Consider two individuals (1 and 2) who wish to have an arranged marriage. Each individual i can apply effort to make sure that the desired arranged marriage actually takes place. Note that effort here is a proxy for a variety of possible tasks that include, but are not limited to, the placement of newspaper advertisements and profiles in online matrimonial sites, information gathering and sharing with parents, relatives, and friends, and the hiring of one or more matchmakers. Let e i ≥ 0 denote the effort applied by individual i . We suppose that this effort is chosen from the closed set [ 0 , 5 ] ².

Applying effort is costly to individual i and therefore this application gives rise to disutility. This disutility can be described by the strictly convex function d i ( e i ) = e i 2 / 4 . The reader may want to think of the effort e i as the number of hours required to either personally engage in or have others undertake one or more of the tasks delineated in the preceding paragraph.

The gross utility of each individual depends on both the effort applied by himself or herself and on the effort applied by the other individual. Specifically, if individuals i and j choose efforts e i and e j respectively then the gross utility to individual i is given by

U i ( e i , e j ) = { 0 if e i < 1 2 if e i ≥ 1 and e j < 2 e i ⋅ e j if e i ≥ 1 and e j ≥ 2 } . (1)

With this background out of the way, let us now stipulate the net utility function of individual i as a function of his or her effort e i in three separate cases. Note that the net utility function is the difference between the gross utility and the disutility functions specified earlier in this section. In the first case e j < 2 , in the second case e j = 2 , and in the third case e j = 4 .

Some thought tells us that in the first case in which the j t h individual’s effort e j < 2 , the net utility function of individual i is

u i ( e i , e j ) = { 0 − e i 2 / 4 if e i < 1 2 − e i 2 / 4 if e i ≥ 1 } . (2)

Similarly, in the case where the j t h individual’s effort e j = 2 , the net utility function of individual i is

u i ( e i , e j ) = { 0 − e i 2 / 4 if e i < 1 2 e i − e i 2 / 4 if e i ≥ 1 } . (3)

Finally, in the case where the j t h individual’s effort e j = 4 , the net utility function of individual i is given by

u i ( e i , e j ) = { 0 − e i 2 / 4 if e i < 1 4 e i − e i 2 / 4 if e i ≥ 1 } . (4)

Inspecting the three net utility functions for individual i given in Equations (2) through (4), we can immediately draw two unambiguous conclusions. First, in the effort range e i ∈ [ 0 , 1 ) , the net utility for the i t h individual is the same in all three cases. Second, in the effort range e i ∈ [ 1 , 5 ] , the three net utility functions look very different. Specifically, in the e j < 2 case, the net utility function for individual i slopes downward from the point ( e i , e j ) = ( 1 , 1.75 ) . In contrast, when e j = 2 the net utility function slopes upward gently from the point ( e i , e j ) = ( 1 , 1.75 ) Finally, when e j = 4 the net utility function slopes upward rapidly from the point ( e i , e j ) = ( 1 , 3.75 ) . We now proceed to compute the i t h individual’s best response or reaction function.

Some thought and straightforward algebra together tell us that when the j t h individual’s effort e j < 2 , individual i ′ s best response is to choose e i = 1 . Similarly, we can reason that when the j t h individual’s effort e j < 4 individual i ′ s best response is to choose e i = 5 which is clearly a corner solution. When the j t h individual’s effort e j = 2 , the i t h individual’s best response is obtained by solving

max e i ∈ [ 0 , 5 ] ( 2 e i − e i 2 4 ) . (5)

The first order necessary condition for an optimum and the optimal value of e i are

2 − e i / 2 = 0 ⇒ e i = 4. (6)

Given the structure of our problem, we can deduce that as e j increases above 2, the best response of the i t h individual will increase above 4 until it eventually hits the corner solution of e i = 5 . In the case where individual i ′ s best response lies between 4 and 5, this individual maximizes his or her net utility. This means that (s)he solves

max e i ∈ [ 0 , 5 ] ( e i ⋅ e j − e i 2 4 ) . (7)

Differentiating the maximand in Equation (7), the first order necessary condition for an optimum and the best response function we seek are given by

e j − e i / 2 = 0 ⇒ e i ( e j ) = 2 e j . (8)

Now observe that for any individual j effort e j ∈ [ 2 , 2.5 ] , the best response of the i t h individual lies in the range [ 4 , 5 ] . Similarly, for any individual j effort e j > 2.5 the best response of the i t h individual is still the corner solution in which effort e i = 5 Pulling all of the above discussion in section 2.3 together, we infer that individual i ′ s best response function in full generality is

e i ( e j ) = { 1 if e j < 2 2 e j if e j ∈ [ 2 , 2.5 ] 5 if e j > 2.5 } . (9)

Suppose we have two identical pairs of individuals of the sort we have been analyzing thus far in sections 2.2 and 2.3. These two pairs are located in two different nations. In the first (second) nation, arranged marriages are uncommon (common) and therefore it is relatively difficult (straightforward) for the two individuals to apply and coordinate effort. In this setting, we want to study the application of effort and an implication of this application by these two pairs of individuals who wish to have an arranged marriage.

To fix ideas, let us think of Sweden as an example of a country where arranged marriages are uncommon and India as a country where arranged marriages are common. Then, it is not surprising that for individuals wishing to have an arranged marriage, effort application and coordination in “arranged marriage unfriendly” Sweden is much more difficult than it is in “arranged marriage friendly” India. Given this state of affairs, we now want to use our theoretical analysis in sections 2.1 to 2.3 to explain whether it is likely that the net utility of the pair wishing to have an arranged marriage in Sweden is much lower than the corresponding net utility of the pair in India.

The discussion in Section 2.3 in general and the best response function in Equation (9) in particular tells us that in the arranged marriage effort application game that we are studying in this note, there are two pure strategy Nash equilibria. In the first Nash equilibrium we have ( e i * , e j * ) = ( 1 , 1 ) and in the second Nash equilibrium, we get ( e i * * , e j * * ) = ( 5 , 5 ) . Straightforward computations tell us that in the first or ( e i * , e j * ) = ( 1 , 1 ) Nash equilibrium, the net utility of each of the two individuals wishing to have an arranged marriage is u i * = 1.75 . In contrast, in the second or ( e i * * , e j * * ) = ( 5 , 5 ) Nash equilibrium, the corresponding net utility to the two individuals is u i * * = 18.75 .

So, our analysis shows that it is certainly likely that in arranged marriage unfriendly Sweden, individuals wishing to have an arranged marriage will obtain net utility that is much lower than the net utility obtained by identical individuals in arranged marriage friendly India. We can think of this result as an example of a situation in which self- fulfilling expectations give rise to two equilibria with high and low net utilities to the two relevant pairs of individuals.

In the arranged marriage effort application game that we are analyzing, our analysis thus far shows clearly that the effort application of individual i is increasing in the effort application of individual j . In game-theoretic language, this means that the effort application choices of the two individuals are strategic complements.³ In fact, it is this complementarity that gives rise to two equilibria with very different net utilities to the two relevant pair of individuals in Sweden and in India. This completes our analysis of the arranged marriage effort application game.