[PEN-L:4733] Re: query -- collective goods problem

["Curtis Moore" <cmoore@xxxxxxxxxxx>]

Jim -

     In _Science News_, May 4, 1996, p.284, Ivars Peterson wrote
an article with the title "Formulas for Fairness: Applying the
math of cake cutting to conflict resolution_."  The article
reports on the recent work of political scientist Steven J.Brams
of New York University and mathematician Alan D. Taylor of Union
College in Schenectady, N.Y.  The article is two pages in length
and most libraries carry the recent copies of _Sci News_ in their
reference sections, at least my neighborhood library in S.F.
does.

     Let me give two brief quotes from this article:

>From the side-bar:

QUOTE

Envyfree Cake Division

  Suppose Alice, Bob, and Carol want to
divide up a cake. Alice starts by cut-
ting it into three pieces that look
equal to her. If Bob views one piece as
being largest, he trims it to look equal
to the piece he sees as second largest.
This leaves one trimmed piece and two
untrimmed pieces. Carol chooses one of
the three pieces. Bob picks next and must
take the trimmed piece if it's available.
Alice gets the last piece.
  By choosing first, Carol can't lose
because she picks the piece she likes
best. Bob can't lose because he can
choose one of the two pieces he made sure
were tied for largest. Alice ends up
with one of the two untrimmed pieces,
both of which are better in her eyes than
the one Bob trimmed.
  This procedure can then be repeated
with the trimmings until the crumbs are
so small that no one cares anymore.
  For four players, the cake cutting has
to start with an extra piece: Alice must
slice the cake into five pieces. The
extra piece ensures that no player is
forced into taking second best. The
number of extra pieces escalates for more
people, the researchers discovered. For
example, nine pieces are needed for five
players, 17 pieces for six, and 2 (to the
power n-2) + 1 for n players.
  It's interesting to note that after
World War II, Great Britain, France, the
United States, and the Soviet Union
divided Germany into four zones of
occupation, with Berlin, which fell
within the Soviet zone, as a valuable
"trimming" that was itself divided into
four zones.         - I. Peterson

284                 SCIENCE NEWS, VOL.149
                              MAY 4, 1996


END QUOTE

Here's the final paragraphs that contain
your latest reference:

QUOTE

  Brams, Taylor, and others continue to
explore a variety of mathematical ques-
tions that have arisen concerning fair
division. For example, they are still
looking for a reasonable point allocation
system for three or more players.
  In cake-cutting schemes, Taylor is
intrigued by the fact that strategies
involving three people are quite
straightforward. But stepping up to four
represents a tremendous increase in
complexity, with higher numbers only
slightly more complicated.

[Comment: So, if the problem can be
solved for four, then large numbers such
as the population of the U.S. don't
present a real difficulty, if I read
this right?]

  "Mathematically, it indicates there's
something there that we don't fully
understand in going from three to four,"
Taylor suggests.
  Brams and Taylor recognize that their
methods don't offer complete, perfect
solutions to human problems.
  "The search for better procedures,
which make the achievement of fairness
not just an outcome but a process as
well, will go on," Brams and Taylor write
in _Fair Division: From Cake Cutting to
Dispute Resolution_(Cambridge University
Press, 1996). "What cannot wait is apply-
ing our knowledge, primitive as it is, to
problems of fair division that cry out
for better and more durable solutions in
realistic settings."
  This approach represents a modest step
toward satisfying a plea made by
economist Herbert A. Simon of Carnegie
Mellon University in Pittsburgh. "If I
were to select a research problem
without regard to scientific
feasibility," Simon wrote in his 1991
autobiography, "it would be that of
finding out how to persuade human beings
to design and play games that all can
win."
                                      285

END QUOTE


Comment:  The "Prisoners' Dilemma" has been around for quite some
time.  This cake division problem caught my attention because I
had never heard of it before and I saw that it must be related
to the prisoner problem.  I wonder, but do not know, whether the
authors discuss the relationship between these problems in their
book.  I like the statement of the problem in terms of the cake
cutting analogy better than in terms of the prisoners' dilemma,
for some possibly deep seated psychological reason??  Or is there
a more objective reason?  I don't know.

Curtis Moore


--- Original message ---


Date:          Mon, 17 Jun 1996 12:57:51 -0700 (PDT)
Reply-to:      pen-l@xxxxxxxxxxxxxxxxxxxxxxxxx
From:          JDevine@xxxxxxxxxxxxxxx
Subject:       [PEN-L:4698] query -- collective goods problem

It's pretty well known that in a repeated two-person Prisoners'
Dilemma game, the two participants can learn how to cooperate to
avoid the dire fate of them both confessing. My reading says that
this conclusion does NOT generalize to a game with many people
(the collective goods problem) unless one abandons the assumption
that people are individualistic.

Is my conclusion correct? what is the most up-to-date source?

in pen-l solidarity,

Jim Devine   jdevine@xxxxxxxxxxxxxxx
Econ. Dept., Loyola Marymount Univ., 7900 Loyola Blvd.
Los Angeles, CA 90045-8410 USA
310/338-2948 (daytime, during workweek); FAX: 310/338-1950
"It takes a busload of faith to get by." -- Lou Reed.