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Historically, most competitive games have been, more or less, "symmetrical" games --games like chess or go, where each player starts out with exactly the same powers at his fingertips. This is nice, because at least then you know the powers are balanced between the two players. However, if you were designing chess, you'd still have to worry about the individual pieces being balanced. Yes, the queen is quite obviously more powerful than the rook, but the downside is that there's only one of her.
The idea of games with symmetrical forces has largely fallen out of favor in the era of video games, both for good and bad reasons. Some of the worst reasons sound like "players expect asymmetrical forces" or "more stuff equals better game". Some of the best reasons come from people like game designer and Street Fighter tournament player David Sirlin. In his article Balancing Multiplayer Games, he describes a scale, with symmetrical games on the left and asymmetrical games on the right.
RTSes are usually on the far right, fighting games not quite as far to the right. FPS games often are located toward the left, with class-based shooters being exceptions. He's careful to note that it isn't a qualitative statement to say that something is on either side; it's merely a matter of preference and how it works within the context of the game. However, I can say with confidence that the farther to the right on that scale your game goes, the more difficult balancing your game will be.
Compounding the problem of balance in games is the fact that a game can actually be balanced at the "professional" level, but imbalanced at the "intermediate" level. Meaning, sometimes a strategy will be way too powerful -- a dominant strategy -- at all levels of play except for the very highest.
Ideally, you want your game to be balanced at all levels, but for a moden video game of even average complexity, that's asking a ton. It's best to figure out what you want your game to be, and then focus your balance efforts towards that.
If you want your game to be competitively played, then focus most on the top-level players. Talk to the best players -- people who play tournaments (if you are so lucky to have tournaments for your game). If you have a high-score leader-board, contact the top people on the list and pick their brain. However, if you want it to be more of a "casual game", then hand the game to random people, even people who've never played a video game before, and see what they do with it. Tailor your balancing work around your audience.
To balance a game at very low levels of play -- for something like a party game, for example -- you need to make sure that everything in the game seems equally useful on the players' first few tries. It is less important, in this example, that after hundreds and hundreds of plays, everything still holds up. If a dominant strategy emerges after hundreds of plays in Mario Party, that's not much of a concern. Just so long as a dominant strategy isn't emerging after the first dozen or so plays, that's probably enough for such a game.
At the end of the day, you're balancing your game because you want to preserve your game's choices, and you want your game to be fair. This is true for a multiplayer game or a single player game, but there are some differences. In a competitive game, the spotlight will be shining much more brightly on the job you did of balancing the game, if for no other reason than because people will be looking for a way to get out of admitting they lost fair and square. But make no mistake -- if a single player game is to have any replay value, it must be well-balanced.
One thing I've noticed in my study of board games, is that they tend to have a much longer life than most video games. The main reason for this is, they are simply more balanced. Chess has survived for hundreds of years; go has survived even longer. The famous board game Settlers of Catan was published in 1995 and is still as popular as ever! Most video games, especially of the last 10 to 15 years, just simply are not balanced enough to stand a test of significant time.
So why are so many games so poorly balanced these days? There are many reasons, but the largest ones are very clear. Above all else, it's that we live in a world of "more is more". It's in our cultural DNA at this time in history that the more stuff in a game, the better the game must be.
In reality, this is not the case. Games are a delicate, intricate web-like machinery of cogs and pulleys, and throwing one new cog into the mix can cause the whole contraption to grind to a halt. We should be building our games with as few elements as is possible to create the experience we wish, while reducing the chance of the machine falling apart, but instead, a quick glance at the back of the box announces proudly that this new game contains "8,000 moving parts".
Another reason behind all the imbalance we see today might be, ironically, the prevalence of post-release patches, and even mods. The ability for developers to send out patches every couple of weeks seems to be a double-edged sword at times. Obviously, it's a good thing that developers are able to fix balance problems after the game has been released.
However, I can't help but wonder if there's a greatly reduced need to, at release or at any point at all, just "get it right". When you're developing for a Super Nintendo game and the code you're sending out to manufacturers will essentially be chiseled in stone, I think that sets a fire under developers.
Mods may cause a unique problem, in that a game can actually end up getting credit for being balanced, even when it's only the case because of player-created mods. An example of this is the game Elder Scrolls IV: Oblivion. Anyone who knows me knows how much I hate this game -- I can't ever seem to shut up about it. But what's really interesting is that most of the people who defend the game to me, when pressed, reveal that they actually haven't been playing Oblivion. They've been playing a heavily modded game, a game which is no longer the game Bethesda Softworks authored.
I'm not at all saying that mods are a bad thing, or that people shouldn't make them -- I love mods, actually. I think that it's important to realize the difference between your game, and the game that people modded your game into. House rules also fall into this category -- game designers should remember that when they're adding house rules, they're no longer playing the same game.