What opening move should I prefer in Hansa Teutonica with 3, 4, or 5 players, and why?
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As the first player in a 4 or 5 player game, I like to play a merchant (disk) and a trader (cube), one on each road into the action city. In the crowd I play with, these roads are highly contended, so I will likely have 3 additional pieces on the board before my next turn. This will leave me in good position to become the third player to get a third action, or perhaps the first to improve the movement or refresh skill, or to get the "remove 3 pieces" bonus marker. In short, I find it maximizes my flexibility while getting additional pieces onto the board.
While contending for the action city is a classic opening move, it by no means is the only way to go. I recently won a 4 player game using the policy of non-competition, and worked on creating as big of a network as possible, as that has the most bonus. In fact, I only gained one additional action the whole game, which balanced out because I didn't waste all those actions contending for the action city.
If you go first, then this question makes some sense, but if you're not first player, the correct opening move will depending both on the other players' actions and their play-style.