Simple games that involve psychological factors?
I cannot think of a better game to match what you describe than Diplomacy. It requires seven players, and it is lengthy. You can speed it up by having all the negotiation happen at the table in front of everyone, but that changes the nature of the game. However, there are plenty of places on the internet where you can play "by email" for free (for example, The Diplomatic Pouch) -- there, you can enter games all by yourself, but the time taken to play a game typically spreads out over days, weeks, or months (because you spend a few days emailing negotiation letters, before each time you "submit orders" for the next turn's actions).

Diplomacy's rules do include strategy, and the rules are quite simple. The game has been around for fifty years, so there are lots of online resources describing various strategies and tips and so forth.

Note that BoardGameGeek has a Negotiation style category for boardgames, and there are lots of other games listed in it; many would be suitable for fewer players and less playing time, but Diplomacy is pretty much the prototypical experience for what you describe in my opinion.