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.