About balanced game.

About balanced game.

Balancing is a wide subject, and I don't mean to cover every of its aspect. Specifically, balancing difficulty and balancing for fun won't be considered. This article will try to provide a few model of games which are balanced. If your game seems to be like one of these, it is possible to balance it. Otherwise, the good news is : perfect balancing is not a necessity.

There is two way a game could be balanced: in a mathematical sense, or de facto. An instance of the first kind, Head or Tail, is obviously balanced, because each player have the same probability to win than the other. If the coin end to be on it's edge, it would be a tie. The other kind is games like Hex : it is a game mathematically imbalanced because the first player is known to have an optimal strategy (it has been proved, period). But in fact, nobody knows what this strategy is! Moreover, it is known (again, proved) that there is no draw in hex. So this game is de facto balanced because of it's complexity and it's near symmetry, and it will be as long as it's optimal strategy remain unknown.
A variant is to have an ever changing game, such that the best strategy is always changing too. This kind of balancing strategy is already in use, somehow, in the card game Magic The gathering. Their business model consist of printing new set of card every 3 or 4 months, and promoting the use of no more than 2 years old cards in tournament. This leads to an unstable meta-game.

For instance, in a Quake 2 death match, every player have the same probability of spawning at some point, the same move set, etc, resulting in a symmetric game. NB : the map doesn't need to be symmetric: only players' options must be the same. Of course, a team mode like "capture the flag" require a symmetric map. If the map is nearly symmetrical, it's good enough (de facto balancing, really close to perfect mathematical one).

Some game could be perfectly symmetrical, but not in an obvious way. Let's consider a simple one : player TANK has 100 health points and deals 10 damage per seconds; player DD has 50 HP and a 20 dps weapon. Obviously, they are attacking each other, and die if their HP <= 0. They gonna kill each other in 5 seconds, and it will always be a tie, except if damages are randomized: 5 to 15 and 10 to 30 for instance. 
In fact, such a game has an obvious rating of the strenght of a character: strength = hp * dps. When everyone have the same strenght, the game is balanced. 
So TANK's strength is 10*100 = 1000, and DD's one is 20*50 = 1000. This game is in fact symmetric. By the way, if you can calculate such value in an asymmetric game, I bet the game is simple enough to not be suitable for competitive play. Or you made a weird version of quake, in which the first player has double damage and his opponent has double health and friendly fire is disabled (<= guess why).

Such game tends to always have an equilibrium. If we add more strategies, ie, things to do, or symbols (for instance, the dynamite), it will only modify the equilibrium. But that's true only if players have exactly the same set of strategies, and this is the definition of a symmetric game. Oops.
In fact, when players have a different set of options, the RPS-condition may result in an equilibrium in favor of one player. I mean, one may win twice more often than his opponents. It could seems bad, but honestly, not all games are that balanced. Often, designers rely on the meta-game to balance things out. I mean there is some kind of RPS condition at some point, such as level 70 rogue beats level 70 mage beats level 70 warrior beats level 70 rogue. The end of a 1 minutes fight is determined by the class a player chose 3 weeks before! Game with "build" included in the actual fight doesn't suffer from this, that's why Starcraft is trivially better balanced than WoW.
A game is some kind of rock paper scissor if players make blind choice, ie they choose to play this or this without knowing what their opponent is going to do. Otherwise, it's just an "everything has a counter" kind of game.  

Balancing a game require only that the states "player 1 has won" and  "player 2 has won" have the same probability to be reached. Everything from the start to the next-to-last states could be utterly imbalanced. For instance, Blizzard has found during the Starcraft 2 beta that in Terran versus Protoss (a subgame of starcraft), protoss tend to win 70% of the time when they got a (pack of) powerfull units called carrier. For instance, if the state "Protoss made carrier" occur 10% of the time (my hypothesis), it means protoss achieve victory 7% of the time because of carrier, and Terran only 3%, which is +2% in favor of protoss. If in the remaining 90% of the game, protoss win 43% and terran 47%, the game would be balanced, according to blizzard's statistics (and my hypothesis).

Exemple of local imbalance and balance at the end.

This graph illustrate the idea of local imbalance : protoss has 70% chance of victory from the carrier node, still only 50% from the start. Probability (in percent) used here mean some degree of uncertainty more than randomness.

I have never seen anything like that. The idea is to make a game "simple enough" such as an AI algorithm could brute force it, and find imbalance if any. Basically, it limits the designers to make game as complicated as checkers, or chess if aiming for a de facto balance (if chess is balanced has never been proved). This doesn't mean "any turn based game", far from it.

Analog to drawing the main path of a maze, then adding paths to dead ends or loops. Build a winning strategy first, then hide it with tons of more or less useless options. It would be a top down design, where everything has to be under control, wich is quite unrealistic. Also, the main path should provide the 50 / 50 balance, so the gameplay must rely on luck or blind move, like in rock paper scissor (a blind move is better because players will not complain to the designer about their lack of luck). Dead end should be something like 40 / 60, 40 for the one who deviate (unconsciously) from the main path. But this kind of design could be extremely vulnerable to bugs. 

Balancing a game for fairness could be anything from trivial to über-complex. Depending on the game, business model, etc different degree of balance are needed. Before tweaking variables, it may be useful to check if the game could be reasonably balanced in some way.
Usually, near perfect symmetry + any complexity works well (go, chess, quake...) ;
Blind move (rock paper scissor, poker...) bring the insurance anyone could achieve victory ;
Game which include a build stage tend to be balanced only if build is done during the fight time;  
Renewing in a regular way avoid the game to be deciphered ;
Complexity bring uncertainty, and it's not a bad thing.