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The rulebook says:
If the highest scores are tied at the end of the game, the tied player who has had the fewest turns wins the game.
What advantages does the first player have over the other players?
Do these advantages overcome the detriment of losing in a tie when the other player has had fewer turns?
What effect does the number of players have on the advantage/disadvantage?
Yes, it is certainly advantageous to go first in Dominion.
There are some good threads about first player advantage on the DominionStrategy forums.
Here is what Donald X thinks about first player advantage. Donald X actually underrates first player advantage based on faulty understanding of optimal play with a disadvantage (his "proof" seems to require symmetric play, which is not optimal under the asymmetry that first player advantage can induce) and hence he understates the advantage.
Here is an expert player stating the reasons for first player advantage.
There is also copious data supporting the idea that first player has an advantage. Here is some data analysis I did that corrects for second player is better confounding problem @Jefromi mentions. It says that first player advantage is worth about 2 or 3 points overall.
There are many advantages to going first.
Your 3rd turn is safe from attacks. For instance in a game where people will discard down to 3 "every" turn, you will get 3 turns of playing with 5 cards, while everyone else will get 2. This advantage can be very significant as dominion lead can snowball.
You will have the most chances to give away curses.
You have first dibs at all piles. If any pile of good cards runs out you have a chance to have the most.
You can take a province and end the game denying everyone more turns. So in a game where everyone is tied after an equal number of turns with 1 province left, you can take a province and win.
The tiebreaker in no way offsets this.
I would recommend letting everyone take the same number of turns, and letting players buy virtual cards from depleted piles on the last round. Even with this and the tie breaker i do not feel like it funny offsets the first turn advantage.
The only thing that may help more than going first is 3/4 vs 5/2 but that is fixed pretty easily
Some simulator data with simple mirror strategies might help to make this more visible:
Sea Hag is the biggest offender: if both players go for Sea Hag, the first player will win almost twice as many games as the second player and in 4-player the first player will win almost 3 times as many games as the player in the 4th seat. Cutpurse's cumulative effect is clearly evident when we add more players. (Tact/Vault is the double Tactician Vault deck and HP/Baron is the Hunting Party/Baron deck)
The obvious thought is that if you scored the same number of points with one fewer turn, you did "better" (and thus deserve to win the tiebreak).
