OT - Hungry lions puzzle

[QuikSand]

Consider a large, enclosed pen which contains ten hungry lions and one poor, defenseless sheep. The lions, as you might imagine, might like to eat the sheep. However, there’s a catch. Whenever a lion eats anything, be becomes very tired (think Thanksgiving) and he himself becomes vulnerable to being eaten – and hungry lions aren’t beneath eating one of their own. So, of course, there is a puzzle in all this.

Assume that each lion would have these ranked preferences:

-Best of all would be to eat (either the sheep or another lion) and no longer be hungry
-Second best would be to remain hungry, but to stay alive
-And worst would be to eat but then, in turn, be eaten himself

For purposes of this puzzle, we will assume that each lion is completely logical and aware of all the information above. (Not just hungry lions, but smart ones, apparently)

What will happen to the sheep?

[QuikSand]

Originally posted by Ronnie Dobbs2
I assume there's a 1:1 ratio on eating? i.e. the lions can't all agree to share the sheep.

Correct. Eating process necessarily involves one eater and one eaten.

[Bee]

The sheep will be cut up into 10 pieces.

Edit: well that's not fair! The lions should really be smart enough to split the sheep.

[QuikSand]

Originally posted by Ronnie Dobbs2
Oooh... I think I just figured it out

Let's find out. Did you get... 2/3?

[Daimyo]

I guess the sheep will be eaten. A couple of notes:

If there are infinite lions the chance of being eaten if you eat the sheep is probably pretty large since you only need one lion out of that infinite pool to take a risk (even if would be a bad risk). If there are only two lions the chance of being eaten if you eat the sheep is probably 100% as there would be no risk for the other lion. Therefore, with a very large number of lions or a very small number the payoff of eating the sheep is very low.

With something like 9 other lions it is sort of in the middle... there aren't enough other lions to assume one will take the risk and eat you and there aren't so few lions to minimize the risk for the other lions to eat you. (although if I were in that situation I'd go hungry... I'm just too risk-adverse and this reminds me a lot of the "value of money" post from the other day)

[Huckleberry]

The sheep will eat 9 of the lions and then the last lion will eat the sheep. It's one of those crazy killer sheep.

[albionmoonlight]

Thinking out loud--

1 lion, 1 sheep--he eats the sheep

2 lions, 1 sheep--If lion A eats the sheep, then lion B will eat him. So both lions will want to be lion B. If you eat the sheep, then you will be eaten. If you don't eat the sheep, then you will be hungry. Neither lion eats the sheep.

3 lions, 1 sheep--If lion A eats the sheep, then he becomes the sheep in the two lions example above (i.e. not eaten). So it would seem to be to a lion's advantage to eat the sheep. Sheep eaten by the fastest lion.

4 lions--If lion A eats the sheep, then he becomes the sheep in the 3 lion example above (i.e. eaten), so no lion will eat the sheep.

I am noting a pattern here. Because 10 is an even number, I will say that no lion eats the sheep.

[mckerney]

The sheep will eat all of the lions.

The lions getting tired after eating is a sign that they've gone soft. It's like some trained police attack dog that gets let out into the wild and ends up being eaten by a deer or something. These weak lions provide easy prey for sheep. Did I mention that the sheep is a total bad ass?

So therefore there is obviously no chance of the sheep being eaten.

[Maple Leafs]

Lions, eh? I'd say the sheep rushes for 120 yards and two TDs.

[John Galt]

I think this puzzle, like most variations of the prisoner's dilemma, doesn't have a universally agreed upon answer. Still, I think no one would ever eat the sheep if there are more than 1 lion.

Edit: and by "universally agreed upon" I mean exactly that. Even understanding the problem, some people will take outlier positions that are usually aggressively reckless or hopelessly idealistic. That isn't to say that there is not one "better" answer. I'm not sure mine is, but I think the even/odds theory is not right either.

[QuikSand]

Well, despite the caveat from John Galt, the answer that I was seeking was indeed the RD2/alb "odds/evens" solution from above.

And, if you isolate the "one eater, one eaten" matter, I don't see any real weakness in the logic there. But perhaps I'm just not being sufficiently prickly about it.

[Radii]

I pretty much agree with John Galt here. Any lion that eats a sheep will then be subject to be eaten himself. Therefore, any lion that eats the sheep will be putting itself into its least desired position. All of the lions would agree to take option #2 on the preference list and not eat and remain hungry.

The only other possible argument I can think of is that one bold lion will eat the sheep, knowing that if the lions start eating other lions, they will all be eaten but one, and taking the risk that no lion will be willing to start that chain reaction... but that is more of a risk/reward/probability puzzle than the one I perceive to have been posed...

it seems painfully simple to me that the sheep will live. I wonder how I'm wrong and what I've missed?

[QuikSand]

Originally posted by Radii
I pretty much agree with John Galt here. Any lion that eats a sheep will then be subject to be eaten himself. Therefore, any lion that eats the sheep will be putting itself into its least desired position. All of the lions would agree to take option #2 on the preference list and not eat and remain hungry.

The only other possible argument I can think of is that one bold lion will eat the sheep, knowing that if the lions start eating other lions, they will all be eaten but one, and taking the risk that no lion will be willing to start that chain reaction... but that is more of a risk/reward/probability puzzle than the one I perceive to have been posed...

I think it is fair to look at this as a step-by-step logic puzzle, rather than trying to get into deep lion psychology.

Rule #1 - If there is only one lion left, and he gets a chance to eat, he will. Period.

Rule #2 - If there are exactly two lions left, and they get a chance to eat, they will not, because of rule #1.

Rule #3 - If there are 3 lions left, and they get a chance to eat, they will all try to be the one to eat, because eating would then invoke rule #2.

...and so on and so forth. Odds yes, evens no. The puzzle specified ten lions, therefore they will not try to eat. Had it specificed nine lions, they would try to eat the sheep, and one (the fastets, presumably) would do so.

[John Galt]

Originally posted by QuikSand
Well, despite the caveat from John Galt, the answer that I was seeking was indeed the RD2/alb "odds/evens" solution from above.

And, if you isolate the "one eater, one eaten" matter, I don't see any real weakness in the logic there. But perhaps I'm just not being sufficiently prickly about it.

That solution, while possible because of the "perfect information" part of your puzzle, is far from a certainty. A risk averse lion seems just as likely to avoid entering the fray, because although his information is "perfect" that doesn't mean his belief that other lions will reach the same conclusion is "perfect." I think that it requires a leap of faith (albeit a logical one - if that's possible) to eat the sheep with other lions around and thus, a risk averse lion could settle for the intermediate outcome rather than risking the worse.

[John Galt]

dola, I'm also not sure why you believe the even/odds rule continues at higher levels. I have no doubt it they could invoke it at the 3 lion level, but as they get further removed from a near certain scenario (2 lions), I don't think they will believe other lions have reached the same conclusion. I'm willing to say a 3+ lion scenario would cause eating (although I still think it depends on how you weight the various outcomes - risking death is pretty bad compared to the other outcomes), but each higher odd number seems to decrease the odds.

[Huckleberry]

Yeah, you said they were all completely logical and aware of all the information above, but you didn't say they were all aware that the other lions were all completely logical.

Besides, if I'm in a room with 9 other starving animals, I think it's logical to assume that at least one of them is going to break the rules and eat just because they're starving.

But this is a good puzzle in all its many forms.

[primelord]

Originally posted by John Galt
I think this puzzle, like most variations of the prisoner's dilemma, doesn't have a universally agreed upon answer. Still, I think no one would ever eat the sheep if there are more than 1 lion.

Edit: and by "universally agreed upon" I mean exactly that. Even understanding the problem, some people will take outlier positions that are usually aggressively reckless or hopelessly idealistic. That isn't to say that there is not one "better" answer. I'm not sure mine is, but I think the even/odds theory is not right either.

But then again you don't think there is only a 1/8 chance of goign broke flipping that unfair coin either.

[John Galt]

Originally posted by primelord
But then again you don't think there is only a 1/8 chance of goign broke flipping that unfair coin either.

True, but in that puzzle I was playing more of a devil's advocate because I like talking about infinity. When it comes to game theory, I'll stand my ground even against Big Blue.

[HornedFrog Purple]

no one wants to discuss the psychological impact on the last lion knowing his life will soon come to an end in solitude with the lack of a food supply... there is a 2/3 chance he will off himself

[primelord]

Originally posted by HornedFrog Purple
no one wants to discuss the psychological impact on the last lion knowing his life will soon come to an end in solitude with the lack of a food supply... there is a 2/3 chance he will off himself

Extra credit for HFP.

[QuikSand]

Originally posted by Huckleberry
Yeah, you said they were all completely logical and aware of all the information above, but you didn't say they were all aware that the other lions were all completely logical.

Okay - fair point of logic. techncally, I should have stated that clearly as a given.

Besides, if I'm in a room with 9 other starving animals, I think it's logical to assume that at least one of them is going to break the rules and eat just because they're starving.

Oh, but you blew it. You made a good logical point, and then summarily dumped every bit of logic right into the trash by resorting to "wild animals" and such. Aw... we were doing so well.

[albionmoonlight]

FWIW, my fiancee has the same problem with these types of puzzles--some people cannot get over the "assume that everyone is a purely logical actor with access to all the information given and knowledge and confidence that all of the other actors are completely logical actors with access to all of the information given." This assumption, of course, has very little to do with the real world. If you do logic puzzles of this nature enough, it becomes such an inherent assumption that you don't even think about it any more (which is why, I suspect, Q didn't fully explain it in the beginning). In order to play the game on the game's terms you have to be able to make the above assumption. If you are ever trapped IRL in a room with hungry lions or cannibals, I suggest that you don't assume that they are all logical actors. If, however, you are playing one of these puzzles, you need to just accept that it is a logic puzzle and suspend disbelief when the analogy to the "real world" breaks down at the level of psychology.

[John Galt]

Originally posted by albionmoonlight
FWIW, my fiancee has the same problem with these types of puzzles--some people cannot get over the "assume that everyone is a purely logical actor with access to all the information given and knowledge and confidence that all of the other actors are completely logical actors with access to all of the information given." This assumption, of course, has very little to do with the real world. If you do logic puzzles of this nature enough, it becomes such an inherent assumption that you don't even think about it any more (which is why, I suspect, Q didn't fully explain it in the beginning). In order to play the game on the game's terms you have to be able to make the above assumption. If you are ever trapped IRL in a room with hungry lions or cannibals, I suggest that you don't assume that they are all logical actors. If, however, you are playing one of these puzzles, you need to just accept that it is a logic puzzle and suspend disbelief when the analogy to the "real world" breaks down at the level of psychology.

Fair enough. I do think, however, that your built in assumption that this a logic puzzle and not a game theory one is unfounded. QS may have intended it that way, but without specification, this problem mirrors a great number of game theory problems derived from the prisoner's dilemma. In those cases, even assuming perfect knowledge, I stand by my answer.

[Huckleberry]

You may consider it dumping logic in the trash, but it's quite the opposite. You appear to be trying to remove reality from your puzzles, but that takes away lots of the interest I have in them.

It's perfectly logical to consider the situation from the standpoint of how will the lions react. And lions are wired to survive. And at some point, they will eat the sheep and take their chances with the other lions rather than just sit there and die anyway. That is also a completely logical decision. The lion must decide - should I not eat the sheep and be 100% certain of starving to death, or should I eat the sheep and be 99% certain that another lion will eat me?

Anyway, the puzzle was answered given the limited scope. I was just extending it to reality if you were to actually prepare the situation and were actually able to infuse logical thought processes into the lions.

[Maple Leafs]

Originally posted by albionmoonlight
FWIW, my fiancee has the same problem with these types of puzzles--some people cannot get over the "assume that everyone is a purely logical actor with access to all the information given and knowledge and confidence that all of the other actors are completely logical actors with access to all of the information given."

My wife can't get over it either. Not for puzzles, I mean in everyday life.

Wife: My friend was supposed to call me today, but she didn't. Clearly she's trying to tell me that our friendship isn't important to her anymore.

Me: Wouldn't it make more sense to assume that it just slipped her mind?

Wife: She probably planned it all out for weeks, just so that I'd feel bad.

Me: OK, assume that everyone is a purely logical actor ...

Wife: You're not supporting me! (breaks down in racking sobs)

Me: (shrugs, turns on TV)

[korme]

I smell logic. I don't like logic. - The Afoci

Anyways, I agree with Galt. If I am one of 10 lions, I eat the sheep, knowing that no lion dare eat me and later become a victim himself.

[John Galt]

Originally posted by Shorty3281
I smell logic. I don't like logic. - The Afoci

Anyways, I agree with Galt. If I am one of 10 lions, I eat the sheep, knowing that no lion dare eat me and later become a victim himself.

He agrees with me, yet reaches the exact opposite answer. Shorty, lay off the pipe.

[korme]

"I'm willing to say a 3+ lion scenario would cause eating...."

Isn't that saying the same thing?

[John Galt]

Originally posted by Shorty3281
"I'm willing to say a 3+ lion scenario would cause eating...."

Isn't that saying the same thing?

Hmmmm . . . Ok, I could see how you could read my answer that way even without your pot-haze. I was actually saying that my original answer (no eating if more than one) would allow for possible eating, but that the risk of eating decreases at each higher level. I'm inclined to believe in risk averse behavior when death is on the line.

[korme]

Hmm, I figured you meant "the risk of eating decreases at each higher level" was meant for the lions, i.e., it is less likely a lion will get eaten as the number of lions gets bigger.

"Ok, I could see how you could read my answer that way even without your pot-haze."

LOL.

[mckerney]

I still like my answer. It uses made up logic.

[Bad-example]

Those poor lions are STARVING! Someone please toss them a few christians.

[QuikSand]

Originally posted by Huckleberry
The lion must decide - should I not eat the sheep and be 100% certain of starving to death, or should I eat the sheep and be 99% certain that another lion will eat me?

Which is fine and all - but clearly this is a different puzzle than the one you were given.

In this puzzle, you were very, very clearly given the ranked order of outcomes for each purely logical lion. You, instead, want to substitute your own preferences for them -- deciding that "going hungry" os worse than "taking a risk." The point of the puzzle is that if you analyze it step by step, there is no 99% likely event -- each and every step of the process is absolutely certain to happen as forecasted, because each actor is one of pure logic.

I'm sorry (not really) that I used confusing words like "lions" and "sheep" in this logic puzzle -- maybe it would have been easier toi grasp if I had used nonsense words to describe each actor... or had made this a cyber puzzle involving blips and beeps?

[Bonegavel]

I still don't see how any lion population over 1 would eat anything. If they are all logical and know that they will be eaten if they eat, they won't eat because that, in turn, would lead to their being eaten and would, in turn, put them in the worst category of eating and then being eaten.

Lion > 1 = sheep lives
Lion = 1 = lamb sandwhich (assuming there is bread in there)

[Huckleberry]

QS -

And I've told you twice now that once your puzzle was solved I moved onto expanding it to a more real-life situation. Sorry if you're having trouble understanding that.

[QuikSand]

Originally posted by Huckleberry
And I've told you twice now that once your puzzle was solved I moved onto expanding it to a more real-life situation. Sorry if you're having trouble understanding that.

I count once, and I re-checked to be sure... I'm not trying to be insensitive to your argument.

It's perfectly fine if you would rather grapple with a puzzle like "what would a bunch of hungry lions do if they were locked in a pen with one helpless sheep?" My only point from above is to clarify that doing so is clearly elimiating the original puzzle here, and replacing it with something else (which I happen to find a lot less challenging).

Sorry if you find my arguments frustrating -- I recognize that you are moving onward to some other sort of construct, I just don't really see the point. It's a logic puzzle, that's all. Real lions each real sheep, I think we all kinda understand that.

[Bonegavel]

Quik,

what is the flaw with my solution (pasted below for easy access) ?

I still don't see how any lion population over 1 would eat anything. If they are all logical and know that they will be eaten if they eat, they won't eat because that, in turn, would lead to their being eaten and would, in turn, put them in the worst category of eating and then being eaten.

Lion > 1 = sheep lives
Lion = 1 = lamb sandwhich (assuming there is bread in there)

[albionmoonlight]

I was looking this puzzle up to tell to a friend today. I had forgotten the direction that this thread took.

Lions and sheep are serious business, apparently.

[QuikSand]

Apparently so.

[MikeVic]

Originally Posted by Maple Leafs

Lions, eh? I'd say the sheep rushes for 120 yards and two TDs.

Now that's funny!

[QuikSand]

Not sure if this actualy was going to generate any new interest... but it's been pulled from the archives and bumped just in case.

Ahhh... the good old days.

[John Galt]

The strange thing is that I don't even remember having participated in this thread. Memory is a fickle mistress.

[Maple Leafs]

Originally Posted by John Galt

The strange thing is that I don't even remember having participated in this thread. Memory is a fickle mistress.

So having reread it, you now see where you were wrong, yes?

[John Galt]

Originally Posted by Maple Leafs

So having reread it, you now see where you were wrong, yes?

No. In addition to finding myself to be the most eloquent and thoughtful poster in the thread, I find all of my arguments to be convincing and persuasive.

[Huckleberry]

I also didn't remember this thread at all. But I recall now that if you dress QS up as a lion and put him in a cage with 8 hungry lions he will quickly eat the sheep. Then as he is devoured by the fastest lion he will protest "But it's not logical!!! Owwweeeee!!!"

[Arctus]

Originally Posted by QuikSand

Consider a large, enclosed pen which contains ten hungry lions and one poor, defenseless sheep. The lions, as you might imagine, might like to eat the sheep. However, there’s a catch. Whenever a lion eats anything, be becomes very tired (think Thanksgiving) and he himself becomes vulnerable to being eaten – and hungry lions aren’t beneath eating one of their own. So, of course, there is a puzzle in all this.

Assume that each lion would have these ranked preferences:

-Best of all would be to eat (either the sheep or another lion) and no longer be hungry
-Second best would be to remain hungry, but to stay alive
-And worst would be to eat but then, in turn, be eaten himself

For purposes of this puzzle, we will assume that each lion is completely logical and aware of all the information above. (Not just hungry lions, but smart ones, apparently)

What will happen to the sheep?

The even/odd solution only works if:

A lack of food makes the lions hungry, but they will not starve to death.

or

The smart lions are not quite smart enough to figure out that they will eventually starve if they do not eat.


Otherwise the scenario presents a valid fourth condition for which the lions' preference is not addressed. This means an additional assumption will be required to solve the puzzle.

I understand the purely logical point of the exercise, but things tend to fall apart a bit if an implied/inferred condition is not addressed.

I believe it was this extra condition that was causing the back and forth between the even/odd solution and the nothing is eaten in any case solution.

[QuikSand]

Originally Posted by Arctus

The even/odd solution only works if:

A lack of food makes the lions hungry, but they will not starve to death.

or

The smart lions are not quite smart enough to figure out that they will eventually starve if they do not eat.

I again disagree. You introduce the concept of "eventually" here, but the framework of the puzzle does not give you that right. We have a very specific order of possible outcomes, and a specific priority in assigning preferences.

I conceded the point earlier in this thread (apparently, since I too hardly remembered this puzzle at all) that one more piece of information is probably necessary -- that each lion knows that all other players are similarly informed and logial actors. Add that, and I still would argue the odds/evens logic is just fine.

[QuikSand]

With the slight clarification (underscored below, for clarity) that the relevant count is "healthy lions" in several references -- this is the explanation that works best for me.

Originally Posted by QuikSand

I think it is fair to look at this as a step-by-step logic puzzle, rather than trying to get into deep lion psychology.

Rule #1 - If there is only one healthy lion left, and he gets a chance to eat, he will. Period.

Rule #2 - If there are exactly two healthy lions left, and they get a chance to eat, they will not, because of rule #1.

Rule #3 - If there are esxactly three healthy lions left, and they get a chance to eat, they will all try to be the one to eat, because eating would then invoke rule #2.

...and so on and so forth. Odds yes, evens no. The puzzle specified ten lions, therefore they will not try to eat. Had it specificed nine lions, they would try to eat the sheep, and one (the fastets, presumably) would do so.

[QuikSand]

Originally Posted by Arctus

The even/odd solution only works if:

A lack of food makes the lions hungry, but they will not starve to death.

To me, on a second reading, this seems to essentially be a paraphrasing of the ordered list of conditions given in the original puzzle (which you quoted for convenience).

Assume that each lion would have these ranked preferences:

-Best of all would be to eat (either the sheep or another lion) and no longer be hungry
-Second best would be to remain hungry, but to stay alive
-And worst would be to eat but then, in turn, be eaten himself

So... setting aside your second condition, it seems to me that the original puzzle satisfied your setup just fine.

[RendeR]

So whats the bloody right answer?