How much of a drop will a mob voluntarily walk off of in Minecraft?

Solution 1:

A pig will voluntarily walk off an edge 3 blocks high, but not an edge 4 blocks high.

To investigate this, I started with @xfs's proof that a pig will not jump off blocks in a 9-patch arrangement if the drop is too high. I then constructed the following apparatus to find the maximum distance a pig would voluntarily jump off:

As you can see, a 2 block drop is fine

A 3 block drop is also fine

At a 4 block drop the pigs no longer jump off voluntarily

Conclusion: The highest drop pigs will drop off voluntarily is 3 blocks

Solution 2:

A quick experiment.

1. None of them fell off
2. All of them fell off at the inner corners
3. All of them quickly fell off at sign posts.

Solution 3:

From what I have seen, most mobs wont wander off a two block high cliff unless they have seen you. It's very difficult to say since if you get too close their behavior changes. It could also be different on a mob by mob basis. I have read that cows will ignore cliffs entirely and plummet to their doom!