Welcome to the Sea of Echoes

Have you ever been stalked by someone and couldnt make them stop?  On the other hand, have you ever been hammering at someone in a relationship only to be consistently turned away?

Of course you have.  While these effects are most noticeable in strong emotional relationships, there is an emotional or relationship “distance” between any two people.  We are continually updating this emotional distance with each interaction, moving closer or farther away from our partners in a kind of constant dance. 

But what factors determine the equilibrium point in a relationship?  The following figure provides some insight.  Here we have a graph of happiness versus emotional distance for two people, blue and yellow.    Where they wind up depends on their style of cooperation and the similarity or dissimilarity of their utility functions.

In the simplest case if Blue and Yellow don’t cooperate at all - Blue constantly pushes toward point “A”, while Yellow tries to reach its maximal happiness at “B”.   The situation is constantly unstable with effort expended by Blue to move closer and by Yellow to move away.  Typically Blue might wind up feeling rejected while Yellow may begin to feel harassed.  How sad for Blue and Yellow!

If Blue and Yellow could cooperate they might choose to maximize their total happiness and wind up somewhere on the thin green line near “C”.  This Utilitarian view is simple, but notice how Blue’s happiness has a much larger effect on the outcome.   Also it requires knowledge of the complete utility curve of both individuals and way of “summing” happiness.

An alternative view due to Rawls would maximize the happiness of the least happy person, resulting in the equilibrium at the crossover point “D”.  They are also equally happy at this point, which might make it easier to reach in practice since Blue and Yellow can compare their level of happiness (or sadness) directly. 

A simple rule for cooperation emerges: compare notes and yield to the person who is most vulnerable.

Things do not stay simple when we expand our view to three parties.  Imagine that Yellow now must choose to allocate time between Blue and a new task Purple. 

If we use Rawlseyan metrics, we wind up in the equilibrium at “C” which leaves yellow satisfied, but Blue and Purple are much less happy.  The only thing that makes this stable is that Blue and Purple would choose this point anyway if they were cooperating.

If a Utilitarian approach is used then a solution much closer to “A” or “B” is possible, where one party is left out altogether.  The utility function then has modes which represent combinations of happy and less happy people.  A formulation like this generalized to many pairings makes for a very interesting optimization problem indeed, though perhaps strict matching (as in the “stable marriage problem”) or distance is not the right take. 

How’s this instead:

Inputs:

• N people
• A total capacity for attention Ci over all people.
• A matrix of utility functions U(i,j,g,r) which is the percieved utility of person i receiving r units of “attention” from person j when they are investing “g” units in the opposite direction.

Output:

• A symmetic (or asymmetric?) matrix A where Aij is the amount of attention from person i to j
• constraint Sum of Aij over j <= Ci
• objective: maximize the sum U(i,j,Aji,Aij) over i,j
• alternate objective: solve the Rawlseyan condition locally

Perhaps more intersting than solving a given optimization problem is how these utility functions are set up.  For example Yellow in the example below is much more likely to be happy than either Blue or Purple.

For example, you may wish to consider what might happen in these cases:

“Love Triangle”

“Family and Friends”