Here are my personal recommendations under which circumstances to play with which player counts:
1 player (solo):
The solo version is of the “challenge your own score” type. It is useful for people who want to practice the rules or who want to increase their tactical and strategic skills at the game. Also, since you can adjust the difficulty, it’s nice for people who like extreme challenges.
This is best for experienced players who don’t mind the extra work of directing a third player (the “puppet”). The main advantage of this player count is that you can concentrate on one opponent and since the element of luck is very low, this becomes a chess-like match, in which you can plan your moves ahead.
This player count is best if you want to have a multi-player game in the shortest amount of time possible and with the least downtime. The disadvantage is that sometimes you will not be able to trade since the two other players are already trading with each other.
This player count is a good balance between the amount of player interaction and the duration of the game. You have can a significant amount of player interaction without the game taking too long. You will also usually have a trading partner (since even number of players).
Playing with this player count is like halfway between the epic experience of 6 players (see below) and the brevity of 4 players. Trading is not so problematic as with 3 players though, since there often is a player who cannot trade anyway for other reasons.
You can play at this player count if you want an epic game with lots of player interaction. This can take about 5 hours though.
Even player counts have the advantage that usually everyone can find a trading partner.
Lower players counts have the advantage that the game is faster.