Straight forward really: What are the strengths and weaknesses of each Prize? What play styles do they complement well?
Also, apart from the obvious (late in the game), when should I take a Duchy instead?
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Trusty Steed and Followers are usually the top, and choosing which is pretty situation-dependent. Bag of Gold is always good, and in most cases I'll take Duchies (or not reveal my Province) rather than clutter my deck with Diadem and Princess.
When Followers shines, it's great. Discarding and cursing is incredibly harsh, but the penalty of taking an Estate can really hurt. If you're lucky enough to get an early Tournament-Province activation, I would hold off on Followers unless your engine is robust against the Estates. The Estates are obviously more tolerable if you've got a Baron or two (which is a great way to get to that early Province) or a decent way to trash them (Bishop!) with enough +card and +action to be reasonably confident of getting it done. If an opponent is behind in getting an engine set up, Followers can make it near impossible for them to catch up, but if +card and +action are plentiful and your opponent's deck is in better shape than yours, the Estates could end up hurting you more than your opponent.
Also ask yourself if the attack will be effective. If other attack cards are in the pool, either cursers or discarders, Followers' strength is redundant. Defense-wise, if your opponents have Moats or Lighthouses, you still take an Estate, while Watchtower or Jack-of-All Trades will just laugh at the attack.
This is my go-to. Originally I liked Followers best, but the Estates too often get me down. +2 Cards +2 Actions isn't so amazing, but it's never something to complain about. Especially if there's other +2 Action cards, Trusty Steed is excellent. And, unlike Followers, the earlier you can get it the better. In most games, I'll try to pick this one up first, and then hope that I get Followers too.
Like Followers, however, it's not good if it's just redundant. If you've got plenty of +card +action, you don't need the Trusty Steed, get something else instead. However, if your deck is just starting to really work, Trusty Steed can help just about any engine get rolling. As long as you have an action when you draw it, Trusty Steed can always do something useful.
Bag of Gold is good, not because it gets you a Gold, but because that Gold goes on top of your deck. This is best in a non-trashy, attack-filled game. If your deck is big and things are moving slowly, putting that Gold on top is fantastic. If the trashing options are poor, you're not drawing so much, and the Curses are almost gone due to a Witch, then you don't want Followers (little point without more Curses), a Trusty Steed may be most useful for its Silvers, but the Bag of Gold will be welcome, maybe even the top choice.
Do you have a bunch of Fishing Villages? Maybe your stack of Cities are activated? Get the Diadem! There's some situations where it will be great, and you'll know because you always know if you're ending your turn with a bunch of extra actions. If you usually have >=3 actions left, go for it.
Related rant: Bridge is a fantastic card, but it's power is only apparent when you buy multiple cards. If you play a single Bridge and use only one buy, it's equivalent to a Silver that you wasted an action on. If you buy two cards, it's equivalent to a Gold (with a buy, but costing an action--pretty good). However, with 2 Bridges in play, buying 2 cards and they're worth $3 each, buy 3 and they're $4 each...
Princess, if you buy a single card, is Silver that you spent an action to use. Buying two cards, it gets better, but don't bother with her unless you are regularly hitting $10, with an extra action, so that you can get a Province and something else worthwhile. Of course, if you're hitting $12 or more and already have the +buy, then Princess doesn't have much appeal.
In a two-player Big Money game, you should start buying Duchies instead of Gold when half the Provinces are gone. With more players, you should start even earlier. I'd put the utility of most prizes about on par with a Gold. In a normally paced game, when there are half (or slightly more) of the Provinces left, you should be getting ready for the game to end soon, and that means maximizing your score with Duchies. A Diadem you use once or twice could get you an extra Province, or it might not. The Duchy is money in the bank. Try to estimate how many shuffles you have left; if you'll only get a chance to play a prize once or twice, the Duchy may be the smart bet.
A fair amount of the time, Followers is going to be the strongest prize. If the board doesn't have another way to give Curses, and you are getting to choose a prize relatively quickly, giving out the only Curses in the game should have a pretty big impact. Also, bringing your opponent down to a 3-card hand will make it more difficult for him to buy those important Provinces.
After that, Trusty Steed tends to be quite strong. Be sure to think about all the options that it offers; it's quite common to just choose +2 Cards +2 Actions, but that in itself isn't all that great... it's a level 2 City, and Cities are only good when you get a lot of them. But all the options have their place.
All the others are more situational:
Princess can be necessary if there's no other +Buy card. Trusty Steed can be necessary if there's no other +2 Actions card.
I don't think I'd ever take Bag of Gold or Diadem as the first prize. But either of them can be good if other prizes are already gone. Bag of Gold is good mostly in a game towards the end when you need a little more boost to get the last or last couple Provinces. I rarely see Diadem worth more than 2 or 3, and when it is, you probably already have a deck with enough money anyway. However, I'll often choose it because I need that extra 2 or 3 this turn.