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\section*{Introduction}

\footnote{   10 October 2005.  Eric Rasmusen, Erasmuse@indiana.edu.
Http://www.rasmusen.org/GI.    }

\noindent
{\bf History}

 \nopagebreak

\noindent
 Not so long ago, the scoffer could say that econometrics and game theory were
like Japan and Argentina. In the late 1940s both disciplines and both economies
were full of promise, poised for rapid growth and ready to make a profound
impact on the world. We all know what happened to the economies of Japan and
Argentina. Of the disciplines, econometrics became an inseparable part of
economics, while game theory languished as a subdiscipline, interesting to its
specialists but ignored by the profession as a whole. The specialists in game
theory were generally mathematicians, who cared about definitions and proofs
rather than applying the methods to economic problems. Game theorists took pride
in the diversity of disciplines to which their theory could be applied, but in
none had it become indispensable.

         In the 1970s, the analogy with Argentina broke down. At the same time
that Argentina was inviting back Juan Peron, economists were beginning to
discover what they could achieve by combining game theory with the structure of
complex economic situations. Innovation in theory and application was especially
useful for situations with asymmetric information and a temporal sequence of
actions, the two major themes of this book. During the 1980s, game theory became
dramatically more important to mainstream economics.  Indeed, it seemed to be
swallowing up microeconomics  just as econometrics had swallowed up empirical
economics.

        Game theory is generally considered to have begun with the publication
of von Neumann \& Morgenstern's {\it The Theory of Games and Economic Behaviour}
in 1944.  Although very little of the game theory in that thick volume is
relevant to the present book, it introduced the idea that conflict could be
mathematically analyzed and provided  the terminology with which to do it. The
development of the  ``Prisoner's Dilemma'' (Tucker [unpub]) and Nash's papers on
the definition and existence of equilibrium (Nash [1950b, 1951]) laid the
foundations for modern noncooperative game theory.  At the same time,
cooperative game theory reached important results in papers by Nash (1950a) and
Shapley (1953b) on bargaining games and Gillies (1953) and Shapley (1953a) on
the core.

 By 1953 virtually all the game theory that was to be used by economists for the
next 20 years had been developed.  Until the mid 1970s, game theory remained an
autonomous field with little relevance to mainstream economics, important
exceptions being Schelling's 1960 book, {\it The Strategy of Conflict}, which
introduced the focal point, and a series of papers (of which Debreu \& Scarf
[1963] is typical) that showed the relationship of the core of a game to the
general equilibrium of an economy.

         In the 1970s, information became the focus of many models as economists
started to put emphasis on individuals who act rationally but with limited
information. When attention was given to individual agents, the time ordering in
which they carried out actions began to be explicitly incorporated. With this
addition, games had enough structure to reach interesting and non-obvious
results.  Important ``toolbox'' references include the earlier but long-
unapplied articles of Selten (1965) (on perfectness) and Harsanyi (1967) (on
incomplete information), the papers by Selten (1975) and Kreps \& Wilson (1982b)
extending perfectness, and the article by Kreps, Milgrom, Roberts  \& Wilson
(1982) on incomplete information in repeated games.  Most of the applications in
the present book were developed after 1975, and the flow of research shows no
sign of diminishing.

\bigskip \noindent {\bf Game Theory's Method}

\noindent Game theory has been successful in recent years because it fits so
well into the new methodology of economics. In the past, macroeconomists started
with broad behavioral relationships like the consumption function, and
microeconomists often started with precise but irrational behavioral assumptions
such as sales maximization. Now all economists start with primitive assumptions
about the utility functions, production functions, and endowments of the actors
in the models (to which must often be added the available information).  The
reason is that it is usually easier to judge whether primitive assumptions are
sensible  than to evaluate high-level assumptions about behavior.  Having
accepted the primitive assumptions, the modeller figures out what happens when
the actors maximize their utility subject to the constraints imposed by their
information, endowments, and production functions.  This is exactly the paradigm
of game theory: the modeller assigns payoff functions and strategy sets to his
players  and sees what happens when they pick strategies to maximize their
payoffs.  The approach is a combination of the ``Maximization Subject to
Constraints'' of MIT and the ``No Free Lunch'' of Chicago.  We shall see,
however, that game theory relies only on the spirit of these two approaches: it
has moved away from maximization by calculus, and inefficient allocations are
common. The players act rationally, but the consequences are often bizarre,
which makes application to a world of intelligent men and ludicrous outcomes
appropriate.

\bigskip \noindent {\bf   Exemplifying Theory}

\noindent Along with the trend towards primitive assumptions and maximizing
behavior has been a trend toward simplicity. I called this ``no-fat modelling''
in the First Edition, but  the term ``exemplifying theory'' from Fisher (1989)
is more apt. This has also been called ``modelling by example'' or ``MIT-style
theory.''    A more smoothly flowing   name, but   immodest in its double
meaning, is ``exemplary theory.''   The heart of the approach is to discover the
simplest assumptions needed to generate an interesting conclusion--- the
starkest, barest model that has the desired result. This desired result is the
answer to some relatively narrow question.  Could education be just a signal of
ability? Why might bid-ask spreads exist?  Is predatory pricing ever rational?

  The modeller starts with a vague idea such as ``People go to college to show
they're smart.'' He then models the idea formally in a simple way. The idea
might survive intact; it might be found formally meaningless; it might survive
with qualifications; or its opposite might turn out to be true.  The modeller
then uses the model to come up with precise propositions, whose proofs may tell
him still more about the idea.  After the proofs, he goes back to thinking in
words, trying to understand more than whether the proofs are mathematically
correct.

   Good theory of any kind uses Occam's razor, which cuts out superfluous
explanations,  and the {\it ceteris paribus} assumption, which  restricts
attention to one issue at a time. Exemplifying theory goes a step further by
providing, in the theory, only a narrow answer to the question. As Fisher says,
``Exemplifying theory does not tell us what {\it must } happen. Rather it tells
us what {\it can} happen.''

In the same vein, at Chicago I have heard the style called ``Stories That Might
be True.'' This is not destructive criticism  if  the modeller is modest, since
there are also a great many ``Stories That Can't Be True,'' which  are often
used as the basis for decisions in business and government. Just as the modeller
should feel he has done a good day's work if he has eliminated most outcomes as
equilibria in his model, even if multiple equilibria remain, so he should feel
useful if he has ruled out certain explanations for how the world works, even if
multiple plausible models remain.    The aim should be to come up with one or
more stories  that might apply to a  particular situation  and then try to sort
out which story  gives the best explanation. In this, economics combines the
deductive reasoning of mathematics with the analogical reasoning of law.

  A critic of the   mathematical approach in biology has compared it to an
hourglass (Slatkin [1980]).  First, a broad and important problem is introduced.
Second, it is reduced to a very special but tractable model that hopes to
capture its essence. Finally, in the most perilous part of the process, the
results are expanded to apply to the original problem.  Exemplifying theory does
the same thing.

  The process is one of setting up ``If-Then'' statements, whether in words or
symbols.  To apply such statements, their premises and conclusions need to be
verified, either by casual or careful empiricism.  If the required assumptions
seem contrived or the assumptions and implications contradict reality, the idea
should be discarded.  If ``reality'' is not immediately obvious and data is
available, econometric tests may help show whether the model is valid.
Predictions can be made about future events, but that is not usually the primary
motivation: most of us are more interested in explaining and understanding than
predicting.

  The method just described is close to how, according to Lakatos (1976),
mathematical theorems are developed. It contrasts sharply with the common view
that the researcher starts with a hypothesis and proves or disproves it.
Instead, the process of proof helps show how the hypothesis should be
formulated.

   An important part of   exemplifying theory is what Kreps \& Spence (1984)
have called ``blackboxing'': treating unimportant subcomponents of a model in a
cursory way.  The game ``Entry for Buyout'' of section 15.4, for example, asks
whether a new entrant would be bought out by the industry's incumbent producer,
something that depends on duopoly pricing and bargaining.  Both pricing and
bargaining are complicated games in themselves, but if the modeller does not
wish to deflect attention to those topics  he can use the simple Nash and
Cournot solutions to those games and go on to analyze buyout. If the entire
focus of the model were duopoly pricing, then using the Cournot solution would
be open to attack, but as a simplifying assumption, rather than one that
``drives'' the model, it is acceptable.

 Despite the style's drive towards simplicity, a certain amount of formalism and
mathematics is  required to pin down the modeller's thoughts.   Exemplifying
theory treads a middle path between mathematical generality and nonmathematical
vagueness.  Both alternatives will complain that exemplifying theory is too
narrow. But beware of calls for more  ``rich,''   ``complex,''  or ``textured''
descriptions; these often lead to theory which is either too incoherent or too
incomprehensible to be applied to real situations.

Some readers will think that exemplifying theory uses too little mathematical
technique, but others, especially noneconomists, will think it uses too much.
Intelligent laymen have objected to the amount of mathematics in economics since
at least the 1880s, when George Bernard Shaw said that as a boy he (1) let
someone assume that $a=b$, (2) permitted several steps of algebra, and (3) found
he had accepted a proof that $1 = 2$.  Forever after, Shaw distrusted
assumptions and algebra. Despite the effort to achieve simplicity (or perhaps
because of it), mathematics is essential to   exemplifying theory. The
conclusions can be retranslated into words, but rarely can they be found by
verbal reasoning. The economist Wicksteed put this nicely in his reply to Shaw's
criticism:

\begin{quotation} \noindent Mr Shaw arrived at the sapient conclusion that there
``was a screw loose somewhere''--- not in his own reasoning powers, but---``in
the algebraic art''; and thenceforth renounced mathematical reasoning in favour
of the literary method which enables a clever man to follow equally fallacious
arguments to equally absurd conclusions {\it without seeing that they are
absurd}.  This is the exact difference between the mathematical and literary
treatment of the pure theory of political economy. (Wicksteed [1885] p. 732)
\end{quotation}

  In   exemplifying theory, one can still rig a model to achieve a wide range of
results, but it must be rigged by making strange primitive assumptions. Everyone
familiar with the style knows that the place to look for the source of
suspicious results is the description at the start of the model.  If that
description is not clear, the reader deduces that the model's counterintuitive
results arise from bad assumptions concealed in poor writing. Clarity is
therefore important, and the somewhat inelegant Players-Actions-Payoffs
presentation used  in this book is useful not only for helping the writer, but
for persuading the reader.

\bigskip \noindent {\bf This Book's Style}

\noindent Substance and style are closely related. The difference between a good
model and a bad  one is not just whether the essence of the situation is
captured, but also how much froth covers the essence. In this book, I have tried
to make the games as simple as possible.  They often, for example, allow each
player a choice of only two actions.  Our intuition works best with such models,
and continuous actions are technically more troublesome.  Other assumptions,
such as zero production costs, rely on trained intuition. To the layman, the
assumption that output is costless seems very strong, but a little experience
with these models teaches that it is the constancy of the marginal cost that
usually matters, not its level.

  What matters more than what a model says is what we understand it to say.
Just as an article written in Sanskrit is useless to me, so is one that is
excessively mathematical or poorly written, no matter how rigorous it seems to
the author. Such an article leaves me with some new belief about its subject,
but that belief is not sharp, or precisely correct. Overprecision in sending a
message creates imprecision when it is received, because  precision is not
clarity.  The result of an attempt to be mathematically precise is sometimes to
overwhelm the reader,  in the same way   that someone who requests  the answer
to a simple question in the discovery process of a lawsuit is overwhelmed when
the other side responds with 70 boxes of tangentially related documents. The
quality of the author's input should be judged not by some abstract standard but
by the output in terms of reader processing cost and  understanding.

  In this spirit,  I have tried to simplify the structure and notation of models
while giving credit to their original authors, but I must ask pardon of anyone
whose model has been oversimplified or distorted, or whose model I have
inadvertently replicated without  crediting them.   In trying to be
understandable, I have taken risks with respect to accuracy.  My hope is that
the impression left in the readers' minds will be more accurate than if a style
more cautious and obscure had left them to devise their own errors.
My major strength is in   boiling down difficult models into simple games that
still capture the essence of the models' ideas. My major weakness is  in
sliding  past technical points, some  unimportant, some important. I apologize
in advance for the mistakes I am sure this book contains, but I hope that they
are orthogonal to  the mistakes of other  books, and obvious enough not to lead
the reader  far astray.


 Readers may be surprised to find occasional references to newspaper and
magazine articles in this book.  I hope these references will be reminders that
models ought eventually to be applied to specific facts, and that a great many
interesting situations are waiting for our analysis. The principal-agent problem
is not found only in back issues of {\it Econometrica}: it can be found on the
front page of today's {\it Wall Street Journal} if one knows what to look for.

     I make the occasional joke here and there. Game theory is a subject
intrinsically full of paradox and surprise. I want to emphasize, though, that I
take game theory seriously, in the same way   Chicago economists  say that they
take price theory seriously. It is not just an academic artform: people do
choose actions deliberately and trade off one good against another, and game
theory will help you understand how they do that. If it did not, I would not
advise you to study such a difficult subject.  There are much more elegant
fields in mathematics  from the  aesthetic point of view. As it is, I think it
is important that every educated person have some contact with the ideas in this
book, just as they should have some  contact with the basic principles of price
theory.

  I have been forced to exercise more discretion over definitions than I had
hoped.  Many concepts have been defined on an article-by-article basis in the
literature, with no consistency and little attention to euphony or usefulness.
Other concepts, such as ``asymmetric information'' and ``incomplete
information,'' have been considered so basic as to not need definition, and
hence have been used in contradictory ways. I use existing terms whenever
possible, and synonyms are listed.

 I have often  named the players Smith and Jones  so that the reader's memory
will be less taxed in remembering which is a player and which is a time period.
I hope also to reinforce the idea that a model is a story made precise; we begin
with Smith and Jones, even if we quickly descend to $s$ and $j$.  Keeping this
in mind, the modeller is less likely to build mathematically correct models with
absurd action sets, and his descriptions are more pleasant to read. In the same
vein, labelling a curve ``$U=83$'' sacrifices no generality: the phrase ``$U =
83$ {\rm and} $U =66$'' has virtually the same content as ``$U = \alpha$ {\rm
and} $U = \beta$, {\rm where} $\alpha > \beta$,'' but uses less short-term
memory.

 A danger of this approach is that readers may not appreciate the complexity of
some of the material.  While journal articles make the material seem harder than
it is, this approach makes it seem easier (a statement that can be true even if
readers find this book difficult).  The better the author does his job, the
worse this problem becomes.  Keynes (1933) says of Alfred Marshall's {\it
Principles},

\begin{quotation} \noindent The lack of emphasis and of strong light and shade,
the sedulous rubbing away of rough edges and salients and projections, until
what is most novel can appear as trite, allows the reader to pass too easily
through. Like a duck leaving water, he can escape from this douche of ideas with
scarce a wetting.  The difficulties are concealed; the most ticklish problems
are solved in footnotes; a pregnant and original judgement is dressed up as a
platitude.
  \end{quotation}

\noindent This book may well be subject to the same criticism, but   I have
tried to face up to difficult points, and the    problems at the end of each
chapter will help to avoid making the reader's progress too easy. Only a certain
amount of understanding can be expected from a book, however. The efficient way
to learn how to do research is to start doing it, not to read about it, and
after reading this book, if not before, many readers will want to build their
own models. My purpose here is to show them the big picture, to help them
understand the models intuitively, and   give them a feel for the modelling
process.

\bigskip

 \begin{small} \noindent {\bf  NOTES} \begin{itemize} \item Perhaps the most
important contribution of von Neumann \& Morgenstern (1944) is the theory of
expected utility (see section 2.3).  Although they developed the theory because
they needed it to find the equilibria of games, it is today heavily used in all
branches of economics.  In game theory proper, they contributed the framework to
describe games, and the concept of mixed strategies (see section 3.1). A good
historical discussion is Shubik (1992) in the Weintraub volume mentioned in the
next note.

\item A number of good books on the history of game theory have appeared in
recent years.   Norman Macrae's  {\it John von Neumann}  and Sylvia Nasar's {\it
A Beautiful Mind} (on John Nash) are  extraordinarily  good biographies of
founding fathers, while  {\it Eminent Economists: Their Life Philosophies} and
{\it Passion and Craft: Economists at Work}, edited by Michael Szenberg, and
{\it Toward a History of Game Theory}, edited by Roy Weintraub,    contain
autobiographical essays by many scholars who use game theory, including  Shubik,
Riker,  Dixit, Varian, and Myerson. Dimand and Dimand's  {\it     A History of
Game Theory}, the first volume of which appeared in 1996,  is a more  intensive
look at the intellectual history of the field.   See also Myerson (1999).

\item For  articles  from the  history of mathematical economics,  see the
collection by Baumol \& Goldfeld (1968),   Dimand and Dimand's 1997 {\it The
Foundations of Game Theory} in three volumes, and Kuhn (1997).
  Collections of more recent articles include  Rasmusen (2001), Binmore \&
Dasgupta (1986),   Diamond \& Rothschild (1978) on information economics,
Klemperer (2000)  on auctions,  and the immense Rubinstein (1990).

\item On method, see the dialogue by Lakatos (1976), or Davis, Marchisotto \&
Hersh (1981), Chapter 6 of which is a shorter dialogue in the same style.
Friedman (1953) is the classic essay on a different methodology: evaluating a
model by testing its predictions. Kreps \& Spence (1984) is a discussion of
exemplifying theory.

\item Because style and substance are so closely linked,  how one writes  is
important. For advice on writing, see McCloskey (1985, 1987) (on economics),
  Bowersock (1985) (on footnotes), Fowler
(1965), Fowler \& Fowler (1949), Halmos (1970) (on mathematical writing),
Rasmusen (2000), Strunk \& White (1959), Weiner (1984), and Wydick (1978).

\item {\bf A fallacious proof that 1=2.} Suppose that $a=b$.  Then $ab=b^2$ and
$ab - b^2 = a^2 - b^2.$ Factoring the last equation gives us $b(a-b) = (a+b)(a-
b)$, which can be simplified to $b = a+b$. But then, using our initial
assumption, $b = 2b$ and $1=2$. (The fallacy is division by zero.)
 
 \end{itemize}
 \end{small}

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