EPTAssume you are in a locked room with 9 other people (10 total). Everyone is wearing a heavy coat, and sitting in a circle of chairs facing each other, but more than arms' length away from anyone else in the room and also has their legs strapped to the chair (so you can't move).
You know:
6 people in this room have guns hidden in their coats.
3 of the guns are loaded with 1 real bullet, and 3 of the guns are loaded with 1 blank.
Everyone (including you) is a crack shot and has 100% accuracy.
Whether you have a gun or not.
In exactly 1 hour, the oxygen will run out in this room.
The door will not open to this room until three people have died.
You do not know:
Who has guns.
Whether your gun has a real bullet or not.
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What would you do if you had a gun?
What would you do if you didn't have a gun?
EDIT: Ok this question is a maximize expected gain/minimize expected loss question, so it's not about teamwork or altruism (at least that's how I approached it, since this came on the tail end of a bunch of game theory and stats questions). So just try to figure out how you would stay alive, although the altruism answers are really interesting they're probably more suited toward a nonprofit, banking, or consulting job than a trading job.
EDIT#2:
On February 23 2009 14:10 Jonoman92 wrote:
How did they ask you this in an interview? Hand you a sheet of paper explaiing it and then asking what you would do? Seems kind of dumb to me.
How did they ask you this in an interview? Hand you a sheet of paper explaiing it and then asking what you would do? Seems kind of dumb to me.
Yea they did give me a sheet of paper to do the question. Two assumptions I made were that time was discrete and people could reacted after 1 second. My solution was
+ Show Spoiler +
assumptions: time divided in 3600 intervals. People cannot react simultaneously. Aim in interval t, shoot in interval t+1.
We are trying to preserve our own life.
How many deaths necessary? 3 deaths
How many people? 10 people: 6 gunmen and 4 innocents
How many bullets? 3 real 3 fake
axioms
1. If someone shoots, you know he can't shoot back anymore (1 bullet per gun)
2. If someone shoots, he/she does not know a priori whether he will kill for certain (real/fake)
3. Any optimal solution must include possibility of all 3 bullets hitting all 3 people
4. Any optimal solution for yourself at any time t in T will be adopted by all other gunmen at the same time t in T
5. Any optimal solution for gunman must fulfill the "suicide rule" (p(survival if you shoot yourself))
--> Then
split into gunmen and innocents
w/ gun:
take out gun. This is important so that no one shoots you because they know you could have a crucial real bullet. Do not point it at someone with a gun, because then the person with the gun will naturally point back at you, and you lower everyone's chances of survival and therefore they have incentive to shoot you. Hence every gunman will either aim at innocent or aim in the air.
If you think optimal solution is to aim at innocent, you have to assume that everyone else does so too.
Since you shoot after you aim, everyone will adjust their aim by switching targets randomly (optimal solution, proof upon request) until 2 people have 1 gun on them and 2 people have 2 guns on them, people aim by keeping aim on target if they are only one aiming at target, switching aim if target is doubled, and not switching aim to a target with only 1 aimer, and also switching aim randomly if the aiming is 3-1-1-1. After 3597 repeats of this aiming cycle, the probability of optimal aiming with 6 not being achieved with random brownian motion is less than 1 in 10 billion.
Then with 6 people shooting at 4 innocents, the chance of doubling up a real bullet is about 7/9. Hence that is not optimal because it violates suicide rule.
If you aim at no one, every person with a gun who is pointed at someone has a 1/18 chance of getting killed after everyone else has shot their bullets. If they shoot you when you have a gun pointed at no one, then they have a 1/6 chance (1/2 chance of bullet times 2/6 chance that you have one of 2 remaining bullets) of killing themselves.
ALSO
If you aim at no one, you increase your chances of survival. This is because with 6 guns aimed at 4 people, there is a chance that 2 real bullets will end up hitting the same person. If you save a bullet, you have a probability of adding to everyone's chances by potentially not wasting a bullet. So you need to add this probability to your own probability, and to the negative 1/18 that each gunman has by not shooting you for not aiming at one of the 4 innocents. This P is bigger than 1/18 (proof upon request).
Same goes for the 2nd gunman to point his gun in the air. His P(benefit by reducing wasted shot) is greater than the P(chance he might backstab defenseless gunman in t+1).
Hence you won't be killed for aiming gun in the air, and you will increase your own chances of survival.
So therefore 2 people will try to aim their guns in the air, but because you guys will make this decision simultaneously, then everyone will aim guns in the air.
Then everyone will swing guns down to 4 people. Then everyone will swing guns down in next cycle t. However, since this is repeated game, then people will realize this is retarded and leads to death, hence they will adopt a cycle of random movement with 1/3rd chance of pointing in air and 2/3rd chance of pointing at a random target.
Optimal is 4/2 wave, eg 4 shots in 1st wave to 4 targets and 2 in second wave to second 2. The chances that this solution will not happen when everyone is picking randomly is less than 1 in 40 billion across 3597 trials.
Once you get 4/2 wave it is easy... just shoot if you are in 4 wave and shoot one of the survivors if you are in 2 wave. The 4 gunmen are not going to care about the chance that you might cap them because there is still a higher chance they would have died from 2 bullets striking the same person.
Everyone with guns will do this (remember you assume everyone is rational and play GTO (game theory optimal)).
w/out gun:
You will be fired upon at least once and have a small chance of fired upon twice. However, since you do not have any weapons, you have no choices to make and you are effectively a null economic actor, hence any actions you make are irrelevant and will not affect GTO solution. So if I were you, I'd go down having some fun, maybe by singing Strangers in the Night or something.
