The Poisoned Lion
In this one: Interview questions, and the bogus logic of a Goldman Sachs quant
This is an interview question from Goldman Sachs :
Q: "In a room there are 253 lions and one steak. The steak is poisoned, so that if a Lion eats the steak, it will fall asleep. While asleep, the other lions will treat it the same as they would treat the steak. If another lion eats the poisoned lion, it too will fall asleep. Each of the lions is intelligent, and aware of the poison and its effects. Do any of the lions eat the steak?"
Give it a try! The stated answer is below the image.
A: “An odd number of lions so one will eat the steak, if there was an even number of lions they will not eat the steak” (answer was listed here on www.efinancialcareers.com)
The supposed correct answer is an example of an inductive fallacy: If there is one lion it will eat the steak, because it knows there are no other lions to eat it. If there are an even number of lions then none will eat the steak because there are always other lions around to eat them and they are intelligent enough to know this. So under these assumptions the generalised answer is : If there are an odd number of lions then one will eat the steak leaving an even number, and if there is an even number of lions then they will not eat the steak/sleeping lion.
Sounds super smart doesn’t it?
Here’s a picture of lions eating a small zebra. You can barely see the zebra.
Lions don’t queue - This is common knowledge, and immediately calls out what else is suspect in the rest of the question?
All roads lead to Rome
The first step of the supposed answer to the puzzle may be correct, in that if there’s one lion, and it has nothing else to fear except being eaten by another lion, then perhaps if it’s quite happy to be indefinitely poisoned into sleep, and maybe it is hungry enough to eat the steak, then… it *might* eat the steak.
But, what else do we need to assume to make the rest of the supposed answer work out? If there are many lions, and they are all cannibals and hungry enough to eat another lion, and they can all tell when a given lion is poisoned as opposed to simply sleeping… and if they all wait in an orderly line to eat the entirety of the first sleeping (poisoned) lion, and they do not share it, nor leave any remainder for any other lion to eat (otherwise the number of poisoned lions will exponentially increase), and that the lions must eat in increasing order of size, otherwise the next lion can not fit the current sleeping lion and any previous lions it has eaten inside, then … given this incredible tower of flimsy induction, then the supposed answer is correct. But that was quite the journey of supposition wasn’t it?
For the question to work out as answered then you are expected to infer that:
Rational cannibals : Lions are cannibals, and are obligated to eat if any food is presented the moment it is presented, but they are also super rational and have great self control so will only eat if and only if it is safe to do so.
Single hazard : Lions only fear being eaten whilst sleeping due to poisoning. Greed/Starvation or being eaten whilst awake or sleeping normally is apparently not an issue for these cannibal lions.
Identical IQ : Lions are intelligent enough to know the inductive reasoning of other lions will protect them from being eaten, all lions are identically capable mentally and so just as in the Beauty Game (Nash equilibrium) it can rely on that to calculate that the other lions will not eat it. If one lion is a bit dim, then it might miscalculate the induction, and all bets are off.
Monotonically increasing physical size: Lions can eat other lions entirely, without leaving a trace of the eaten lion. The consequent accumulation of mass means that each lion must arrange themselves in order of gastric capacity such that each will be larger than the previous one.
Non-dilutable poison: Despite the increase in mass the poison remains as effective no matter how many lions it has been eaten by.
So that list of absurdities is absolutely acceptable, but at the same time you will fail the to get the answer if you were to infer any of the following:
Non-Queuing lions : Lions eat simultaneously (ask a zebra), and even though you already know this you have to ignore that. For the question to work out as answered it must be the case that lions eat sequentially.
Collaboration? Our super intelligent lions have to be incapable of collaboration for the answer to work, because if they all work together they can either all have some steak, or can swap future steaks in exchange for protection whilst they eat this steak - If they can infer and extrapolate other lions actions and intentions, then why not?
Discerning palates : For the question to work you have to arrive at the conclusion that sleeping lions are inedible, but sleeping poisoned lions are delicious. You must not infer that sleeping lions that are not poisoned will be eaten, even though it states that sleeping lions are eaten in the premise of the question. This matters because lions spend a lot of time asleep, and all except the last one to snooze will all end up getting eaten if sleeping lions are eaten. It then might eat the steak depending on whether it is still hungry after eating 252 lions.
Size does not matter? The answer only works if the size of the lion does not affect the rate or outcome of the poisoning. If the larger lions are unaffected by the poison, or the poison can be diluted by mixing with the other lions then all bets are off.
Size does matter? We depended upon a monotonically increasing procession of gastric capability to ensure the lions could all eat the correct sequence. But this adds another consequence - If a lion can not eat a lion that is bigger than itself then the largest lion will eat the steak knowing that no other lion can eat the entirety of it when it sleeps, and the fractional remainder will poison a second lion leading to an asynchronous branching of the probability space that increases the complexity of the consumption problem beyond the limit of the lion (or human) cognitive capability. This means that since no lion can correctly calculate whether it will be eaten or not, the inductive dependency is broken and the largest lion can rely on the uncertainty to eat the steak.
But if all lions end up eating each other, then for the selected answer to work out then you must not question whether it is logical that the 252nd lion might be unable to eat the 251 lions that are now sleeping/eaten.
If any single part of this sequence is broken, then the whole construct comes down. If fractional poisoning is a thing then the poison never loses efficacy but the lions must eat the entirety of the poisoned object to maintain the logic of the answer.
Without going beyond the evidence of the question you already have two routes that lead to the steak always being eaten.
If a Fractional dose is an effective poison –forces-> Total consumption to prevent fractional remainder leading to exponential increase of poisoned objects –forces-> Eat in ascending order of gastric capacity –which means-> Largest lion can always eat the steak in safety
If a Fractional dose is not an effective poison –means that-> Partial consumption and sharing of objects would prevent poisoning or reduce it to a low probability for any individual lion –which means-> all lions would be rational to share any poisoned object –which means-> the lions eat the steak
So what does all that mean?
It means it has nothing at all to do with what you answer, the data you have is utterly insufficient to answer the question asked. So why ask it? What they are looking for is an impressive, coherent and structured narrative. The kind of thing you could use to sell CLOs to an investment committee: You can eat the steak and all the lions as long as the Fed’s got your back.