The following method can be used for any game in which victory is determined solely by some point scheme and in which turn order may be important. So it is not limited to Puerto Rico.

The bidding system itself comes from Philippe Keyaerts. It provides a wonderfully elegant and efficient way of resolving several auctions simultaneously. It would not surprise me to see this method used in other games, just as Keyaerts' method of handicapping open selections from his game Vinci found its way (in modified form) into Puerto Rico.

Having read the previous objections to the blind bidding method for determining turn order (and agreeing with most of them), let me suggest something different: using Evo-style bidding. Player A begins by bidding zero or more VPs for any of the five positions. Each player then bids for one of the positions in turn. If they bid on a position that another player has bid on, their bid must exceed the old one. The original bidder will then have to bid again when their turn comes around. The bidding ends when all five players have active bids, which means that all five positions have bids and the player order is determined. Each player subtracts the amount of his final bid from his score at the end of the game.

Using this method, a fair price for each of the turn positions can be determined without any surprises and without using randomizing elements. The players bidding first do have a slight advantage, but it is very slight--they only set the value of one position and the advantage is only real if everyone agrees on the true worth of that position, which seems unlikely. If you've established a clear pecking order in your gaming group, let the better players bid last.