**The PROBLEM VECTOR**
__An Introduction to the Problem Vector__

The Problem Vector or PV is a concept that uses mathematics as a basis for describing the way people subjectively approach their life problems (concerns, worries, frustrations et cetera). My inspiration for the concept came when I realised that it doesn't seem to matter what *objective* problems an individual has- but what *subjective* problems they have. In other words, it's not what is actually wrong with people's lives that drives their mood/behaviour; it is what they *think* is wrong.

For example, someone who has just had a minor car accident that is not their fault is likely to be extremely angry and upset with the party whose fault it is. They would not be calm and understanding on the basis that objectively, their life is not ruined or even majorly affected.

It is important to note that people can sometimes see things more objectively than others. Often, after experiencing some difficulty, a person might say: 'Well, it could be worse...' or 'At least there's food on the table...' and the like. These people have gained some level of objectiveness as to the value of their problems in macro or 'global' terms. Any such objectiveness must be taken into consideration when developing a complete model to describe an individual's approach to his/her problems.

**The Two Problem Model - with no Objectivity**

The simplest way to explore the PV is to take the situation of two people faced with two problems. A model with any less individuals or problems is either fairly trivial- or degenerate in the sense that it produces meaningless results. Also, this model is to contain no objectivity: people do not appreciate, or are not aware of the *real* relative extent of their problems.

*Problem Readings (PRs)*

Problem readings or Problem Vector Entries (PVEs) are numerical measurements of someone's problems. They are cardinal, not merely ordinal. In this sense, PR's can be distinguished from measures of 'utility' in economics. Someone who obtains a utility of 6 from consuming a given good values it more than consumption of some other good from which they obtain a utility of 3. Nothing more can actually be said about the extent of this preference- merely that one is preferred to the other. With PVs, someone who has a PR of 6 finds that problem twice as important as another problem with an entry of 3. The entries can be compared mathematically because they are cardinal.

Now consider a problem vector for individual A denoted < w , x >_{A} - where w is the PR for problem one and x is the PR for problem two {w,x>0}. Similarly, individual B will also have a problem vector denoted < y , z >_{B} where y is the PR for problem one (the same problem one as faced by A) and z is the problem vector for problem two (the same problem two as faced by A) {y,z>0}. Now note if there is no objectivity, then it is reasonable to assume that these two individual's both see there problems as being equally important. Such an assumption may not seem realistic- but this case is one with no objectivity. We have to imagine two individuals, A and B who are completely unaware, in any sense, of the relative importance of the problems they face as a whole. They know only that they have problems and they know the relative importance of these problems to themselves only.

*The Magnitude of a PV*

The concept of the 'extent of someone's problems' as a whole has been mentioned, but how might such a variable be measured? Well the answer is simple: problem magnitude. That is, a measure of someone's problems is the magnitude of their problem vector. Recall that the magnitude of a vector is the squareroot of the sums of the squares of its entries. e.g. mag(< 3 , 4>) = sqrt(3*3 + 4*4) = sqrt(25) = 5 (mag always +ve)

From the above discussion, then, if there is no objectivity, people will have problem vectors with the same magnitude.

*Objective and Subjective PV Analysis*

When it is said that two people are objectively or subjectively approaching their problems, this will affect the magnitude of their problem vector. Since this model is concerned with no objectivity, we can say that the two individuals involved are perfectly subjective and the magnitudes of their problem vectors are equal.

*Worked Example of a Purely Subjective PV Problem*

Consider Anne and Bob- two house-mates who live very simple lives. They both face two problems: the loud music playing next door and the fact that the television has broken down. The thing is that Anne and Bob rate these problems differently, Anne cares more about the tv than does Bob and Bob cares more about the music next door than does Anne. Say Anne thinks the tv problem is three times as important as the music, and Bob thinks the music is two times as important as the tv. Find Anne's and Bob's PVs.

Let p_{A} = {w , x}_{A} denote Anne's problem vector; and

let p_{B} = {y , z}_{B} denote Bob's problem vector.

Let the first entry denote the loud music PR and the second denote the tv PR for each person respectively.

Assume that both Anne and Bob are perfectly subjective individuals.

Now we simply solve the following equations:

It makes sense to let the magnitude of each problem vector equal some arbitrary amount e.g. 100, since we know that mag(`p`_{A}) = mag(`p`_{B}).

so, 3w = x ..............................................................................................(1)

and, sqrt(w^{2} + x^{2}) = 100 .....................................(2)

=> sqrt(w^{2} + 9*w^{2}) = 100

=> 10*w^{2} = 10000

=> w^{2} = 1000, and since x>0

=> w = 10sqrt(10), x = 30sqrt(10)

And now, y = 2z........................................................................................(3)

and, sqrt(y^{2} + z^{2}) = 100 .....................................(4)

=> sqrt(4*z^{2} + z^{2}) = 100

=> 5*z^{2} = 10000

=> z^{2} = 2000

=> z = 10sqrt(20), y = 20sqrt(20);

Hence, p_{A} = {31.623 , 94.868}

p_{B} = {89.443 , 44.742}

And this is the problem solved to 3 dec pl.

**Notes**: Interesting to note is the fact that Anne cares about the broken tv only about 6% more than Bob cares about the music while Bob cares almost 50% more about the tv than Anne cares about the music. Just because they have the same magnitudes, does not mean that they are to approach their problems in the same way. Here, Bob is clearly someone who is more likely to "worry" than Anne.

**Measuring Individual Worry Factors**

A Worry Factor is a measurement of the extent to which an individual will worry. Above we have established by reasoning that Bob appears to be someone more worried than Anne.

Definition: Let `f(q,r,s,t)`_{i:j} denote the "worry" factor for an individual i compared to an individual j subject to problems with PRs that are q and s, r and t respectively, so that

p_{i} = {q , r} and p_{j} = {s , t} then:

`f(q,r,s,t)`_{i:j} = (q*t)/(r*s)

and `f(q,r,s,t)`_{j:i} = (s*r)/(t*q)

Hence, `f(q,r,s,t)`_{i:j}*`f(q,r,s,t)`_{j:i} = 1

In the example of Anne and Bob:

`f(w,x,y,z)`_{A:B} = (31.623*44.742)/(94.868*89.443) = 0.167

`f(w,x,y,z)`_{B:A} = (89.443*94.868)/(44.742*31.623) = 5.997 = 1/0.167

The factors show that Bob is more of a "worrier" than Anne.

*The Conflict Factor*

One can also now measure the Conflict Factor (CF) which uses the extent to which the two individual's PRs differ in order to determine the likelihood of conflict resulting from opposing interests. **It is simply defined as the sum of the squares of the worry factors minus 2**

That is, CF = (`f(w,x,y,z)`_{A:B})^{2} + (`f(w,x,y,z)`_{B:A})^{2} - 2

Simple Example: Consider the simple case in which two individuals have identical PR's for two given problems; say {2 , 2}. Then they have respective worry factors (2*2)/(2*2) = 1. And this means they have conflict factor 1 + 1 - 2 = 0. However, minimising x^{2} + y^{2} subject to x*y = 1 occurs when x = y = 1 (Can be proven using La Grangian techniques). Hence, the amount of conflict is minimized when the worry factors are equal. That is, when both individuals are worried equally about the same things. This makes sense and it results in people with equal problems to have a zero CF.

For Anne and Bob:

CF = 0.167^{2} + 5.997^{2} - 2 = 34 approx

and this is significantly larger than the minimum possible value for CF which is of course, 0. There is, therefore, a positive probability of conflict between Anne and Bob.

**Summary**

I developed the idea for PVs, PRs, WF's and CF's when considering the general behaviour of people when it came to their problems. Whether you are a refugee fleeing persecution, or an ordinary housewife dealing with the day to day conflict of modern society, we are all subject to problems that we consider only on a subjective scale. While in a western society, a person's biggest problem might be that they are too busy and don't have time for a vacation; in a third world country, one's greatest problem might be that they can't find any food to eat or fresh water to drink. On an objective scale these problems are significantly different. However, as the PV analysis attempts to describe, 'your biggest problem is your biggest problem' and your response to it (whether it be significant objectively) will not differ substantially from the response of a third world person to his/her biggest problem,

Of course, the analysis is not complete, by any means. People do not necessarily look at their problems purely subjectively. They may see the situations of others around them that allow them to better judge the true importance of their problems. Any more accurate model, would have to take such objectiveness into account. But for simple analysis, the model achieves the aim of demonstrating problems and the relative approaches of different people to them. Next time someone you know tries to tell you that you shouldn't complain because you are far better off than a citizen of a less fortunate country, simply explain to them the notion of subjectivity and the concept of constant Problem Vector magnitudes.