Most philosophers think that this sort of time travel scenario is impossible: that is, the scenario is in fact inconsistent. There is no way our world could be, in which time travel like that occurs. What these philosophers object to is the idea that someone could travel into the past and change that time from being one way, to being some other way. This, they hold, simply makes no sense. Change is always change with respect to something: usually it's with respect to time (you and I change by being one way at one time, and a different way at some other time). But there is no dimension in which a moment in time can change.

If one accepts this widely held view about the past, then all sorts of interesting questions are raised about the nature of decision and agency. The first thought you are probably having is that if the past cannot be changed, then we can make no sense of time travel at all. After all, when a time traveller steps out of the time machine she is thereby changing the past: she is changing it from being a time when she was not there qua time traveller, to being a time at which she is there qua time traveller. Even stepping on an ant will amount to changing the past, even if only in a small way. 

Here is where philosophers want to distinguish between changing the past, and causally interacting with the past. Changing the past is altering some moment in time from being one in which x happened, to being one in which x did not happen. Causally interacting with the past is being in the past, and being part of the events that unfold, and causing various things to happen in the past. Why is the second not the same as the first?

Surely if I causally interact with the past, I thereby change it. No. Suppose that Billy travels back in time and meets the young Hitler and tries to assassinate him. Billy fails in his quest - we know he fails, because Hitler did, in fact, go on to become leader of Germany. This is a fact about the past, and as such is something that will never be changed. But Billy causally interacts with Hitler when he talks to him, and when he tries to kill him. Indeed, for all we know Billy tries to kill Hitler multiple times, and it may be that it is these repeated assassination attempts that significantly impair Hitler's psychology to the point that he becomes a sociopath and heads down the road that we all know he takes. If that were so, then Billy's time travelling would, as it turns out, be one of the causes of the second world war. Billy has not changed the past in this case, he has merely made it the case that the past is the way we now know it to have been. What philosophers call consistent time travel stories (time travel where the traveller does not change the past, but instead is part of the reason why the past is the way it is) can be found represented in popular culture by films like 12 monkeys and one of the Harry Potter films that features time travel.

The crucial thought is that when someone travels back in time, what they do in the past has just the same sort of causal effects as what you and I do in the present. If a time traveller drops a glass on a hard floor, it most likely shatters. It is just that the things that the traveller does makes the past the way that it is. If the time traveller shatters a glass, then it has always been the case, ever since that moment in time, that a glass was shattered at that moment in time. The past is not changed from being one in which no glass shatters, to being one in which the glass does shatter.

Here is where the whole time travel scenario becomes interesting from the point of view of thinking about decisions and agency. For the time traveller, unlike you and I, already knows a whole bunch of things about the past moment to which he travels, and also knows much about what will happen after that moment. Some of this knowledge might come from memories, and some from historical record.

Suppose that you remember a tragic day when you were 6 when you left the gate of your house open and your dog escaped and was lost forever. That was a terrible time for you, so you decide to travel back in time to that day. On the assumption that you cannot change what happened in the past, there is an interesting question about what the older wiser time travelling you ought to decide to do when you go back in time. You know that the gate was left open. So you know that there is no point deciding to go back in time in order to prevent yourself from leaving the gate open. That action is doomed to failure, since you clearly remember that the gate was left open. So it would be irrational of you to travel back in time with the intent of closing the gate. You also know that the dog escaped, never to be seen again. So it would be irrational to try and prevent the dog from having escaped. But you do not know what happened to your dog. You have always feared that something bad might have befallen the dog. So here is a perfectly rational plan. You travel back in time and wait until your younger self leaves the gate open and the dog escapes. Then you grab the dog as it decamps from the yard, and take it to live with a nice family in the next town.

That action is consistent with everything you know about the past, so it is a perfectly rational plan for you to formulate before you get into the time machine. Indeed, let us suppose that you travel back in time and succeed in your endeavours. You now feel much better, knowing what became of the dog. Moreover, your actions as a time travel explain why the dog was never seen again: it was living with a new family a town away. You did not change the past by your visit, you were responsible for the disappearance of the dog.

What is interesting about this case is that only some decisions you might make about what you will do in the past are ones that are rational. It would be silly of you to decide to prevent the dog from having disappeared, given that you know that the dog did disappear. You have good reason to think that the dog did disappear, because not only do you remember this happening, but so does everyone else in your family. But now consider a case in which your knowledge is rather less certain. The moa were large flightless birds that lived in New Zealand and were hunted to extinction by the Maori. That there are no more moa in New Zealand is beyond doubt, and that there were hunted by the Maori is beyond debate. But suppose you are a huge fan of these flightless birds, and you have a time machine. What can you do about their plight?