An event happens. There is a cause. Do I have free will in how I respond to it? That depends: Am I bound to give a specific response? (ie: is my response predetermined by the inputs) Or am I free to chose my own response (ie: Am I free to chose my response, regardless of the inputs?).
Which is how Free Will works, yes...
In a causal-universe, give me the same event an infinite number of times, and I will give the same response an infinite number of times.
A universe with causality may stil be devoid of free will. However, a universe without causality...
In a non-causal universe, give me the same event an infinite number of times, and I will give an infinite number of different responses.
But here's the catch. In a non-causal universe, the thing which causes your reaction can be you, in the future. In which case you are constrained to react this way, because if you do not, the you of the future that causes this reaction will never be able to cause this reaction, and therefore, you will not be able to react. Thus, free will does not exist.
I'll put my atom on the shelf and leave it sit for a few years. At some point, those four nucleons will find themselves just a bit further away from the nucleus (credits to Quantum Tunneling) and... an alpha particle is born. An event without a cause. No action was taken on the atom, it decided all on its own, without external inputs, to decay into Radon-222.
No, the alpha particle is born because the atom decayed. The event has a cause, one that comes before the event. That's what causality means. With a closed time-like curve, causality can be violated: An item or information can return to its own past, and therefore, could cause a paradox, if not for the fact that this already happened. But if this already happened, then everything is already predetermined, and you cannot change it.
Assume for a moment I can send information both forward and backward in time. Lets say on January 1st, 2012 I send a message to myself on January 1st, 2011, containing the day-by-day stock prices. So starting on January 1st, 2011, I start trading on the market, and quickly become wealthy. My mere act of trading causes the present to deviate from what my future-self reported to me.
Ah, but here's the catch: Some time before January 3rd, 2011 (1st and 2nd are weekends and the market isn't open), you receive a message from a mysterious stranger claiming to be yourself from the future. Or maybe not so mysterious, if you, like me, have an authentication protocol you use to identify yourself from the future. Either way, for whatever reason, you follow this guy's advice, which somehow seems curiously resistant to everything you do, even if for some reason you attempt to derail the plan out of pure contrariness, which probably just ends up making you less rich than you could have gotten had you not done that. Sometime in the future, however, you discover the ability to transmit messages to the past. At that point you realize you MUST transmit that message, or else none of this would have happened. But it DID happen, so you can't NOT do so. Even if you attempt to defy destiny and not transmit the message, it ends up happening anyway...just like it did. Because in a non-casual universe, FREE WILL DOES NOT EXIST. Your actions, including going back in time, are predetermined by events in the future.
A series of alternate flows that flipflop: I get rich in the market, and lack motivation to invent time travel, thus I don't send the information back in time
Flipflopping universes is "many worlds", though: In some worlds you are motivated to invent time travel, in others you are not. This is not "true" time travel. In TRUE time travel, you are motivated to do it, or perhaps do so on accident without motivation, or perhaps someone does it for you and you merely exploit it, specifically because you ALREADY DID, in your future-future self's past. In response to the fact that you DID receive this help from the future, you become motivated to invent, discover, or maybe just stumble upon, time travel. You can't avoid this, because it already happened. The fact that the universe is non-causal results in predestination, which necessarily implies a lack of free will, because you don't choose your actions, they're already determined. The ability to "change" the "past" is moving into "many worlds" territory in which you are altering *A* past, but not your OWN.
Time doesn't really have an arrow--our language only assumes it does. Causality is only conserved because we use the measure of entropy to measure time. That's a trap, because entropy increases with time, but entropy is not time. You need to be able to think in four dimensions.
Time may not necessarily have an arrow, but we don't know what the hell it DOES have. Without causality or the "many worlds" escape hatch, though, time is like a book that is already written and then posted onto the Internet: Unalterable and indelible, even if you set the book on fire, and indeed, attempting to set the book on fire makes it stick around forever.