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The higher a player's number (i.e. the later they get to move), the better their chance of victory.  It's become almost impossible for Player 1 to win against larger numbers of players.  More often than not, the “last mover” will have conquered the rest of the galaxy while you're still securing your hold on your first conquest.

To verify that it's not just me being a bad loser, run a few tournaments between 4 or more equal computer players.  (Well, 2 will do but it's easier to spot the pattern with more).  Check out the end-of-game statistics display, and click on any of the column headings to sort from “weakest” to “strongest.” By any of these orderings, the order of the listing will be very close to Player 1 — Player 2 — Player 3 etc, as in the initial ordering.  With some room for chance, the higher a player's number the better they do.

Reproducible: Always

Steps to Reproduce:
Play a game involving similarly-skilled players (such as, for demo purposes, identical computer players).
Actual Results:  
Running 50 tournaments between two “Computer Normal” players, I found that Player 2 won 64% of the time, which means that its chances of victory are almost double those of Player 1.  (The effect is more obvious with larger numbers of players).

Expected Results:  
Equally-skilled players should have similar chances of winning.  In my tournament example, the outcome was two standard deviations away from the expectation of 50% — putting the likelihood of this happening in a fair game at 5% or so.

I think this happened when the minimum travel time for a fleet was dropped from 2 turns to 1 turn, so that you can send off an attack to an adjacent planet in one turn and own it next turn.