Some people love slot machines. They love sliding their coin in the slot, watching the wheels spin and the machine ding out the results. Some people love chess. They love looking at the board, enjoying the satisfaction of besting their opponent based solely on skill. For board game enthusiasts, the dirty middle is the answer.
Luck has always been the bane of board gamers, at least in America. The most popular board games have always been largely luck-dependent. Monopoly, Candyland, Chutes and Ladders, Risk, Uno, Sorry!, The Game Of Life--all of these games depend significantly on luck. If the dice don't go your way or you don't draw the right cards, it doesn't matter how well you play--you're just plain not going to win. Modern board games try to avoid this. A better gaming experience--so it goes--is when players have control of their destinies on the board.
Modern board game designers try and reduce the amount of luck as much as possible. This trend--which appears to be abating--probably reached its pinnacle with Puerto Rico, where the only luck was in the appearance of various foods for choosing. Other than that, each player had the same opportunities, at least based on their previous transactions.
Of course, having no luck at all can also be a problem. Luckless games tend to be monotonous and repetitive. Aside from chess and go--and possible games like Diplomacy and the GIPF project, which have centuries of history to back them up, successful luckless games don't come around very often. To create a genuinely decent luckless game, there can't be a dominant strategy that works every time--something very hard to pull off and still have the game have no luck and still be fun.
So most designer games try to minimize luck, and even American-style and war games attempt to curb the luck factor. Can luck be both an enhancement to a game without also breaking it?
There are a few theories. Now, before we go much further, I know full well that there are plenty of philosophical ideas about what constitutes luck, with rival theories landing on either 1) Everything is luck and 2) Nothing is luck. Both are wrong, and while I have my opinion, I am more or less going to keep this restricted to just standard definitions of luck.
The Law of Averages: When you are rolling dice and/or drawing cards, the results are important...but each individual incidence becomes less important the more of them there are. If you roll three sixes in Risk, it sucks--but the chances of you rolling sixes each and every time you attack are infinitesimally small. (And, no, the Brazil-to-West Africa route is not rigged.) Getting bad luck isn't so bad when you know you are going to have plenty of opportunities to make up for it. Now, each incidence of luck can be important--that roll to defend a worthless territory went your way, but the crucial game-winning roll got flubbed--but on average it won't. This, to some, is a mark of good balance. Overall, all players will average the same results, so the game is balanced in that regard; however, since individual rolls will occasionally have big impacts, there is still a level of risk involved.
The Long-Term Suspense Factor: If a game more or less came down to both players rolling one die and the higher player wins, no one would play it. If there are a hundred rolls, it's still a bad game, but at least there is an element of suspense--even though roll 100 has the same weight as roll 1. Likewise, in a game of 1960, it often comes down to a handful of swing states to determine who wins, and this is chosen by pulling a few cubes out of a bag. If they're red, hello Tricky Dick; if they're blue, welcome to Camelot. Some people argue that the entire game comes down to essentially a coin flip. This is looking at it the wrong way; every single choice you made got you to that point at the end of the game. How you played determined whether you were fighting over California and Texas or whether it was Maine and South Dakota, and whether you both had five cubes in the bag or you had two while they had eight. While luck is the determining factor, players are framing the entire situation in their favor.
Risk Management: This is a valid way of taking essentially complete luck-based mechanics and converting them into strategic decisions. Poker is the most well-known of these; players manage their hand, which is drawn completely by luck, and then uses bets and/or draws to "manage" it. Many card-driven wargames also use this. If you are dealt an awful hand, use the rules of the game to minimize the damage, and good hands can be made great with proper card play.
Beginner's Luck: Some games randomize the starting positions, but everything else is luckless. This forces the game to be fresh each time you play, but still adheres to the purity of fairness. Making sure the initial setup is balanced takes some skill, however.
Universality: This is when there is luck in a game, but it's theoretically available to everyone. The Settlers of Catan uses this; if a 5 is rolled, for example, each player with a building next to a 5 gets a benefit. While players aren't equally affected, there is equality of opportunity--anyone can build next to a 5, if they choose to. Event cards often have the same effect.
Other Players: This is where the philosophers start getting into it with one another. The actions of other players is sometimes considered "luck." I, personally do not; each other player is making a conscious decision, and this is not luck, even if that player makes their decision based on a coin flip. However, the outcome can often seem the same--you could be hit by three sides at once, for example, or nothing. Of course, I believe that managing the other players within the rules of the game is part of the game, and not some completely random element. But the similarities are close enough that I can see using the same tactics against other players you are uncertain of the results for as, say, a set of dice.
Of course, with all of these ways to make luck not as much luck, what is good for a board game design? I tend to enjoy games where outcomes are uncertain, but as a player I can define what my results are going to be--for example, in Risk, the luck in battles is always going to be the same odds, but you have the option to determine when and where to attack, and for how long. Sometimes designing this is easy, and sometimes it is not. A skillful designer will be able to effectively reach that balance.