My recommendation for a lower budget build is FX-6300. Same price bracket as i3. Consider the direction where game developers have been going and probably will be going in the future (better multithreading) and that sony confirmed an 8-core CPU for PS4. Like I said, I wouldn't bother with a dual-core or a triple.
I'd agree that if the option is present (e.g. if the budget allows), a quad-core is a stronger choice, but of the three presented (plus the Ivy Bridge i3), the later i3's are better. They are hyperthreaded, which can help.
Out of those choices I agree that i3 is best. But if you're buying the CPU as new and paying a retail price I'd say there's no point even considering the i3 over FX-6300 as far as performance goes. Power consumption for a HTPC type of small enclosure is another matter but for atleast midsized desktop case with decent ventilation 6300 is the best option at the price range.