Not Reporting A Crime

[Elysium]

Well, we had learned about the "reporting a crime" scenario in Game Theory. Here was the representation of that situation:

1. Let's say that each person has the same preferences with regard to this case.

2. Let's say that you derive 2 units of satisfaction (hereafter "utils") from knowing that the crime was reported.

3. Let's say that it costs you 1 util to make the report yourself.

4. Let's say that there are N number of people who witnessed the crime and could report it.

If you decide to report the crime, you can be in 1 of 2 situations:

- You reported the crime, and someone else has as well. Your "payoff" here is 2 - 1 = 1. (the crime was reported, and it cost you effort to make the call yourself).

- You reported the crime, and no one else has. Your payoff is 1 as well. (same as before.)

If you decide to not report the crime, you are in 1 of 2 situations:

- You didn't report the crime, and someone else has. Your payoff is 2 (the crime is reported, you didn't have to do it).

- You didn't report the crime and no one has. Your payoff is 0 (no one reported, so you get nothing).

After some derivation, you arrive at this equation:

probability that ONE PERSON will report the crime is (1 - 1/(2^(1/(N-1)))).

Therefore, the probability that NO ONE will report the crime is 1/(2^(1/(N-1))) * N = 1/(2^(N/(N-1))).

If you take a look at that, and make N approach infinite, you see that the probability that no one (i.e. every single person will choose to not report the crime) will approach 1/2. You have a probability 1/2 that the crime will be not reported at all, which is pretty darn high.

There you go with the notion of "diffusion of responsibility."

[softwareNerd]

You have a probability 1/2 that the crime will be not reported at all, which is pretty darn high.

But, it is only high because you have assumed it is high: i.e., you assumed the figures 0,1 and 2. [Well, the zero was not really an assumption.]

I'm not clear as to what assumptions go into the calculation of probability? Are these people acting rationally? randomly? in ignorance of the action of others? with an assumption about the actions of others? what?

[DavidOdden]

Game Theory: ugh. I don't even know where to start. Well...

1. Let's say that each person has the same preferences with regard to this case.

That's a rabidly implausible assumption. What can't we just stipulate the assumption that no people commit crimes? Or, at least, pick some range of values and salt the population with various numbers of people with various values.

2. Let's say that you derive 2 units of satisfaction (hereafter "utils") from knowing that the crime was reported.

This too is implausible. It seems like an arbitrary number (I assume you can have 2.3 units as well as 1.87 units and so on) which in itself is not a problem: but, the value has to be computed recursively since that value is not arbitrarily plucked out of a sphincter, but comes from a computation of.... knowledge that the agent caused justice.

3. Let's say that it costs you 1 util to make the report yourself.

Again, more realistically, it is a range of values: the mean will be low, the thin tail of seriously detrimental cost wil be vanishingly narrow.

4. Let's say that there are N number of people who witnessed the crime and could report it.

Even this is a grotesque oversimplification. The bottom line is justice: of those N people, less than N can produce justice (lousy witnesses, for example, do not result in justice and can result in miscarriage of justice).

And finally, the equation utterly fails because the bottom line is not "satisfaction". I turn in credit-card thieves not because I get "satisfaction" but because I recognise that they are a direct threat to my existence. This factor -- recognising threat to existence -- needs to get lots of points if you're gonna mathematically model it. The fact of the matter is that when people see that a person is a murderer, understand that they themselves could easily be murdered, and understand that they live in a just society with a legal system that does not allow murderers to go unpunished, the chances of the crime being unreported are virtually nil.

Edited April 12, 2006 by DavidOdden

[Hal]

The probability that ONE PERSON will report the crime is (1 - 1/(2^(1/(N-1)))).

Therefore, the probability that NO ONE will report the crime is 1/(2^(1/(N-1))) * N = 1/(2^(N/(N-1))).

Leaving aside the problems relating to the oversimplifications in your assumptions, I find these equations strange. I dont really know what your working was, but if the probability that one person will report it is (1 - 1/(2^(1/(N-1)))), then it folows that the probability that either noone or more than one peson reports it is 1/(2^(1/(N-1))). But you then go on to say that the probabality noone reports is N times this number, which seems intuitively wrong. I'm also not noticing any binomial coefficients which is surprising - if the probability that a given person reports it is p, then the probability that only one person reports it should be N*p*(1-p)^(N-1), and its not clear how you can get the expression you mention from this.

Finally, 1/(2^(1/(N-1))) * N = 1/(2^(N/(N-1))) isnt generally true (for N=0, you get 0=1).

edit: Also when N is large, 1/(2^(1/(N-1))) * N is order(N), and hence tends to infinity at a linear rate as N->infinity, not 1/2 (although the expression you claimed it was equal to does -> 1/2).

Edited April 12, 2006 by Hal

[Hal]

I've read it again, and I cant work out how you got your answer. I'd be interested in reading the working, if you want to post it.

edit: actually your answer seems wrong, however you got it. If I'm the only person who witnesses the crime (N=1), and my expected utility from reporting is higher than my expected utility from not reporting, then probability of it not being reported should be 0. However your expression doesnt even give an answer for this, because you end up trying to divide by 0 (you have 2^(N/N-1) which is 2^(1/0) here)

Edited April 12, 2006 by Hal

[Elysium]

First of all, I do also agree that a lot of Game Theory is pretty dumb in the sheer number of assumptions it makes to make its games work. However, with regard to DavidOdden's post, I think this depiction of not reporting a crime does make sense in its payoffs and its setup; it, unlike a lot of Rand's works, is designed to capture people as they currently are, not their philosophical ideals. For example, I could argue that most people DON'T recognize that not reporting a crime means a possible DIRECT threat to their own values, and thus, only assign it a "2" payoff. Likewise, I could argue that a lot of people are really lazy; it really hurts them to report a crime, so much so that it takes up 1/2 of the actual payoff of having the crime reported in the first place. That there are N number of homogeneous people in this example really just makes it a lot easier to do the calculation.

Anyway, that said, here is the process that we were taught:

The process by which you will choose whether you will report a crime or not is based on, according to this model, the notion of "mixed strategy." More specifically, this means that you will divide reporting and not reporting a crime into two probabilities: P (report crime), and 1 - P (don't report the crime). You want to choose the P's in such a way that you will make others "indifferent" to your personal decision, the logic being that "if I am always known to report things, then others have an incentive not to do it themselves, because they know that I will. Therefore, the best approach for me to is to mix it up so that they will know that their payoff whether they report or not will stay the same regardless of whether I report or not." In short, you are playing a glorified guessing game with all other witnesses to the crime.

So your goal is to make it so that other people's payoffs expected payoffs will remain the same if they report or if they don't. Therefore, assuming the 0, 1, and 2 payoffs I had given earlier, we have:

If you decide to report the crime, you can be in 1 of 2 situations:

- You reported the crime, and someone else has as well. Your "payoff" here is 2 - 1 = 1. (the crime was reported, and it cost you effort to make the call yourself).

- You reported the crime, and no one else has. Your payoff is 1 as well. (same as before.)

If you decide to not report the crime, you are in 1 of 2 situations:

- You didn't report the crime, and someone else has. Your payoff is 2 (the crime is reported, you didn't have to do it).

- You didn't report the crime and no one has. Your payoff is 0 (no one reported, so you get nothing).

1*p(someone else reports it) + 1*p(no one else reports it) = 0*p(no one else reports) + 2*p(someone else reports)

Solve it out, and you get p(others report) = p(others don't report). Your probabilities are, therefore, p = 1/2, (1-p) = 1/2.

Now you have N people, each represented in this same way.

p(not a single person reports) = (1 - p1)(1 - p2)*...*(1 - pn), each person's probability of not reporting multiplied together until you reach N people.

This is (1 - p)^(n-1). Set this equal to 1/2, because that is what I derived earlier for p(you don't report).

(1 - p)^(n-1) = 1/2.

Solve for p again. This new p is the p(each individual person will not report).

1 - p = (1/2)^(1/(n-1))

p = 1 - (1/2)^(1/(n-1))

Everyone will call with this probability if they want others to be indifferent.

1 - that p will get you the probability that each witness will NOT report.

1 - [1 - (1/2)^(1/(n-1))]

= (1/2)^(1/(n-1))

Take this to the Nth power (for all people), and you have the probability that NO ONE (as in, not any single 1 of them) will report:

(1/2)^(n/(n-1)).

Now increase that N to near infinite (showing that there is a growing number of witnesses), and the probability approaches 1/2.

I acknowledge that I mighta made an error somewhere in there, but I do know that I had the right answer with the right procedure understood a few weeks ago.

If you want to change your payoffs such that the benefit of having a crime reported is exorbitantly high compared to the "cost" of reporting it, then you will, of course, arrive at an answer that will reflect a more "Objectivist" outlook on crimes, specifically, that O'ists will all be much more inclined to report the crime than 1/2.

And with regard to Hal's question that you cannot have (1/0), keep in mind that Game Theory implies having 2 or more players, otherwise, it's not Game Theory, but just one person making a decision.

... Whew. That took too long. >_>

Edited April 12, 2006 by Elysium

[DavidOdden]

unlike a lot of Rand's works, is designed to capture people as they currently are, not their philosophical ideals. For example, I could argue that most people DON'T recognize that not reporting a crime means a possible DIRECT threat to their own values, and thus, only assign it a "2" payoff.

My point is that, even just taken as social science, this isn't a realistic model because is suppresses variation. I'm not sure whether I would accept the claim that the majority of people in, say, the US don't understand that leaving crime unreported could be a threat to their values; it could be 45%, or 65%, or 20% -- whatever it is, you're talking about a fact about people's beliefs and not an arbitrary assumption where you can make up numbers. The model is clearly wrong in presuming a completely homogenous population, and the question is, what would happen if say 10% of the population put a really high value on reporting crimes, like a trillion. I understand that these simplifying assumptions make it easy to do the calculation, but they also make the calculation invalid. The glorified guessing game strategy may be correct for a very small percentage of the population -- game theorists, I suppose -- so the question is whether such an equation is supposed to model anything other than the behavior of game theorists.

[Hal]

The glorified guessing game strategy may be correct for a very small percentage of the population -- game theorists, I suppose -- so the question is whether such an equation is supposed to model anything other than the behavior of game theorists.

Game theorists often try to wave away the problems relating to the lack of applicability of their work by claiming that what they are modelling is how people ought to behave, rather than how they do behave. However I've never encountered any particularly convincing arguments as to why anyone 'ought' to behave according to the axioms of utility theory. Personally I find game/utility theory, and neo-classical economics in general, to be fairly silly, although it is useful for problems where the compexity of the real world can be ignored (like analysing board games! Combinatiorial game theory has produced some pretty cool techniques for studying Go endgames, although interesting enough, even here its only applicable to the more 'simple' parts of the game and doesnt really shine any light on most of it).

Elysium - thanks for posting the working, I'll read over it tomorrow.

Edited April 12, 2006 by Hal

[Elysium]

I will mention that I agree with David and Hal on the notion that a lot of Game Theory tends to assume that actions are "the way people should act," yet very little actually does. However, in its defense, it was primarily a tool deviced in the Cold War for thinking up nuclear strategy. You'll note that the basic 2 x 2 matrix of the Prisoner's Dilemma is a marvelous representation of the nuclear dilemma (e.g. if we launch a nuke, what does the enemy do, etc). I think that a lot of the errors of its ways arise when it attempts to overapply itself to too many situations where it's not just a one-off life-or-death situation, where relationships are fostered between participants, and credibility can be molded. Credibility and relationships really comes into play with Mixed Strategy Game Theory, and even then, a lot of it is still iffy in terms of the assumptions and adherence to reality.

Edited April 15, 2006 by Elysium

[Illuminaughty]

Its not. You're erroneously thinking that law enforcement officers are automatically moral. That the system isn't intrinsically flawed. That legal equates to moral. Any system of laws where someone can get life in a cage for selling marijuana cannot begin to claim morality.