An "unbalanced" though symmetric game

I've observed here that symmetric games are usually perfectly balanced, except that there may be an advantage in order of movement. This is why, in many of my symmetric games, I've tried to eliminate order of movement or at least associate it with some factor that players have a chance to control.

Chess, for example, is a symmetric game with a big advantage to first-mover (white). Other games may have an advantage for last-mover. When I playtest a symmetric game with a set turn order, I try to record the score by move order so that I can look for patterns of advantage.

Recently playing a four player Wii game involving Olympic events (I don't recall the name of the game), I saw a symmetric game that gave a big advantage to later movers. This is not so much inherent in the game as inherent in the situation, where none of the players had played before, and some had not played the Wii before. So as we played we had to figure out the different controls for each event, and how we could succeed. Those who played early in turn order were disadvantaged because they had not seen as many attempts by all the players as those who played later.

The solution would be to randomize turn order. So the player who goes first in the first round of an event, might go third in the next round, then second, and so forth. I'd suspect, though, that Nintendo would respond that this would confuse the players, so just go with the disadvantage.

Once the players are familiar with the event's controls, the advantage is still with those who go later, as they have some idea of how much they have to do to win the event, which tells them how much risk to take. Here I might decide that in each round after the first, the players play in order of the standings, so at least the last-mover would be the player in last place.