Paying for a common wall

Avodah
1

Trying to figure out the sevara behind AhS CM 158:5 left me confused.
And since I have no chavrusah, I'm yet again trying it out on you.

Reuvein ends up with a churvah completely surrounded by Shimon's churvos.
Shimon build fences around three sides, since the 4 side is still open
to reshus harabim, Reuvein gets no utility and doesn't have to chip in.
However, when Shimon puts up the fourth wall and Re'uvein uses them in
some way (demonstrating a desire to have walls), R has to pay his share
of all four walls.

The part I don't get is the definition of "his share".

If Shimon's borders are 10x10, and Re'uvein's are 5x5, they divide by
area. Re'uvein has 1/4 of the total area (25 sq amos out of 100), so
I would have thought his fair share would be 25% of the price. But the
AhS says 1/8 -- apparently splitting by area, and then again splitting
into two shares.

That's the real problem I had.

I was also wondering why they're splitting based on utility rather
than avoided cost. IOW, why is it based on area guarded? Wouldn't
it make at least as much sense to divide the cost that Shimon saved
Reu'vein?

After all, someone who benefits your field without being asked is
paid the lesser of either the costs they invested or the value they
added to you. And the value would be the price of the fence around
R's 5x5 field.

So why is it divided by the ratio of their areas (divided by 2), rather
than by the ratio of their perimeters?

GCT!
-Micha

2

3/4 of Shimon's fences benefit nobody but him, so he bears 100% of their cost. 1/4 of Shimon's fences benefit both of them equally, so they share that cost.

3

: >If Shimon's borders are 10x10, and Re'uvein's are 5x5, they divide by
: >area. Re'uvein has 1/4 of the total area (25 sq amos out of 100), so
: >I would have thought his fair share would be 25% of the price. But the
: >AhS says 1/8 -- apparently splitting by area, and then again splitting
: >into two shares.
: >
: >That's the real problem I had.

: 3/4 of Shimon's fences benefit nobody but him, so he bears 100% of
: their cost. 1/4 of Shimon's fences benefit both of them equally, so
: they share that cost.

This ties my questions together.

The proportion is by area. The amount of fence both are using would be
an issue of how much perimeter Re'uvein would have needed.

If we are measuring by area, we are presumably dividing by amount of
utility. Not splitting how much of the fence is sharing, but determining
how large is each party's share, where they share the full fence. Which is why I thought of halving
the ratio as a second accounting for the sharing.

Here's how I would have thought sharing the amount of fence both use /
need would go:
Shimon made 10 x 4 = 40 amos of fence of which Reuvein needed 5 x 2 = 20
amos. So, it would be 1/2 of the fence that both are benefiting equally,
and a 1/2, and not 1/4, that would therefore be shared equally. Making
Re'uven pay 1/2 of 1/2 = 1/4 of the cost.

(Not because of area; that's just a coincidence because in the example,
Re'uvein's property's length is 1/2 of Shimon's.)

Going on a tangent for a bit: And if we look at how much Re'uvein needed,
rather than how much he would have posted for just himself, we could say
that Reuvein is benefiting from 100% of the fence. He didn't have to pay
if Shimon's fence still left his property exposed to reshus harabim. And
the fence wouldn't be a closed loop if not for all 40 amos.

When a town is afraid of pillaging, the community can levee a tax to pay
for a wall. They bill the wealthy and those who live near the wall the
most, as they are the most likely targets and get the most benefit. (This
division is only if the threat is theft. If it were an army capable of
rape and murder too, then the cost is divided equally. As having more
money or being in a home closer to the edge of town doesn't make it
measurably more likely to be a victim if the army arrives.)

So, it seems that sharing is based on proportion of utility, not
proportion of service used. Justifying dividing by area. The person with
more square amos of wheat benefits more from the of the fence.

In total, there would be 10x10 = 100 sq amos
Reuvein has 5x5 = 25 sq amos = 1/4 of total
Shimon has 10x10 - 5x5 = 75 sq amos = 3/4 of total

Where do you get the 25% that both are sharing?

GCT and :-)@@ii!
-Micha