TITLE: Understanding the solution of a riddle about lions and sheep.
QUESTION [17 upvotes]: I heard a riddle once, which goes like this:
There are N lions and 1 sheep in a field. All the lions really want to eat the sheep, but the problem is that if a lion eats a sheep, it becomes a sheep. A lion would rather stay a lion than be eaten by another lion. (There is no other way for a lion to die than to become a sheep and then be eaten). 
I was presented with this solution:
If there were 1 lion and 1 sheep, then the lion would simply eat the sheep. 
If there were 2 lions and 1 sheep, then no lion would eat the sheep, because if one of them would, it would surely be eaten by the other lion afterwards. 
If there were 3 lions, then one of the lions could safely eat the sheep, because it would turn in to the scenario with 2 lions, where no one can eat.
Continuing this argument, the conclusion is as follows:

If there is an even number of lions, then nothing happens.
If there is an odd number of lions, then any lion could safely eat the sheep. 

But to me this seems utterly absurd. I think this is similar to the Unexpected Hanging Paradox (Link: http://en.wikipedia.org/wiki/Unexpected_hanging_paradox). I might have forgotten some assumptions, and those assumptions might actually solve this problem. 
Is there a fault in the argument which I haven't discovered? Does anyone have any insights? Is the argument sound?

REPLY [5 votes]: None of the answers posted here are actually fully correct.
Let us use the formulation of the problem as posted on the braingle website: http://www.braingle.com/brainteasers/teaser.php?id=9026&comm=0 
The backward induction solution to the problem (where the sheep survives if the number of lions is even, and gets eaten if the number of lions is odd) can only be applied if lions have common knowledge of lion rationality. It is not enough for lions to just be "infinitely logical, smart, and completely aware of their surroundings". It is even not enough for lions to know that all other lions are rational! Common knowledge is much stronger than that, and means that everyone knows that everyone knows that everyone knows ... etc. etc.
Only then can we start applying backward induction! And the original problem formulation makes no such statement about common knowledge of rationality - which is a very strong statement to make!
Note that when I used the word "knowledge" above, I used a very strict definition of knowledge - knowledge is a true belief, which holds regardless of any new information becoming available.
Merely having common belief in rationality is not enough for a backward induction solution here! Imagine a situation with 4 lions: what happens if the 4th lion decides to eat the sheep, contrary to what the backward induction solution says he should do? This will invalidate the common belief of all the other lions in common rationality, which was the prerequisite for the backward induction solution! So lion 3 can no longer assume that lion 2 will never eat the sheep - after all, lion 4 just did! And since above all, lions do not want to be eaten, lion 3 will no longer eat the sheep. Turns out the behavior of lion 4 was rational after all, assuming lions did not have common knowledge in rationality at the start!
This explains why a lot of people feel unsatisfied with the backward induction solution to this problem, and why they feel that this problem is a paradox. They are completely right to be unsatisfied! Common knowledge in rationality is an extremely strong condition, and is unrealistic in most practical scenarios. And in fact in this problem, even common belief in rationality is not stated, merely the rationality of each individual lion, which completely invalidates the backward induction solution - even with only 2 lions, lion 2 cannot reason about what 1 lion would do if he doesn't hold a belief or knowledge of that lion's rationality (which cannot automatically be assumed)!