Not to exclude anyone, but I like to collect fresh impressions and I can't do that if you've read threads like this in advance.
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So, the name "bargaining power" is only so descriptive, and it's a bit easy to confuse with influence if you are new to the game.
Bargaining Power: This is the amount of current political clout, or favors you can call in, or in general social currency you have, at any given time. You gain this as you do missions, and you cash it in in exchange for things you want later on. Typically for political deals, but sometimes in a barter sort of fashion. It is not money, as you are not paying anyone off. This doesn't buy you goods or anything like that. Rather... it convinces people to do things.
As Keith put it at one point (to paraphrase), it's "the number of kneecaps you can break at the moment." To some degree BP represents the amount of respect you've gained -- but from that angle cashing it in and then earning it again doesn't work. In a lot of ways it does represent your "political capital," but that sounds really odd and it's not just something you use in politics. You can use BP to convince mercenaries of stuff, and you can use BP earned with one race to convince another race of stuff.
In a very very real sense, it's that sort of barter system of "you do this for me, and I can get you what you want, because I know a guy" sort of system. That's what this is representing, just in an abstract way. Your computer handles the details of getting what the other party wants, let's say -- you just see the number.
Changing this to money would be simpler in some ways, but I think it sends the wrong message. That would make sense for dealing with the mercenaries, but when you're convincing governments to do this or that, it makes less sense. I don't want this to be like you're bribing government officials -- there's already a mechanic for that, anyway. This is rather more like lobbying, minus the bribery part that can go with that. (Although, some of the "pork barrel" deals actually kind of do mirror what this represents, in some ways).
So one big thing that this could be called is Favors. Or really, Favor Points, though that sounds odd. Though not as odd as having 506 Favors gained from a mission, and then spending 2000 favors to get the Skylaxians to do something.
And having the Skylaxians or whoever do things for you as a "favor" seems a bit too pacifist. You're negotiating with them to do stuff, really. So Negotiation Power might actually be a really good name for this, now that I think about it. That basically says the same thing as Bargaining Power, but without sounding like Influence... sort of.
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So, what is Influence? If you're not fully up on that, it's basically a -100 to 100 meter that rates how much general... influence... you have with a race. It's how well they think of you, and how much they are willing to listen to you, practically speaking.
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While we're at it, Attitude is kind of the inter-race version of Influence. It's a unidirectional thing where each race has a -100 to 100 scale of how it feels about each other race, regardless of how the other races feel about them. You might hate me, but I actually like you. Until you declare war, after which I hate you some, and you burn with fury. Etc.
My feeling is that both attitude and influence are extremely aptly named and are easy to explain; I don't have any expectations of changing those terms. Bargaining Power might wind up staying in the end, but since it is something that is conceptually complex, and since it sounds like influence in some ways, I thought I'd pick your respective brains and see what you thought.
So... thoughts?