# 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?

Note by Pi Han Goh
6 years ago

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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.

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".

- 6 years 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

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:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 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)

- 6 years 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.

- 6 years ago

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

- 6 years 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.

- 6 years 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.

- 6 years 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.)

- 6 years ago

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.

- 6 years 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.

- 5 years, 12 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.

- 6 years ago

- 5 years, 1 month ago

Choosing 6 points is the Nash equilibrium

- 6 years 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

Found the redditor. :D

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

- 6 years ago

2

- 6 years 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

- 6 years ago

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

- 5 years, 12 months ago

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

- 6 years ago

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

- 6 years 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

- 5 years, 10 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.

- 6 years 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.

- 5 years, 12 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

- 5 years, 11 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.

- 5 years, 11 months ago

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

- 5 years, 11 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.

- 5 years, 8 months ago

Yes! For shits and giggles!

- 5 years, 8 months ago

6 points

- 6 years ago

no 2 points are better

- 6 years ago

why 6 ???

- 6 years ago

2 points.

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

- 6 years ago

6 points

- 6 years 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.

- 6 years 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

- 6 years ago

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

- 6 years 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..

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".

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.

- 6 years ago

I thik its 2 marks

- 6 years ago

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

- 6 years ago

6 points

- 6 years ago

6 points

- 6 years 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.

- 6 years 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

- 5 years, 12 months ago

6

- 6 years 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.

- 5 years, 12 months ago

6

- 5 years, 12 months ago

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

- 5 years, 12 months ago

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

- 5 years, 12 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!!!

- 5 years, 11 months ago

I would not choose

- 5 years, 11 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.

- 5 years, 10 months ago

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

- 3 years, 1 month 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.

- 2 years, 9 months ago

6 so that I can ensure fairness.

- 1 year, 5 months ago

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

- 1 year ago

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

- 12 months 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.

- 5 years, 11 months ago

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