The player taking the first turn in Splendor receives the following advantages, regardless of player count:
- The ability to reserve any single development card before any other player, during the first round of turns
- The ability to acquire the correct gem tokens to purchase any single, specific lowest tier (i.e., green-backed) development card in round 3, should no other player reserve that card during rounds 1 and 2
- The ability to acquire, in aggregate, at least 2 of a single specific gem color during the first 2 rounds.
- The ability to receive a noble first, if the requirements are met, prior to any other player also meeting those requirements on their turn, during the same round.
In theory the advantages provide an opportunity to enact a strategy given the visible development cards and nobles, e.g., "get the development cards corresponding to the most common color among nobles." In practice, the strategy may not be feasible for the entirety of the game as new development cards are revealed and other players acquire them. Also consider that a player may take advantage #1 to the exclusion of advantages #3 and 4 (ceding those advantages to player 2) and vice versa. Certainly, as some games play out, the actions taken by the first player during the first few rounds may look sub-optimal in retrospect!
The player taking the first turn in Splendor receives no initial disadvantages. However, possibly based on player count and remaining gem token count, the first player may have to take a sub-optimal action starting on the third round. That is tougher to exactly quantify given the gem token counts at different player counts (and of course specific player actions). Here is one possible scenario, which illustrates the minimum non-joker gem token counts the first player can expect at the start of the 3rd round:
Now for (more?) anecdotal evidence: I've played over a hundred games of Splendor at 3 and 4 player counts, and I've never felt like player order mattered much. Although going last has caused me to make contrarian token picks in the hope that I get lucky when new development cards are revealed. I have won games vs good players even when I was the last player. I have also lost games badly when I was the first player.