Sunday, May 06, 2012

Is there a thing called a stable marriage and relation ?? - Part 1

Disclaimer: This is more of a mathematical and logical analysis of marriage. This is for rational audience who can enjoy a piece of logic with out any involvement of emotion. Or rather some logical analysis of how relations, trust etc involuntarily get biased by numbers and probability. Normal (non engneer, non scientific-mathematical audience might find it a boring read. But even others might find it as boring! :D )
  1.  All men and woman are straight
  2. all men and woman appear in the "desired" list of each other with some weight. e.g. Some men may be 99% desirable for woman X and some may be just 0.001% for X and vice versa. But every one appears on the desirable list of each other.
  3. We have not taken into account the possibility of preference change after they are coupled. (this will over complicate the situation :P) (this will be covered in part 2 of this research.)
An unstable marriage: When given a married pair, X-a and Y-b, if man X prefers another woman 'b' more than his current wife 'a' and woman b prefers X more than her current man Y, then X-b is called a unstable pair. 
Any pair (X-Y) when not an element of the "unstable" set is said to be a stable pair.
Lets see:
Assume we have a sample set of 26 males marked as A,B,C,D....Z and a sample of 26 females marked as a,b,c,d....z.
Ask each of the man and woman to form a list desired partner in the order of precedence.
For example the list may look something like this:
Male                                             Female
-----                                             --------
A: b,s,d,c....                                 a: C,D,B,Z....
B: b,a,c,t....                                  b: A,P,C,W...
C:a,b,c,r....                                  c: B,D,S,T....   
(This just an example combination shown here)

Now our job is to find how many "stable pair" can be formed which can result in a happy and trustworthy marriage. One of such approach as it would occur to mind first is to  go on forming pairs based on the preference list. Call a pair when a mutual inclusion of male,female exist in their preference list high on precedence. And swap it when encounter a yet higher mutual precedence. Example: Initially (B-c) will be paired but later their partners can be swapped and they may form a more stable pair like (C-a) and (B-c)
This seems working here but it may lead to re-swap and hence form an infinite loop on a bigger data set.
So lets first solve the problem at hand to form stable couples set:
pick an unpaired male X and the list female x on his list(first preference)
remove x from his list and check if x is already paired
if already paired then check if X appeared on a higher precedence then x's current partner in her list (if x is unpaired then pair up X-x for now)
          if yes then pair up x with X and continue with the next male.
else if X does not appear on a higher precedence on x's list as compared to her current partner then pick next female on X's preference list.
Now so far so good! We feel we can actually form a stable marriage and stay happy(works for an arranged marriage). But in a real world we don't run such an exercise while considering to get married. So I tried to just have some data around it and apply for game theory to see what might be the trend look like if we consider married people or people newly in love and assign a weight to their trust and relation how would it look like. I asked a few people to answer a few questions:
1. What their dream partner attributes look like
2. How often they end up admiring some one from opposite gender cause he/she matched with some of the attributes they admire are present in them?
3. How many such people are around them in acquaintance circle who can match their criteria?
4. What is the attribute precedence list look like for them?
and few other question which were little private and optional for them to answer like their sexual behavior and compromising point of "more-moral-ego-based-attribute" to a "more-materialistic-based-attribute" Like a happy charming fellow's charm compromised to someone else's salary etc.

One I had the answers to these questions. My job was to do a little research to find out from their Facebook behavior what they tend to like + how many guys at least I think match their criteria. And once let us say paired, what are the emotional, geographical exposure they will have to the other potential matches we had found earlier. All these taken into consideration with the futuristic projection of the change in the "desired attributes" in the partner and other potential candidates makes a recipe for stable and trustworthy relation! Lets take an example first of Male A and female 'c':

Male A say is paired with female 'c'. Male A's criteria looked like
- Good looks, decently educated, fair skinned, lovable, belonging to a reputed family background, easy going, working
- With these criteria he got married to female 'c'.  Attributes female c had in reality - good looks, fair skin, a well earning day job, higher educated, independent mind and lifestyle, easy going.
- 'c' had criteria of a match as - presentable, tall, secure job, good sense of humor, dependable background with handsome earnings, extrovert.
Attributes male A had in reality - Good looks, tall, handsome earning, not that dependable background, extrovert with a big friend circle, high educated, secure but frequently travelling job.
Female 'c' has exposure to matching criteria males in job, from past friends, in society she lives.
Male A has exposure to matching criteria females in cities he travels to, from past friends, in society he lives, in parties etc (as he is extrovert and has a huge friend group).

Let us say these criteria changed in both partner's after few years the female stopped working or male lost job, male/female or both deteriorated in looks, easy going became tough given social complexities.

In such a situation the probability of choosing another partner becomes prominent. Lets see how prominent by applying some game theory to it. And lets see should one break the trust in a situation like above?

N.B. (I will post a follow up on this today. Either will update the same blog or post a new one on top of this. As its growing big, am breaking it into two. :) till then keep wondering and let me know your guess to the above question in the comments.



amruta said...

kindly post the 2nd part. waiting for a long time.

abinash said...

Amruta: Hey thanks for the reminder, and sorry for the delay. Will post it ASAP. (read as latest by this weekend) :) Do let me know if you like the second part.