If we think about it, small settlements start with no crime. When someone tries to steal something or kill an innocent person, he would get killed by the rest of the people. So, at what point of growth of a settlement will a large crime net start developing? Even better, when will crime start being a profitable action worth to take the risks for?
At a population of 150 people.
Because 150 is "Dunbar's number". It's the limit of how many people on person can have a social connection to.
There are two kinds of crime.
First there is crime committed for emotional reason. You slept with my wife, I punch you in the face (assault and battery). This kind of crime happens because of social connections, so it can happen in communities of any size. It could happen when we three would be stranded on an island.
But you were talking about the second kind of crime: Professional crime which is committed for economical reasons. I need money, So I go to the marketplace and pick some pockets. This kind of crime is easier to commit on people you don't know personally, because not feeling empathy for the victims of said crime makes it easier to justify it emotionally.
Also, when everyone knows everyone, they also know everyones source of income. It would be impossible to live of crime under this condition. Sure, one could choose to isolate themselves from the rest of the group, but then that person would be suspect just for that. Money keeps disappearing, and that one guy who talks with nobody and doesn't seem to have any job doesn't seem to run out of cash? Something must be wrong about him, better investigate. But when the community exceeds Dunbar's number, you get used to interacting with people you don't personally know, so it might be possible for the lone thief to coexist with the rest of the population without standing out.
Another aspect you should keep in mind is that the biggest catalyst for professional crime is social inequality. It's no coincidence that in any larger society, those demographics of the population with the lowest income are also those with the highest crime rate. People on the lower end of society will turn to a life of crime because of jealousy or out of desperation. So the higher the social disparity in your society, the more likely that organized crime emerges.
I think once the settlement reaches the point where not everyone knows one another, crime would become more of an issue. In a small, tight knit social community, there would be a lot of peer pressure to fit in and adhere to the the socially correct behaviors.
I think criminals being killed is a pretty big assumption to make. Most likely, the greater threat would be exile (murder may be the exception here). For minor crimes, social shaming would be another large factor. In general, how crimes are treated will depend a lot more on the justice system or the religious or moral beliefs of the community.
People will always find crime profitable, it's just a question of how much they can get away with. In cities the opportunities are much greater and anonymity gives you protection. In small settlements, everyone knows each other and there's a lot less trouble you can get into.
I'd also question the assumption that small settlements start with no crime (which also depends on the definition of crime). Ever hear stories about sordid going-ons or murders in small towns? True, if you look at the police report section of small town newspapers, it tends to be mainly people growing weed in their parents backyard or driving while drunk. But a smaller population in general will produce fewer crimes than a larger population.
Also, other factors would play into the growth of a crime net. Look at the Prohibition era in the United States - the ban on alcohol gave rise to the growth of mobs and gangs, who made a tremendous profit. Are there specific laws that large numbers of people want to break? Is there a large demand for an illegal good? Are certain crimes not prosecuted and considered "safe" to commit?
If we think about it, small settlements start with no crime.
I think this is an extremely questionable assertion. I'd accept that it starts with small crimes: stealing the milk, minor bullying, etc. but I doubt you can gather any collective of humans without there being bad behaviour. It tends not to be dealt with through judicial means if it occurs at the lower end and may thus be not considered crime but it's always there.
What happens as the settlement increases is that potential for anonymity, potential for unearned rewards and the potential for factions or individuals within the settlement to come into conflict increase. This increase drives an increase in both the frequency and severity of the crimes.
This was a comment, but it got a bit too long. I agree with CoolCurry that anonymity is a very likely way this could start; in fact, it was my first thought as well. Just look at people who act anonymously on the internet (or better, don't).
But as a counterpoint to that, you can also have crimes that result from feuds between two rival groups (e.g. Montagues & Capulets). In this case, everyone knows full well who the two sides are, but neither side can let go of some grudge. This could be about literally anything that two people can disagree over: money, property, politics, religion, behavior, a marriage… If you firmly believe that you're "right" about something, and the other guy is "wrong", then you may be willing to do whatever it takes to correct the perceived wrong, even if that means taking action outside of the law. Even if everyone knows who you are.
Crime does not exist per se...It is a social construct, meaning that it is kind of artificial. And like any other, or at least the most of, social phenomena it is hard to find "math formula" for explaining it.Our society is based on things which are allowed and things which are forbiden.Things which are forbidden become crime. Now the question is how, where and who made list for allowed and forbidden things? This is the point where the connections with settlement size appear. When settlement starts growing everybody, basically have same amount of wealth, everyone are producing goods for personal use, roughly everyone have the same manpower (family members).As the population is getting bigger and bigger new people are behind "old" one, because they have to start from scratch. At some point that is not possible anymore; they are forced to work for earlier established families.Now, not all of the "old" families can afford to pay help, depending on many things, some of them are random.Gradualy, those who can hire help are getting richer and richer, while the others are getting poorer. Now wealthier want to protect their wealth and possition.Thats how the laws were made. Of course you can guess who wrote them? Now that we have law we have a criminal. So short answer to your question is criminal starts when social segregation appear, and when small group of people succeed to force their interests to whole community.
Trivia: There is a Tibetan legend on how all of evil came to our world, and how crime started: In the beginning people looked like Gods, they were made of light, good and honest. They were eating something like a corn which was growing from the earth. Every person had its own plant, and every day one new fruit would appear on every plant, and one fruit was enough to get them through the day. One day on someone’s tree 2 fruits have appeared; he picked both and eat them. Next morning there was no fruit on his plant, (because day before he got 2 I guess), so he went to some other plant and stole a fruit. Since owner of that fruit would stay hungry, and then he had to take fruit from someone's plant. The rest we all know :)
Means, motive and opportunity.
Means x Motive x Opportunity = Chance of crime
All three factors must exist for a crime to take place. All considerations must be examined through these three factors.
when will crime start being a profitable action worth to take the risks for?
Risk can be mostly assessed as a question of motive. Regardless of the level of risk, is there sufficient motive to deem that risk acceptable?
What influences motive? The pay-off of accomplishing the crime without punishment, and/or the impact of not accomplishing the crime. Carrot and/or pointy stick.
Let's look at a straight-forward pointy stick motivation. Hunger. Is hunger a direct factor of population size? No. It's a matter of the societies ability to feed the masses. In a utopia, you could feed millions, the hunger motive never surfaces. But for a dozen people in isolation with limited supplies, there's pretty quickly motive to steal food from each other, or kill and eat one another.
Population x Chance of hunger x Moral compass factor = Motive
What would I consider to be a moral compass factor? Well there's some people that regardless of motive, just won't do it. It's their up-bringing, it's their morals.
Back to the other factors though, regardless of motivation, a crime isn't going to happen if the would-be protagonist doesn't also have the means and the opportunity. Under incarceration or some other police-state, means and opportunity are closely controlled in an effort to mitigate what may be a high-motive situation for the population.
at what point of growth of a settlement...
If growth is to be simplified as population size, then consider it only (per above), as one factor in the overall calculation.
Crime starts in settlements of at least 2 people.
Take a two year old with a one year old. Left unsupervised, the two year old will kill the one year old.
Or take another violent crime, assault and battery, but no killings. Practically in every family, children will assault and batter each other. And of course, children will also steal from each other.
The parents will be the ones to either condone or punish such behavior. So in that sense, the parents decide whether something is a crime, or not a crime. And that decision is made internally and enforced internally by the parent(s), or babysitter, or teacher/principal.
Unfortunately, parents do not keep paper records of their children's crimes. Plus, it's not like they really need to do that. Those incidents are dealt internally anyway.
I do not know the answer, but...
Both top-rated answers rely on group size to keep members non-anonymous. I don't think that is enough.
I know that social sciences is not a real hard science, but the "Dunbar's number" rule is disproved by the population of Pitcairn Islands. With a population of less than 100 (currently 56) they had pretty bad sexual assaults - worst per capita in whole world. Non-anonymity, or religion, did not prevented that.
Unless you count those assaults as "boys would be boys".
Donner party got stuck in snow in winter on Oregon trail and resorted to cannibalism to survive. Was it crime?
How many counter-examples are needed to disprove a "rule" in sociology as invalid?
So your answer might need better definition of acceptable/unacceptable crime, and cultural expectations.
My guess is that number is as low as two - depending on circumstances.
I would argue that "crime" already starts with having multiple people around, but then it is not called a "crime", but simply socially inacceptable, and dealt with differently.
For example, imagine there are few families, a member of one family commits a "crime" against someone from another family. As there will be much intermarriage and connection, this will lead to larger family disputes (e.g. aunt doesn't visit anymore, because the guy treated someone in her family badly), and perhaps eventually to feuds.
Only when the society becomes large enough is there a need for a more formal way of dealing with such issues, and only then is a "crime" properly defined and treated as such (in early societies, there was much more often a form of compensation, even for losing limbs or killing someone, eventually, exile or even death).