Last year I wrote an article entitled Games Are Not Supposed To Be Fun . I got a lot of flak for that article, so it's only natural that I push the same buttons with a similar idea, namely that games do not have to be fair.
Fairness in games is already not as immutable as it may seem.
What exactly makes a game fair, anyway?
When you sit down to play a game, there's an unstated hope that the game is equally winnable by either player. For games of skill, that's hardly likely. The player with the greater skill has an advantage; but we call that a fair advantage. What about the player who's more tired?
For games of luck, it may be theoretically possible to start the game from a fair position, yet by the end of the first random occurrence, the game is no longer fair. Now one person is in a better position to win than the other, possibly or probably through no fault of his or her own. Is that fair? Of course not. But it's a special case of unfairness, which is called them's the breaks.
We regularly accept the idea of unfairness in our competitions and games, yet we still cling to an illusion that games must be, somehow, presented as fair.
Well, actually we don't.
The odds are in the house's favor, but people still gamble. In this case, the reason people are willing to accept playing in an unfair game is two-fold.
First of all, we grudgingly accept the fact that the entity providing the gambling venue is entitled to be compensated for their outlay. In other words, the situation is one of unequal stakes. The casino laid out more money before coming to play, and therefore deserves a better percentage chance of winning.
Second of all, the thrill of winning in gambling is related to the the odds being against us. If we flip coins, it's not thrilling to win exactly half the time, but it is to win slightly more. Simply breaking even at a casino is an accomplishment. This situation is one of fighting the odds. Accepting the unequal nature of our winning conditions, we feel an accomplishment for whatever gains we achieve.
War games can be built around the principle of fighting the odds, and undoubtedly I'll hear about a bunch that already are. If you know that one side has only a 40% chance of winning the game, you're not going to complain about the unfairness of the game, you're going to relish the opportunity to beat the odds.
In traditional RPGs, one player plays the Game Master (GM), and the others play Player Characters (PCs). The GM can do whatever he or she wants, including kill the characters, define the encounters, and so on.
This is accepted because the enjoyment for the GM comes from an entirely different source than the enjoyment of the PCs; they are effectively playing two separate, intertwined games. For the GM, the creation and narration are storytelling arts with captive listeners. Meanwhile, the players are trying to be clever, or they are gambling.
The GM is supposed to not run the game unfairly for the players. Each player should have approximately equal opportunities to succeed within the game, and the group should have a fair chance against whatever opposition the GM throws at them.
The situation of GM and players is one of differing roles. Players with more experience in a game, familiarity with otherwise hidden elements, or with control over the game play voluntarily hinder themselves in order to ensure that the other players have a good play experience. The person in this position is the game leader. The other players are simply the players.
Ladder card games are often deliberately unfair to some of the participants. For instance, President  is game where you derive benefits to your position based on how well you succeeded in the previous game. The President (who won the last game) is given some better cards from the hand of the Peasant (who lost the last game). This serves to make it easier for the President to win the current game, perpetuating the win/loss states. How unfair!
The first reason why people enjoy playing with this mechanism is that it is a situation of fighting the odds. If you can succeed as the Peasant, and it's possible, the victory is sweet, especially as you get to knock the President off his or her throne.
The second reason that this works is that it is a situation of role play. The players are role playing the unfair advantages and disadvantages that have occurred in past societies, in fact that they see occurring in their very lives. And so, although the situation for one or more players is patently unfair, the unfairness is subsumed to the sense of story being enacted.
Which brings me to art, or the message possibilities of games. Whether games are meant to be played for entertainment, or whether games are installed in a museum and meant to convey a message without regard for fun or repeatable play, a game can convey powerful messages by virtue of the unfair positions allotted to the players.
A peasant feels like a peasant when he lacks the resources, advantages, and hopes that a prince enjoys. Unequal doesn't necessarily mean unfair, but also doesn't require it.
The principle problem with unfair positions are when dealing with young children, or with people who are expecting fairness, or who equate fairness with fun. If you know what you're getting into, and there's challenge and at least possibility, the endeavor can be intense, vivid, and fun. That's because you measure your success in these situations against the expected outcome of someone in your situation, not whether you ultimately win the game. 
One More Thing
All of the above explorations about the nature of unfairness in games assume a certain egalitarianism between the players. In other words, the two people playing the game come to the board as equals, regardless of their actual status in real life.
Our culture rebels against the idea that a rich person should gain an advantage over a poor one if they play a game together (excepting the ability to pay for better training). But this is not always the case; and even if it were always the case, it is possible to envision a situation where it was not the case.
Consider a situation where the nobles gather once a year to play games with the peasants. The nobles play at an "unfair" advantage, but not egregiously so. Any peasant who can best the noble at the game wins something. Or in a company, bosses and employees sit down to play games once a month, where the bosses also get certain advantages.
We may group these situations as ones of unequal stakes. That is, the nobles offer up an ante, while the peasants don't. No one is forced to play the game, so we could accept the situation as fair, even though the game itself is not.
Now consider a game where your starting advantages are based on the first letter of your last name. People with a last name starting with "B" have an in-game advantage over people whose last name starts with "G". Wherever you go, the advantage or disadvantage goes with you.
Again, no one forces you to play the game. And likely, those with an advantage would offer bigger stakes to those without it, if stakes are involved, in order to make the game "fair". If they don't, the players at a disadvantage could get their enjoyment out of fighting the odds. It could work.
If the advantage were given to those of a certain race or religion, we're talking a different scenario; now we're entering into the field of art, politics, and morality. This idea will offend some people - it might offend me - but it could also spark ideas, if done right. Likely as not, the game wouldn't get much play, or those who played it would ignore the parts of the rules that offended them.
But this just goes to show how a deliberate introduction of unfair elements can be entertained when designing a game.
 "Not supposed to be fun" is in answer to those who tell me that "games are supposed to be fun"; the emphasis is on the word "not". The games you play at your game group, with your friends or family, or in nearly every situation must be fun, because you play them for entertainment. But that doesn't mean that games have to be fun; a game can be created for non-entertainment purposes. For example, games used entirely to raise awareness about an issue, without any intentions of having them be replayed, can be considered games, even if they are not fun.
 Similarly Tichu for 5+ players, and The Great Dalmutti.
 See my articles on alternatives to winning and losing in games for more information on this subject (sidebar of my blog)
P.S. An article with some similar thoughts
As you said, wargames are a very good example. The wars and battles of history were not always balanced. Often, one side had a distinct advantage over the other. My favorite game like this is Commands and Colors: Ancients. Many of the scenarios for this game are unbalanced. The rules suggest playing to victory, then switching sides and playing again. But my friend and I almost never do this. If we have the time, we prefer to explore a different scenario.
Traditional RPGs were concerned both with party balance (that is, fairness among the players) and game balance (fairness between the party and the environment, or the players and GM). A number of modern storytelling RPGs, though, toss this approach completely out the window. Some are not even about besting the odds. In these games, players know ahead of time that their characters are destined to die or go insane. The fun is in developing an interesting or dramatic story.
I'm still unclear on how you define "fair" in this context. You seem to equate fairness with gameplay symmetry from your examples; does this mean that an asymmetric game (e.g. OGRE/GEV, or Scotland Yard) cannot be fair, even if it is balanced?
The other potential definition of "fair" is that all players have an equal chance of winning, perhaps after controlling for other factors like skill. But even then you run into edge conditions. Is Scotland Yard "fair" if Mr. X wins 50% of the time (since it's him as one team against one other team), or 20% of the time (since it's one player against four)?
Ultimately, I think that "fair" comes down to player perception: does the player's preconceived notion of their chances of winning match their actual chance? This covers all of your examples. Gambling is okay even though odds favor the house, because the player understands the odds and wins about as often as the odds would predict. (A player who misunderstands the odds of a casino game may very well feel that they got "cheated" and then stop playing.) For games of skill, a player who understands their skill relative to their opponents' can realistically know their chances of winning ahead of time -- in this case, the only time someone calls "not fair" is when some minor random element ends up swinging the game in a way perceived as being too unlikely.
RPGs are an interesting case. Consider a sadistic GM who decides he wants to kill off the entire party. If the players perceive the death as arbitrary and unavoidable, they'll call unfair. But if the GM drops enough hints to make the players *think* that their death was their own fault (for example, by not paying attention to a few dropped hints that there was a trap in the room) then they will accept the outcome as fair -- even if the GM had no intention of letting them escape alive :)
David, thanks for the comments.
Ian, I don't think exact symmetry is required for fairness. But, as you say, players must acceptt the aseymmetry as fair or unfair, before starting.
Regarding Scotland Yard, whenever I refer to players, I mean sides; there are only two sides in SY, so fair must mean 50/50.
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