# Where should you sit in a circular pecking order?

1. Oct 3, 2012

### lugita15

I have a question that I can't quite categorize, which is why I'm putting it in the General Math section, and even the question title doesn't do a very good job of describing it. So here goes: We have a game involving a group of N people seated in a circle. Each person has their own "magic number" K, which is some natural number that's unique to them. The game starts with the first person saying the number "1", the second person saying the number "2", etc. all the way around the circle, and we just keep going around the circle again and again counting higher and higher. There isn't any activity in the game other than people saying numbers. The game continues for a total of M moves, where M is a number that none of the players know in advance, and then it ends (it's like musical chairs, where people keep moving around until the music abruptly stops). Finally, at the end the winners and losers are determined at follows: if you ever said a number that was higher than your magic number, you lose. Otherwise, you win.

So to sum up, you know your magic number K and the number of people playing N. The only thing you have control over is where to sit, i.e. do you want to be the first player, the second player, etc. Also, although you don't know in advance exactly what the total number of moves M is going to be, you do know some rough information about it, perhaps a probability distribution, or the likely confidence interval that M/N is within. So given this, how do you choose where to sit in order to maximize your probability of being a winner?

I apologize if this question is vague or convoluted. As I said, I don't really know what subject of mathematics this falls under, but although it's fairly abstract, it's a problem that's inspired by various real-life situations I've encountered. I'm happy to clarify anything that's confusing in my description.

Any help would be greatly appreciated.

2. Oct 3, 2012

Well I think it depends on if you know something specific about M/N or don't know anything about it.

Is that ratio close to 1? 3? .5? I think I would have a different answer for every single different value of the probability distribution.

If you know absolutely nothing about that distribution though, I would think being the first person would be the best. If I'm understanding it correctly, K (magic number) is a randomly assigned natural number. If it's randomly assigned and you don't want to say a number higher than it, well then it only makes sense to be the 1st person. This is how I'm thinking about it:

If M=N and every single person said a number, well the average number somebody said is the sum of 1+2+3....N all divided by N. Give N the value 10. That's 55/10 = 5.5. All of this is quite obvoius, saying that being the first person, you will always say the lowest number in a single loop.

Now I understand multiple loops around the circle can be made. Of course, if N=10 and M=11, the first person says the highest number of everyone. But if M is completely random, you'll find that the 1st person has the lowest probability of saying the highest number. IF M IS COMPLETELY RANDOM. Some pretty complicated math could show this, I'm not sure if I want to spend the time trying to do it though.....

But like I said, if M/N isn't completely random and is known to be a probability distrubution close to some number, then the whole problem changes completely and gets more complicated.

EDIT please tell me if I'm thinking of this in the wrong way

3. Oct 3, 2012

### lugita15

spaderdabomb, let me give you the specific information I know in the real-life situations that inspired this. M/N tends to be between 1 and 2. It is certainly almost always above 1, and it often goes pretty close to 2. Also, K/N tends to range from a little under 1 to something significantly less than 2, although in principle it could go as high as 2.

4. Oct 4, 2012

### Norwegian

This has nothing to do with probabilities or ratios. If you make sure that your sequence of numbers to count includes K, you have obviously given yourself the best chances to win.

5. Oct 4, 2012

### Alcatrace IV

If M is unspecified in a given set of extractions, then any logistical coordinate is as good as any other. Really, there is no range of optimality without a set of reference for the value(s) of M. Even if M was only defined as a numerical range or even an algebraic formulation of some concoction, then we would have a context for judgement and decision; instead, we are left with only a platform to project our undulating ambivalence. If M turns could be 5 turns or any other number and if the numbers are registered at an arbitrary imbalance, then you can't make an informed gamble on what possible selections you should wager upon under duress of inspection, now can you? In example, if M was undetermined beforehand, then it wouldn't matter where you would sit, assuming a finite set of operative delegations. Meaningful rationality is a localized protocol and demands circumstantial criteria. An estimation without demarcation of error is effectively meaningless; calculations without the marker of units cannot be interpreted realistically. Likewise, the problem at issue as beset with an absence of specification is merely a semantics argument as opposed to a mathematical concundrum capable of being solved in terms of educated parametres.

If M was specified by a certain precept, such as M*N/2 or MN, then the best bet would be to go sit in the beginning, at the first seat, if M > than the number of attendees. That way, with continuation of rounds, you would always have the lowest number for accounting in a given loop of sequences. Otherwise, I would have to aim for the books in saying that, in the case of N>M, to sit in the last seat (or any seat ahead of the caller identity M) in order to avoid all means of taxation by compromising.

On another foot, I am new here (a highschool freshman, in case you wanted some context) and wish everyone a nice time! :D

6. Oct 5, 2012

### Norwegian

A most sophisticated troll has entered the building. I wish you a pleasant stay, and hope to see you around. Please balance your act accordingly as the house police may strike without warning.

7. Oct 5, 2012

### Alcatrace IV

Well, subtle innuendos in the art of trolling are one of my favourite vocations

However, as I do not (ostensibly) have a sufficiently advanced background in the technical formality of the language you are anxiously aquainted with on a regular basis, I wasn't so much trying to troll you as to allude nuances by contingently analogous contextualization and thereafter, explanatory elaboration. Remember, i am only a highschool freshman - i don't quite understand the higher level maths you might have negotiated in per nonconsensual stipulation

Could you please inform me as to the manner of my supposed deficiencies in adrressing the issue at hand?

On the one hand, what I contributed was perfectly reasonable. On the other hand, your empty criticism was nothing but a deflating emission of annoyance as repudiating your respectful guests by improper demeanor

I would enjoy elucidating my points ... if only you would tell me what I need to do in order to conform

Last edited: Oct 5, 2012