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I help designers improve their games.
I've seen three different strategies, one of which appears to be better than the others.
(A) 15-point focus
First, get a cart so you can hold 2 gold. Then collect gold 2 at a time to ramp up to about 6 gem-carrying-sacks. Focus on fulfilling 15 pt commissions when able. When unable, focus on diversity of colors.
This strategy benefits from the most efficient point engines (15 point cards + diversity). It is also very robust against luck/memory when collecting commissions, since it usually has gems lying around and can fulfill almost any commission it buys. Similarly, it is robust against other players pursuing the same strategy, since you can don't rely on 15-pointers, you just profit from them.
(B) Gold & Silver
Try to get sets of 3 silver missions, then deliver them all at once. When unable, get gold missions. You'll likely get the workshop for both, giving you workshop bonuses and (later) snagging other colors when the silver/gold craftsmen run out.
The workshop bonuses are not as efficient as diversity bonuses, but in theory the relative ease of finishing gold/silver missions makes up for that.
The drawback is that you need to find sets of silver missions. There is no silver mine on the road where you turn in silver missions, thus you really need to be turning in sets of 3 at a time, to keep up with players who are doing colored gem missions. This leaves you a bit at the whim of luck when it comes to turning pages to find the right missions. IF you end up getting unlucky and have to turn too many pages, then you need an extra silver mine stop to be able to fulfill the missions, which costs you 2 turns, not 1, since you'll always cross a city along the way.
Worse, if two people do this, you will definitely both lose. In contrast, the other strategies suffer less from doubling up.
So, this approach looks good in theory, but it's unreliability catches up with it over the course of the game.
(C) Rush for diversity
Don't buy a lot of extra sacks, and make sure you have at least one 'wild' cart. Start fulfilling missions right away. You'll be the first to each stack, and the first to getting all colors. Diversity bonuses (10 points for each color) is very efficient.
Sadly, getting to a color first is not that important. What's important is to not get there last. There are actually plenty of each tile to go around in 3-4 player games. It anything, your rush will randomly hurt an opponent, but won't really help you. That's a good strategy for second place, but not a good plan for winning.
A seems to be the strongest strategy and the most reliable, and thus it seems to dominate barring notable swings of luck. When multiple people play A, they somewhat dilute each other, but not that much. So, either A wins, the 2nd/3rd person to pursue A is kingmaking, or everyone plays A. None of those outcomes are very satisfying, so I hope I'm missing something.
I think the game would be better with fewer craftsmen in each color (to help C be more viable) and more points for gold/silver missions (to help B be more efficient).
Caveat: This analysis is based on 3 player games, but I suspect the issues with B will only worsen with more players.