Valentine day special

Did you ever think that there can be an economic analysis of saying ‘I love you’ to a girl/guy? Well… here I’m trying to build up a model which determines the cases under which a girl/guy will say these magic words to a guy/girl.

Suppose that person ‘A' is going to say these magic words to person ‘B’ then ‘A’ can expect one of the two replies; either a ‘yes’ or a ‘no’. Essentially the probabilities of any of these replies lie in the range of 0 to 1. The probability of ‘yes’ and ‘no’ depend upon the specific circumstances and are endogenous to the model following the fact that A can perform certain acts and put efforts in order to increase the probability of ‘yes’ and reduce the probability of ‘no’. Putting any effort on the part of ‘A’ is a cost to ‘A’. To simplify the calculation assume this cost to be zero. The probabilities of ‘yes’ and ‘no’ are known to ‘A’ and are ‘p’ and ‘1-p’ respectively.

Now assume that the payoff that A receives from a ‘yes’ is ‘H’ and the payoff from a ‘no’ is ‘-T-N’. The negative payoff from a ‘no’ can be separated into two different components. The payoff from not having B as his/her partner is (-T) and (-N) is negative payoff from a ‘no’ because, say his ego gets hurt.

Indeed we can expect H to be non-negative and T and N to be non-positive. A non-negative H implies that A is happy and gets some positive utility if he gets a ‘yes’ and certainly doesn’t get negative utility. Similarly, a non-positive T and N imply that ‘A’ gets negative utility if ‘no’ and in certain cases where ‘A’ is not at all affected by a ‘no’; T and N both are equal to zero.

‘A’ will then decide to say these magic words to ‘B’ if and only the expected payoff from saying these magic words is greater than or equal to from that of not saying these words. So we compute the expected payoff from saying these magic words

E (say) = pH + (1-p) (-T-N)

On the other hand, the expected payoff from not telling these words to B will be

E (don’t say) = (-T)

The right hand side of the above expression is simple to derive. If ‘A’ is not going to say these words to B; with probability 1 he gets no response and receives a payoff of (-T). Therefore, A will say these magic words to B if and only if

pH + (1-p) (-T-N) ≥ (-T)

or, pH –T –N + pT + pN ≥ (-T)

or, p (H+T+N) ≥ N

or, P ≥ {(N)}/(H+T+N)

Thus A will decide to say these magic words to B if the probability of a ‘yes’ is greater than or equal to the expression in the right hand side above. The equation suggests that value of ‘N’ is crucial.

Some interesting cases:

Case 1. When (-N) = 0;

Did you ever wonder there are guys who will say these words to each and every girl just at every possible opportunity? These are the guys who don’t worry at all about the negative response from the girl. In other words, for them (-N) as well as (-T) both are zero but since H is still positive, for them expected payoff from saying these words is always more than the expected payoff from otherwise.

Notice that even those who get negative payoff from not having the person as partner will say these words to their desired ones as long as a ‘no’ does not add to their negative payoff from not having the person as their partner, in equation form it’s just because the numerator (-N) is zero. So even if probability of a ‘yes’ is slightly positive or even zero that is, p=0; they will take the chance.

Case 2. There are people who can’t digest a ‘no’. For them to say these magic words, ‘p’ has to be very close to 1. In equation terms

N = H+T+N

Equivalently H is equal to (-T). Recall T has a negative sign.

Important points

Here we allow ‘p’ to be negative as well which is practically not possible however a negative ‘p’ should imply that A is going to say these magic words to B with certainty.

Examples,

A. The ‘p’ can only be negative when H is greater than the sum of T and N. Since the numerator is always negative. This implies that positive payoff from a ‘yes’ is very large and hence A must convey these words to B. This is a result which one would arrive at intuitively.

B. The second case pertains to the situation when absolute value of T is very large. What does intuition say in this particular case? Of course A should go ahead and tell this to B since the negative payoff from a ‘no’ is very large. The equation tells the same thing, if T is large enough compared to H in absolute sense, a smaller ‘p’ is needed to say these magic words.

In fact the difficulty in saying these magic words arises only when absolute value of N is very large compared to absolute values of H and T. This is also intuitive since those who are too egoistic or give much more value to social sanctions will go ahead only when they are almost sure about a ‘yes’. In this case ‘p’ will be near equal to 1 since we also expect H and T to be correlated. The more pleasure you get from getting the company of person, you also miss him/her badly.

So if you are mulling over whether or not you should tell these magic words to the person you love, calculate the probability with which you should tell her and compare it with the probability that you think he/she is going to give a positive response.

I would like to have your comments on this post.If you find some computational error please let me know. Thanks.