Seems straightforward to me. McGonagall knows that she does not recall being handed 100 Galleons an hour ago, so the three states of the world with high probability are: 1) She is not in a universe where she will hand Harry 1 Galleon, 2) She is in a universe where she hands Harry 1 Galleon and Harry breaks the agreement, or 3) She is in a universe where she hands Harry 1 Galleon and Harry keeps the agreement in a way that leaves her unable to recall this happening. By not handing Harry a Galleon, she will ensure that she is in universe 1. By handing Harry a Galleon, she will find herself in universe 2 or 3. She should therefore give Harry a Galleon if she judges it less than 99 times more likely that Harry will break the agreement than fulfil it in a way consistent with her experience.
As Harry has access to a time machine, he doesn't need to decide to give her 100 Galleons before he gets the 1 Galleon, so the situation is quite different to one based on predicting her actions, as Omega does in the Counterfactual Mugging. Rather it has most of the properties of the forward-time version of the gambit: "If you give me 1 Galleon now, I'll hand you 100 Galleons in one hour", except that McGonagall has a big piece of evidence that the promise will be broken, namely that she doesn't remember it being kept.
Update: Please post new comments in the latest HPMOR discussion thread, now in the discussion section, since this thread and its first few successors have grown unwieldy (direct links: two, three, four, five, six, seven).
As many of you already know, Eliezer Yudkowsky is writing a Harry Potter fanfic, Harry Potter and the Methods of Rationality, starring a rationalist Harry Potter with ambitions to transform the world by bringing the rationalist/scientific method to magic. But of course a more powerful Potter requires a more challenging wizarding world, and ... well, you can see for yourself how that plays out.
This thread is for discussion of anything related to the story, including insights, confusions, questions, speculation, jokes, discussion of rationality issues raised in the story, attempts at fanfic spinoffs, comments about related fanfictions, and meta-discussion about the fact that Eliezer Yudkowsky is writing Harry Potter fan-fiction (presumably as a means of raising the sanity waterline).
I'm making this a top-level post to create a centralized location for that discussion, since I'm guessing people have things to say (I know I do) and there isn't a great place to put them. fanfiction.net has a different set of users (plus no threading or karma), the main discussion here has been in an old open thread which has petered out and is already near the unwieldy size that would call for a top-level post, and we've had discussions come up in a few other places. So let's have that discussion here.
Comments here will obviously be full of spoilers, and I don't think it makes sense to rot13 the whole thread, so consider this a spoiler warning: this thread contains unrot13'd spoilers for Harry Potter and the Methods of Rationality up to the current chapter and for the original Harry Potter series. Please continue to use rot13 for spoilers to other works of fiction, or if you have insider knowledge of future chapters of Harry Potter and the Methods of Rationality.
A suggestion: mention at the top of your comment which chapter you're commenting on, or what chapter you're up to, so that people can understand the context of your comment even after more chapters have been posted. This can also help people avoid reading spoilers for a new chapter before they realize that there is a new chapter.