Some other games have second mover advantage, or last mover advantage, or other advantages given depending on the tactics of the game and how it is structured. The real time boardgame Icehouse, for example, has noticeable advantages for acting first or quickly, but also big advantages to the person with the last remaining pieces to use. The interplay of first mover advantage and last mover advantage is part of what makes Icehouse's play strategic and interesting.
But for most board games, the designer wants to balance the sides and not immediately guarantee that the first player (or last player or whatever) wins automatically.
But the thing that occurred to me is this: what is bad for boardgame design can be good for roleplaying game design. When I'm designing an RPG, I don't necessarily want the sides to be equally likely to win. When my paladin goes down into the nest of orcs looking for treasure, I don't want a 50% chance of needing to roll up a new character. If I'm making a Star Wars game, I don't want there to be a 50% chance that Han Solo is killed by an average stormtrooper, or even by a nominally equally skilled bounty hunter. In general, the desired outcome is for play to be unbalanced in favor of the PCs. (It's also quite possible to deliberately engineer the game to be unbalanced against the PCs, if your game is a horror game or a tragedy.) So in 4th edition Dungeons and Dragons, a typical adventuring party fighting an equal level combat should rarely result in a Total Party Kill. In Dogs int he Vineyard, if all the PCs agree on a course of action, there's only a slim chance that the NPCs could oppose them. These imbalances are deliberate and function to help the games do their jobs.
First mover advantage is most likely to come up in a game where the system provides meaningful tactical choices. In 3rd and 4th ed D&D, there are a variety of tactical decisions a player makes each round of combat, and if the other side goes first they can limit or discourage certain courses of action. The advantage of going first is recognized as existing and thus characters can spend resources to boost their initiative scores in hope of going first. In Ganakagok, the conflict rules create a substantial last-mover advantage, so the going last is given as a reward for doing well on your initial roll (intermediate narration is a way of shifting around who goes last). I'm not sure, but I'm pretty sure that there is a meaningful firs tmover advantage in the conflict rules for Dogs in the Vineyard, what with the moving dice around and picking what to raise with, what to see with, etc. In any game with a complicated tactical aspect to it, it would pay to see whether the first layer or the last tends to win conflicts and structure things accordingly.
It's also possible to have a first mover advantage even in relatively mechanics light games. If we all narrate in parts of the setting, but I get to do so first, then you are constrained by what I have already established. (There's a microfiction in the Nobilis 2nd ed book where Satan claims that everyone in the world gets exactly what they want. When asked why there is pain and suffering and eternal damnation in Hell, Satan responds "Perhaps I get to go first.") In Fiasco, if I start the first scene, I have some ability to focus events in one way or another. Later scenes can try plot twists and unexpected motivations or try flashbacks to before my first scene, but what we saw in my initial scene will still be true on some level. In A Penny For My Thoughts, the second person presenting a scenario has some advantage over the first, because they can take the first player's idea and build on it or they can totally reject it and make an alternative scenario. Ganakagok uses this fictional first mover effect quite effectively: Players are ordered based on the age of their PCs. Younger PCs will act first, and explore an unknown world and get into trouble. Older PCs will be more constrained in what they can introduce, because the world has already been fleshed out, and the older PCs will be drawn into the conflicts of the young folks. The differences in acting first or last reinforce how each character relates to the fictional setting, even though there are few to no mechanics in play (in this part of Ganakagok, anyway). It is also arguable that later rounds of narration have their own sort of advantage, since they have more fictional material to build on. But it is clear that narrating first has greater freedom than later narration, which is a unique sort of power. Who gets to narrate first has an effect on play that the designer can use to channel play one way or another.
So to sum up: What is a problem for boardgame design can be effectively used to structure a roleplaying game in the way a designer wants it to go. By providing the first move to the GM or to a player, the designer can influence (but not guarantee) the outcome of an interaction. First mover advantage is quite likely to show up any time that players make players are making tactical or strategic decisions. But it also occurs on a fictional level, where the first narrator has a different sort of power from later narrators. Careful use of the ordering in which players and GMs act in a roleplaying game can influence the final game's product.