Committed Compatibilty Coefficient

Necessity is the mother of invention,they say. 

Have you ever been one of those , where you did not know why you were in a certain place at a certain time? Did you ever feel out of place just because the rest of the people around you are talking on the phone with their girlfriends and/or boyfriends? Were you there ever , when all the logic in the world seemed to fail , where a "couple" was concerned? Did you ever see any of your previously respected friends, lose all the respect you had for them after they became committed? Or were you in a dilemma ever, if you should join a particular group of people , in their social plans, trying to think whether you would gel in or not?

Well, I have been in plenty of these scenarios, and if you have ever been single in your life,surrounded by "committed"/"its complicated" kind of people, you would know how it feels. I was once wondering if I should accept a particular social invitation, and pondering over this on my bus ride home. Thats when I thought, why not have a measurable figure of merit to make this decision.?Its too taxing on my small "single" brain anyway! One thing led to another, and after hours of unfunded research, and studying of special cases, I came up with C3- the Committed Compatibility Coefficient. 

The following text is extremely readable and demands no prerequisites except the will to understand the truth. And fractions, which you learnt in 5th grade. Examples covering the most common cases are also published to aid the reader in understanding the concepts faster. 

C3  =  1 - P

C3 is actually the compatibility coefficient for "singles",and obviously, P is the coefficient for the complimentary set, the "committed" people. In most cases, C3 is what matters most, but just to be clear, in a rare case of population inversion, where there are less committed people in a group of many single people, they also, theoretically feel left out, and might want to calculate this "P" factor.

P is defined by 

P =   C             where,

      ----

       T

C : Number of committed people in the group

T: Total number of people in the group.

Now you must be thinking, this is basic! But more is coming, and it is necessary to establish the foundation.
Lets take an example. Let there be 5 people in a social group. Let their names me Tom ,Dick,Harry, and Sita ,Gita. Let 3 people out of this group be committed. Say Dick, Harry and Sita are committed and Tom ,Gita are single. Hence P would be 3/5 and hence C3 would be 1 - 3/5 = 2/5.  This should be the compatibility coefficient for Tom and Gita to the group.  3/5 i.e P would be the compatibility of Dick, Harry and Sita  to the group.

It is easy to see, and even intuitive to imagine practically, that if 4 people were committed, lets say Gita too was committed, then Tom would have felt more weird in that group. This "more weird" can be easily seen through C3, as now, P would be 4/5 and C3 would be 1/5. Now 1/5 is definitely smaller than previous C3 which was 2/5 . Hence the compatibility of Tom went down, as expected.  

Also interesting to see is that P went up , which agrees with the other practical observation, that the compatibility of "committed" people goes up when they are in similar crowd. Essentially because everyone seems to be doing the same stupid things, and no one seems to mind anyone else. They are in their own world.

Now there is one more effect to be modeled in the formula.  We all know that if , out of the committed people in a group, there is a "couple" there itself, the uncomfortness of the "single" person further increases. This is because, the "couple" behave, like they are a separate entity in itself. Mathematically, it adds a virtual person to the group, and to the number of total committed people too. Or, to put it simply, the P factor gets modified to 

P =   C + Nc

       --------

      T + Nc

Nc: Number of "couples" or "pairs" in the group.

C and T mean the same thing as previously.

Example again. Lets now assume, that except Tom, everyone is committed. So , the basic formula gives the C3 value for Tom as 1/5 as seen above. But what if Harry and Sita are committed? The incompatibility of Tom to the group has to go up. Nc = 1 in this case. Hence the new P factor is (4+1)/(5+1) = 5/6 and the new "corrected" C3 value for Tom is 1 - P = 1/6.  It is easy to see that the corrected C3 value 1/6 is less than the basic first order approximation of 1/5, which agrees with the practical observations that Tom must be feeling more out of place.

Corollary: If a social group involves finite number of distinct couples, each one of the social group is completely compatible to the rest of the group.

This can be easily verified, even by first approximation. But, this proves that the second order approximation is correct too. A group involving only couples means its a  group with even number of people. Lets say 6. Hence 3 couples. So , according to first approximation, the P factor is 6/6 which is "1". Remember , that for "committed" category, the compatibility is given by P and not C3. Now, by the revised formula, P is again (6+ 3/6+3) = 1 !

You must have observed how married /committed people plan picnics and get-togethers together. There is hardly any "single" person involved.And that is the best thing to do. Each person in that group feels comfortable, with he/she being concerned only about his/her partner , who , obviously is also there. Also, in all these cases,  C3 will be tending to zero . Hence , such a social invitation would turn out to be pretty dangerous for "single" people. They should avoid C3 values below .2 under all circumstance, or it might leave deep lasting scars on their mind, with consequences as ghastly as they permanently being "single".

Also, logically, it makes sense for singles to go in for a social group of C3 values more than .5, as there are chances of other opposite sex(same sex for some) single persons being in the group. 

And this brings us to the next degree of approximation. And this is interesting.Imagine a situation, where in a group, a set of people are not really seeing each other, but one of them wants something to work out between them .He/she might have , lets say, a crush on the other. In these cases, the person convinces himself/herself that he/she might be compatible to the group even if they are not. For e.g  if in the set of 5 people mentioned,  Tom has a crush on Sita.  Even though he might find the group boring,he still might go in for a movie plan, because he wants to have a chance with Sita. So, assuming Harry and Gita are a couple, Dick is committed, Sita and Tom are single , by the second order approximation which was explained earlier, the C3 value for Tom was 1 - (3+1)/(5+1) =2/6 = 1/3 .

The revised P factor now is 

P =  C + Nc -Ni          where

      -------------

         T + Nc

Ni is the number of people in the group you are "interested" in. 

Interested could mean different things for different people .Rest of the things mean the same.

So the new P is (3 +1 -1)/(5+1) = 3/6 and the new C3 values for Tom is 1/2 . Wow..the compatibility of Tom to the group did increase, and he might actually go for the movie! In real life too..Tom would have. It is important to note that, the denominator still has T + Nc term, because unlike in previous case, the total number of people did not virtually change. This P factor holds as long as P is positive. If Ni becomes more than C+Nc, then ideally, C3 factor should be assumed to be 1. Readers are left to interpret this on their own. 

I think this level of approximation is enough for the "single" layman  out there, waiting for some answers from the heavens above. A lot more research is going on, because as you already know and have experienced, relationships are amazingly complex. There might be cases of a social group where there is a couple, and one of the persons of the couple is attracted to someone else in the group who is single . Take that! And what if they are of the same sex. Beat that! Modelling all that in the C3 values will take years of funded research. Till then , this amateur work, done solely with the intention of serving the public, with noblest of intentions, should suffice. Any question , suggestions, corrections to the modelling are welcome. We all want to together make this world a better place for "single" people. My roommate Gagan, has been my best critic. He might be instrumental in taking this research to the next level. My thanks to him.

So to summarize, the C3 value which is the compatibility coefficient for a "single" person to the group is 

C3 = 1 -P

P =  C + Nc -Ni

      -------------

        T + Nc

T:   Total number of people in the social group.

C:   Total committed people in the social group.

Nc: Number of couples in the social group 

Ni:  Number of people you are "interested" in , in the social group.

P:  Committed person's compatibility to the social group.

So, singles, go walk into this world, with your head held high, and the C3 value calculated.

Nachiket Mehta

Single

14th Feb 2009