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Game Balance, brought to you by SCIENCE!

July 5, 2011

For a long time when thinking about game balance I fell into the trap of thinking that when something was “perfectly balanced” then all the strategies available to the player were equally viable. If they weren’t all viable, then what was the use of having those strategies available at all? This soon turned into somewhat of an existential crisis: Oh dear, games that are well balanced are actually devoid of interesting decisions – NOTHING MATTERS! RUN FOR THE HILLS! Silly me. All it takes is one look at a game like Starcraft to know that balanced games aren’t brainless, but for some reason I couldn’t accept it until I figured out a quantitative justification. Lame, right?

Well, you’re going to roll your eyes again when I tell you what finally calmed my mind: quantum mechanics. Yes, that quantum mechanics – the one where things really don’t behave at all like we expect up here in bulkland. Specifically the field of statistical mechanics, which acts as the link between thermodynamics and quantum mechanics, can tell us a lot about game balance. Statistical mechanics is fun (and eventually instructive) because we can take all the billions of quantum “microstates” in a system and say, “Well, I think we’ll just do some averages and call it good enough.” Using magic/math we can define “macrostates” such as work, heat, and free energy that describe the overall, yet abstracted, effect of the “statistical ensemble,” basically the total collection of the microstates.

Are you confused yet? How does this have anything to do with game balance? Well, let’s start with this: both statistical mechanics and game balance are about probability distributions, like so:

Oh dear. Boooooriiinnggg. Sorry.

Ok, ok, ok. Let’s give this some context:

                                                

 That’s better. GET THAT DRAGONITE, DISTRO!

Let’s assume you’re playing a competitive game versus an opponent. Each of you will employ a strategy to gain advantage over each other by spending resources. These resources could be anything: money, time, units, et cetera. Ideally, your strategy will gain you the most advantage by using the fewest resources – we’ll call this “total success.” (Substitute “great success” if you are a Borat fan.) The strategy that gives you the least advantage (or negative if you want to include it) for the most resources we will call “total defeat.” Between these two extremes there will be a continuum of situations, and smack dab in the very mediocre center we will define our “zero” point where the resources spent exactly equals the advantage gained. Because we are defining success and defeat as advantage minus resources, it will go from a negative number to a positive number. To make this simple, I’ve normalized each end at -1 and 1 to represent total defeat and total success:

I admit that advantage is a rather abstract concept – basically it embodies whatever will lead the player to victory. The easiest visualizer is victory points in a boardgame or capital that will help the player gain victory points. In a game like Starcraft, advantage could simply mean killing a unit, destroying a building, or even gaining good positioning that will multiply advantage in the future (advantage gained in this case could be defined as the sum off all surplus advantage gained in the future by the move). The definition of advantage will shift from game to game depending on the mechanics. Either way, a dissertation you will not find here. Just accept that there’s some not-quite-tangible thing called advantage – something you “know when you see it.”

Let’s fill in the continuum so we have something to talk about:

There we are. Now let’s use the language of statistical mechanics to describe this situation. For this given strategy, there are a number of game conditions that can affect the outcome of our strategy. This includes the opponent’s strategy, any luck elements, and pretty much every single other thing going on in the game. Depending on the game state, your ONE actual result will fall somewhere in one of the two probability curves. That direct result we will refer to as a “microstate.” If the game state is different when you use the same strategy again, you will end up with a different resultant microstate (ex. +0.75 vs. -0.5).

The entire probability distribution shown here can be called a “statistical ensemble” as it is in statistical mechanics, because it is an ensemble of every single potential result from using this strategy. Finally, we will use this statistical ensemble to look at what kind of strategy this is. What we see here is an equally large chance of a terrible defeat and a glorious victory. All in all, this strategy could be called a “risky” strategy or a strategy with a “hard counter.” These names define the “macrostate” of the strategy – how it is described in a nutshell. Just like we can say something is “hot” or “cold” by looking at the statistical ensemble of atomic motion, we can say a strategy is “risky” by looking at the statistical ensemble of results. If both curves were closer to the center, it could be called a “safe” strategy.

We can also think of strategies and resources like inputs into a function, where the function is the game condition and the output is the advantage gained. If the game is completely deterministic, then given perfect information of the game state each player could choose their optimum strategy and be right every time, as long as their strategies aren’t dependent on what the other player chooses.

Let’s go back to the existential crisis I referenced at the beginning. Remember how I was being dumb? By just looking at the distributions, it’s pretty easy to say “Well, this strategy will work 50% of the time, and it will work really well. The other half of the time it’ll work terribly. Seems simple enough to me. Boring, even.” The secret here is that these distributions are made up of strategies that are irrevocably affected by the game state. It is not enough to have a strategy; you must be able to accurately assess the game state and your opponent’s strategy to push your “microstate” into the best result possibly allowed by the statistical ensemble.

Now let’s talk about the actual balance of these distributions. At first glance it may be obvious that you want the areas on both sides of the zero mark to be the same, but that isn’t exactly right. Imagine we had a distribution like this:

There’s a good chance of total succes, and there’s an equally good chance of a minor defeat across ALL possible game states. Obviously unbalanced. The area of the curves do matter, but so does the position of the curve. If you have a defined curve closer to the zero point, it matters less to the player because the payoff is smaller, conversely if you have the same area closer to +1 it matters a lot because the payoff is much greater. Essentially, we want the summation of all probability-payoff products (PPPs) across all states on each side to equal each other. We would do this pretty easily using integrals. Again, though – not a math class. Sorry, you can do that on your own time. We have a number of options for balancing this – just to name a few, we could change the likelihood of the total success:

We could shift one or both of the curves to match:

We could change the shape of the curves:

The key point here is that strategies that are sure wins should have low payoffs, and strategies that are risky or hard to execute should have high payoffs. This is commonly called the risk vs. reward balance. If risk did not balance reward, everyone would simply choose the optimal strategy and be done with it. Properly balanced strategies will at times appear to be overpowered or underpowered (cue angry fanboys raging on internet forums across the lands), but if it is truly balanced then the statistical ensemble will have balanced PPP sets.

Based on your and your opponent’s skill level, these distributions will morph and shift, but generally the metagame will bring them back into equilibrium. If a strategy that has a counter begins to be used more and more at a particular level of play, people will begin to counter it more often, thus bringing it back into equilibrium. However, if the strategy has no counter, then it is up to the developers to adjust it so that it does not become the sole dominant strategy. In games that have a strong metagame (League of Legends or Magic: The Gathering, for example) the distributions will be constantly shifting depending upon the hot strategies at the time or people getting adjusted to game changes. Ideally, though, the curves on each side of the zero should hover around negating each other.

So when is a game “overbalanced,” then? Basically, when a player is unable to exert proper control over the result of his strategy, or if he feels like the balancing mechanics in the game are making his decisions meaningless. One overbalance-inducing mechanic is the preponderance of luck (a “luck driven” game). Over the long term, approaching all game-states, the game would begin to look balanced. In reality it is overbalanced because the player doesn’t really have ownership of the results because luck, a factor out of his control and difficult to predict, was the driving force behind the outcome.

Similarly, a game where all the strategies have been designed to be useful most or all of the time will be overbalanced because again the player retains little to no ownership over the results because he could have made any other decision and achieved the same end. A “balanced” game would instead give the player the ability to make mistakes by using a strategy that may be useful in some situations, but in the current game state it is not. A balanced game is all about letting players make mistakes and feel the consequences, big or small, but at the same time not feel cheated.

The degree to which overbalancing a game is “safe” is definitely dependent upon the context. For lighter party or family games (Clownshoe Romp), overbalancing is actually encouraged to prevent frustration, keep players in the game, and promote a casual, fun gaming experience. For heavier, more competitive games (Grognard’s Revenge 40,000), overbalancing is a death-kiss because players will get bored of the game  easily, and bemoan the game as not being skill-based even if there aren’t random elements built in.

So, we solved an existential crisis, talked a bit about game design, and had a good time. All in a day’s work, right?

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From → Mind Games

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