[yotamarker]

you are in a classroom of 24 students. suddenly a roach walks in.
if no one kills it you know it will annoy all of you throughout the day and even might breed or bring about more roaches.
who ever kills the roach will soil their shoes and waste some energy chasing it.(lets just assume you all have brand new shoes).

roach killer gain : 1-c : deadroach(gain)-shoes(cost)
standbye gain : 1
##########################################
case no one kills : -1 for all

[frankinstien]

you are in a classroom of 24 students. suddenly a roach walks in.
if no one kills it you know it will annoy all of you throughout the day and even might breed or bring about more roaches.
who ever kills the roach will soil their shoes and waste some energy chasing it.(lets just assume you all have brand new shoes).

roach killer gain : 1-c : deadroach(gain)-shoes(cost)
standbye gain : 1
##########################################
case no one kills : -1 for all

There is one parameter missing, which is intrinsically biological and that is territorial control. In other words, the problem of the roach breeding and producing more annoying and disease-carrying roaches is dependent on whether or not its the individual's personal property,  where the consequences of not killing the roach will affect the person personally.  So a classroom is not viewed as a person's personal space so they don't care about the presence of the roach, it's the school's problem.

[HS]

Others might be grateful, giving you a plus one, or 24, depending on the number of people effected. But a single theory trying to model a system seems untenable, likely to reduce itself into absurdity like Ouroboros. Keeping the vehicle of society rolling requires an interaction of opposite forces within the engine. Energy imbalance at smaller scales indicates energy in balance at larger scales, rolling the whole thing forwards through mechanisms of spin, rhythm, and oscillation, indicative of something living.

This counter force to the first theory can take “my way” or the “high way”. That is, on one hand you can circumvent such laws by being extremely selfish. One of the fun things about having a theory of the “game” is the ability to occasionally point your middle finger at it and do things which contradict it just to demonstrate that it’s not the boss of you. So, you quash the roach, and there is mush rejoicing in the community.

On the other hand, you can be very self less. You can relinquish your grip on the idea of you as an entity that’s very separate from your ecosystem, then your concept of self is able to expand to include the whole shebang, leading you to take actions which are best for the entire system. Which often ends up including your narrower definition of self.   

So maybe the answer is to base actions on the interplay of several modes of personality. The rational the selfish and the selfless. But then, for the modes of personality to become sustainable, you’d need modes of non personality. An emergency is not the time to be frantically juggling game theory, personal advantages, and altruism. You’d have to suspend those higher-level calculations for the time being and calmly deal with physics.

[WriterOfMinds]

I catch the roach in a glass and take him outside, which requires no soiling of my shoes. I saved his life and made everyone else happy; I gain 2 points from the resulting personal satisfaction.

But I suppose what you really wanted to know is how to look at the mechanics of the game, so let's take its skin off and consider only: what to choose in a competitive game where the possible scores are 1-c, 1, or -1?

Consider the case in which nobody is currently taking the "1-c" option. Everybody is currently in danger of scoring a -1. If your only goal is to maximize your own score, then the ideal choice for you depends on how big c is.  If (1-c) is greater than -1, changing your strategy will improve your score, and you should do it.  Sure, you will only end up with 1-c while everybody else gets 1, but 1-c is better than the -1 that you were going to get.  If 1-c is less than or equal to -1, changing your strategy will not improve your score, so you might as well not.

In technical terms, I think all players choosing -1 is a Nash equilibrium of the game if c >= 2.  If c < 2, any player choosing 1-c and all other players choosing +1 is a Nash equilibrium.

If you play multiple rounds of the game, then (supposing that 1-c is greater than -23) the ideal for both you and the group is for all players to take turns picking the 1-c option. That way you pay the cost in one round out of 24 and score +1 on every other round. If communication is allowed, you and the other players can set up an agreement to do this; if it is not, you can just try to implement it and assume everyone else is smart enough to catch on. If one player tries to cheat the agreement (doesn't take the 1-c option when it's their turn to do so), the other players can "punish" them by also refusing to take the 1-c option on their turn.

[yotamarker]

I catch the roach in a glass and take him outside, which requires no soiling of my shoes. I saved his life and made everyone else happy; I gain 2 points from the resulting personal satisfaction.

But I suppose what you really wanted to know is how to look at the mechanics of the game, so let's take its skin off and consider only: what to choose in a competitive game where the possible scores are 1-c, 1, or -1?

Consider the case in which nobody is currently taking the "1-c" option. Everybody is currently in danger of scoring a -1. If your only goal is to maximize your own score, then the ideal choice for you depends on how big c is.  If (1-c) is greater than -1, changing your strategy will improve your score, and you should do it.  Sure, you will only end up with 1-c while everybody else gets 1, but 1-c is better than the -1 that you were going to get.  If 1-c is less than or equal to -1, changing your strategy will not improve your score, so you might as well not.

In technical terms, I think all players choosing -1 is a Nash equilibrium of the game if c >= 2.  If c < 2, any player choosing 1-c and all other players choosing +1 is a Nash equilibrium.

If you play multiple rounds of the game, then (supposing that 1-c is greater than -23) the ideal for both you and the group is for all players to take turns picking the 1-c option. That way you pay the cost in one round out of 24 and score +1 on every other round. If communication is allowed, you and the other players can set up an agreement to do this; if it is not, you can just try to implement it and assume everyone else is smart enough to catch on. If one player tries to cheat the agreement (doesn't take the 1-c option when it's their turn to do so), the other players can "punish" them by also refusing to take the 1-c option on their turn.

I agree the solution is wait to see if anyone else kills it, if not than kill it yourself, but, as far as game theory goes,
how does it predict if more students mean a higher or lesser probability someone else would do it ?

if you assume high : you can also assume the other players assume it is high and therefore its low ?
if you assume it is low than you can assume the other players assume it is low than it is high. so you realy cant tell

[WriterOfMinds]

I agree the solution is wait to see if anyone else kills it, if not than kill it yourself ...

That's not what I said the solution was. The solution is to try to do the deed yourself if c < 2, and to not do it at all if c >= 2.
I wasn't really considering waiting as a possible option. If you introduce that, then you have to decide how long to wait, which brings in additional complications.

how does it predict if more students mean a higher or lesser probability someone else would do it ?

The game theory paradigm I'm using assumes that the students are all rational agents. They're all going to pick the best possible strategy. In this particular game, adding more players does not change the best possible strategy for any player, so it neither increases nor decreases the probability that someone besides you will take the c-1 option.

In real life, humans are not always rational agents ... you have to consider things like the Bystander Effect. But that's more of a matter for psychology than for game theory.

if you assume high : you can also assume the other players assume it is high and therefore its low ?
if you assume it is low than you can assume the other players assume it is low than it is high. so you realy cant tell

The point of the Nash Equilibrium concept is to shortcut these "they think that I think that they think that ..." loops.  I suggest doing some further reading on it.