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Page Five'
Friday7 September 25, 1 9 7 0 THE MICHIGAN DAILY Page Five
Prisoner's
Dilemma: An answer in game
theory?
A FEATURE REVIEW BY STAN KAPLOWITZ
Anatol Rapoport and Albert
M. Chammah, PRISONER'S
DILEMMA, University of
Michigan Press, $2.95 paper-
back.
In the late 1950's, a group of
social scientists, convinced that
war was not so much a clash
between good and evil as a con-
sequence of the existing inter-
national system, formed a peace
research movement. This niove-
ment was based on the supposi-
tion that if the mechanisms
which cause conflict were bet-
ter understood, the international
system might be changed so as
to make war less likely.
Prisoner's Dilemma by Rapo-
port ad Chammah (which was
first published in 1965 and
whose appearance in paperback
this year is the occasion for this
review) is one of the best and
most important works to be
stimulated by this movement. It
analyzes conflict and ,coopera-
tion through the scientific
method of controlled experi-
ment, logical rigor and mathe-
matical precision. One clearly
cannot do experiments on the
international system. Instead,
experiments were done on pairs
e of subjects playing a game
which represents the kind of
choices relevant to conflict and
cooperation in larger social sys-
tems.
The name Prisoner's Dilemma
comes from the following anec-
dote. Two prisoners held in-
communicado are charged with
the same crime. If neither con-
fesses, there is not enough evi-
dence for conviction and both
are set free. If both confess
they are given the same moder-
ate sentence. If only one coh-
fesses, he is rewarded for hav-
ing turned state's evidence and
is not only set free but given an
additional reward. In this case,
the prisoner who holds out is
convicted and is given an es-
pecially severe sentence.
At this point I should bring
in the concept of game theory.
Contrary to the connotations of
the name, game theory is not
-especially concerned with sports
or parlor games. It is a theory
which looks at the underlying
structure of a decision problem
and attempts to find a rational
procedure for making a deci-
sion'
In game theoretic terms we
can represent the prisoners di-
lemma with the following game
matrix, where the prisoners will
now be called players.
~~
To explain the matrix above,
erich player has a chioice be-
tween C co-operation ie. not
4confessing) and D (defection
,,{N e confessing)r J.Y Tha meansy. ~h
there areifour possible out
comes, each one represented by
a box in the matrix. The sym-
w hic th e p lay ers rece v fo r a
particular outcome. The letter
Sbefore the comma in figure 1
is the payoff to player 1 and the
letter after the comma is the
pa yoff toS playe 2 . Ea.lte
has crti numerical valu
more desirale the out omet
th player e eivingd cibe
the C cotio e cried on
ther aedoerreouirelefot-
lowing relationship among the
payoffs: T is greater than R is
greater than P is greater than S.
A simple matrix fulfilling the
necessary conditions is as fol-
' ~
~ .
~~
.ows;
t -. * * .
If we look at figure 2, then
we notice that regardless of
what player 2 does, player 1 will,
receive a higher payoff by
choosing C. In the terminology
of game theory, D is the domi-
nating strategy for player 1.
Since the situation is symmetric
the same clearly holds true for
player 2. But if both players
are concerned only with their
own self interest and rationally
make the choice, D, which best
serves that self-interest, they
will both receive -1. If on the
other hand, they had both made
the "irrational choice, C, they
would have each received ±1.
It is this then, which is the core
of the dilemma.
Therefore, unlike Adam
Smith's vision of a society in
which men who rationally pur-
sue their individual interests
are "led by an invisible hand"
to act in the best interest of all,
in Prisoner's Dilemma the in-
visibleshand leads a collection
of rationally self-interested peo-
ple to an unhappy fate.
If the prisoner's dilemma
were merely the figment of the
imagination of some science fic-
tion writer, R a p o p o r t and
Chammah would not have done
the studies reported in this
book. The reason the book Pris-
oneris Dilemma is worth writing
is precisily because so many
problems in this work have ele-
ments of an actual prisoner's
dilemma.
Take the following examples:
1) Two countries must each de-
cide whether or not to arm.
Both nations would be better off
spending their resources on im-
proved living conditions than on
armaments. But if one arms
and the other doesn't, the arm-
ed nation can subjugate the un-
armed. In this case the armed
nation would receive its best
possible outcome and the un-
armed nation would receive its
worst possible outcome. If both
are rational and choose the
dominating strategy, both arm
and both loose. 2) The problem
of whether or not to pollute is
in many ways a commonplace
prisoner's dilemma. To simplify
the matter we can regard each
industrial firm as having the
choice of polluting or not pol-
luting. It is in the individual
interest of each firm to pollute
because the firm saves money
by polluting, whereas the social
cost of the pollution is spread
across a large section of the
population. Therefore, those re-
sponsible for any decision to
pollute suffer only a small frac-
tion of the total damage caused
by that decision. But if every-
one pollutes to his heart's con-
tent, an area eventually be-
comes unliveable and everyone
is worse of f.
The experimental study de-
scribed in this book had pairs
of subjects presented with a
prisoner's dilemma matrix and
had them play the game for a
few hundred trials. Each player
had to make his choice without
consulting his partner and with-
out k n o w i n g his partner's
choice.
When prisoner's dilemma is
played for many successive
trials (not just once which we
have assumed in past discus-
sion) it might seem that D is
not necessarily the "rational"
choice. For if player 1 always
plays D he can expect that after
a while player 2 will do likewise
in which case player 1 will do
worse than if he had played ac-
cording to the principle "if you
co-operate with me I'll co-oper-
ate with you."
But as it turns out, if the
number of trials is known and
finite with two players who are
r a t i o n a 11 y and strategically
sophisticated and assuming the
other to be likewise, they will
always play D. To see this, let
us assume that the game will
be played for 100 trials. Since
both players are rational, each
knows that the only reason to
ever play C is to avoid provoking
his partner to retaliate by
choosing D subsequently. Player
1 might reason as follows.
"Clearly, I should play D on
trial 100, since player 2 will not
be able to retaliate after that
trial. But player 2 also knows
this so he also will play D on
trial 100. Therefore, no matter
what I do on trial 99, the out-
come on trial 100 will be DD.
In this case I should choose D
on trial 99 and since player 2
also realizes this, he too will
choose D. Since the outcome of
trial 99 will definitely be DD
then I should choose D on trial
9 and so should he." This rea-
soning ran be extended all the
way back to trial 1 so that each
player will always choose D.
"Confronted with this para-
dox," say the authors, game
theoreticians have no answer.
Ordinary mortals, h o w e v e r,
when playing prisoner's dilem-
ma many times in succession
hardly ever play DD one hun-
dren per cent of the time. To
be sure long stretches of DD
d6 occur but so do long stretch-
es of CC choices, Evidently,"
t h ey add,. "the run-of-the-
mill players are not strategical-
ly sophisticated enough to re-
alize that strategy DD is the
only rationally defensible strate-
gy. and this intellectual short-
coming saves them from losing."
Not all results can be regard-
ed as simply common sense. To
give an interesting and some-
what significant example of a
surprising result, in most con-
ditions, subjects had the matrix
in front of them as they played.
In the concealed matrix condi-
tion the matrix was not display-
ed, though as in other condi-
tions they were told the outcome
and payoffs from each trial.
The authors had expected a
tingent propensities indicate
someth.ing which we've already
seen evidence of - that fear of
being the lone co-operator is a
more important cause of the
DD lock-in than is the hope of
being the lone defector. More-
over the severe games (those
with the least co-operation) are
severe because subjects tend to
be more afraid to take the risk
necessary to break out of the
DD lock-in.
While the first section of the
book is devoted to discussing
the game and analyzing the
can be briefly summarized. 1)
To create a certain model of a
process is not to say the pro-
cess works like this but rather
to say, "Let us assume it works
like this, find out the conse-
quences of our assumptions and
see how well they agree with +
the data. The degree to which
the predictions of t h e model
agree with some of the observ-
ed features of our data will give
us an idea as to what extent
and in what ways the model has +
or has not captured what is go- +
ing on in the process.
2) All of the models present-
ed are dealing with the proba-
bility of a co-operative response '
from an average subject in a
particular population. As such
they attempt to predict regular-
ities in gross data and probabil-
ities of individual behavior but
they very carefully avoid guar-
anteeing a particular sequence +
of choices for anyone. The kind
of variation permitted by the
laws of probability maintains
the notion of causality in peo-
ple's behavior while at the same
time allowing for the kind of
unpredictability and freedom
that many people feel is such an
important part of human na-
ture.
In the next section of the book
the models are tested and while
none seems to fully capture the
process, the time the authors
spent developing and testing the
models and the time a reader
would spend reading them is by
no means wasted. At the time
this book was written these mo-
dels were fairly new and had
been subjected to few tests. Two
of them, the stochastic learning
model and the markov model
seemed, from other studies, to
be especially promising. More-
over the authors were able to
learn some substantive informa-
tion from the models. (eg: the
main reason the markov model
fails is that there is a greater
tendency for a CC lock in then
the model predicts). Finally, the
state of mathematical model
building has not, to my knowl-
edge, progressed greatly beyond
these kinds of models. Hence
those who want to learn how
models can be built and tested
can learn much about the state
of the field from this book.
While these specific kinds of
models are still important for .
mathematical model building, I
would agree with those who say
that they have now been shown
to be " sufficiently inadequate.
Mathematical social scientists
and social science-ish mathe-
maticians should , be trying to
construct new models. In par-
ticular, they should try to in-
vent new kinds of mathematics
more appropriate to the espec-
ially complex phenonmena of
human behavior.
It seems that nowadays there
is such a proliferation of pub-
lished material that it is im-
possible to read more than a
small fraction of it. Prisoner's
Dilemma, however is a b o o k
which many people should in-
clude in that s m a 11 fraction
which they read. One group of
people for whom this book is
worthwhile is those (whether in
the social sciences or not) who
are interested in the basis of
conflict and the promise of co-
operation. Even with minimal
formal mathematical training,
such a person can understand
most of the book (except the
chapters devoted to models) if
he can handle numbers a n d
read tables.
The other major group for
whom this book is of value is
people in the social sciences and
in mathematics who are inter-
ested in the potentialities and
limitations of mathematics for
studying human behavior. This
is by no means the only book
in which mathematics is sub-
stantially used by social scien-
tists, but it is one of the best.
It pursues a particular problem
in unusual depth, applying
mathematics to many different
aspects. For example, it looks at
the behavior of the process over
a long time period as well as the
effects of the immediately pre-
ceding trial. More importantly,
the book is extremely well-
written (as is the case with oth-
er works which Anatol Rapoport
has authored and co-authored),
Its style is rigorous and at the
same time both clear and inter-
esting. Among the important
features which increase the
readability of the book are the
heuristic and intuitive interpre-
tations which often accompany
mathematical derivations and
presentation of data.
Events of the p a s t several
years have made many social
scientists less enthusiastic about
peace research than w a s the
case in the early 1960's. This
change stems largely from the
fact t h a t the peace research
movement was based on undue
optimism about the willingness
of the U.S. government to ap-
ply the findings of peace re-
search towards building a more
peaceful world. For this reason,
many people interested in peace
research a r e trying to orient
their research towards findings
which can be of use to institu-
tions which are seriously com-
mitted to peace. Other peace re-
searcherss having become con-
vinced that American institu-
tions stand in the way of peace
have switched to research aim-
ed at understanding and ulti-
mately changing powerful in-
stitutions in American society.
Even though Prisoner's Di-
lemma is part of the earlier,
more naive, stage of peace re-
search, and is in t h a t sense
somewhat dated, it is not use-
less or irrelevant. The kind of
research movement that it was
part of was necessary to show
us that we were naive and per-
haps, if we can change the in-
stitutions of this country, these
findings may yet be applied to-
ward creating a less violent
world.
To sum up - it ain't quick
reading but its a damn good
book - interesting, informative,
relevant and thought provoking.
b
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RESERVATIONS
FOR ROSH
HASHANAH
MEALS WILL
NOT BE ACCEPTED
AFTER MONDAY THE 28th
RESERVE NOW
,.. .
greater degree of co-operation
in the concealed matrix condi-
tion on the grounds that sub-
jects who saw the matrix would
be more likely to be aware that
D is the dominating strategy.
In fact the opposite was the
case, suggesting that the pre-
dominant function of the ma-
trix is to make obvious the pos-
sibilities for m u t u a l gain
through co-operation. This sug-
gests that fear of being shafted
(receiving S) is a more impor-
tant motivation for people to
play D than is desire to profit
at the expense of ones partner
(by receiving T).
Aside from looking at the
gross data, which gives overall
rates of co-operation for differ-
ent conditions, the authors show
that much can be learned by
dissecting the data in various
ways. Not surprisingly, a strong
positive correlation (greater
than .9) is found between the
p e r c e n t a g e of co-operative
choices of the two partners.
These correlations clearly dem-
onstrate that the process is not
simply determined by the char-
acteristics of the individual
players but that there are strong
interaction effects.
One way of further under-
standing what is happening is
to look at the contingent pro-
pensities. These are a set of
probabilities, each one being a
probability of a C response after
a particular outcome on the
previous trial. Examining these
contingent propensities can be
quite intriguing and helps us
to understand the cause of the
DD lock-in, in which both play-
ers continuously lose. The con-
<Ramparts Magazine, 1967
data, the next section is devoted
to mathematical theory. Those
who believe that use of con-
trolled experiment and quanti-
tative methods for exploring the
inner workings of human beings
seem invalid and perhaps even
blasphemous will find this part
of the book particularly dis-
agreeable. This section attempts
to go beyond looking for regular-
ities in the data and tries doing
something which, if successful,
would constitute a much strong-
er, more universal way of un-
derstanding the problem. That
is the construction of mathe-
matical theories, which are at-
tempts to describe and predict
the entire process with a few
simple assumptions and para-
meters.
While some may be repelled
by this attempt, I believe that
most readers will find it inter-
esting, instructive, and non-of-
fensive. Since all but the first
chapter of this section is highly
mathematical, this may sub-
stantially limit the amount that
can be understood by the non-
mathematical person. On the
other hand, the book is suffi-
ciently clear and interesting
that while a non-mathematical
person will not be able to fol-
low derivations, if he really
thinks about what he is reading
and has some understanding of
probability, he might be sur-
prised at how many of the re-
sults and concepts he is able
to understand.
While those who can follow
the mathematics will learn most
from this section in that they
will see the applications of the
k i n d of models which social
scientists often use, even t h e
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person who 1reads only the in- t VAINMAI-KAI iHEATRE IFOpnel V
Today's Writers . . . troductory chapter of this see- 1-4270400
Stan Kaplowitz, a graduate tion will find it quite enlighten- 340 Maynard St., Ann Arbor Intersection of Plymouth and Farmington Rds. in Lvon a
student in sociology, has a ing in that the authors give a
strong interest in the potential good discussion of the purpose - . .-
for using mathematics as a key and limitations of models. This
to understanding problems in discussion is quite important
the social sciences. His own both for people who are involv-
work gives testimony to his ed in model building and for U of M School of M usc presents its 2nd annual
long acquaintanship with the outsiders who view the endeavor
studies of Anatol Rapoport. with awe or suspicion.
A few of the author's points
TTOctober 2 Hill Auditoriu 8:00 p.m.
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