Game theory time: Lions and the Lamb

[b0rk]

Here's a fun one:

A sheep sits in the middle of a field surrounded by n lions (where n is some number greater than 10). Any lion can eat the sheep but, as each lion knows, it would become so tired that it would be as defenseless as a sheep itself--easy prey for another hungry lion who would in turn become tired and defenseless. The lions are all hungry (equally so for lamb or lion meat), super rational, but naturally not suicidal. Does the nearest lion, or any lion for that matter, pounce on the tasty ovine?

For those who enjoyed the hand shaking puzzle (http://www.redflagdeals.com/forums/showthread.php?t=229423), here's a problem from the mid-term exam of my undergraduate Economics game-theory class...

[Tshute]

What the heck..? I vote that 1 lion eats a lamb, knowing the rest are not stupid enough to eat him. :|

I don't get it.

[gaurav_fhs]

Hey this applies as much to Biology and sociology as it does to economics. Game theory rocks :twisted:

[b0rk]

Actually, belive it or not, I was asked this on my University of Toronto exam..I'm not surprised that not many of you will be able to figure this out as the class average was quite high so the prof had to 'play' around with our averages ;)

It's ok not to get the 'right' answer for 'full marks' but at least you tried! :D

[Tshute]

Every lion will probably just give'r even with the consequences. That's my vote.

[gaurav_fhs]

Well you can include co-operativity I assume. Why should merely 1 lion attack the sheep? Or for that matter only 1 lion attack the weakest lion?

Oooh how about putting in a clause-- if a lion waits too long, it dies of hunger. :twisted:

[b0rk]

Well you can include co-operativity I assume. Why should merely 1 lion attack the sheep? Or for that matter only 1 lion attack the weakest lion?

Oooh how about putting in a clause-- if a lion waits too long, it dies of hunger. :twisted:

LOL, that's what many students wrote on their exam papers, and guess what? 0/10! It's encouraging to see that you're making assumptions, but making assumptions on exams will not give you 10/10 ;)

[Tshute]

LOL, that's what many students wrote on their exam papers, and guess what? 0/10! It's encouraging to see that you're making assumptions, but making assumptions on exams will not give you 10/10 ;)
I don't understand how there can be a right and wrong answer for this though. If you ran multiple experiments like this, different outcomes are bound to happen.

[b0rk]

I don't understand how there can be a right and wrong answer for this though. If you ran multiple experiments like this, different outcomes are bound to happen.

You can also apply this question to CS and ENG students, as they typically think in algorithms and are good at applying themselves to theoretical questions. My ENGSCI friend answered it in 2 mins :confused: . Go figure.

[chchchigga]

I vote that a lion goes for it - revising...if n is odd. If n is even, no one goes.
Since for an even n, i.e. 4 lions, lion 4 knows that lion 3 will go for it knowing that lion 2 won't go for it leaving lion 1 with a free meal.

[d_jedi]

Here's a fun one:



For those who enjoyed the hand shaking puzzle (http://www.redflagdeals.com/forums/showthread.php?t=229423), here's a problem from the mid-term exam of my undergraduate Economics game-theory class...
I would say no. Each wants the other lions to eat first - such that the probability it gets eaten is less (in each case, a lion has 10 targets to choose from - but the earlier a lion chooses to eat, the more chances it has of being eaten by other lions).

Edit: By simple probability, we know the chances of getting eaten for each lion is as follows:
Lion Survival chance
1 0.387420489
2 0.43046721
3 0.4782969
4 0.531441
5 0.59049
6 0.6561
7 0.729
8 0.81
9 0.9
10 1

So it is advantageous for each lion to wait. Hence, no lion eats, hoping the others will do so before.

[gaurav_fhs]

OK seeing how all the lions are super rational and not suicidal (and yet donot want to co-operate), I'd say they all starve to death.

Although I'm pretty sure, I'm missing a link :)

PS: I'm assuming in chchchigga's case, one lion will eat the lion that ate the sheep hoping the others wont eat him and so on...until the last lion (most patient dude) remains.

[Tshute]

I say the lions ignore the sheep and just turn on each other right away (although in nature this would NEVER happen).

[ainsane]

I took game theory too :) Great course... except most people had no idea what was going on.

I'll let others take a stab at it.

[Mems]

Hmm, I'm only 16 so don't laugh at my answer. :(

I believe that a lion will go for it.

-Mems

[Barton03]

I took game theory too :) Great course... except most people had no idea what was going on.

I'll let others take a stab at it.

The anticipation is killing me .. I'm holding my breath for your answer ;)

[AnimeEd]

I say that the smartest (one which took game theory) lion would eat the sheep. Even if a lion would enjoy sheep meat and lion meat equally, let's assume that attacking a sheep and attacking a lion are different (or else there won't be a sheep in the first place, there would simply be a tired lion).
Anyways, if the first lion attack, then there would be a tired lion. Now, the another lion COULD attack the tired lion, but since attacking a lion is different that attacking a sheep, the first lion being attack would also mean the second lion would be attacked also (assuming the all the lions are smart to the same degree). Since there are so many lions, no lion would ever attack because they would know that they would never become the last lion.

Of course, this all goes out the window if the lions are hungry enough that they would die anyways if they didn't eat.

[bnb]

Yeah.... I agree ^... the nearest lion will eat the lamb knowing that no other lions will eat him for fear of getting tired and being eaten in turn.

[mrpurrhead]

if there are n lions and 1 sheep, why would the lions just sit there and eat some chump sheep? wouldnt some of them leave the scene in search of a better meal? maybe the sheep leaves and no lion attacks.. maybe the sheep being naturally suicidal decides he HATES one specific lion so he ATTACKS and the LION responds by killing the sheep and another lion eats him.. SHEEP are very cunning and insance animals, we all know this to be true... Why are there sheep and LIONS in the same place??? are there sheep in Africa? Is this a zoo, maybe the sheep is protected by a fence - haha, stupid LIONS...

[gman]

If a lion does not eat, it will die.
If a lion eat, it has chance not to be eaten because the other lions may be fear to be eaten.

[Casanova]

nearest lion will eat the sheep, the second one in the chain will not eat it since it has less chance of survival as it will be more tired (it will eat the equivalent of a lion + a sheep) so no one else will eat!

Alternatively, if my assumption above is not valid, then all lions will wait the max before they die of hunger and then at the last moment, the closest one will pounce on the sheep. By the time he is done eating, all other poor souls are dead! :cheesygri

[PrimoTurbo]

This question is unrealistic and completely flawed therefore there is no accurate answer, you can play with the numbers and create theories but all you are really doing is answering an assumption some has setup and unless you think the way they do then you will be labeled as being wrong.

First of all, if a lion attacks a lamb why would it be tired? Lambs run at a fairly slow speed, a mature lion can easily take out a lamb.

Second of all, since when do lions and lambs share the same environment? You will not find lambs in Africa roaming around.

Third of all, lions are cooperative creatures if they are of the same family. You will not see different types of lions in one area, they have social structures in which one lion is usually at the top and the rest abide him.

Finally, if you could some how set this up then all of these lions would simply attack each other and forget about the lamb. Since hunger is not their number one priority, the threat of another is. Some will die, others will run away, until finally you will have one lion who will attack the lamb if it has not ran away.

[FearSonic]

They all band together and just jump that stupid sheep. God, don't you people learn anything from gangs? They come in groups. So they're all alpha males, so what, they just rip the thing apart and take parts. If it isn't enough, they scuffle for it. Last lion standing wins. Winner!

[gman]

They all band together and just jump that stupid sheep. God, don't you people learn anything from gangs? They come in groups. So they're all alpha males, so what, they just rip the thing apart and take parts. If it isn't enough, they scuffle for it. Last lion standing wins. Winner!

Male lion does not hunt. Female does the job.

[FearSonic]

Male lion does not hunt. Female does the job.

So you caught me on a technicality. They'll defend each other's territory and last lion standing eats the lamb, but he'll really be too tired to fight the lamb.. so the lamb will eat the lion.

Up next.. Geriatric Animals.. on FOX!

[AnimeEd]

I don't think you guys get the point of game theory, it's not about the lion or the sheep, it's about finding the most advantageous answer
so stop trying to find flaws in the question and bend the rules, that's too easy!

[Dragonrana]

Q# The lions are all hungry (equally so for lamb or lion meat), super rational, but naturally not suicidal. Does the nearest lion, or any lion for that matter, pounce on the tasty ovine?


Let me try and take a stab at this.....

The answer to this question is based on the number of lions in the circle.
Depending if it is odd or even. Let's say N= Lions and the constant C= Lamb

If N=1, C=1
Let say that there is one lion and one lamb, therefore the lamb get's eaten by the lion, since there is no threat of the lion being eaten after getting tired.

If N=2, C=1
If there are two lions and one lamb, the lamb is safe because if one lion tries to devour the lamb he puts himself in danger of tiring and risk of being eaten.

If N=3, C=1
If there are three lions and one lamb. One lion will eat the lamb and the other two lions will look at each other and realize that they don't want to be eaten by the other so the new "lamb" in the middle is now safe.

Therefore, this will work for any "N" quantity of lions. :cheesygri

[mrpurrhead]

OK, here is an answer based on the information given.

Two lions decide to share the meal and protect each other from attack, assuming that two tired lions in cahoots with one another might be able to fend off the attack of the other lions.

One question n > 10, does 10 have to be base 10? If base 2 it is > 2 (10 in binary is 2 in decimal).. so there could be 1 lamb, 3 lions...

[ainsane]

One question n > 10, does 10 have to be base 10? If base 2 it is > 2 (10 in binary is 2 in decimal).. so there could be 1 lamb, 3 lions...

Don't be stupid. Who the hell talks in binary in a situation like this. And a question like this usually implies that the lions have no contact with each other so they can't be in cahoots.

[AnimeEd]

don't get ainsane mad guys, or he'll throw 30 random pictures at you

[b0rk]

Q# The lions are all hungry (equally so for lamb or lion meat), super rational, but naturally not suicidal. Does the nearest lion, or any lion for that matter, pounce on the tasty ovine?


Let me try and take a stab at this.....

The answer to this question is based on the number of lions in the circle.
Depending if it is odd or even. Let's say N= Lions and the constant C= Lamb

If N=1, C=1
Let say that there is one lion and one lamb, therefore the lamb get's eaten by the lion, since there is no threat of the lion being eaten after getting tired.

If N=2, C=1
If there are two lions and one lamb, the lamb is safe because if one lion tries to devour the lamb he puts himself in danger of tiring and risk of being eaten.

If N=3, C=1
If there are three lions and one lamb. One lion will eat the lamb and the other two lions will look at each other and realize that they don't want to be eaten by the other so the new "lamb" in the middle is now safe.

Therefore, this will work for any "N" quantity of lions. :cheesygri

Wow, you should be the prof! No one in the class got the right answer, but YOU DID!! Awesome work.

But your solution looks eerily like a Google answer..;)

[ainsane]

don't get ainsane mad guys, or he'll throw 30 random pictures at you

lmao

http://koyasunomiko.com/quizzies/furuba/yuki.jpg

[chchchigga]

Wow, you should be the prof! No one in the class got the right answer, but YOU DID!! Awesome work.

But your solution looks eerily like a Google answer..;)

lol...that's the answer i gave except if you extend it to N=4 you know that number 4 isn't gonna go for it since number 3 knows he's free to go after four without worrying about two and so this situation will alternate for every odd and even number.

[spike-spiegel]

Wow, you should be the prof! No one in the class got the right answer, but YOU DID!! Awesome work.

But your solution looks eerily like a Google answer..;)

Does the N=3,C=1 answer apply if N = >10?

[SiGnoDx]

use backwards induction :)

[spike-spiegel]

use backwards induction :)

so I guess it's whichever gets it first that wins. lucky lion :lol:

[goffeebeans]

I dunno.. found the answer obvious even without any calculations. Only thing that is remotely difficult is just making up a formula to prove yourself. Just that the N > 10 throws you off.

handshake one seemed to require quite a bit more deductive reasoning to get the answer.

[d_jedi]

Wow, you should be the prof! No one in the class got the right answer, but YOU DID!! Awesome work.

But your solution looks eerily like a Google answer..;)
D'oh. It would have helped if I read the question right.. I thought there were 10 lions and 10 sheep. Whoops.

[epiphony]

For the guy who went as far as N=3 and then concluded that it must follow that it works for all N, that's wrong. You've only given an example up until N=3, and not for N>10 as was asked.

I see the problem as being a chain of lions in line to eat one lamb. It makes more sense this way than a circle of lions around a lamb.

You have to use backwards induction and start from the Nth lion. If you're the Nth lion, it is ALWAYS optimal to eat the lion before you, simply because there is no threat after you eat him. If you are the N-1th lion, there is a threat of being eaten. A CREDIBLE THEAT, since there is no other lion after that. To make a long story short, in the end, lion 1 always eats the lamb, and lion 2 does not eat the lion, because the threat of lion 1 being eaten by the second lion is a NON-CREDIBLE THREAT. It does not depend on N after N>4 (I think anyway, I didn't bother to double check).

[b0rk]

It depends if N is even or odd.

[goffeebeans]

Does it even matter if N is even or odd, the question states that the lion's are super rational and non suicidal. You're assuming that one super rational lion will knowingly go ahead and eat the lion that ate the sheep, only to be eaten by another lion. So that all these rational lions will continue to eat each other until theres only one. If you assume this, than all lions will all come to the same conclusion and want to be the last lion, hence nobody will want to eat the lion that ate the sheep.