I like how the game Achron answers this question. Achron is a real time strategy game in which you can send units back or forward in time, and doing so costs more resources the farther in time you want to go.
The fascinating part is how the game's internal time engine resolves paradoxes. It can actually resolve the grandfather paradox, and it does so in a quite ingenious manner.
Let's say I build a barracks, from which I can train soldier units. The barracks finishes building at T=5.
Next, I simultaneously build a time machine and a soldier. The soldier finishes at T=10 and the time machine finishes at T=20. Now, I use some resources to send the soldier back in time to T=0.
The soldier waits there until T=5 when the barracks is built. I then instruct him to destroy the barracks which takes him 4 seconds, completing at T=9, one second before it created the soldier. I have now initiated the grandfather paradox.
What happens now is called a time wave. At T=10, the barracks no longer exists and thus it could not have trained the soldier that now stands in front of its ruins. This creates a time wave which takes 10 seconds to propogate. It's like an echo of time where what actually happened catches up to what should have happened. 237
However, during the ten seconds the soldier is still standing there. Why? Because it wasn't until T=20 that the soldier went back in time and altered the course of events. The first moment where something different happened was T=10, ten seconds before T=20, where the time wave began. So for those ten seconds, nothing happens.
Now, at T=20, the time wave catches up with us. The soldier vanishes because he destroyed the barracks that created him.
But wait! Here's the interesting bit. We are now in what I would call phase two of a time wave oscillation. Waves (sine and cosine for example) oscillate continuously between two distinct states. What has happened now is that the soldier that destroyed the barracks at T=9 no longer exists, and therefore could not have destroyed the barracks in this branch of time. It doesn't even appear at T=0, where it came from the future.
So ten seconds after T=0 when the soldier no longer appears, the time wave propagates again, and now the barracks reappears at T=5 when it was originally created, and there is no soldier around to destroy it. At T=10, the barracks still exists and finishes training a soldier just as it did before the time wave hit, and we are back at phase one.
So we continue to oscillate between two conflicting series of events until some other event (like the opponent winning the game, or becoming bored and quitting) occurs to jar us out of the time wave.
Having recently watched Doctor Strange, this also reminds me of
! the ending where Doctor Strange creates a time loop and defeats the being that rules the dark universe.
In this model of time, the answer is yes: you can alter the present by altering the future. My example was of sending a unit into the past, but you could just as easily send a unit into the future and create a time wave or paradox that way. 283