Tuesday, 8 May 2018

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.



               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.


               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.


               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 7th province doesn’t matter so much – it’s the 5th and the 4th 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.

Tuesday, 18 April 2017

Amonkhet Draft Quantitative Analysis

After some time, I'm back again to break down some of the numbers relating to a new Magic: the Gathering limited format. Per normal, I'll be dishing out the numbers of certain classes of cards (on a per-draft basis) to try to help everyone get a better picture of what archetypes are supported, against which ones are not. (Big reason this can be useful is that some of the archetypes are really constructed plants - and I don't mean Sylvan Caryatid - in terms of being loaded at high rarity).

This time, I'd like to make a special shout out to the fine folks at https://scryfall.com/ , which made putting this together FAR easier than it has been in the past.

In terms of the numbers themselves, it's a pretty normal "big" set. 101 commons, 80 uncommons, 53 rares, and 15 mythics. This leads us to .099, .0375, .0165, and .0083 of any particular card of that rarity, respectively, per pack. This gets multiplied by 24 packs to get a per-draft average. If you want to know a about a sealed, you'd divide that by 4. (I'll note that due to the way print runs happen, I think there's one common with a slightly different incidence rate, but there's little way at this point to know which that is; I'm also ignoring foils here, since I'm not sure how that replacement works, so that would slightly increase non-commons and decrease commons; these are all very small differences, but I wanted to mention them in the interest of full disclosure).


One of the first things I always want to look at in any format is how much mana-fixing there is. This helps us figure out how many colors we can be playing, how much you'd have to work for extra colors, how easy it is to splash, how much contempt you should have for picking multicolor cards early, etc.

Amonkhet has 4 common mana fixers, 1 uncommon, and 9 rares (I'm not including Vizier of the Menagerie, which only fixes for creatures). This leads us to a total of an average of 14 pieces of fixing per draft. Typically you want something like 4-8 pieces of fixing to play a third color, which means you'd need roughly half (or maybe a little under) of the fixing in the draft - seems possible, but you'd have to work for it. But let's drill a bit deeper. Painted Bluffs is a common fixer that could go in any deck, but not one you'd want to. Cascading Cataracts and Pyramid of the Pantheon are similar, but at rare. The cycling lands are probably going to be quite hard to pick up if you don't open them, and in any case will only fix your mana if you just happen to be the right colors. This leaves us with Evolving Wilds as the only good, reliable fixer for any colors, which is a place we've been pretty often before. Additionally in this set, though, we're back to having noticeably more fixing in Green exactly - Oashra Cultivator and Gift of Paradise at common, Spring of Spring//Mind at Uncommon, and a couple different rares all add up to make Green the color of fixing again. It's worth noting that these are generally a bit overpriced from what we'd expect (3 mana Rampant Growth seems to be the norm here), but will get the job done in a pinch. And importantly, splashing multiple colors seems only marginally harder than splashing one, and actually easier than trying to be fully 3 colors.


Sure, cycling is a theme of the set. But just how present is it? EVERYWHERE. There are fully 20 Commons, 10 Uncommons, and 8 Rares with the popular returning mechanic, leading to an average of 59.7 cards per draft! This means even the average player will end up with 7-8 of these cards in their pool. And some of those won't be in the right colors, and some will be unplayable (though the option to cycle means very few will be embarrassingly bad). But even if your normal half-the-cards you draft end up in your deck, you're still looking at about 4 per player. Which means if you crack open a Drake Haven, and you actually prioritize these cards a bit, you should really be able to have plenty of enablers to turn that card on. I'll also note here that most of these cards that care about cycling also trigger off of other forms of discard, of which there are 14 in the set - bringing you to an even healthier number of enablers. So you shouldn't really have problems in 'getting there' with those kinds of cards.

How many such rewards are there? Well, if you also include cards like Shadow of the Grave and Sacred Excavation, which don't trigger off cycling per se, but definitely care about the mechanic, you end up with 3 commons, 6 uncommons, and 5 rares, for a total of 14.5 per draft. So not all that many. When you factor in that a lot of these are at higher rarity, and several of the commons give mediocre bonuses, I don't think this is an archetype you should expect to see in every draft pod. But it is something you can go with if you get the right card(s) early. And worth noting that this is centred in blue and black particularly, also with some presence in red.

Lastly, because cycling is something that happens from the hand, at instant speed, and is on lots of cards, if your opponent has something like Hekma Sentinels or Pitiless Vizier, keep in mind that they basically have threat-of-activation on activated abilities - since most any card in hand could be a combat trick with card advantage. So value that accordingly in the draft, and play round or bluff it accordingly in gameplay.


Embalm appears on 5 commons, 4 uncommons, 5 rares, and a mythic. It is centred mostly in white, with strong representation in blue as well, and the smallest sprinkles in Red and Green. In total, you can expect 17.7 Embalm creatures to show up on average in a draft. Because of the color imbalance, you can expect white and blue drafters to probably have a few each (WU drafters a bit more than that even), but not at all a strongly themed deck.


This leads us right to Zombies, which seem to be the tribe du jour on Amonkhet. Apart from the Embalm cards (all of which make white zombie tokens when embalmed), there are 28.75 other zombies per draft in the set, (including cards which make multiple zombie tokens, like Liliana or her Mastery, once each for their rarity). Altogether, that makes a total of 46.4 - definitely less than cycling, but more than about anything else you're going to find. Especially important is that these other zombies are all white and/or black, so that when you combine the embalm in, you get the most Zombies in white, followed by black and blue, and very few in red or green. 

But the bigger story here is the pay-offs for zombies. There appear to be quite a few in the set. But the problem is that, like with the cycling bonuses, they're focused at higher rarities. 2 commons, 4 uncommons, and 2 rares leaves you with only 9.1 zombie bonus cards per draft (I didn't count the Liliana ultimate here, full disclosure). So this is somewhat like the BW Lifegain theme from Oath of the Gatewatch - sometimes it will come up, but you can have decks even in those colors where it doesn't really.


Something that people have been noticing throughout the spoilers is that there seems to be a subtheme of cards, mostly in black and red, which care about having few cards in hand - specifically, many of them are improved when you get to having 0-1 cards in hand. People have dubbed this "Heckbent" as a lite version of the Hellbent (no cards in hand) keyword from Dissension. But this is really a constructed-slanted mechanic - 1 common, 2 rares, and a mythic have that text, plus an extra uncommon that's huge but shrunken for each card in hand. Don't count on this in limited.

-1/-1 Counters

Instead of the near-ubiquitous run of +1/+1 counter mechanics we've had over the last few years, this block returns us for the first time since Scars block to -1/-1 counters. These are fairly prevalent in the set, with 26.7 cards per draft that give them out. These are primarily in black and green, with a bit in red. And it's especially worth noting that many of these cards actually have you putting the counters on your own creatures, at least at first (many of those in turn have ways for you to take them off later).

How many cards care about these kinds of counters is, as often, the bigger question. The answer in this case is 12.4 per draft (this follows some logical progression on what counts as "caring about" - I'm not counting here Exemplar of Strength, but of course I am counting Nest of Scarabs). This is definitely the kind of thing which again, doesn't look terribly supported, but again, is something you probably will see from time to time.


These cards are known perhaps more descriptively as Split Flashback cards. And while for constructed, the thing to look out for is that they mostly look priced for limited, the thing to know from a drafter's perspective is that these are all at high rarity. They only exist in 3 cycles - enemy-colored split uncommons, allied-color split rares, and same-color split rares. This leads to only 8.5 per draft, and especially spread throughout the colors - don't expect to see an aftermath deck in any way shape or form across the lifetime of the format. In other words, just evaluate these cards at face value.


Exert is a mechanic that allows you to choose at the time one of your creatures attacks to have it not untap in your next untap step. In return, you get some sort of bonus right now. These cards obviously promote attacking, and in general, racing. There are 23.4 such cards per draft. The bonuses for exerting come off of a couple uncommon red cards which pay you out whenever you exert any creature, as well as a couple of cards which give you some bonus for having tapped creatures. Again though, these are really small potatoes - the cards should be evaluated really on their faces far more than for synergies.


Thursday, 2 February 2017

Last Brews Pre-PT AER: Upgrades and Miscellany

The Pro Tour is coming up in a few hours, so I'm going to power through the last of the brews I have now pretty rapid-fire. Starting with updated versions of some old standard decks.

https://www.mtggoldfish.com/deck/551364#online is my update to Aetherworks Marvel. Sultai are the colors I like here now. Noxious Gearhulk and World Breaker take the place of pretty good hits that are also castable. Aethertide Whale is a hit that lets you spin again. Renegade Map helps make Glint-Nest Crane much better (though its primary purpose is still to find the Marvel). And Ulamog is the biggest, best hit.

https://www.mtggoldfish.com/deck/548534#online is a very unfocused Bring to Light deck. It's certainly possible to get to 5 colors now. And there's quite a decent number of hits you can go for. But you want most of them to be fairly high in mana cost, but you need things to do early on.... well, I do think the Expertises help the deck, but I'm not hugely convinced this deck is quite where you need to be. But something to be aware of. Note that if this would help you Saheeli Combo, I could easily see that being very good, but a careful reading shows that BtL can't grab Planeswalkers, and only getting Guardian just isn't enough.

https://www.mtggoldfish.com/deck/548550#online is a look at a green ramp deck. Splashing colorless is pretty cheap, and Rishkar's Expertise is a Big Game. Possibly even more important is that Emrakul as the de facto big top end doesn't invalidate you nearly as much as it used to. Lots of mana plus lots of ways to use it makes the deck overall look pretty good, but you do have to be wary of highly synergystic decks (most notably combo decks), so probably those would need to fall out of favor, or you need to up your interaction a little, for it to be good. If it's just Copycat, then probably upping your Walking Ballista count can get you there.

https://www.mtggoldfish.com/deck/547510#online is an update to the deck I'd been calling "Jolly Green Giants". Woodland Wanderer feels so much better now that it doesn't have to face down Reflector Mage. And Rishkar's Expertise is pretty good in a deck looking to make 6/6s and 8/8s (though you don't want to cast the Wanderer off of it). Also, lots of the deck works with Winding Constrictor (man, that card is just fantastic). Quite possibly there should be a mix of more Anguished Unmaking in the main deck; the card seems pretty well positioned right now, with not so many super fast, powerful aggro decks.

https://www.mtggoldfish.com/deck/551391#online is the update of my RB control deck. The biggest gains it gets are from the removal of Copter and Emrakul, which squeezed it out either side. But Shock and Fatal Push help a bit, and Yahenni's Expertise is real. Probably not busted enough to be best, but just a solid collection of good cards can't be too bad.

Last but not least, Metalwork Colossus: https://www.mtggoldfish.com/deck/547677#online

Trophy Mage helps us a little bit here, as does Scrap Trawler. But the main improvements are, of course, the Improvise Cards. Spire of Industry shows up as well, and Walking Ballista being fetchable in this deck is pretty nice - I only have one copy main, but between Inventor's Fair, Sanctum of Ugin, and Glint-Nest Crane, you have pretty good chance to find it. Also the Crane is exceptional in this deck - there are approximately a bajillion (I think it's 29?) artifacts. And it can crew Aethersphere Harvester to keep you alive - possibly there should be more of those (swapping a Caravan for one seems good for a lot of metagames).

Emerging into the new Standard

With the banning of Emrakul, we have a new adjustment in terms of finding the best top-end in the standard format. One way we might try to go about doing that is to look at the next on the list of everyone's favorite cost-reduced Eldrazi: Emerge creatures. In particular, Elder Deep-Fiend has a combination of bulk and tempo that just might thread the needle into being the Mistbind Clique-esque threat that people were predicting when it was first released. Additionally, Distended Mindbender seems pretty well positioned, with most decks playing pretty important cards in both the larger and smaller slots. And in all these cases, the loss of Emrakul means much less chance of Summary Dismissal happening. Flashing back a Kozilek's return also seems pretty potent in the format in general at this point (though it still matches up pretty poorly against Torrential Gearhulk).

My first attempt to get back in this area threw 'the kitchen sink' into the deck:

This list was probably too unfocused to be any good. So I split this up into two decks - one focused on Zombies: https://www.mtggoldfish.com/deck/548961#online

One on getting value off of Trophy Mage: https://www.mtggoldfish.com/deck/548951#online
The nice thing there is that Trophy Mage gets you multiple 3s to emerge off of, for lots and lots of value:

My favorite, of course though, is my pick for the sweetest, and that has to be going off with Emrakul's Influence: https://www.mtggoldfish.com/deck/549046#online

All I really want to do is sac a Wretched Gryff to a Wretched Gryff, paying U to draw 3 cards. Is that really too much to ask for? This one is probably quite bad, because it's low on interaction and quite slow. But you do have a toolbox of interactive emergers, and if you can get a little time... well, you can really go off.

Wednesday, 1 February 2017

More Sweet and Artifacty Standard Brews

Time for the sweetest deck I've brewed so far:

 The basic idea is to have Inspiring Statuary in play, with either Trail of Evidence or Metallurgic Summonings, then 'go off' by casting card draw spell after card draw spell for only the blue mana. Pore Over the Pages gets you your U replenished, and the 1-of Paradox Engine allows you to reuse all that gosh-darn mana. There are a number of win conditions - Summonings has to be the main one, but you can mill them out with Fleeting Memories, or alt-win off Mechanized Production as well.

The main weakness of the deck is that it's slooooow and not terribly interactive. The sideboard is built to help that a lot, but it's still probably a bridge too far to be great except in a very warped metagame.

Next, we have several decks that are abusing the improvise mechanics and/or.... EGGS.

The main idea here is that if you can get a Foundry Inspector in play, you have 16 cards that cost 0 and draw a card on dying. Scrap Trawler and Pia's Revolution add to the value party, and Ravenous Intruder is your Atog of free sacrificio. Reckless Fireweaver and Key to the City allow you to turn these into victory. The main concern I have here is that we don't have enough redundancy and, though the combo pieces more or less function independently, so you don't need all of them, I'm not sure if they're good enough by themselves. The mana base is also very confusing, because you have a bajillion cantripping eggs.

https://www.mtggoldfish.com/deck/550198#online version is even jankier, playing Inspiring Statuary with Main-deck Release the Gremlins for hilarity - tap 4 lands and 3 eggs, draw 3 make 3 2/2s seems decent.

https://www.mtggoldfish.com/deck/547660#online is UB and more of a value approach to a similar kind of deck, and https://www.mtggoldfish.com/deck/550209#online is similar to that, but goes bigger.

Maybe my most hopeful for the archetype in general is this: https://www.mtggoldfish.com/deck/551382#online

I'm pretty sure Herald of Anguish is the real deal. RB Gives us lots of sac outlets to give us some reliability there. Scrapheap Scrounger allows us to have lots of fodder over and over. Unlicensed Disintegration is just a very good removal spell. We get some pretty decent ability to go grindy out of the board as well.