Emergent Mathematics pt.2

Tactical Asymptote

I've explained the difference between tactics and strategies previously. A tactic is a general plan of action while a strategy is a specific one. Obviously, the key modifiers "general" and "specific" are somewhat vague here. It works best to conceptualize tactics and strategies as existing on a continuous spectrum that runs from the most general tactics to the most specific strategies. On the extreme tactical end, it's clear why such plans are tactics. "Keep moving" and "Stay Alert" are examples of the most vague plans of action. Notice how no information about the specific game/objective are communicated. I could "keep moving" in a racing game, fighting game, FPSs, RPG, or even Wii Fit.

On the extreme strategy end of the spectrum, an example would be something like... In Super Smash Brothers Melee playing as Kirby in a matchup against Falco on Final Destination if you have acquired Falco's powers, stand anywhere between 1 and 3 Kirby lengths away from the edge of the stage and use a SHFFed laser to force the opponent into a defensive position. Take this opportunity to go for a grab and throw Falco off the stage. Immediately follow up with a approaching dash-short-hop 2x lasers to prevent Falco from using the Falco Phantasm, Falco Fire, or getting much height off his remaining jump. Regardless if Falco avoids the lasers or gets hit, you should be close enough to reverse hammer (side+B) to drop Falco even lower beneath the stage preventing any successful Falco Phantasms for recovery. From there Falco Fire is his only option. Proceed to throw out 1-2 meaty back+airs and finish Falco off with a strong back+air saving a few jumps to return to the stage. If Falco tries to spike you with his side+B, because you're Kirby you can meteor cancel the spike using a spare jump and still recover. 

Notice the specificity! I could get even more specific covering even more emergent options and including more detailed data (like frame data), but I think the point is clear. It's easy to identify the most extreme tactics and strategies. But I wondered if we could pinpoint the exact level of specificity where a tactic becomes a strategy. Is this even possible? Am I merely jumping into a black hole of semantics? The basic idea I'm pursuing isn't simple. The point where a tactic becomes a strategy I'm trying to identify represents a specific piece of knowledge necessary for developing or using strategy. As long as you do not have this knowledge, as long as you're lacking this particular skill, no matter how much more specific knowledge you gain in any other area, you'll never quite reach or move past threshold into the zone of strategy. Mathematically, this concept reminded me of asymptotes. Thus I've coined the term tactical asymptote for this theoretical threshold between tactics and strategy.

When I first thought up a way to measure the tactical asymptote, I figured I could find a point where simply reacting to the in-game conditions would not be enough to win. This idea was, if tactics are general and strategies are specific, then tactics rely on STM and strategies require LTM (at least for games with more than 7+2 complexities/rules; ie. most games). A player who only reacts to a situation fails to plan ahead or anticipate because these plans require knowing specific things about the challenge and its possibilities. If we could find a point in a challenge where simply reacting will not result in victory, then we've found the location of the tactical asymptote. Or so I thought.

We need an approach that is a bit more sophisticated than simply looking at whether an adpative tactician succeeds or fails. First of all, we can't really be sure what a player reacts to or why. Did I react to Concrete Man's jump? His missile? Or did I react to something outside of the game because I'm tense? Isn't nervousness a result of anticipating possibilities? Therefore reacting off of nerves can indicate increasingly specific, strategic considerations. Furthermore, a chess grand master can react to a novice's opening and crush him/her using a memorized, specific, text book strategy. The difference between strategy and tactics has nothing to do with reflexes, playstyles, or when one decides to act. The difference is solely within the specificity of the plan.

I decided to frame the investigation to analyze one case at a time. By narrowing our focus to one game and a specific level of the metagame or a specific interplay barrier, we have a much clearer way to break down and rank the all the potential knowledge at play. Because an interplay barrier is a strategy (specific plan) that one player can force on an opponent, we can clearly label the knowledge needed to counter the interplay barrier as strategy, and everything else as tactical play or useless information. If the opponent doesn't have the knoweldge or the other skills to overcome the barrier, no level of tactical play or reasonable level of luck will produce a victory. Finally, because we're setting a reference point at a location within a game's metagame, we don't have to worry about future discoveries that may change the context of player actions. We're merely looking for the specific knowledge necessary to overcome a specific and significant opposing strategy.

Competing in multiplayer games typically requires lots of knowledge and strategy. The better you understand the constantly morphing metagame, the better strategies you can employ on the battlefield. For the sake of illustration, it would be better for us to look at a single player game because the challenge is fixed. Let's look at Mega Man 10 bosses on hard difficulty while only using the M.Buster, the default weapon. I wrote this article series about MM10 and its bosses as I attempted to earn every achievement in the game including the aptly titled "Mr. Perfect." Needless to say, I needed a complete strategy to overcome some of the challenges. A complete strategy takes into account all possibilities and includes counter strategies that when executed correctly have a 100% chance of success. So, if the goal is to beat a boss without taking any damage and only using the M.Buster, we know that anything sub a 100% complete strategy is incomplete. Using an incomplete strategy relies on tactics to fill in the gaps. In this case, because the bosses are so difficult, anything short of an 100% compete strategy is just rolling the dice and will probably end in failure. Knowing the set of specific knowledge needed to develop these complete strategies, I can confidently identifywhich plans fall on the tactical side and which fall on the strategic side.

Even in the Mega Man examples above, there's more variation and strategy one can use after crossing over the tactical threshold. Just because you can avoid getting hit doesn't mean there isn't more strategic room to figure out how to more efficiently take out the boss. There's also room to develop strategies on how to escape dire situations if you mess up executing the complete strategy. 

Being able to grasp the concept of tactical asymptotes and even map out specific cases gives us great insight to the relative importance of various gameplay complexities. Because we can determine the specific knowledge needed to reliably overcome or master a challenge, we can create stronger criteria to evaluate difficulty design. If the player needs to develop certain skills to do better at point C, we can look at points A and B to see what they teach. Because strategies and tactical asymptotes deal with specific complexities, it's difficult to extrapolate the results to give us insight on anything outside the specific case. In other words, understanding the tactical-strategic spectrum for Advance Wars tells us nothing about the tactical-strategic design of Star Craft. 

In part 3, we're looking at pigeonhole mathematics.