Predict: What to name

Last night on Jarvis Yu’s stream (check it out at twitch.tv/jarvisyu , it’s a good watch) a situation arose where he was priced into playing a blind Predict. He named a four-of without much thought as to which four-of to pick, and then some discussion ensued as to what the best play was.

TLDR: in a situation like that, where you're not likely to see a very large percentage of your deck by the time the game ends, the difference is really tiny. But let's look at some concrete maths, because I like this kind of little math puzzle.

First of all, let's simplify the situation to assume that there aren't particular graveyard effects (e.g. graveyard eldrazi shuffle triggers, flashback, etc.); let's also assume that we can narrow down the options to one card that's "good" and one that's "bad" (presumably the best and worst 4-ofs left in the deck). The first thing I want to do is show that, very concretely, there are some situations where naming the good card is optimal, and some situations where naming the bad card is optimal.

Scenario A: Your opponent is empty-handed. You've just untapped with a billion mana and only the predict in hand. A Sulfuric Vortex trigger just went on the stack which will kill you when it resolves. The only cards left in your deck are 2 Lightning Bolts and 2 Shocks. Your opponent is at 3 life. 

If you name 

Lightning bolt: 

You have a 50% chance of guessing right, in which case you win on the spot (any 2 spells are lethal)

You have a 50% chance of guessing wrong. In this case a shock gets milled, and you draw 1 card out of a deck of Bolt/Bolt/Shock. This gives you a 2/3 chance of winning.

Overall, you win 5/6 of the time.

Shock:

You have a 50% chance of guessing right, in which case you win on the spot.

You have a 50% chance of guessing wrong, in which case you mill a Bolt, and draw 1 out of Bolt/Shock/Shock. This gives you a 1/3 chance of winning.

Overall you win 4/6=2/3 of the time.

So we can see in this scenario, it's better to name the better card (Lightning Bolt), which increases your win rate in this case by 1/6.

Scenario B: Exactly the same as Scenario A, except now your opponent is at 6 life instead of 3.

If you name 

Lightning bolt: 

You have a 50% chance of guessing right, in which case you draw 2 from shock/shock/bolt, which can't win. You lose.

You have a 50% chance of guessing wrong, in which case you draw 1 card, can't burn them out, and lose for sure.

Overall, you win none of the time.

Shock:

You have a 50% chance of guessing right, in which case you draw 2 from Bolt/Bolt/Shock. You need the shock to be on bottom to win, which only happens 1/3 of the time, so you win 1/3 of these games.

You have a 50% chance of guessing wrong, in which case you draw 1 card, can't burn them out, and lose for sure.

Overall you win 1/6 of the time.

So in this version, it's better to name the worse card, which increases your win rate by... well the same 1/6 as it happens.

 

 

 

So we've established at least that scenarios can exist where it matters significantly, and that sometimes you want to name a better card, and sometimes you want to name a worse card. Let's quickly check a couple more cases similar to these; if instead the opponent had 4 life, you just need to draw 2 spells, any 2 spells, so it doesn't matter - you need to predict right to win, it's 50/50 either way. If they're on 5 life, you need to draw either bolt/bolt or bolt/shock; you want to name Shock here, which gives you a 50% chance to win (whereas naming Bolt would give you only a 1/3 chance).

 But these scenarios are pretty artificial. What about other scenarios?

Scenario C: You're digging for one specific card which will win you the game, and you need to find it right now. Let's say you're down to 8 cards again, 4 Winners and 4 Losers.

If you name 

Winner: 

Guessing right gets you a 5/7 chance to win, and guessing wrong gets you a 4/7 chance to win, for an overall win rate of 9/14

Loser:

Guessing right gets you a 6/7 chance to win, and guessing wrong gets you a 3/7 chance to win, for an overall win rate of 9/14

So here it doesn't matter, which matches Jarvis's original intuition

Scenario D: Same as C, but we add 32 other 1-of blank pieces of cardboard to the deck. You still need to find one of the 4 copies of Winner to win the game. Now in this case, 32/40 of the time, a card besides one of your options is on top, and it didn't matter at all which of the two cards you named. So we only need to look at the 20% of the time where it actually matters.

If you name

Winner: 

Guessing right gets you a 15.0% chance to win, and guessing wrong (with Loser on top) gets you a 10.3% chance to win, for a win rate of 12.6% in the 20% of cases it matters

Loser:

Guessing right gets you a 19.7% chance to win, and guessing wrong gets you a 7.7% chance to win, for a win rate of 13.7% in the 20% of cases it matters.

Given that these scenarios overall are only 20% of games, we can see that it the difference is just over 0.2% to the overall win rate. But it's better in this case to predict a bad card than the one you're looking for, since two looks at the better deck has a bigger impact than the flipside one look, even though the chances of guessing right are very small.

Other scenarios get more complicated, and I don't want to go in to that kind of depth. The long and short of it seems to be:

• The better the position you're in, the more likely you are to want to name a good card, so that you're improving how bad the worst-case scenario is
• The worse the position you're in, the more likely you are to want to name a bad card, to maximize your best case scenario
• The bigger your deck tends to be, especially relative to the number of cards you'd be happy to draw, the more you want to probably name the bad card
• Naming a good card is likely to be best mostly in cases where something like half your deck (or, especially, more than this) is a good draw. 
• Mostly the differences are going to be tiny unless your deck is tiny, which... shouldn't come up very much. The most likely case for this is probably if you're playing against some kind of mill deck(?). A farfetched situation that might actually arise is some kind of doomsday deck where you need to fetch a land out of the doomsday pile. I would love to see that happen, but I'm not holding my breath
  

WAR Limited Analysis: Part II

Unfortunately, I've been sick most of the past week, so this analysis may not be quite as detailed as I'd hoped. Nevertheless, we move forward!

Also, a special shout-out to Scryfall (https://scryfall.com/), which has been immensely helpful in putting this analysis together.


Final numbers for Amass, +1/+1 counters, and Proliferate

Per draft of WAR, you can expect 32.5 Amass cards to be opened; 11.7 of these are mono blue, 9.3 mono black, 8.0 mono red, and the rest multicolored. So if you have an opponent in two of these colors, you should expect them to likely have several Amass cards. But if they're in only one of those colors, they'll probably only have a few. It's also good to note that almost all of these cards are Amass 1 or Amass 2.

In terms of non-amass cards that produce +1/+1 counters, you can expect 12.8 of those to be opened per draft in White, 8.4 in Blue, 0.9 in Black, 6.0 in Red, and 12.1 in Green. So in the proliferate colors, there's lots of cards that are going to enable proliferation (note that blue also gets to count the Amass cards as noted above, so they're actually in first here). On the other hand, this is not so many cards that you can expect to have just tons of different permanents to proliferate onto at any point - one target will happen, two will be common, and you'll be quite happy to manage getting three.

Speaking of Proliferate, it ends up on 5.7 cards per draft in both White and Blue, and 8.4 in Green. So, don't expect to be able to build a deck around proliferating over and over again - unless you get one of the cards which singlehandedly pumps trigger after trigger out, you're more likely to get one, or maybe two over the course of a game, even in these colors.


Is spellslinger a real strategy?
As is so often the case in these sets, Blue/Red's theme seems to be "spellslinger", i.e. it wants you to play lots of instants and sorceries. Slightly confusingly, in this set in particular, some of the cards in this direction point you towards those particular types, but some care about noncreature spells more generally. And I expect most decks in this format to have a few noncreatures which aren't in these types (mainly planeswalkers, though there's some playable enchantment-based removal as well).

So what are the numbers? Well, on the non-creature side, there's 5.1 monored, 4.2 monoblue, and .9 hybrit Izzet cards per draft. And on the Instant/Sorcery side, we're at 4.4 blue, 1.5 red, and 1.3 Izzet gold cards. Overall, if you're completely alone in your lane, you might be able to scrape together a deck based around these.... but I wouldn't really bank on it. I think the biggest way to get into this deck is to open a good rare that's on theme and then pick up another couple early - but don't be trying to get payoffs later, it's just not likely enough to happen.

Of note, the red cards here also work with the red-white (even less supported) subtheme of pumping your own stuff - I don't think both of those decks can exist at the same table, though obviously you can build decks in these colors that don't exactly follow those themes.



Mana Fixing, or lack thereof
There are 10.4 pieces of mana fixing per draft which are colorless (i.e., lands or artifacts); you get access to an additional 6.6 if you are base green. This is actually a reasonable amount of fixing... but it's a LOT less than we've seen in the last couple of sets set on Ravnica (or actually, any of the Ravnica sets). So five color decks will be nigh impossible to make... three color decks are even going to be very ambitious. Especially if you aren't green, you'd need basically all the fixing at the table. Plus, since most of the fixing isn't in lands, you would end up with like half your spells just being dedicated to fixing, and I just don't see the payoff being worth it. (Sorry, Niv-Mizzet).

Having said that, splashing seems very plausible. It's definitely not to the point where you would say that splashing some spells from a third color is free, by any stretch - you still have to work a bit to get your mana to get there, like normal - but if you have a reason, you should be able to find something to get you there most of the time (provided it's not like, halfway through pack three already or something).



Creature Sizing
How big are the creatures in the format? Obviously it's a little bit tough to tell just by looking at a list of cards, since there are questions of playability, plus a lot of +1/+1 counters running around and affecting the sizing.

But if we look at everything, just on the base stats, then in terms of power, there's a massive hump at 2 power. There are nearly as many creatures with exactly 2 power (58.1 per draft) as there are with greater than 2 power. Moving from 3 power (28.2 creatures per draft) to 4 (21.1 creatures per draft), there's not nearly as big of a drop off. Per normal, not many creature get to the 5+ power range (12.4), so don't be super surprised if your opponent has one of those, but it won't be often they have multiples.

On the Toughness side of the equation, things are more spread out. 31.4 creature per draft have 1 toughness, so most of your opponents will have a target for your ping effect to hit (although in many of these cases, you would need to time it precisely, as a few of these creatures grow from an ETB counter, and some others are unplayable... so be ready to sideboard around this situation one way or another, which ia fairly common problem if we're honest). 50.2 creatures opened have 2 toughness, and 43.4 have 3. This is the point where the biggest drop-off is, with only 20.4 creatures per draft having 4 toughness, with an additional 16.5 at 5 toughness, and 7.9 at 6.

In the hopes of finding a "magic toughness" or sizing in general, it's also important to look at the toughness-based removal the set provides (damage or -N toughness). 6.4 such cards per draft punish 1 toughness, 8.2 on 2 toughness, 4.6 on 3 toughness, 5.1 take care of creatures with 4 toughness, and 2.3 (1 common) deal with creatures having 5 toughness or less.

Based on looking at this, I doubt that there really well be any "magic size" for creatures in the set, though I guess that most things with 5 toughness will be fairly hard to take out using a single card, especially if that card isn't one of the few premium removal spells in white or black that don't care about size at all (or the fight-like spells in green which are pretty close)



Final thoughts

Overall, The biggest thing about this set is that it looks much closer to a 'normal set' to me than we've had in a while. Well, except for having a couple of planeswalkers per deck, which is, I guess, a pretty significant difference. But the fixing numbers, creature sizing, and for the most part lack of cohesive on-rails plan for each color combination makes things mostly more block-and-tackle. Or more, uh... I feel like there should be a better metaphor which doesn't draw a parallel to a sport which is virtually exclusively played in a single country. Anyway, I digress.

Take good cards, probably don't splash, realize your opponent will have a couple planeswalkers, but also realize they won't be the be-all and end-all. Try to have board presence. And most of all, have fun! It's a new set, that's what they're for.


Hopefully I'll have time to get a moxiously early pick-order list generated before the end of the week, with some notes about specific cards, but we'll see...

WAR of the Spark Limited Analysis: Part I

War of the Spark previews have started, and in between brewing new Standard (and Modern... and Vintage...) decks, I'm also thinking about my favorite format - booster draft. And one of the fun things come this set is that the Mythic Championship for the set (these things used to be called Pro Tours) will effectively be a pre-release, meaning that going into it, nobody will have been able to play a sanctioned tournament. This makes preparation, prognostication, and full-set evaluation even more important than normal. (Obviously, it's possible to proxy some of these up for playtesting and do mock drafts and such, if you have the resources - mostly time and friends - for that, and I would recommend this if you're actually playing in the tournament). Anyway, I doubt (m?)any pros are going to read this, but on the off chance (and because it will help me and readers in our own low-stakes events), I figured I'd jump back in to the limited analysis game after a good period off.


A quick disclaimer - of course, not all of the cards are out yet, so some of this can still change slightly.

Mechanics

Planeswalkers

Planeswalkers are an apparent theme of the set (kind of), with many more than we've ever seen before. However, there are some twists, which mean that evaluating them is going to be different from normal - which is a bit problematic, given that they're already often on the harder end to evaluate. However, in limited, almost all prior planeswalkers were designed in a way such that they were very, very good, so evaluation didn't matter that much. The changes of this set mean that's no longer the case.

What do we know?
There are going to be 36 Planeswalker cards in the set, 20 at uncommon, 13 at rare, and 3 at mythic. (There is an additional Mythic as the Buy-a-Box Promo, but it's not in any packs and so doesn't change limited). Moreover, barring foils, there is going to be exactly one Planeswalker in every pack. And while we don't know exactly the process they're using to ensure that, or how that affects the collation, I think it's a fairly safe bet to assume that the Uncommon:Rare:Mythic ratios will be about the same as on normal cards, which leaves me thinking that at a normal 8-player booster draft, you'll expect to have opened about 18 Uncommon 'walkers, a bit more than 5 Rares, and just over 1/2 of a Mythic. We should also note that exactly 1 per pack means each player on average will draft 3, and because we should expect some of unplayed, I think we can expect that most decks will have about 2 Planeswalkers on average (possibly a little more), though with some significant variation.

Static and Triggered abilities
Something new to this set is that all the Planeswalkers have a static and/or triggered ability, in addition to one or more traditional loyalty abilities. (And yes, I know, there have been some commander PWs that had things like "Can be your commander" as static abilities, but whatever). The value of these abilities appear to differ pretty significantly, and can change the value of the planeswalker to being mostly an attackable/burnable enchantment, if it's most of the power. Sometimes, though, it looks like it will be mostly an afterthought.

Evaluations:
In general, in evaluating these cards, there's a few cases I think are worth keeping in mind.

1. How good is this card if I'm behind on board, and it more or less dies right away?
2. How good is this card if I'm ahead on board or in a stalled board state?

These two scenarios help define floor and ceiling for the card.




Uncommons
We know that the 20 uncommon planeswalkers in the set all have only their static/triggered ability and a single loyalty ability, which removes loyalty. The big thing to note here is that, unlike previous Planeswalkers, these cards aren't going to be able to chunk out huge amounts of card advantage by activating their loyalty abilities turn after turn, if left unchecked. The exception, of course, is if the static ability is particularly strong.


Rares:
These have two loyalty abilities (one plus and one minus) along with the static ability. Sometimes, there's an ultimate, sometimes not, but in general, all of these are going to generate significant card advantage for you if they get to stick around, so it's really all a question of how well they protect themselves, and/or how good they are if you can't protect them.

Mythics:
These appear to be pretty close to the more traditional, yeah-they're-just-busted designs we're used to.  

Amass

Amass is a new keyword ability, such that if a card has Amass N, you put N +1/+1 counters on an Army you control; if you don't control any armies, then you make a 0/0 black Zombie Army creature token first. Note here that you can almost never have more than one army at a time (the only way which looks to be possible is gaining control of your opponent's army while already having one yourself).

If you evaluate these cards in a vacuum, they're going to look better, probably than they will be in practice. This is because if you have lots of amass cards, you aren't getting extra bodies every time, but more often just getting them only once. And in general, it's better to get your N stats on a new, extra body, than it is to add them to an existing one without choice. But that comparison deserves some further analysis.

If you're putting counters on an existing army, it's very much like a basic aura that pumps your dude. This has the distinct advantage of giving those stats effective haste, but is significantly worse against unconditional removal and bounce. Most importantly, of course, is the impact on creature sizing. But this is hard to work out in the abstract - is a 4/4 better than a pair of 2/2s? Depends on both boards. In general, there is, though, value in simply having the largest creature around, particularly defensively, as it greatly discourages attacks. So in general, the biggest drawback of having to put your eggs all in one basket is down to these interactive spells which don't care about the size of a creature.


Something that's very important to note is that Amass is localized to only the Grixis colours (blue, black, and red). So if your opponent is two of these colors, they're likely to have a lot more amass than if they are one, and if they're GW, they probably won't have any. Because for the most part Amass seems better to me if you have some, but not too much (a la Delve), I suspect that this would make you slightly prefer to be exactly one of these colors.

Proliferate

Proliferate is a really hard mechanic to judge. For the most part, we're dealing with two kinds of counters, +1/+1 and Loyalty (there's at least one card with a Charge Counter, but it looks at this point as thought it might be only one, and it's a rare). Loyalty counters' value varies a great deal - it might get you a whole extra use of one of your uncommon planeswalkers, let it survive an extra attack, or do basically nothing. Extra +1/+1 counters are nice, but for the most part aren't worth a card until you start to get 3-4 or so of them. Now, there are lots of cards with +1/+1 counters in this set, and every Amass card also counts to some extent (though having 8 amass cards isn't going to help you much in getting multiple proliferate targets), but I imagine that unless the format ends up leading to lots of board stalls, that simply being able to proliferate isn't going to be worth a full card very often. Fortunately, it looks like for at least most of the cards in the set, Proliferate isn't the main point of the card, but an extra bonus. And getting even 1-2 counters as a bonus on top of an already close to playable card seems like a good deal.

Note that proliferate is localized to the Bant colours (green, white, and blue, plus one rare land), which means, just like in the case of Amass, you should expect much more of it from players in these colors, and none at all from BR players. Note that blue is the only color which overlaps both of these here, so that's going to be more likely for you to proliferate onto your Armies. Overall, this isn't really a huge thing, but a small adjustment to keep in mind both in the draft as well as in gameplay.







Please join me again soon where I will break down some of the numbers more precisely, to see if we can find out things like critical creature sizing and sub-theme prevalence. And eventually, evaluations of specific cards.

Opportunity Denial

Opportunity Denial – A Disruption Evaluation Framework

A glaring hole exists, I believe, in the game strategy literature surrounding the evaluation of hindering your opponents’ strategies and goals. As a concept, it’s obviously known, and it’s known to have value, but the amount of value it has is poorly understood. There’s no framework for knowing how to compare it to advancing your own game plan. In this article, I seek to fill that void.

Opportunity Denial

               The basic concept of these evaluations is something I call “Opportunity Denial”. Effectively, it can be summed up as: “The value of thwarting your opponents’ goal is equal to the difference between the value of the goal you have stopped and the opponents’ next best goal”. Effectively, it’s the flipside of opportunity cost – the value of your own prospective choice is tempered by the value of the next best option in that case, and the value of your denial works the same here. In short, denial is more valuable when your opponent doesn’t have any other good options, and less valuable the closer your opponent’s best unstopped option gets to being as good as what you’re preventing.

Examples

Chess

               Chess is a game where this concept is already fairly well understood (albeit, not by this name). The biggest case of this is with space. A space advantage is often referenced as a good thing, though not often explained. Why is it valuable? Because it constricts your opponent’s pieces – generally, they can’t go to squares attacked by your pawns, and it’s hard to get behind enemy pawns safely, so having advanced pawns means your opponents have fewer squares for their pieces. This is a big denial insofar as they don’t have enough good squares for their pieces, which means that as more pieces get traded off, the less you are denying them, as they have more options per piece overall, and so your space advantage is good for less and less.

               A similar situation presents itself in deep endgames with Kings blocking each other. The opposition is a big deal because it lets you deny your opponent the opportunity to advance. As long as the opponent has another piece to move, then this doesn’t matter as much, but as the number of other pieces goes down, the more zugzwang comes into play, and the opposition matters a lot more.

Magic

               In Magic, the concept gets referred to as playing on a different axis. For instance, you can imagine a limited deck of 20 Plains and 30 Swords to Plowshares. Such a deck is never going to lose to most limited decks you’ll come across, which must win only through a pretty limited number of creatures beating you down. However, once your opponents bring any other kind of way of winning – a bigger deck to deck you out, a hexproof creature, a non-creature threat like a planeswalker, etc, then you’re just cold. Often in limited, winning though other means isn’t really viable, so you might be fine (assuming you can guard against them boarding in a hundred extra basic lands and milling you out that way). But in constructed, this is a very bad idea. This is because even though you’ve shut down the creature plan hard, you are only denying them on one axis, and there will be decks with other axes. This concept is exemplified even more by cards that do this on their own, like Moat or Ensnaring Bridge. These cards can take care of creatures pretty well, but they aren’t exactly busting a lot of formats. Part of this is because those cards can be answered, but a big part is that they don’t cover everything. Bridge needed a deck like Lantern Control, which completes the lock by stopping alternatives from getting in hand, in order to really make a huge mark.

               This is why most control decks end up playing Counterspells – a counter can answer basically any spell. Even in these cases, there are some things you can’t answer – too many spells per turn, uncounterable spells, lands – which is why particularly in the older formats, with lots of options, pure control decks don’t end up doing super well all that often, and also why they tend to do particularly poorly in wide open new formats, because they don’t have a narrow list of threats such that they know exactly what they want to answer.

Ticket to Ride

               Generally placing trains, or picking up certain colors of cards, simply to block your opponent isn’t a great strategy. This is because they can usually just go for something else, besides what you blocked, and be in totally fine shape. The closer you get to the end of the game, or the more you’re sure they have some particular route they need to complete, the more it can start to become reasonable.

Multiplayer Games, generally

               In this case, your opponents’ collectively are analogous to one opponent in a 2-player game. And in this setting, attacking a single opponent tends to be a poor strategy, precisely because each of the other players is unaffected, so your collective opponents’ next best plan – in this case, beating you with more or less any other player – is hurt relatively little. Where it becomes more reasonable is, predictably, when that particular opponent you’re attacking is much ahead of everyone else.

Dominion

               The most obvious case of this in Dominion is Contraband. Contraband isn’t such a good card usually, because you give your opponent the power to deny you. That ability is reasonably powerful, because at some point, there’s usually going to be a specific card you need – Victory cards if nothing else – at which point Contraband is pretty useless to you. And besides this, there are almost always other options which are nearly as good as a $3 +buy treasure for 5 anyway (and usually, stronger).

               However, the concept comes up in many other situations more commonly. There are a couple of other cards(/landmarks/events) which are pretty direct in this respect.

Take, for instance, the Landmark Defiled Shrine. With N tokens on it, buying a curse is exactly like buying a victory card worth N-1 points, right? So if there are, let’s say, seven counters, then it’s the same as  buying a (0-cost) Province? Not exactly. First of all, there’s an issue about piles running out – usually buying a province will hasten the end of the game moreso than buying a curse (though I guess that’s not always true). Moreover, though, there’s some amount of denial to each play. When you get the curse, the points leave Defiled Shrine, meaning that you’re effectively stopping your opponent from making the same play on their next turn. Some people say that this is like a 12 point swing. But when we look at this under the paradigm of opportunity denial, we can see that this is not the case. First of all, you haven’t denied them anything if they weren’t going to buy a curse anyway. But even if they were, they now get to spend that buy on something else, whatever the next best thing was. So it comes out to the full 12 point swing only in the case where they were otherwise doing nothing with the buy.

Let’s compare that to buying a Province. Every province you get is a province your opponent can’t get in the long run. But getting a province now doesn’t do much in terms of the overall number they can get until the game is about to end. Is buying a province, therefore, a 12 point swing? No, it isn’t either. First of all, your opponent may not be going for provinces at all – if they have access to VP tokens, or alt victory cards, or some other way of winning the game, then it doesn’t make much difference. Additionally, while buying the province is a long term denial of the Nth province (where N is how many remained before you bought it, plus how many they have right now), that only tends to matter as N gets low. In other words, denying them the 7^th province doesn’t matter so much – it’s the 5^th and the 4^th where it starts to become pertinent. And the fastest way to deny them those may not be to buy one straightaway.

The same logic from the Province case actually applies to any pile that is running out. Think about a case where there's only one pile of villages, and generally the best deck to go for is some kind of draw-your-deck-using-terminals-then-play-a-bunch-of-payload thing, which is often the case. In such a situation, having more of the villages means you can play more actions - more draw cards, as well as more terminal payload cards. Fantastic. But is it worth it? It's easy to imagine a situation where, let's say the fifth village will eventually move you from two provinces per turn to three. And you already have five, so you're set there, but there's one left, and you're trying to figure out whether to deny your opponent. Let's also assume that it will cost you a turn to get the village (because if it's free, then obviously you should do it). In this case, the answer is pretty clear that you should not bother with denial - you're costing yourself a turn, and your opponent will get to cut some gains (one less village and a bit less payload, since they can't support it), which means they're actually getting off the ground faster. Between all that, you might still be ahead, but it's hard to imagine you'll be more ahead than if you just went for your own greening phase. The more interesting question comes up when it flips you from single province turns to double. This reduces time from greening start to four provinces by two turn cycles. Spending time on the village which is superfluous for you costs you one turn, and them not needing to build as much means they can cut this one village, along with probably one draw card and about two to three payload cards. One thing extra for you plus 4-5 for your opponent looks like more than enough turn cycles, but we have to remember that probably some of these things get bought on the same turn anyway, and the extra village does also help your reliability (probably more so than the extra cards hurt it). So all in all... it's actually a close call, and depends on the specifics. But certainly the value over not denying isn't super high.