Generally (I hope I get this explanation right), the PRNGs on all players' computers are the same algorithm, with the same seed, executing the same player comma

Generally (I hope I get this explanation right), the PRNGs on all players' computers are the same algorithm, with the same seed, executing the same player commands in the same order. So as soon as the game starts, you can predict what all future calls to random() will return. However, you can't predict the other players' actions, and those actions can insert an arbitrary (and unpredictable) number of random() calls between now and when your next action executes, so practically there is no gameplay advantage you can get from that avenue (AFAIK).