Game Like A Statistician: Cave Story’s Stone Shower

For my own enjoyment, I’m going to take some (temporary) space out and talk about gaming, and how sports metrics changed it for me in the past couple of years.

Recently, I’ve jumped on the bandwagons of a football writer, Bill Barnwell, who works probabilities into coverage of a sport where teams typically used them incorrectly or not at all when not nicknamed Riverboat Ron. Since I tend to play video games a lot more than I play sports, I’ve absorbed a certain way of thinking prevalent throughout his work – though execution ultimately makes the difference, one’s chances of victory* can be increased by playing in unorthodox ways even if you’re not the best at them.

Inefficiencies as they are typically discussed in the game of football tend to rear their head when a coach takes the traditional, “safe” option and ends up slow cooking his team instead of taking a chance to actually dive through the fire and come out of the other side alive. After reading these articles for a while, and dealing with one particular hallway in Daisuke Amaya’s Cave Story, I realized I was leaving a good chunk of my time and lives on the table by being too afraid to take damage.

The particular level I’m talking about is one straight hallway, the second section of the 3-part Sanctuary area. This level has two ways it can inflict damage – indestructible falling blocks that hit for 10HP, and flying enemies that do 5HP of damage. Take a total of more than 55HP worth of damage in that section, and it’s game over (for the purposes of simplicity, I’m ignoring the minor impact of hearts that can be grabbed in the middle of the hallway). In each case, when hit, one gets roughly 2 seconds of mercy invincibility. So how does one play to get the best chances of making it to the end of the segment intact and/or with the greatest average amount of hit points? For a true veteran player who can dodge 100% of the blocks 100% of the time, this isn’t a complex problem – just dodge all the blocks! For the rest of us, or at least for me, it’s still tempting to play that way, aiming for the dodge, even though we get hit some of the time. Taking damage is bad, period, right? Not necessarily. Because of the complex motions of both the Butes and the blocks, there are a very few venues available to the player to take in dodging completely. And because the blocks fall randomly, it’s entirely possibly to hit a snag on a region where it’s impossible to dodge, an outcome all the more likely if you’ve just gassed your booster in an attempt to break free.

If it takes 30 seconds to make it through the hallway, and player X’s dodging is only good enough to keep the rate at which the blocks hit him to 1 for every 3 seconds (close to what I’ve been clocking in that segment), then he’ll make it to the end of the hallway unscathed an average of (2/3)^30=.0005% of the time. The odds of making it to the end with 1 hit are (2/3)^28=.0001% (factoring in 2 seconds for each mercy invincibility he gets). With 2 hits, (2/3)^26=.0026%, with 3, (2/3)^24=.0059%, with 4, (2/3)^22=.01%, and, with the last non-fatal 5, .03%. All told, his chances of survival with that level of dodging skill are close to nil, less than 5 hundredths of a percent.

Now let’s try mixing things up a bit. Instead of trying to avoid the blocks for the entire 30 seconds, let’s take the same level of block-dodge skill, but intentionally take some hits from the less-punishing Butes. In the maximum case, he can take 10 hits from them (something that’s easier to do than avoid them, though still not an entirely trivial skill) while still progressing at the same rate and having the same dodge skill, and his chances of survival rocket up to 1.7%. That’s still a dismal number, but it’s literally more than 30 times better than what this player had before. What the numbers say varies significantly by how exactly good a player is at dodging, but the odds are 11% versus 2% for a .8 chance of avoiding damage per second and 35% versus 30% for a .9 chance of avoiding damage per second. At some elite point of block-dodging skill, as one would guess, the odds of survival get better for the elite dodge-em-all playstyle. Specifically, one has to be able to dodge blocks such that one takes fewer than 0.085 hits per second for the no-damage route to yield superior chances of survival.

While this particular section has an area where you can grind health/xp/what have you, the next portion only has a fixed number of hearts with which one can recover. So getting to that with a good starting value is moderately important, and preferable if you’re good enough to dodge consistently. It’s just that many players are not that good, and thus stand a better chance of conquering the area by letting the lesser damage happen and reaping the harvest of free time that comes with it. Even if a player only has a 2% chance of beating the rest of the level with that 5HP total, compared to, say, a 10% chance for 25HP, if they’re getting to that point 10 times as often, their fastest path to victory lies in taking some unorthodox risks.

This is hardly the only time in the game that such a problem shows up. There’s a boss in the bonus Wind Fortress level where one has the opportunity to repeatedly take one damage from a barrier that prevents one from leaving the arena, gain the mercy protection and avoid a number of attackers doing triple that amount of damage. Make it to the boss with 15HP or more, and doing so effectively cuts one’s odds of defeat to zero with good shooting. Fight head-on with the same amount of HP and there’s a decent chance your bar will hit zero before the boss’ does.

It’s also not the only argument in favor of a damage-as-insurance playstyle. Because when players run a level in games, they aren’t just losing, they’re learning how to avoid failure. So, if one isn’t yet to the point where they’ve figured out in theory how to beat the five-phase boss at the end of this level, there’s a lot of hidden value to be gained from getting there and simply fighting a losing battle for the experience with whatever form they end up being done in by. Of course, the same argument can be applied to increased block-dodging skill; choosing the right method is a matter of recognizing when you’ve plateaued on the skill level you’re at.

This philosophy has just been useful to me at times, and I hope it might be useful to some others. That said, I believe that games are entertainment, and should be played for fun. This column outlines the basics of one style of play, and is by no means the gospel for how to run your personal sessions. Go have fun, whatever you do.

*There’s a stat in sports that’s called win probability, the odds of a team winning a game in any given situation, and it’s really refined for football and baseball both.

3 thoughts on “Game Like A Statistician: Cave Story’s Stone Shower

  1. Pingback: ICYMI: Highlight Posts of 2014 | Animetics

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