Three of the four policies you can choose as your first policy in Civ V have the primary benefit of generating more culture to use unlocking future policies. I know I've played games in the past where my first three policy choices were actually all three of the openers under the plan of using the early culture boosts to ramp into more policies. Unfortunately one of the factors determining how much culture it takes to unlock a policy is how many policies you already have. So these minor culture boosts may or may not actually speed anything up. And even if they do speed things up in the short term it may well end up hurting you, especially since one of the early policies eventually does absolutely nothing.
But that's just a gut feeling... I want to actually look at what's going on with the policy costs and the different early policies and see just how big a boost they end up giving. Do note that even if it may not be much of a boost, if any at all, they do all unlock future policies and have secondary bonuses along with them so they can still be fine choices. Just likely not all three of them!
Tradition gives you 3 extra culture per turn in your capital city and 'greatly increases border expansion' though I've been searching for how much it reduces it and haven't been impressed. It sounds like you normally pay an extra (10*(t-1))^1.1 extra culture for tile t but with tradition that changes to (10*(t-1))^1.075. That's not nothing, but it's not greatly increased. No idea if that's actually the case; I may need to start up a game and hold off on taking tradition until late and see what it seems to go. It also lets you build the Hanging Gardens wonder.
Liberty gives you 1 extra culture per turn in all of your cities and lets you build the Pyramids wonder. This is worse than tradition in terms of getting more policies until you have more than 3 cities, but I think there's benefit to getting the culture growth going in your other cities to get their initial border expansions in quicker.
Honour gives you 33% more combat strength against barbarians and gives you culture for killing barbarians equal to the barbarian's strength value. This can range from being worthless if you have no barbarians to kill (like on an island map when your cities can see all of the land tiles) to being a pretty big number if you have barbarians to farm for culture. It also lets you build the Statue of Zeus which is good for conquering other people.
The cost for your next policy is a function of how many cities you own and how many policies you have. The function is c(25+21.03*p^1.7)(1+.3(n-1)) where n is the number of cities you own, p is the number of the policy you're buying, and c is a constant based on the difficulty settings and game speed. Right now in my current multiplayer game I have 3 policies chosen, but I didn't take tradition. (I bee-lined directly to the free settler in the liberty tree because my starting city spot sucked.) Would I have been better off grabbing tradition instead?
The combined cost for 4 policies would have been 220 culture. For 3 policies would have been 110 culture. In order for tradition to have paid for itself it would have needed to take me 37 turns to get to this point. It was more like 20 turns. And I wouldn't have gained 3 culture for all of those turns, since some of them I actually had liberty instead so I'd only be gaining 2 culture per turn with tradition first until I got enough to pick up liberty too.
Ok, well, in this game where I really wanted that first settler policy it was definitely right to skip tradition. But maybe I should go pick it up now? Let's look ahead a ways into the future. If I have, say, 4 cities each with a monument then I'll be rocking 13 culture per turn. Getting 3 more from tradition would take that to 16 culture per turn, or a 23% boost. If I look to the area of the table where I have 4 cities and 8 policies the cost difference for a 9th policy is 22%. So at that point it would be pretty much break even if I had a free tradition to start the game or didn't. But I've already played a bunch without it...
So it really feels like grabbing tradition at this point is just wrong in terms of getting more policies. I still might want to get it if I want some of the other policies in the tree (monarchy is pretty sweet) or if I want to ramp up my border growth. Or maybe I want to build the Hanging Gardens! (I did a quick test and my second border growth cost 10 instead of 15 which is way better than what that one formula on the internet told me. I don't know what's up with that, but knocking a third off is pretty significant if it keep bubbling forward.
But how about honour? It's the one I really care about. Robb apparently had a settler kidnapped by barbarians and I jokingly suggested he should grab honour to go beat them up. 33% increased combat strength is a pretty big deal, after all, and his city is stuck until he goes and frees his settler. But is he culture gain from honour viable? Tradition is worth 3 per turn and I don't want it... Could I want honour? It is important to note that this is a multiplayer game with no AI which makes it easier to keep barbarian encampments around to farm up more culture.
I did some searching and can find nothing to tell me how often units spawn out of a barbarian encampment. I feel like I've seen 3 units spawn out of the 1 camp I've found in this game so far (on top of the 4th that's guarding it) and the camp has probably been around for 15 turns? So one unit per 5 turns is probably a decent enough baseline. The power of the units that spawn depends on the techs learned by people in the game but having an 11 strength spearman doesn't seem too unreasonable? So each encampment generates 2.2 culture per turn. So if you could keep just 2 spearman camps around it would be a fair bit better than tradition in terms of more culture for policies. A lot worse for border growth since I don't think any city gets the culture from the barbarians, though. Getting 3 or more farms going? That's actually a pretty significant culture boost. If 3 culture per turn is break even with 4 cities then farming 3 barbarian camps for 6.6 culture per turn feels like it has to be worthwhile. Plus you get the vision of spawning camps and the 33% power boost to help take them out! And you may even get some city states who like when you kill barbarian units?
Will I be getting honour next? I don't know. I settled right beside my first barbarian camp so I can't farm it (it'll disappear when my borders expand into it I think) but I may be able to set up some other farms? I feel like if I have something really important to head towards then I should probably skip it, but that if I have nothing better to get then it's a fine use of my time. (Some policy trees unlock at higher tech levels so if you wanted to save up to get a bunch of city state or culture related techs fast then picking up a booster now could make sense.)
But I don't think it would be a big mistake. And if my settler had been junked by the barbarians then I think getting honour to help get it back faster would actually be a good idea.