Sympathy represents one party's care for the other.
Every relationship in the
network of care (shown by an arrow, above)
has an accompanying sympathy value, denoted
s.
In accordance with the
basic principles of
altruistic economics, everyone is free to publish whatever
evaluation of sympathy they feel like, in whatever format they like.
The first format for doing this we will introduce (below) is
linear sympathy, after which
we will introduce a more advanced format called
non-linear sympathy.
The simplest modelling of sympathy is to use a single number,
s≥0, so that the greater the
s value, the closer the relationship.
Formally, we define s as
the greatest number of units of a resource
you are prepared to sacrifice to prevent your friend from losing one unit.
For example, if you would work 1 hour only if it saved
your friend spending 2 or more, then s=0.5,
If you are indifferent between spending $5 and letting your friend
lose $50 then s=0.1.
In most cases, 0≤s≤1:
If s=0 then the relationship has no sympathy
(if all the reltaionships have s=0 then the network can be disregarded, and
altruistic economics reduces to the simple, capitalist model of individual self-maximisers.
If s=1 then complete sympathy exists between the parties -
in the sense that, if there were a bilateral relationship with s=1,
both parties would be indifferent to resource flows between them.
To understand the name, let's look at a specific example.
We will consider the resource in question to be time,
and assume s=0.5.
Graphing the time spent by the giver along the bottom (x-axis)
and time saved by the recipient up the side (y-axis),
we have the following indifference function for time spent:
Assigning a single s value to each relationship is a crude approximation,
since sympathy is likely to vary according to the resource involved.
For example, someone who is financially rich but very busy may wish to be
generous with money but not with time. It is easy to see how the linear sympathy model could
be adapted to tackle this issue. A more significant weakness of the linear model is the fact that sympathy
cannot be expected to be constant - being willing and able to spend $10 to save a friend $50 does
not necessarily imply being willing or able to spend $10,000 to save him $50,000. This is the
main motivation for the non-linear sympathy model explained below:
2. Non-Linear Sympathy
A simple approximation (using 2 lines)
Let us now consider a more general and expressive model to represent sympathy.
A non-linear sympathy model automatically provides for the important issue of limits on resources.
We motivate the model by considering the indifference function as outlined above, with resources given
on the x axis, and benefit received on the y axis.
In general, the indifference function is likely to
be a continuous curve. A little thought will also suggest
that, in line with the law of diminishing returns,
its gradient is likely to be increasing.
This also fits from a psychological point of view,
since the more work you do for someone, the more reluctant
you may be expected to be to do more -
altruism has its limits.
Note that, whatever shape the curve has, we can
approximate it as closely as we want
by assuming that it is a set of straight lines.
A close approximation (using 7 lines)
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