Which option would you choose? And why?

Notes:
1. You can't guarantee full cooperation with other students. That is, some students may promise to do something but decides to defect.
2. You don't know the number of students in the classroom. Just make up a number between 10 to 100 in your solution.
3. Scores are not normalized.
4. You must choose one of the two options.

Thoughts:
1. El Farol Bar problem might be relevant.
2. Nash equilibrium and prisoner's dilemma can be considered as well.
3. What happens if you are given a choice to not choose at all? Does it help others if you decides to skip this step?

#Logic

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Comments

Human psychology pushes way more than 10% of the students to pick 6 points, so it's a lost cause and a waste of time to try to figure out "optimum game strategy" as might be devised by others. It's pretty much a sure bet that nobody will get anything, and my choice will likely not matter. I would pick 2 simply because it just might tip the balance in favor at least everybody getting something, and thereby myself as well.

For confirmation, just review the answers already given here.

Edit: A variation of this problem is having the students repeat the experience many times. After a while, they'll settle (consciously or subconsciously) on randomly deciding to go with 6 ten percent of the time or less. This is the basis of evolution of cooperation.

Later Edit: It's interesting to observe that in spite of the fact respondents here are choosing 6 over 2 by a 2 to 1 ratio, still, newcomers pick 6 arguing that "few will pick 6 so I will".

Michael Mendrin - 5 years, 11 months ago

I'm interested in exploring consequences of the long term version of this problem.

The BEST case here if you could guarantee cooperation is $2.4$ points per round. This is if everyone took turns every ten rounds picking $6$ , and the rest of the time picking two: $\frac{9 \times 2 + 6}{10} = 2.4$

However, according to the problem specifications, we can't guarantee cooperation.

What if $10\%$ of the time, someone randomly picked $6$ instead of $2$ ? How close does that get us to the true "optimal" cooperative strategy ( $2.4$ average points)?

$10\%$ gets us to about an average of $\approx1.6$ points per player per round.

So, what do the other percentages look like? I thought I would generate the ones less than $10\%$ :

graph graph

This tells us that our best "random chance of picking 6" is $\approx1.9\%$ .

Here's the code that makes the graph:

Python 3.3:

from random import random
from time import time
import matplotlib.pyplot as plot
from pickle import load, dump
try:
    total, trials = load(open("points.p", "rb"))
except FileNotFoundError:
    total = {}
    trials = {}
def points(chance_of_6):
    picked_6 = 0
    total = 0
    for agent in range(10):
        if random() < chance_of_6:
            total += 6
            picked_6 += 1
            if picked_6 > 1:
                return 0
        else:
            total += 2
    return total
start = 0.0
stop = 0.1
step = 0.001
# initialize statistics
chance_of_6 = start
while chance_of_6 <= stop:
    if chance_of_6 not in total:
        total[chance_of_6] = 0
        trials[chance_of_6] = 0
    chance_of_6 += step
best_percent = 1
best_outcome = 0
# run simulations for a while
end = time() + float(input("How long? (seconds) "))
print("Press ctrl + c to stop.")
while time() < end:
    try:
        chance_of_6 = start
        while chance_of_6 <= stop:
            total[chance_of_6] += points(chance_of_6)
            trials[chance_of_6] += 1
            chance_of_6 += step
    except KeyboardInterrupt:
        print ("\nBreaking loop...")
        break
# make graph and determine best percent
chance_of_6 = start
x = []
y = []
while chance_of_6 <= stop:
    gain_per_player = total[chance_of_6] / 10 / trials[chance_of_6]
    if gain_per_player > best_outcome:
        best_percent = chance_of_6
        best_outcome = gain_per_player
    x.append(chance_of_6)
    y.append(gain_per_player)
    chance_of_6 += step
plot.title("Which option would you choose?")
plot.xlabel("Chance of a player picking 6 (best: {0}%)".format(round(best_percent*100, 3)))
plot.ylabel("Gain per player per round (best: {0})".format(round(best_outcome, 3)))
plot.plot(x, y, 'bo')
# plot optimal strategy
plot.plot([best_percent], [best_outcome], 'y*', markersize = 20)
plot.show()
# save results for later
dump((total, trials), open("points.p", "wb"))

Dependencies:

matplotlib (if you're on Linux: sudo apt-get install python3-matplotlib)

Brock Brown - 5 years, 11 months ago

This is great! I knew that the optimum would be somewhere south of ten percent of the time, but I'm surprised to see that it's significantly less. Just about 2% of the time? This suggests that the best total expected value "for everybody" occurs when virtually everybody makes the altruistic choice. Is it any wonder, then, that societies commonly involve altruistic behavior?

I find this "long term variation" of U of Maryland's variation of the Prisoner's Dilemma a fascinating subject. Most of the time that we hear about the Prisoner's Dilemma, it's a one-time experience, what to do if this is the first time one is faced with such a choice. But in fact in societies, we are faced with such choices repeatedly. It's part of our daily, weekly, monthly, even yearly life. Game theory often presupposes that if players are playing to their self interest only, then they are not aware or paying attention to what others are doing, and are not drawing from past experience. But what if the players are? It's still about self-interest, but it's tempered with experience, an "unspoken collusion" for the best expected individual outcome.

Thanks for your interesting take (and computer simulation) of this problem! A thumbs-up for this one, and I've got you on Follow now.

Michael Mendrin - 5 years, 11 months ago

Some (like me) thought of 'First in, best dressed'.

Sharky Kesa - 5 years, 11 months ago

What are points? Why exactly are points desirable? How much more desirable is $6$ points than $2$ points? You could go with the obvious answer, "well, when you get $6$ you earn $3$ times the utility as you would if you got $2$ points." Okay.

But REALLY, why do you care about points? What purpose do they serve? Does either choice steer the course of your life significantly?

Let's look at a slightly more primitive example:

You and $9$ other people don't have any food to eat. A farmer offers you a choice. You have to choose carefully or you could starve. You can choose $2$ apples or $6$ apples. If more than one of you picks $6$ apples he will decide that you're too greedy and he won't give you anything.

In this context, it's very easy to see the utility that points serve, and that two apples is (probably) the best choice for everyone here. According to The Nature of Human Altruism, in reality humans take a bias towards cooperative behavior, and a simple "rational" self-interested model can't account for this behavior.

More context than "points" is necessary for determining possible rational actions.

Brock Brown - 5 years, 11 months ago

"Humans take a bias towards human behavior". What I argue is that it often takes a while before humans come around to making decisions that are not directly in self-interest. So, for example, when students are asked to pick either 2 or 6 for the first time, too many will pick 6. After a while, they'll stop picking 6 so much. One sees so many such examples in history.

Michael Mendrin - 5 years, 11 months ago

I would have to probably agree that picking $6$ at first is common, but the quickness at which it converges to $\approx 2.4$ depends on the commonness of communication between the agents. For example, before the internet there was really tremendous amounts of xenophobia, and after the creation of the internet there is considerably less xenophobia. This is due to communication and cooperation of agents. (In my opinion.)

Brock Brown - 5 years, 11 months ago

@Brock Brown – Again, what I'm arguing is that there is such a thing as "unplanned cooperation". Communication is not necessarily required, but at least individual experience is required. On the obverse side, such "unplanned cooperation" isn't necessarily always altruistic. For example, intentional price-fixing is often against anti-monopoly laws, but it can happen that the individual enterprises involved can end up "through experience" with price fixing without any actual communication or conspiracy. Since no intent nor planning can be proved, no anti-monopoly laws have been broken.

Michael Mendrin - 5 years, 11 months ago

Well in this scenario, the points are only important if your not very confident in your abilities on the rest of the test. I guess what I'm saying is if this is the first question on the exam, it become a whole less relevant.

Joseph Baez - 5 years, 10 months ago

6 points... I'm a student in Healthcare and don't want peers who skate by. I work very hard and maintain high grades so I don't need the points.

Sean Brennan - 5 years, 10 months ago

I love this answer

Hung Woei Neoh - 5 years ago

Choosing 6 points is the Nash equilibrium

Francisco Rodríguez - 5 years, 11 months ago

100% the students choosing 6 marks is a nash equilibrium if there are more than 10 students. Or all of them choosing 2 if there are less than 10 students

10% of the students choosing 6 and the rest choosing 2 is also an equilibrium

Agnishom Chattopadhyay - 5 years, 11 months ago

Found the redditor. :D

Siddhartha Srivastava - 5 years, 11 months ago

Something is better than nothing, so I would choose 2 points.

Saubia Ansari - 5 years, 11 months ago

Clayton Bennett - 5 years, 11 months ago

It is a case of maximin equilibrium I will choose 2 points as being a rational person it is the logical answer as it. Maximixes my minimum gain

Suchitra Pandey - 5 years, 11 months ago

Maximises your minimum gain.... I like that...

Joseph Baez - 5 years, 10 months ago

This is something that I might try in a random group. Will post later the result of this.

Adeen Shukla - 5 years, 11 months ago

Please do! I'm definitely curious to see how it turns out.

Brock Brown - 5 years, 11 months ago

I did that. With 5 groups of 10 different people in each group . And the results were completely random 3 times, no one chose 6 points and in the other two, once around 5 people chose 6 and once only one (that's ten percent) chose 6 points Maybe I need to try with a larger group

Adeen Shukla - 5 years, 9 months ago

getting anything in this case is unlikely, choosing 6 would be counter intuitive. unless I could advocate for 2, but chose 6. although still I would chose 2, so I can be one of the people who didnt ruin it for every one els.

Martin Shoosterman - 5 years, 11 months ago

I'd choose 2. Benefits everyone. And too risky to choose 6. Or maybe it's just a matter of whether we are selfish or not.

Nirmalee Cr - 5 years, 10 months ago

I would pick 6 only because everyone would read and think everyone picks 6 but then goes for 2 so therefore no one would pick 6 they would all pick 2 so I would pick 6

Kelsey Aldridge - 5 years, 10 months ago

Id pick 6 ..... I mean other peoples logic is everyone will pick 6 so ill pick 2. But everyonenelse thinks the same, so why not 6? Only a handful of people would think of this when it is a test and they are stressed to finish.

Hudson McCallum - 5 years, 10 months ago

I either want 6 points, or no points at all. Go big or go home!

David Cove - 5 years, 9 months ago

People are greedy and use the logic that safety is first so almost nobody will pick 6. That's why most people pick 6; they think they're one of the only ones. In the end, my choice doesn't matter so I may as well choose 2 for shits and giggles.

Ruby Syuukyoku - 5 years, 7 months ago

Yes! For shits and giggles!

Michael Mendrin - 5 years, 7 months ago

6 points

Department 8 - 5 years, 11 months ago

no 2 points are better

Kaustubh Miglani - 5 years, 11 months ago

why 6 ???

Kaustubh Miglani - 5 years, 11 months ago

2 points.

Even if one is allowed to not choose at all, it is still better to choose 2 points.

Raghav Vaidyanathan - 5 years, 11 months ago

6 points

Shashank Rammoorthy - 5 years, 11 months ago

(If same condition happens for others) We think that we will waste our time by selecting 6 then for having no loss of point everyone will think to not loose the point by selecting 6 and will select 2 because they may think that 10% student will surely apy for 6. Then because of this thinking rarely students will choose 6. so we can choose 6.

Shivam Dixit - 5 years, 11 months ago

6 points. Just go for it. Because if more than 10% students picks 6 (which has a higher probability) then noone gets any points. Betting on the odds

Chirag Jung Kunwar - 5 years, 11 months ago

well, i would choose 6 as nobody would choose 6 once they see the 10% thing

Jingyang Tan - 5 years, 11 months ago

I'll choose 2 points. There'll be a little chance of leveling up the poles then. But I think more than 10% will go for 6 points either way..

Tithiprani Chakrabarty - 5 years, 11 months ago

I will definitely choose 6. Because it is possible that most of the students think that "2" is the best answer, and it may left only a few of the students that will choose "6", therefore maybe I will be one of the 10% of the students who choose "6".

Hans Sebastian Mulyawan - 5 years, 11 months ago

The choices are irrelevant. The probability that only less than or equal to 10 percent of students choose 6 might probably happen because people want to be safe and choose 2 instead, so that they think at least they got extra points. Thus choosing 6 will be a better option. That is if not more that 10 percent of the class think the same logic as I do.

If more than 10 percent of the class does have my logic, than even if I pick 2 point, the probability one person's decision to pick 2 can tip the balance to counter the 10 percent is too small. That is, the choices are irrelevant and picking 6 will be more logical.

Billy Sugiarto - 5 years, 11 months ago

I thik its 2 marks

Firdous Fatma - 5 years, 11 months ago

6 points more people are oblighted to choose 2 points because of fear of not getting any points.

Tessa Valentino - 5 years, 11 months ago

6 points

Anbu Srinivasan - 5 years, 11 months ago

6 points

Jermaine Hamilton - 5 years, 11 months ago

If I choose 6 points, then I will get equal or more points than everyone else. If I choose 2 points, then I will get equal or less points than everyone else. Therefore I will choose 6.

Julian Yu - 5 years, 11 months ago

But if more than 10% of the class thought like that, no one would gain any points, which would mean you gain equal points as the rest of the class but would possibly give you less overall points

adam maneely - 5 years, 10 months ago

Omkar Sabne - 5 years, 11 months ago

Choose the 2points. You cannot guarantee everyones cooperation so do your part. Some is better than the possibility of none. It wont help if you skip because then youarnt helping raise the % for the 2 pointers.

Gage Gansen - 5 years, 10 months ago

Mitchell Kale - 5 years, 10 months ago

What if you leave it blank.... How would that change the aspect if yit were possible to leave it blank?

Joseph Baez - 5 years, 10 months ago

I think people would pick 2 points because they don't lose anything if more people pick that

David Pearson - 5 years, 10 months ago

Choose 6 to help ensure that nobody gets extra credits.. don't take the class if you can't ace it on your own!!!

Kyle Wilderson - 5 years, 10 months ago

I would not choose

Amy Marie - 5 years, 10 months ago

Personally, I'd always choose 2, it is better to have some than none at all in this situation so why risk it in the first place.

Steven Pohl - 5 years, 9 months ago

Sir, I have some doubts. Can you please clarify them here

Ram Mohith - 2 years, 12 months ago

I think it is a recursive pattern of thinking people will not pick six points, because they are worried that too many will pick six points, and because other people know that people will not pick six points and therefore pick six points. There is no real answer to this question, because we cannot presume.

Krishna Karthik - 2 years, 7 months ago

6 so that I can ensure fairness.

Saya Suka - 1 year, 4 months ago

most people pick 6 for thepoints but will pick two because of this fact and this process keeps repeating

NSCS 747 - 11 months, 3 weeks ago

I will pick none of the options, I can't get $102 \text{ or } 106$ if all my answers are correct.

Vinayak Srivastava - 10 months, 3 weeks ago

Taking into account greed of others, I'd pick 2pts so I could try to counteract the majority which would more than likely win and nobody gets points.

Nick Blackshear - 5 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$