled together.

p1/p2 (both with 4/3)

tournie/tournie: 58/39
tournie/baron: 57/41
baron/tournie: 55/43

Code: [Select]

<player name="Baron/tournie"
 author="Empathy"
 description="Baron trying to race an early province to block the tournie p1.">
 <type name="Generated"/>
 <type name="Province"/>
 <type name="BigMoney"/>
 <type name="Bot"/>
 <type name="TwoPlayer"/>
 <type name="UserCreated"/>
 <type name="SingleCard"/>
   <buy name="Followers"/>
   <buy name="Trusty_Steed"/>
   <buy name="Princess"/>
   <buy name="Bag_of_Gold"/>
   <buy name="Province"/>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="4.0"/>
      </condition>
   </buy>
   <buy name="Estate">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Gold"/>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="6.0"/>
      </condition>
   </buy>
   <buy name="Baron">
      <condition>
         <left type="countCardsInDeck" attribute="Baron"/>
         <operator type="smallerThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Tournament"/>
   <buy name="Silver"/>
</player>

Agreed, it's a small effect. But I think veto allows you to create (or remove) a lot of such scenarios: therefore, you should be thinking about it while vetoing (I think).

[WanderingWinder]

 
Stef used Baron as a passing example.  I don't think Stef should be required to teach everyone how to build with Baron in order to use the concept in high-level article about risk-taking. 

So first it's a concrete example, now it's a passing example? I agree, it's a passing example. I don't think he needs to explain every card in this article. But I would like some examples of 'I played this risky strategy here, here's why I did it.' If it had that, it could be one of the finest articles I've read (since it's already pretty good).

Seriously, this is one of the things I'm talking about: if this is the right play as second player, I really don't know how it's not the right play as first player, too. If taking the risks are worth it, it's going to be worth it either way.

You literally don't know what he is talking about.  Regardless of any specific example the whole point is precisely "how to think about when the right play is not always the right play and if the risks are worth it."

And my whole point is that he isn't explaining how to think quite right. Because you shouldn't be taking more risks as p2 than p1. Okay, you should, but it's a really really small difference. You'll notice the difference once every hundred games, at most. Probably once every few hundred games. Does this make it un-important? In the scheme of things, yeah. I'm fine talking about it, because it's not nothing. But I think way too much emphasis is being put here on the p1 vs p2 thing. I think prima facie that the difference in positions at the start of a game p1 to p2 is being blown out of proportion here. The larger points that he's trying to make about taking risks, particularly when behind (though I think here only later in the game, where you actually are significantly behind!) are absolutely important, and sort of getting lost with this p1 vs p2 thing.

Example:
Consider two strategies: Geronimoo's optimized tournie bot, and a tournie/baron bot I quickly scrambled together.

p1/p2 (both with 4/3)

tournie/tournie: 58/39
tournie/baron: 57/41
baron/tournie: 55/43

Code: [Select]

<player name="Baron/tournie"
 author="Empathy"
 description="Baron trying to race an early province to block the tournie p1.">
 <type name="Generated"/>
 <type name="Province"/>
 <type name="BigMoney"/>
 <type name="Bot"/>
 <type name="TwoPlayer"/>
 <type name="UserCreated"/>
 <type name="SingleCard"/>
   <buy name="Followers"/>
   <buy name="Trusty_Steed"/>
   <buy name="Princess"/>
   <buy name="Bag_of_Gold"/>
   <buy name="Province"/>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="4.0"/>
      </condition>
   </buy>
   <buy name="Estate">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Gold"/>
   <buy name="Duchy">
      <condition>
         <left type="countCardsInSupply" attribute="Province"/>
         <operator type="smallerOrEqualThan" />
         <right type="constant" attribute="6.0"/>
      </condition>
   </buy>
   <buy name="Baron">
      <condition>
         <left type="countCardsInDeck" attribute="Baron"/>
         <operator type="smallerThan" />
         <right type="constant" attribute="2.0"/>
      </condition>
   </buy>
   <buy name="Tournament"/>
   <buy name="Silver"/>
</player>

Agreed, it's a small effect. But I think veto allows you to create (or remove) a lot of such scenarios: therefore, you should be thinking about it while vetoing (I think).

I find it ironic that you come with simulator bots. The whole point of the article is that you have to adjust to your opponent, something you  can barely take a stab at with simulators, surely without spending LOTS of time at it.

[Empathy]

Simulation is good at answering *very targeted* questions. Here the question was: give an example where there is a mean-variance trade-off happening from a baron opening where p2 and p1 have differing strategies.

Any other comment?

[WanderingWinder]

Simulation is good at answering *very targeted* questions. Here the question was: give an example where there is a mean-variance trade-off happening from a baron opening where p2 and p1 have differing strategies.

Any other comment?

Well, the point is that this is a useless question. There's obviously some cards that benefit a little more than others from being p2 in pre-set simulated strategies. This fails to prove anything meaningful for the game. I should change my behaviour off of the 'optimized' bot (BTW, this tournament bot is NOT optimized) based on whether I'm p1 or p2, more importantly based on where the game stands, and also very importantly based on what you do. Sure, you can get some benefit if they put their heads down and only play some pre-written strategy like a simulator would. The whole point of this article is that you should NOT do that - you have to react to them, meaning that you should stray from your script.
The simulator does NOT answer this question. And while it can sort of be made to, it will take you way way way way too long to do that - you need to program decisions for at least tens of thousands of different circumstances, and probably meaningful differences on hundreds.

[RisingJaguar]

I'm not sure if I just had the superior strategy or just superior luck here (I do get great draws... or at least make great use of my draws? No I definitely got good draws), but the idea is that going different as second player gave me a better chance than the mirror would.  Or at least that's how I feel.  The player going as engine can make a lot of mistakes with their purchases and its 'best' to take the BM approach with player 1. 

An excellent example of what I'm disagreeing with. Pretty sure that the highway route is just better here. But my larger point is... like, if it increases your win% as player two to play the 'riskier' strategy, then it probably also increases your chances as player 1 to play the 'riskier' strategy, at least just looking from the outset. At some point in the games, you do go to playing riskier. But you don't want to plan that from the beginning. Again, 'I can't play the mirror, because that gives me a disadvantage as p2' is wrong thinking. You ALREADY HAVE the disadvantage as player two; playing a worse strategy is generally going to compound that.

This was the first example I thought of and I personally agree that highway on average will probably beat out Wharf BM.  Maybe a little different if its 4/3 as the engine player would need to obtain actual silvers which I think really hurts here but not the point.  This game was 5/2.  I would say this is a good example that $3 leads you to wharf-BM and $4 leads you to my strategy (having options with your opening).  Quasi was probably thinking this after getting $3/$6 with his draws and I suspect I was too. 

I pretty much agree with what you are saying that this was a bad example.  Although I wouldn't say Stef (or people supporting Stef) is to play the out-right worse strategy because that would be stupid.  By riskier, it is to play something  has more variance in the outcome, which often looks much worse than it is.  My intention was to show that higher-variance has its place as 2P (or by 2P, Stef is essentially saying when you are behind and 2P is the most common time you are behind).  I just sucked at it. 

I think that the higher-variance we are trying to find (myself included) is more subtle than we expect.  This could be a difference of a single purchase that turns an engine from low-variance to high-variance.  Something like double terminal vs. terminal and non-terminal (including silver).  Or a mid-to late game purchase. 

With that said, I am curious how often 2P wins mirrors in BM+X games as well as engine games.  I just assumed it was worse than the 55% - 45% of first turn advantage.

[Obi Wan Bonogi]

So first it's a concrete example, now it's a passing example? I agree, it's a passing example.
......
And my whole point is that he isn't explaining how to think quite right. Because you shouldn't be taking more risks as p2 than p1...
.. like, if it increases your win% as player two to play the 'riskier' strategy, then it probably also increases your chances as player 1 to play the 'riskier' strategy,

Yes...it's a concrete example that is mentioned in passing.  As opposed to a vague or unclear example mentioned in passing.  You argue about the most pointless shit I swear.  I'm not sure what makes you come here and try to take an article by the best dominion player we have seen(in my opinion) and turn it on it's head.   It's clear that you are thinking about the cards and their effects in relation to the state of the game very differently.  However I have no doubt that this thinking occurs in your play somewhere on some semantically pleasing level and that you could add to the topic if you wanted to...

Every choice within a line doesn't have a linear flat impact.  Each option will produce results on a spectrum of worst case to best case. His point with Borinion is that the results of these options will be polarized.   The notion of "risk" comes both from how polarized those results are and where their balance lies on this spectrum.  Sometimes your evaluation of state of the the game will lead you to conclude that the choice giving you the best chance to win is to take the option with the most polarized results aka the "riskier" (even though this new "best" chance of winning may still be well below 50%).

He perhaps overemphasizes P1 vs P2 in terms of "ahead vs behind." But he also talks about how important shuffle luck can be in terms of risk-taking and the need for constant reevaluation. 

Finally, in passing I will leave a concrete example of a cute risk-taking option.  This is card's text might as well just say "polarized results."  I don't buy it often but when I do I am almost surely WAY behind.  It is the: Scout.

[Empathy]

But my larger point is... like, if it increases your win% as player two to play the 'riskier' strategy, then it probably also increases your chances as player 1 to play the 'riskier' strategy, at least just looking from the outset.

Well, the point is that this is a useless question.

Maybe, but a question was raised, so I try to answer.

I think way too much emphasis is being put here on the p1 vs p2 thing.

There's obviously some cards that benefit a little more than others from being p2 in pre-set simulated strategies. This fails to prove anything meaningful for the game.

So, would you open tournament or baron? It would be consistent if you were not allowed to make that depend on being p1/p2.

I should change my behaviour off of the 'optimized' bot (BTW, this tournament bot is NOT optimized) based on whether I'm p1 or p2, more importantly based on where the game stands, and also very importantly based on what you do.

Agreed, which is why I only reasoned in terms of openings. The simulation is just a crude measure of the quality of the two openings with respect to p1/p2 position. I am sure that methodology has been used in the past.

Sure, you can get some benefit if they put their heads down and only play some pre-written strategy like a simulator would. The whole point of this article is that you should NOT do that - you have to react to them, meaning that you should stray from your script.
The simulator does NOT answer this question. And while it can sort of be made to, it will take you way way way way too long to do that - you need to program decisions for at least tens of thousands of different circumstances, and probably meaningful differences on hundreds.

A sub-part of the article which you vehemently disagreed with was the impact of the p1/p2 bias on openings (regardless of reactions later in the game). Which is why my two examples are basically purely reactions on opening luck: how to react to a 5/2 vs 3/4 or a p1/p2 position. Reading anything beyond that is a waste of time.

Sorry if this sounds negative, I usually really like what you do WW: but it really feels like you are the one being somewhat vague in this topic. Sorry if we are slow and force you to double-post and expand on your posts a bit.

[ftl]

So I like the article and it makes sense. But on second thought, doesn't it only apply to the very very highest levels, where you can basically be assured that your opponent will play optimally?

Like, at my level (25-30 when I played regularly), I think that either as p1 or p2, I would take whatever strategy I thought had the lowest median number of turns to half the VP (or to the mega-turn, or to the pin, or to the golden deck, or whatever). Like, in the example of the 90/10 game from post http://forum.dominionstrategy.com/index.php?topic=2711.msg43622#msg43622, if I was p2, I'd probably still take the the safe strategy, because I'd expect there's a more than 10% chance that my opponent would just mess something up, or pick a 'safe' strategy that was actually slower than the optimal safe strategy, or something.

I would guess that if I'm not a priori giving the opponent the benefit of the doubt and assuming they'll play optimally, this would still be very applicable to mid-game responses to luck, though. Once I've seen my opponent's strategy and thus can tell whether they're doing and thus respond accordingly. Or am I underestimating p1/p2 disparity?

By the way, I really appreciate Obi Wan and Stef's contributions here. You guys think (or, at least, talk) about this game quite differently than some of the other top players here, and it's very interesting and informative to hear from you guys.

[def]

I'm not sure what makes you come here and try to take an article by the best dominion player we have seen(in my opinion) and turn it on it's head.   

Don't care about who is wrong or right or both in the substantial discussion, but it's not like that solely the fact one of the best players wrote an article means you can't argue about it all. Don't see why you're so absolute here.

[WanderingWinder]

I'm not sure if I just had the superior strategy or just superior luck here (I do get great draws... or at least make great use of my draws? No I definitely got good draws), but the idea is that going different as second player gave me a better chance than the mirror would.  Or at least that's how I feel.  The player going as engine can make a lot of mistakes with their purchases and its 'best' to take the BM approach with player 1. 

An excellent example of what I'm disagreeing with. Pretty sure that the highway route is just better here. But my larger point is... like, if it increases your win% as player two to play the 'riskier' strategy, then it probably also increases your chances as player 1 to play the 'riskier' strategy, at least just looking from the outset. At some point in the games, you do go to playing riskier. But you don't want to plan that from the beginning. Again, 'I can't play the mirror, because that gives me a disadvantage as p2' is wrong thinking. You ALREADY HAVE the disadvantage as player two; playing a worse strategy is generally going to compound that.

This was the first example I thought of and I personally agree that highway on average will probably beat out Wharf BM.  Maybe a little different if its 4/3 as the engine player would need to obtain actual silvers which I think really hurts here but not the point.  This game was 5/2.  I would say this is a good example that $3 leads you to wharf-BM and $4 leads you to my strategy (having options with your opening).  Quasi was probably thinking this after getting $3/$6 with his draws and I suspect I was too. 

I pretty much agree with what you are saying that this was a bad example.  Although I wouldn't say Stef (or people supporting Stef) is to play the out-right worse strategy because that would be stupid.  By riskier, it is to play something  has more variance in the outcome, which often looks much worse than it is.  My intention was to show that higher-variance has its place as 2P (or by 2P, Stef is essentially saying when you are behind and 2P is the most common time you are behind).  I just sucked at it. 

I think that the higher-variance we are trying to find (myself included) is more subtle than we expect.  This could be a difference of a single purchase that turns an engine from low-variance to high-variance.  Something like double terminal vs. terminal and non-terminal (including silver).  Or a mid-to late game purchase. 

With that said, I am curious how often 2P wins mirrors in BM+X games as well as engine games.  I just assumed it was worse than the 55% - 45% of first turn advantage.

Bolding mine. This (the bolded thing) is my biggest point. Because it's absolutely when you're behind that it matters (defining behind is another tricky thing - one which he doesn't do here, but which I cannot do quickly and easily either). I think people get way oversold on how much being second player makes you behind (similar to the way people tend to overrate opening splits). I doubt it's more than 60% of the time. And so I think it's a fantastic point that -Stef- is trying to make, I'm just worried that people will get it too tied to the 1p/2p distinction, rather than the behind/ahead. Apart from being worried that people will take the gambit too far - i.e. something like switching playing treasure map with no enablers a few turns into a dry, BM/smithy matchup because you've hit 2/5/2 or something. Again, that's a bit of an overboard reach, but I think that it's quite possible to take too big a risk. I guess that's something people have to figure out on their own, though.

Edit: Oh, and on your last point, it's hard to get real perfect data on that, but A) depends on the matchup, always, B) best I can tell, which is pretty much more your BM mirrors, 1p advantage is indeed in the ballpark of 55/45.

[WanderingWinder]

So first it's a concrete example, now it's a passing example? I agree, it's a passing example.
......
And my whole point is that he isn't explaining how to think quite right. Because you shouldn't be taking more risks as p2 than p1...
.. like, if it increases your win% as player two to play the 'riskier' strategy, then it probably also increases your chances as player 1 to play the 'riskier' strategy,

Yes...it's a concrete example that is mentioned in passing.  As opposed to a vague or unclear example mentioned in passing.  You argue about the most pointless shit I swear.  I'm not sure what makes you come here and try to take an article by the best dominion player we have seen(in my opinion) and turn it on it's head.   It's clear that you are thinking about the cards and their effects in relation to the state of the game very differently.  However I have no doubt that this thinking occurs in your play somewhere on some semantically pleasing level and that you could add to the topic if you wanted to...

I'm not trying to turn the article on its head. I don't know why it matters who is saying making the argument - I'm only trying to argue the substance, not the person. Seriously, why does that matter? I really don't appreciate you talking so degradingly towards me. Thanks. Oh, and the second part of your paragraph (starting from "It's clear...") - I have no idea what you're trying to say here. Could you re-phrase it? Because I think you're trying to make a point, but in all honesty, I'm not sure what it is. I.e. I don't know what I'm thinking about the cards differently than (than you, than most people, than -Stef-?) I don't know how that's clear at all, because I'm not thinking about the cards when I'm talking here really at all. And your last sentence - I really don't get what it is you're trying to say, though I take it that it's in someway cryptic or insulting based on the ellipsis. Anyway, feel free to clarify for me, the bozo who doesn't get it.

Every choice within a line doesn't have a linear flat impact.

You have already lost me. What line are we talking about here? And how is something within a line not linear? I'm not trying to be pedantic here - I really, truly don't know what it is you're getting at.

Each option will produce results on a spectrum of worst case to best case. His point with Borinion is that the results of these options will be polarized.

What do you mean by polarized? Like, there's two directions, good and bad, and the results point to one or the other? Is that what you mean? Because that's true, you can say it much more simply, and so obvious it doesn't need to be said. So I'm guessing you're getting at something more. But I have no idea what that means. To me, electromagnetic polarization, then magnets, then geography, then bipolar disorder, and none of these things seems to make real clear sense in terms of the thread. So again, maybe I'm just too dumb to figure out your rhetorical style, or you're using vocabulary that's above me, but I don't know what you're trying to say here. Which is why, actually, I argue semantics so much. Because a huge part of the disagreements that go on are simply based on not understanding what each other mean.

The notion of "risk" comes both from how polarized those results are and where their balance lies on this spectrum.  Sometimes your evaluation of state of the the game will lead you to conclude that the choice giving you the best chance to win is to take the option with the most polarized results aka the "riskier" (even though this new "best" chance of winning may still be well below 50%).

Okay, so this makes me think you mean 'riskier' i.e. you stand to lose more and at the same time, stand to gain more, in a more 'polarized' situation. Makes sense I guess. Still, I wish you'd say that, rather than me having to scratch my head to come up with it. Then again, maybe that's the simplest way to put it in your head. Anyway, misunderstandings, again. Semantics are important, again. Please confirm that I understand that right, will you?

He perhaps overemphasizes P1 vs P2 in terms of "ahead vs behind." But he also talks about how important shuffle luck can be in terms of risk-taking and the need for constant reevaluation. 

Totally agree. All I was saying is that I think he does what you're talking about in the first sentence a bit. Not that he's saying anything particularly wrong (once more, it is a good article), just that I think he's emphasizing different things more than he needs to, and other things (not only shuffle luck here), less than he needs to.

Finally, in passing I will leave a concrete example of a cute risk-taking option.  This is card's text might as well just say "polarized results."  I don't buy it often but when I do I am almost surely WAY behind.  It is the: Scout.

Perhaps I've been using this term wrong as well. When I say a concrete example, I mean a specific example. Scout in general is not a specific example. I want you to tell me when to get scout. Because surely it's not only 'when I'm way behind'. Because there's lots of times where you're behind and have the option of taking scout, but don't. Also, give me some examples of when you should ever ever take scout because you're behind. There probably are some very very weird fringe cases, but I don't really see it. Which isn't to say it's not there. It's to say I don't see it. So I'd appreciate it very much if you'd show me.

[WanderingWinder]

But my larger point is... like, if it increases your win% as player two to play the 'riskier' strategy, then it probably also increases your chances as player 1 to play the 'riskier' strategy, at least just looking from the outset.

Well, the point is that this is a useless question.

Maybe, but a question was raised, so I try to answer.

I think way too much emphasis is being put here on the p1 vs p2 thing.

There's obviously some cards that benefit a little more than others from being p2 in pre-set simulated strategies. This fails to prove anything meaningful for the game.

So, would you open tournament or baron? It would be consistent if you were not allowed to make that depend on being p1/p2.

I should change my behaviour off of the 'optimized' bot (BTW, this tournament bot is NOT optimized) based on whether I'm p1 or p2, more importantly based on where the game stands, and also very importantly based on what you do.

Agreed, which is why I only reasoned in terms of openings. The simulation is just a crude measure of the quality of the two openings with respect to p1/p2 position. I am sure that methodology has been used in the past.

Sure, you can get some benefit if they put their heads down and only play some pre-written strategy like a simulator would. The whole point of this article is that you should NOT do that - you have to react to them, meaning that you should stray from your script.
The simulator does NOT answer this question. And while it can sort of be made to, it will take you way way way way too long to do that - you need to program decisions for at least tens of thousands of different circumstances, and probably meaningful differences on hundreds.

A sub-part of the article which you vehemently disagreed with was the impact of the p1/p2 bias on openings (regardless of reactions later in the game). Which is why my two examples are basically purely reactions on opening luck: how to react to a 5/2 vs 3/4 or a p1/p2 position. Reading anything beyond that is a waste of time.

Sorry if this sounds negative, I usually really like what you do WW: but it really feels like you are the one being somewhat vague in this topic. Sorry if we are slow and force you to double-post and expand on your posts a bit.

Ah, but the thing is, I can't tell you what I'd open, because I don't know the rest of the board. I'm saying it's very dependant. I'm also saying that because the game-plans will change form the 'optimized' single-card strategies, using these two simulations definitely does NOT tell you what to open with, either as p1 OR p2. To truly figure it out, you'd need to go through a series of improvements, back-and-forth, until you get to an equilibrium, where any change in the play you make will not increase your chances, given that the other player can then make a corresponding adjustment which will make the series of plays work out better for them than for you, as compared to your other options. This is an extraordinarily complicated process, and one that you basically cannot perfectly do. So you have to go on intuition, experience, ingenuity, and instinct.
My point is that the question your simulations answer is an EXTREMELY narrow one: whether, given the p1/p2 split, and what your opponent is playing if you're p2, what should you play, also given that the only options are to play sticking exactly to the tournament script or exactly to the baron script. But this last assumption is one that you never actually face in a game, which makes your simulation... not so useful in a real game. If you really want to come up with a better strategy, you at the least have to take both of the cards into account in both scripts - what your opponent is doing, how much of a lead they have, what's they're deck look like, do I need to block their tournaments, what's their chances at getting a big turn, etc. etc. Only, don't waste your time, because in the vast majority of cases, a third card is going to be really important as well, not just these two. And then all your work is for naught, because you have to step back and recalculate.

So that's why I don't have any concrete examples here - I just don't have any off the top of my head. I guess I should apologize a bit, because it's slightly hypocritical, given that many of the articles I write don't have so many specific examples. But I do think that an article like this one, on such an important concept, a general concept, a high-level concept, is greatly benefitted by having specific examples to point to in it, where you can say, "here's this line of thinking in action". I don't have any of those off hand. I have wanted to write an article on a similar subject to this, but haven't collected the resources I need yet. I will try to get that done for you. But it will take some time.

[WanderingWinder]

So I like the article and it makes sense. But on second thought, doesn't it only apply to the very very highest levels, where you can basically be assured that your opponent will play optimally?

Not at all! In fact, even at the highest levels of play we have, nobody is really all that close to playing optimally all that often. I know Bonogi agrees with me there, and I would guess -Stef- does too, even though he may well be the person who is closest? But moreover, I actually think this stuff is in some ways MORE important if your opponent is not playing optimally, because the point is that you need to make adjustments to take advantage of that, and give yourself better control over the game.

Like, at my level (25-30 when I played regularly), I think that either as p1 or p2, I would take whatever strategy I thought had the lowest median number of turns to half the VP (or to the mega-turn, or to the pin, or to the golden deck, or whatever). Like, in the example of the 90/10 game from post http://forum.dominionstrategy.com/index.php?topic=2711.msg43622#msg43622, if I was p2, I'd probably still take the the safe strategy, because I'd expect there's a more than 10% chance that my opponent would just mess something up, or pick a 'safe' strategy that was actually slower than the optimal safe strategy, or something.

I would guess that if I'm not a priori giving the opponent the benefit of the doubt and assuming they'll play optimally, this would still be very applicable to mid-game responses to luck, though. Once I've seen my opponent's strategy and thus can tell whether they're doing and thus respond accordingly. Or am I underestimating p1/p2 disparity?

I'm not entirely sure what you're getting at here. But if your opponent deviates from what you expect, you need to adapt and respond accordingly, absolutely. Not really sure what that has to do with p1/p2 disparity. The only reason that that's important is that p1 has the chance at an extra turn which USUALLY gives him some initiative to be ahead (a majority of the time, but not an overwhelming majority). But what you make your adjustments TO is whether you're ahead or behind (and how much you're ahead behind, what's in your deck, what options you have open, etc. etc.), not actually the player seat. The player seat is only important insofar as it will confer an advantage. Well, ok, and player two wins on ties if he takes fewer turns, which comes up sometimes.

[WanderingWinder]

My preferred example for the "p2 likes variance, p1 likes certainty" is the following game:

First to 43 points wins. Each player has two available strategies:

-Safe: 43 points in 20 turns, no matter what
-Risky: 10% chance of getting 43 points in 15 turns, 90% chance of self-destructing and never getting 43 points.

p1 will always play the safe strategy, and p2 the risky one.

I guess I should respond to this, since I meant to earlier but apparently forgot. Shouldn't p2 also always take the safe strategy, because it ensures him the tie? 100% ties is better than 10% win and 90% loss, no? But in any case, this game has very little resemblance to Dominion.

[O]

My preferred example for the "p2 likes variance, p1 likes certainty" is the following game:

First to 43 points wins. Each player has two available strategies:

-Safe: 43 points in 20 turns, no matter what
-Risky: 10% chance of getting 43 points in 15 turns, 90% chance of self-destructing and never getting 43 points.

p1 will always play the safe strategy, and p2 the risky one.

I guess I should respond to this, since I meant to earlier but apparently forgot. Shouldn't p2 also always take the safe strategy, because it ensures him the tie? 100% ties is better than 10% win and 90% loss, no? But in any case, this game has very little resemblance to Dominion.

"first to" means P1 wins, much like buying out provinces (its not in the normal 1-1-1-1-1-1-1-1 that BM games give us, but for 3-3-2 or something that engines can give us..

And it does have a resemblance. People have posted game after game that demonstrates that the second player often should take a riskier strategy. The mathematics is sound in the basic game systems, and the logs demonstrate that those concepts apply to dominion too. (OK, yes, anecedotal evidence does not prove statistical concepts. But you also rejected simulation as a viable method, so you seem to be unreasonable picky here.)

[heron]

My preferred example for the "p2 likes variance, p1 likes certainty" is the following game:

First to 43 points wins. Each player has two available strategies:

-Safe: 43 points in 20 turns, no matter what
-Risky: 10% chance of getting 43 points in 15 turns, 90% chance of self-destructing and never getting 43 points.

p1 will always play the safe strategy, and p2 the risky one.

I guess I should respond to this, since I meant to earlier but apparently forgot. Shouldn't p2 also always take the safe strategy, because it ensures him the tie? 100% ties is better than 10% win and 90% loss, no? But in any case, this game has very little resemblance to Dominion.

"first to" means P1 wins, much like buying out provinces (its not in the normal 1-1-1-1-1-1-1-1 that BM games give us, but for 3-3-2 or something that engines can give us..

And it does have a resemblance. People have posted game after game that demonstrates that the second player often should take a riskier strategy. The mathematics is sound in the basic game systems, and the logs demonstrate that those concepts apply to dominion too. (OK, yes, anecedotal evidence does not prove statistical concepts. But you also rejected simulation as a viable method, so you seem to be unreasonable picky here.)

I'm a bit confused by this example.
If when P1 and P2 both play safe, P1 wins, does that mean that if both P1 and P2 play risky and fail, P1 wins?

If so, I think the P1 should always play risky too, given that the players' strategies are not known to each other.
For example, if P1 plays safe and P2 plays risky, obviously the result is P1 90%, P2 10%.
However, if both players play risky, and P1 wins if both fail, the result is P1 91% P2 9%.

This is why I'm leaning toward WanderingWinder's point of view on some of this—most often, if P2 can increase his win chance by playing riskily, P1 can do the same.

Actually, this is more of a psychological game: P1 can think, well, P2 knows I can force a guaranteed win if I play safe, so he'll play risky, and therefore I will too.
P2 might think: Well, P1 might think [the above thought process], so I'll play safe.

Kind of a rock paper scissors kind of game.

[-Stef-]

Dear readers of this post, I think it's time for the group hug 

It was/is my first article here, I liked writing it, and I like the fact that people respond to it.
I think I can handle a bit of criticism too. Nevertheless, with a little more respect from everyone to everyone, I think we could get a better argument with more solid conclusions, advice and/or revisions on the original post.

So far I understand: actual game logs would be an important addition.
Also: ahead/behind is more important then P1/P2, maybe point that out more clearly.
I'd also like to add a game where my P1 and P2 openings actually are different. Still not sure if the steward/steward versus fishing/steward example can fill this role. Will work it out though.

And finally: I make tons of mistakes. Not a single game without any, and complicated engines probably at least one each turn. Dominion is way to complex to even think about 'playing perfect'. Tic tac toe I can play perfect, but that's about the most complicated game that statement applies to.

[WanderingWinder]

My preferred example for the "p2 likes variance, p1 likes certainty" is the following game:

First to 43 points wins. Each player has two available strategies:

-Safe: 43 points in 20 turns, no matter what
-Risky: 10% chance of getting 43 points in 15 turns, 90% chance of self-destructing and never getting 43 points.

p1 will always play the safe strategy, and p2 the risky one.

I guess I should respond to this, since I meant to earlier but apparently forgot. Shouldn't p2 also always take the safe strategy, because it ensures him the tie? 100% ties is better than 10% win and 90% loss, no? But in any case, this game has very little resemblance to Dominion.

"first to" means P1 wins, much like buying out provinces (its not in the normal 1-1-1-1-1-1-1-1 that BM games give us, but for 3-3-2 or something that engines can give us..

Ok. Makes sense. Thanks.

And it does have a resemblance.

Sure, it has SOME resemblance. Hey, poker has some resemblances, too, and in poker, the later player gets more information, which gives him (or her) a serious advantage. And that's also true in dominion. But it doesn't mean there's a second-player advantage in dominion...

People have posted game after game that demonstrates that the second player often should take a riskier strategy.

Other games, not dominion. And those games have really big and significant differences. You don't have such clear-cut options in dominion as you do in Stef's game. More importantly, p1 is not guaranteed an extra turn in dominion. More importantly still, dominion is much longer. Most importantly, he's not applying the full logical tree. Yes, you do take more risks when behind. Who's arguing that? The questions are how much does p1/p2 have to do with this, and to what extent do you play the riskier options? And these games have enough differences that you can't just say 'oh it's the same.' Empathy's game is much much different, because in VIRTUALLY no dominion game can you guarantee ending exactly on a timetable (big exception is something like the golden deck, though even here there are some adjustments to make...), and there's usually not such a big gambit option either. Things are much more subtle than that.

The mathematics is sound in the basic game systems,

Yeah, but incomplete. Actually I think I've posted the most mathematics here.

and the logs demonstrate that those concepts apply to dominion too.

What logs? Where do you see logs that show this? Where? I don't see any. Give me some logs. What I want is logs!

(OK, yes, anecedotal evidence does not prove statistical concepts. But you also rejected simulation as a viable method, so you seem to be unreasonable picky here.)

I'm not being unreasonably picky. I'm quite reaonably saying that there's no quick way to show what is right in dominion. It's a wonderful, terribly complex game, with lots and lots and lots of different paths to take, which makes it extremely difficult and complicated to come out at the right equilibrium. Indeed, if we narrow it down to one kingdom of ten cards, we can probably model out all the reasonable choices someone can make, and find the 'best way' of playing that kingdom, all the choices you should take, how to react to anything your opponent might do. It's like tic-tac-toe, except that instead of a 3x3 board with only a handful of choices, you have dozens of choices, or more, every turn, for a non-fixed number of turns, which can quite easily reach into the 30s. So let's conservatively say that there's 5 reasonable choices per turn (I submit that there are almost always more), and say 20 turns you need to prepare for (again, there's always more than this). Even not really taking into account your opponent's turns, you're almost at 100 trillion possibilities. On just a normal-ish board. So it's pretty complicated, you should not be trying to solve it. So given that, what I want is examples; give me a game, where you used this kind of thinking, and where that was definitely what you needed to do. And I'm sure they're out there, because it really is an important consideration to take in mind. On the other hand, choosing the strongest strategy, before the lead is that big and risk becomes such an important factor, is much more often going to be a bigger deal, at least until you get toward the endgame. But most of all, I want logical reasoning behind 'okay, in this particular situation, here's why you want to do this. There's reasons X, Y, and Z. You're in a bad spot, but you need to get lucky, so you play to let yourself. If you don't, you were probably going to lose anyway. If you do, then great, you stole a game'. Actually having said this, the youtube video at the bottom of my signature is a great example of this. I actually could have done it better, I'm sure, but the point is I kept picking up engine components, particularly things which would let me get more plays of monument in. I was so far behind, I knew the only chance I had would be for the game to somehow go long enough to let me play monument a thousand times to make up enough point difference. I played to get lucky. I could have taken more green earlier, and it would have totally killed my chances, because when he stalls out, I have no way of taking advantage. I'm sure that I lose less in those cases, but it doesn't matter how much you lose by, only if you have more than the other guy. So I played to get lucky, and I did get lucky, so I was able to steal a win. There's an example for you.

[Empathy]

I remembered that I had two mild examples in my article a few weeks ago. Tell me if these fit the bill!

First a game where my opponent opens pirate ship/Xroad, with only black market and tribute as virtual dollars. I counter by loading up on a lot of variance: BM/TM. My idea is that he will trash my coppers, and I'm going to grab a few Xroads and menageries, making a TM collision somewhat feasible (menagerie mitigates the pirate ship attack and the TM signleton, as well as combos with BM). In the mean time, I need enough money to buy the second TM, and there were a few interesting things in the BM pool. I get lucky with the BM, grabbing embargo (on pirate ship) and remake early on. My TM gamble was not successful at first, but it gives me a big money boost on T12 once I have more engine parts. Note that I always only have two TM! It could be argued that I was lucky, or that I just countered his opening, but I thought Xroad/pirate ship was pretty scary, and probably a good opening for p1.

The second game is one of my few wins against WW. It's one of those games I dread to play against him, because it contains all kinds of BMish cards I dislike and he knows how to play well. So I decide to spice up the opening by getting the NB, successfully stealing WW's first silver, with the afterthought that I can always upgrade NB later on, and that it's actually a decent terminal in the BMish start of the game. This initial gamble gives me enough time to build an engine, forcing WW to do the same. I definitely got lucky on this one. Tell me if I'm wrong WW, but if you had opened CY, I would have been toast playing BM!

PS: I also agree that my game is way oversimplified. But it does make the idea come across as to why variance can help p2 more than p1. The question is, as WW points out, whether this effect is important enough to be really taken into account. I have the impression the answer is yes, but that is my opinion. Truth be told, I just like thinking about these concepts during the veto to 'train' myself to think in terms of mean/variance tradeoff. I think most of us agree that it definitely is an important concept in the midgame when there is a clearer leader and lagger.

[RisingJaguar]

I'm not sure if I just had the superior strategy or just superior luck here (I do get great draws... or at least make great use of my draws? No I definitely got good draws), but the idea is that going different as second player gave me a better chance than the mirror would.  Or at least that's how I feel.  The player going as engine can make a lot of mistakes with their purchases and its 'best' to take the BM approach with player 1. 

An excellent example of what I'm disagreeing with. Pretty sure that the highway route is just better here. But my larger point is... like, if it increases your win% as player two to play the 'riskier' strategy, then it probably also increases your chances as player 1 to play the 'riskier' strategy, at least just looking from the outset. At some point in the games, you do go to playing riskier. But you don't want to plan that from the beginning. Again, 'I can't play the mirror, because that gives me a disadvantage as p2' is wrong thinking. You ALREADY HAVE the disadvantage as player two; playing a worse strategy is generally going to compound that.

This was the first example I thought of and I personally agree that highway on average will probably beat out Wharf BM.  Maybe a little different if its 4/3 as the engine player would need to obtain actual silvers which I think really hurts here but not the point.  This game was 5/2.  I would say this is a good example that $3 leads you to wharf-BM and $4 leads you to my strategy (having options with your opening).  Quasi was probably thinking this after getting $3/$6 with his draws and I suspect I was too. 

I pretty much agree with what you are saying that this was a bad example.  Although I wouldn't say Stef (or people supporting Stef) is to play the out-right worse strategy because that would be stupid.  By riskier, it is to play something  has more variance in the outcome, which often looks much worse than it is.  My intention was to show that higher-variance has its place as 2P (or by 2P, Stef is essentially saying when you are behind and 2P is the most common time you are behind).  I just sucked at it. 

I think that the higher-variance we are trying to find (myself included) is more subtle than we expect.  This could be a difference of a single purchase that turns an engine from low-variance to high-variance.  Something like double terminal vs. terminal and non-terminal (including silver).  Or a mid-to late game purchase. 

With that said, I am curious how often 2P wins mirrors in BM+X games as well as engine games.  I just assumed it was worse than the 55% - 45% of first turn advantage.

Bolding mine. This (the bolded thing) is my biggest point. Because it's absolutely when you're behind that it matters (defining behind is another tricky thing - one which he doesn't do here, but which I cannot do quickly and easily either). I think people get way oversold on how much being second player makes you behind (similar to the way people tend to overrate opening splits). I doubt it's more than 60% of the time. And so I think it's a fantastic point that -Stef- is trying to make, I'm just worried that people will get it too tied to the 1p/2p distinction, rather than the behind/ahead. Apart from being worried that people will take the gambit too far - i.e. something like switching playing treasure map with no enablers a few turns into a dry, BM/smithy matchup because you've hit 2/5/2 or something. Again, that's a bit of an overboard reach, but I think that it's quite possible to take too big a risk. I guess that's something people have to figure out on their own, though.

Edit: Oh, and on your last point, it's hard to get real perfect data on that, but A) depends on the matchup, always, B) best I can tell, which is pretty much more your BM mirrors, 1p advantage is indeed in the ballpark of 55/45.

Just to conclude our lovely conversation. 

I agree with your biggest point that's for sure.  Taking risks are when you are behind (especially when you are really behind).  I have a feeling that using 2P is just the easiest way to relate to every player since it is so common.  It also is a great hook, it definitely got me here.  He may have overplayed it, but I'm definitely glad it is in there. 

The idea that in practice, people may be over-extend themselves is a risk but definitely a risk worth taking (eh?).  One, I think people will learn best from their own mistakes and see trying to implement their strategy themselves.  Two, Over-extending themselves is a much better scenario than not being aware of this at all.  People will find their sweet spot whatever it may be. 

This is very hypothetical/abstract in terms of thinking and showing it all in a game log.  I would suspect that one of the many videos would hope to capture someone's exact train of thought and adjusting to a riskier strategy (or even just a riskier purchase).  Regardless, I think this is an incredibly important topic, which we all agree upon. 

Stef, this kind of discussion after your article is a GOOD thing.  Please don't let this discourage you. 

[Obi Wan Bonogi]

I really don't appreciate you talking so degradingly towards me. Thanks.

In regards to the degrading tone Im taking with you this is not coming out of nowhere...you have called me out TWICE(and I haven't been here long) accusing me of "not knowing what I'm talking about" on subjects I felt very confident in knowing what I was talking about.  No one appreciates that. 

I was compelled to Stef's defense because you jumped on the article with half a dozen sniping questions and the accusation that the lack of clarity would cause a lot of readers to conclude "'oh, I have more points than him right now, playing my BM deck, I need to absolutely buy green every step of the way so that I have a big lead before his engine roars into life" which I still think is a JOKE and terribly INSULTING to the quality of the article.

You go on to say "If he can't take some criticism, he really ought not to be writing." Where does that even come from man?

You have managed to spit out half the replies in this thread all while demonstrating very little understanding of the concepts presented. 

To clarify "line" because it might be confusing in that usage: a Line is the path/build/game of a single player.

Polarized results really should make some sense by now given all the context and examples provided. 


 

[cherdano]

Really nice article!

Here is a game where I chose a swingy strategy on purpose:

http://councilroom.com/game?game_id=game-20120528-080750-ea2e2993.html

Cards in supply (key cards in bold):
Black Market, Embargo, Embassy, Highway, Laboratory, Pearl Diver, Potion, Scout, Spice Merchant, Treasure Map, and Vineyard

I am second player and my opponents opens 5/2 and chooses Embassy. An Embassy is not an elite opening, but still a pretty strong one - it is one of the fastest BM type cards, I would guess it's better than Envoy/Silver.
I draw 4/3 and it seems I would be behind on any strategy close to BM, and Labs/Highways are not a strong combination without a reliable +buy. So I decide to open Spice Merchant/Black Market and hope to get lucky underground. In the actual game this works fantastically as I get a tournament on turn 5, and a Hunting party on turn 10. Winning Followers on turn 11 essentially decides the game.

So it worked, and it makes for a good story for this thread. But was my logic correct, and does the story actually hold up? That is still not clear to me. For example, yes I am behind, but Embassy is also quite swingy by itself (for example, there is a huge difference between drawing it in turn 3 or turn 4), so just going for Embassy myself is already a high-variance strategy. Also, maybe my strategy was just good by itself? Even if I don't immediately get lucky via BM, I can hope for some synergy between Labs/Highways with the unreliable +buy from Spice Merchant, the virtual +buy from Black Market, and maybe some actual +buys found in the Black Market.

As usual, it's more complicated in an actual game than in theory...

[DG]

So it worked, and it makes for a good story for this thread. But was my logic correct, and does the story actually hold up?

With highway, vineyards, and embargo there's easily enough in that kingdom to make a black market deck work. I'd suggest that deck evaluation comes first in every game, and only once you're sure of the decks can you start making risk judgements on competitive strategies.


edit- took out the unintentional insult.

[cherdano]

Unfortunately I don't recall the BM deck, but from what I remember it was fairly dull overall. In particulary, hardly any plus buys.

Embargos cut both ways - he is already picking one up on turn 2, and can easily embargo the highways/labs/vineyards.
So I don't think it's as clear-cut as you make it.

[Geronimoo]

The Torturer/Village (any Village) engine might be a good place to start if you're looking for real life example games to illustrate this article. The safe strategy is probably to make sure you have an equal or greater number of Villages as Torturers (this means buying a Village over Torturer at times). The risky strategy is to get the minimum number of Villages (always buy the Torturer if you hit $5 and hope to hit $3 for the Village at opportune times).

[philosophyguy]

In terms of example games, doesn't WW have a video recently where he used a KC/Wharf/Monument comeback after he fell hopelessly behind on green cards? Generally on that board the higher percentage play is more towards money/buy green than "hope to King 3 Monuments every turn," but when you're behind…

[WanderingWinder]

I really don't appreciate you talking so degradingly towards me. Thanks.

In regards to the degrading tone Im taking with you this is not coming out of nowhere...you have called me out TWICE(and I haven't been here long) accusing me of "not knowing what I'm talking about" on subjects I felt very confident in knowing what I was talking about.  No one appreciates that. 

I say what I believe. Now, probably I should have phrased it better - I don't mean it as a slight on you, to say that you never know what you're talking about or anything. What I mean there is "You're absolutely wrong on this issue, and you haven't presented any evidence to back yourself up." I stand by THAT, but I sincerely apologize if you took any offense. I can see how you would, and I'll try to phrase things better in the future. But you also need to understand that just because I think you're saying something that's wrong doesn't mean I think anything less of you. I say wrong stuff all the time. I hope this helps square things away.

I was compelled to Stef's defense because you jumped on the article with half a dozen sniping questions and the accusation that the lack of clarity would cause a lot of readers to conclude "'oh, I have more points than him right now, playing my BM deck, I need to absolutely buy green every step of the way so that I have a big lead before his engine roars into life" which I still think is a JOKE and terribly INSULTING to the quality of the article.

I DO think the lack of clarity will lead readers in the wrong direction. I think we've seen that throughout this thread. It's fine if you think that's a ridiculous assessment, but I disagree. Obviously my words there are a little hyperbolic, as I've said before now, but not so much so. I really think people don't get it, mainly because I've SEEN them suggest stuff like that before. I've read every post on these forums, so I'm going to know that. I presume you haven't, so you won't.

You go on to say "If he can't take some criticism, he really ought not to be writing." Where does that even come from man?

It comes from you implying that I shouldn't be criticizing his article. I don't see any reason not to state where I disagree with him. This is what criticism is. It starts a conversation, and ultimately it leads to a better understanding for all. But you seem to be saying "Well, Stef is really good" (which is true) "and the article is good" (also true) "So we shouldn't say anything about how to improve it". It's only the last part which is wrong here, because the article isn't perfect, not that you'd have any reason to expect it to be, and so other people can help you work toward improvement. You need to be able to handle that criticism, and work toward improving the stuff, or ultimately your work (in this case articles) will have a hard time reaching their full potential.

You have managed to spit out half the replies in this thread all while demonstrating very little understanding of the concepts presented. 

Okay... care to explain the relevance here? I don't think you've demonstrated much of the relevance either - much less than I have. But your purpose isn't to say "oh hahaha, I'm clever, I understand what he's saying, see my mastery of the concept." So I wouldn't expect you to have demonstrated this. Neither is it my purpose, so I don't understand why you'd have expected me to do this either.

To clarify "line" because it might be confusing in that usage: a Line is the path/build/game of a single player.

Polarized results really should make some sense by now given all the context and examples provided. 


 

Thanks for the explanation of line. That was not at ALL in my mind when I read your paragraph, any of the several times I read it. Now it makes much more sense.
On the polarized thing... well, maybe you think it should be clear by now, but it's obvious you thought that before (well, either that or you just wanted to say something cryptic and confusing, but I don't peg you for the kind of person to do that). I'm TELLING you it's not clear to me, so please, explain it.

Edited to fix a quote blocking issue.

[WanderingWinder]

I remembered that I had two mild examples in my article a few weeks ago. Tell me if these fit the bill!

First a game where my opponent opens pirate ship/Xroad, with only black market and tribute as virtual dollars. I counter by loading up on a lot of variance: BM/TM. My idea is that he will trash my coppers, and I'm going to grab a few Xroads and menageries, making a TM collision somewhat feasible (menagerie mitigates the pirate ship attack and the TM signleton, as well as combos with BM). In the mean time, I need enough money to buy the second TM, and there were a few interesting things in the BM pool. I get lucky with the BM, grabbing embargo (on pirate ship) and remake early on. My TM gamble was not successful at first, but it gives me a big money boost on T12 once I have more engine parts. Note that I always only have two TM! It could be argued that I was lucky, or that I just countered his opening, but I thought Xroad/pirate ship was pretty scary, and probably a good opening for p1.

The second game is one of my few wins against WW. It's one of those games I dread to play against him, because it contains all kinds of BMish cards I dislike and he knows how to play well. So I decide to spice up the opening by getting the NB, successfully stealing WW's first silver, with the afterthought that I can always upgrade NB later on, and that it's actually a decent terminal in the BMish start of the game. This initial gamble gives me enough time to build an engine, forcing WW to do the same. I definitely got lucky on this one. Tell me if I'm wrong WW, but if you had opened CY, I would have been toast playing BM!

PS: I also agree that my game is way oversimplified. But it does make the idea come across as to why variance can help p2 more than p1. The question is, as WW points out, whether this effect is important enough to be really taken into account. I have the impression the answer is yes, but that is my opinion. Truth be told, I just like thinking about these concepts during the veto to 'train' myself to think in terms of mean/variance tradeoff. I think most of us agree that it definitely is an important concept in the midgame when there is a clearer leader and lagger.

This is exactly the kind of stuff I'm looking to see. Unfortunately, what I'm seeing here is not so much 'risky play paying off' as 'better play paying off'. First one, I think a BlackMarket-heavy strategy should win. There's basically no reason to to go for the BM, and depending on what's in it, it can be very very good. You can actually get quite the engine going here, and I don't expect crossroads+ship to really be that strong. Ship itself is incredibly weak, to the point of being maybe barely barely better than Big Money if you play it just right? And the village will certainly help it a good bit. But, uh, don't think that's really that strong anyway. In the meantime, your menageries will be beastly, tribute is going to help you, either with money or actions probably, and black market will probably get you the key parts of your engine. I don't like the maps so much, early on anyway, as that gives more fuel to him, and I don't know that you really need them. But maybe they're good too.
Second one. There are lots of strong BM cards, but that doesn't mean it's a BM board. I'm not at all convinced that BM is the strongest option here, largely because of the presence of Noble Brigand. There's most everything an engine wants: good +actions, a source of +buy, pretty good +cards, and an attack to slow a BM opponent down. Now, you did get pretty lucky hitting my initial silver (1/3 chance), but that doesn't mean it's the wrong play. You probably want Brigand anyway. I should have opened either courtyard or Brigand myself. Probably courtyard. Though I'd like to get a brigand pretty quick, I have nothing to steal off the bat. By the way, Noble Brigand is probably the biggest P2 card - in the sims (which of course you can only trust so far), P2 literally has advantage in a pure NB mirror.

[rrenaud]

The thread is long, and I wasn't really checking on it as it developed, so it's possible someone made this point already. But one important thing to recognize about the expectation/variance or expected margin of victory vs probability to win tradeoff is that the longer the game goes, the more the expectation has an advantage.  As the length approaches infinity, the one with largest expectation wins almost surely. This is one reason why it's hard to apply this advice to merely starting at position 2, there is still plenty of game left, and the highest expected margin strategy has a lot of chances to revert to the mean.

This idea is much easier to demonstrate late in the game, but it requires cooking up a complicated scenario (or grabbing the log for such a scenario and carefully reviewing it).

[PitzerMike]

To clarify "line" because it might be confusing in that usage: a Line is the path/build/game of a single player.

Polarized results really should make some sense by now given all the context and examples provided. 

Thanks for the explanation of line. That was not at ALL in my mind when I read your paragraph, any of the several times I read it. Now it makes much more sense.
On the polarized thing... well, maybe you think it should be clear by now, but it's obvious you thought that before (well, either that or you just wanted to say something cryptic and confusing, but I don't peg you for the kind of person to do that). I'm TELLING you it's not clear to me, so please, explain it.

Edited to fix a quote blocking issue.

I'm not a master of the English language but I think what he means with "polarized results" is results that lean themselves to the extreme ends of the spectrum (or poles). For example a card that will produce polarized results is Treasure Map because you'll either get crushed (negative extreme) because you don't manage to get your 2 Treasure Maps together. Or you'll get a comfortable lead if you manage to play both your maps on turn 5 (positive extreme).

On the other hand Silver would be an example of not producing polarized results. I mean with silver you get what you payed for after all.

[Empathy]

To clarify "line" because it might be confusing in that usage: a Line is the path/build/game of a single player.

Polarized results really should make some sense by now given all the context and examples provided. 

Thanks for the explanation of line. That was not at ALL in my mind when I read your paragraph, any of the several times I read it. Now it makes much more sense.
On the polarized thing... well, maybe you think it should be clear by now, but it's obvious you thought that before (well, either that or you just wanted to say something cryptic and confusing, but I don't peg you for the kind of person to do that). I'm TELLING you it's not clear to me, so please, explain it.

Edited to fix a quote blocking issue.

I'm not a master of the English language but I think what he means with "polarized results" is results that lean themselves to the extreme ends of the spectrum (or poles). For example a card that will produce polarized results is Treasure Map because you'll either get crushed (negative extreme) because you don't manage to get your 2 Treasure Maps together. Or you'll get a comfortable lead if you manage to play both your maps on turn 5 (positive extreme).

On the other hand Silver would be an example of not producing polarized results. I mean with silver you get what you payed for after all.

Yes. You can also call it 'heavy tails' or 'non linear behavior'. BM is -by essence- as close to a linear strategy as you can get in dominion. Of course, the inherent structure of the green cards add some non-linearity even to BM (hitting 8 is twice as good as hitting 7), but overall, you just try to ramp up your average moneyness, knowing that the law of large numbers will reward you in the long term for it.

Maybe the best way to visualize it is the vp/turn graph. Do not forget this is a random graph (what you see in simulators is its 'average').

Engines (especially heavy trashing ones or golden decks) tend to have a very narrow number of possibilities for that graph: there is basically the average one, and a little bit of wiggle room (sometimes a lot more towards the end when you start playing risky again).

Big money has a broader range of possible 'greening' paths, but they are very well behaved around the mean indicated by the simulator. Basically, add the standard deviation around each point and you have a perfect picture of the thing.

Then there are one-turn explosions, TM, or other 'risky' or 'messy' strategies (these terms are not necessarily synonymous). Maybe the most counter-intuitive example is alchemist. It gives the false impression of an engine, when really, it often has a much higher chance than other engines of fizzling, leading to a much more 'polarized' greening curve: basically, there is usually around 80% of the weight at the 'typical' shape and 20% around the much lower 'fizzle' shape. It therefore has a lot of downside, while the upside is very limited (so you trade off your upside for a more stable outcome in most cases) Of course, you can try to ramp it up to 90, 99% or whatever, but that often costs time. Hence playing alchemist loops is often a risk (at least, when you are tight with turns). Treasure map is the opposite: it has a lot of upside, but ends up often in the 'down' scenario.

This makes me wonder if the simulator could draw all possible paths. Hopefully all the superposing ones would give a visualization of the probability mass of a possible outcome. Or maybe it'll just look like a mess xD.

PS: before someone tells me that engines fizzle too: yes but once an alchemist loop is broken, there is a lot of auto-correlation, and it fizzles for longer.

[RisingJaguar]

Really nice article!

Here is a game where I chose a swingy strategy on purpose:

http://councilroom.com/game?game_id=game-20120528-080750-ea2e2993.html

Cards in supply (key cards in bold):
Black Market, Embargo, Embassy, Highway, Laboratory, Pearl Diver, Potion, Scout, Spice Merchant, Treasure Map, and Vineyard

I am second player and my opponents opens 5/2 and chooses Embassy. An Embassy is not an elite opening, but still a pretty strong one - it is one of the fastest BM type cards, I would guess it's better than Envoy/Silver.
I draw 4/3 and it seems I would be behind on any strategy close to BM, and Labs/Highways are not a strong combination without a reliable +buy. So I decide to open Spice Merchant/Black Market and hope to get lucky underground. In the actual game this works fantastically as I get a tournament on turn 5, and a Hunting party on turn 10. Winning Followers on turn 11 essentially decides the game.

So it worked, and it makes for a good story for this thread. But was my logic correct, and does the story actually hold up? That is still not clear to me. For example, yes I am behind, but Embassy is also quite swingy by itself (for example, there is a huge difference between drawing it in turn 3 or turn 4), so just going for Embassy myself is already a high-variance strategy. Also, maybe my strategy was just good by itself? Even if I don't immediately get lucky via BM, I can hope for some synergy between Labs/Highways with the unreliable +buy from Spice Merchant, the virtual +buy from Black Market, and maybe some actual +buys found in the Black Market.

As usual, it's more complicated in an actual game than in theory...

You lay out a lot of the thought process I would have in this game.  Creating contingency plans if you grab something huge in the Black market (tournament) or not.  Possibly all those contingency plans you created actually make it better than embassy 5/2.  However, black market is just so high-variance that this is similar to what the argument is generally about.  If your contingent strategies are better than embassy BM, I wouldn't expect it to be a very considerable margin.  I mean if WW actually had a chance to embargo labs early, much much different. 

[Gamer-man]

I'm not sure if this counts as off topic, but i solved -stef-'s game

i will be listing turns as 1-9 so 1st player plays the odd ones, 2nd player plays the even ones

1st player turn 1:  +3
1st player turn 3:  +5 if (losing by 11 or 12) otherwise +3 if (losing by 10 or less) +10 if (losing by 13 or more)
1st player turn 5:  +5 if (losing by 5, 6, 11, or 12), otherwise +3 if (losing by 10 or less) +10 if (losing by 13 or more)
1st player turn 7:  +5 if (losing by 4, 5, or winning by 8, 9, 10), otherwise +3 if (losing by 8 or less), +10 if (losing by 9 or more)
1st player turn 9:  logical, try for the win (behind by 3 or 4 need a 5, behind by 5 or more needs a 10, losing by 2 or less needs 3's)

2nd player turn 2:  +10 if (losing by 10) otherwise +3 [ignores impossible cases]
2nd player turn 4:  +10 if (losing by 6 or more) +3 if (losing by 5 or less)
2nd player turn 6:  +5 if (losing by 1,2,8) otherwise +3 if (tied or winning) +10 if (losing)
2nd player turn 8:  +5 if (losing by 1,2 or winning by 5,6,7),  otherwise +3 if (tied or winning) +10 if (losing)

overall the odds of 1st player winning is 59%

the 5's seem to not effect the overall strategy, and instead are tactical options  due to special cases (if 2nd is losing by 1 or 2, missing a 5 is no worse than missing a 3, but is alot better if it hits)

[Insomniac]

BTW: Forager has a heavy heavy player 2 advantage. (Assuming both players go forager)

[rrwoods]

I'm starting to get convinced the optimal strategy is actually quite complex. I think it uses all 3 options, and also includes the 'turns to go' and 'actual point difference' (not just 'am I behind or ahead'). But I won't go into that - I didn't name it Borinion for nothing.

Well I will :-P

I've attached linked a spreadsheet I used to solve the game.  With optimal play, Player 1 wins 59% of games.

Solving the game is a simple matter of backward induction.  Evaluate turn 9 first; based on the P1 win percentages in turn 9, we can solve P2's optimal play on turn 8, and so forth.

Something interesting:  Not only does the optimal strategy involve all three options, and not only does it involve "turns to go" and "how far ahead/behind am I" (of course it would!)... the strategy is non-monotonic!  As a simple example, in turn three, if player 1 is 2 points behind or 6 points behind, he should choose option A.  But if he's 5 points behind he should choose option B.  (It's impossible for him to be 3 or 4 points behind, so the sheet ignores those possibilities; something it doesn't do beyond turn 3.)

There's a "pocket" at the 5-point deficit mark where player 1 should choose the riskier option, despite that at some higher deficits, he should choose the safer option!  Don't know about you, but to me this is fascinating.  "Borinion" -- maybe to play, but analysis is extremely interesting :-)

In the later turns more pockets appear.  In turn 8 (might as well choose the last player 2 turn since we are talking about taking risks in the player 2 seat), there are two such pockets; one from +5 to +7, and one from -2 to -1.  At -6 to -7 we choose B as well, but at -8 to -13 we choose option C and at -14 or worse it is hopeless.

Is it possible there is something to be learned about Dominion (or games in general) from the non-monotonic nature of this strategy?  Can this revelation be put to any practical use?


EDIT: looks like I was beaten to this by Gamer-man, but our results differ a bit.