Friends and colleagues ask me about resources on convention and related
areas for research and classes somewhat regularly. I thought it might be useful
to collect some references here for those interested studying these areas
further and who could use a few leads. Please note that the brief comments that
accompany many of these references express my personal opinions, and should be
taken as such.

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Convention

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Game Theory

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Equilibrium Selection

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Common Knowledge

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Game Theory in Moral and Political Philosophy

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Computer Software

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Classical Music

Sugden (1999) gives a splendid short overview of analyses of convention,
past and present. In my opinion, those
interested in learning about convention can do no better than to start here.

David Lewis's 1969 monograph *Convention* is the pioneering
contemporary philosophical study of convention. Happily, *Convention* was
reprinted in 2002 and is now once again widely available. Lewis used game
theory as scaffolding for his definition of convention, and it has become
standard to use game theory to analyze convention. Philosophers such as Russell
and Quine argued that languages could not be ultimately conventional in
origin. Much of Lewis’ work in *Convention* is devoted to his attempt to present
a successful conventionalist account of languages. Lewis' work is classic, but Sugden (1986),
Gilbert (1989) Skyrms (1990) and Vanderschraaf (1998) have all argued that in
certain respects his analysis is insufficiently general.

Lewis acknowledged that some of his analysis is foreshadowed in Schelling
(1960) and Hume (1740), which happily are always in print. Schelling's *The
Strategy of Conflict* is one of the acknowledged masterpieces of the social
sciences. One can study Schelling's work easily with no background in
mathematics or the social sciences. Among Schelling's landmark contributions,
he argued for a reorientation of game theory that would examine problems of
coordination more carefully, foreshadowing the revival of work on convention.
Schelling's work sparked several research programs that continue today.

David Hume gave perhaps the earliest explicit analysis of convention in *A
Treatise of Human Nature*. Hume repeats this analysis in abbreviated form in
Appendix 3 of *An Enquiry Concerning the Principles of Morals*. Hume's
treatment of convention predates game theory by more than two centuries.
Nevertheless, Hume's analysis of convention foreshadows a number of
game-theoretic concepts. Some of Hume's observations on the origins of
conventions remain more sophisticated than any of the contemporary discussions
of salience.

More recently, Margaret Gilbert (1989) presented an alternative account of
convention motivated in part by her disagreements with Lewis' account.
Gilbert's account draws upon sociology, whereas most other contemporary
accounts of convention rely more heavily upon economic theory. Gilbert's work
has been under appreciated, perhaps in part because of her difficult writing
style.

**References **

Gilbert, Margaret. 1989, *On Social Facts*.

Hume, David. 1740 (2000). *A
Treatise of Human Nature*. ed. David Fate
Norton and Mary J. Norton.

Hume, David. 1777 (1998). *An
Enquiry Concerning the Principles of Morals*, ed. Tom Beauchamp.

Lewis, David. 1969 (2002). *Convention: A
Philosophical Study*.

Schelling, Thomas. 1960. *The Strategy of Conflict*.

Skyrms, B. 1990. *The Dynamics
of Rational Deliberation*.

Sugden, Robert. 1986. *The Economics of Rights, Co-operation and Welfare*.

Sugden, Robert. 1999. "Conventions", in *The**
New Palgrave Dictionary of Economics and the Law*, ed. Peter Newman.

Vanderschraaf, Peter. 1998. "Knowledge, Equilibrium and
Convention", *Erkenntnis* 49: 337-369.

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Philosophers who want an in-depth introduction to game theory but don't have
time to work through lots of exercises might consult Bicchieri (1993) and
Vanderschraaf (2001). Each of these books gives readable introductory material
for newcomers to game theory as background for the subsequent original
material. Bicchieri's presentation of the introductory material is quite
informal, and somewhat oriented towards refinements of the Nash equilibrium
concept. The presentation in Vanderschraaf (2001) is more detailed and more
formal than in Bicchieri (1993) and covers various generalizations of the Nash
equilibrium concept Bicchieri does not discuss in detail. However, Vanderschraaf (2001) does not
discuss equilibrium refinements.

Most game theory textbooks I have seen tend either to have formidable mathematical prerequisites or to give a "gee whiz" presentation of topics in game theory that teaches readers little of substance. A welcome new exception is Osborne (2004). Osborne’s book is ideal for either a first course in game theory for undergraduate or for self study. The only formal prerequisite for studying this book is elementary algebra. While readers would find some prior exposure to calculus and elementary probability theory helpful, Osborne includes a complete appendix of the mathematical concepts he employs in his text. Osborne gives a lively exposition of the elements of game theory, many exercises that will help consolidate one’s understanding of the theory, and interesting discussions of experimental game theory and important developments in the history of game theory. Another excellent and more advanced textbook is Gintis (2001). Gintis is a problem book, and is designed to teach readers the elements of game theory as they work through Gintis' exercises, which are really more like short research projects. Gintis teaches readers a lot of substantive game theory, but it will take a significant time commitment to study profitably. Gintis also includes fine reviews of the elementary probability theory and differential equation theory needed to study parts of the game theory covered in this work, an invaluable addition since many newcomers to game theory have not previously studied these branches of mathematics.

There are a number of fine graduate level textbooks in game theory that have a de-facto undergraduate real analysis prerequisite (and topology wouldn’t hurt, either!). My favorites, in ascending order of difficulty, are Binmore (1992), Myerson (1991) and Fudenberg and Tirole (1991). Unfortunately, Binmore’s text has fallen out of print, but Myerson and Fudenberg and Tirole remain widely available. Osborne and Rubinstein (1994) is a graduate level text that may prove of special interest to some philosophers. The formalism in Osborne and Rubinstein is quite compact and will be hard going even for readers with a background in game theory. However, this book contains some especially lively and illuminating informal discussion of foundational issues. The authors Osborne and Rubinstein occasionally openly disagree in the text, and when they do they give interesting short debates for readers to study.

Those who study a graduate level text in game theory such as those mentioned here will want access to a text on undergraduate analysis. For those who don’t already have a personal favorite book on analysis, I recommend Stromberg (1981) or Ross (1980). Some of the fundamental results in game theory rely upon fixed point theorems from topology that are not proved in any game theory text I know and are frequently not included in advanced topology textbooks. Readers who would like to see the proofs of these fixed point theorems can find them in Border (1989) and Aliprantis and Border (1999).

Luce and Raiffa's classic *Games and Decisions* (1957) is always a
great book to have at one's elbow when studying
anything with game theoretic content. I don't recommend studying this work in
isolation because their formalism is occasionally cumbersome and their
notations are no longer popular. Moreover, *Games and Decisions* of course
does not discuss developments of the last fifty years. Nevertheless, their
informal discussion of game theoretic concepts is as illuminating today as it
was in the 1950s.

The Prisoners' Dilemma is the single most studied of all games. Kuhn (2003)
gives a simply beautiful overview of the Prisoners' Dilemma and its
philosophical significance.

Readers interested in the history of game theory can find a fine overview of
the development of mathematical game theory up to the 1990s in Aumann (1989). They might also go straight to the classic
sources von Neumann and Morgenstern (1944) and Nash (1950a, 1950b, 1951, 1953). As with Luce and Raiffa, the formalism in these
classic works is both hard to study and avoidable, and the informal discussions
are fascinating. In 2004 Princeton University Press published a 60^{th}
anniversary edition of von Neumann and Morgenstern (1944) that includes as
“extras” some the original revues of this treatise and a new afterward by
editor Ariel Rubinstein. Nash’s
published papers on game theory have been reissued in several in collections in
recent years. One of these collections
that is of special interest is *The Essential John Nash*, which is both affordable and which
includes a facsimile of Nash’s doctoral thesis in addition to his published
articles in game theory. (I don't
recommend obtaining any of the anthologies of "classics" in game
theory that have been marketed in the past few years. These are terribly
expensive and tend not to include all of the most important papers.)

**References**

Aliprantis,
Charalambos and Border, Kim. 1999. *Infinite
Dimensional Analysis : A Hitchhiker's Guide*.

Aumann, Robert. 1989. "Game Theory", in *The**
New Palgrave: Game Theory*, ed. John Eatwell,

Bicchieri, Cristina. 1993. *Rationality and Coordination*.

Binmore, Ken. 1992. *Fun and Games*.

Border, Kim. 1989. *Fixed
Point Theorems with Applications to Economics and Game Theory*.

Fudenberg, Drew and Tirole, Jean. 1991. *Game
Theory*.

Gintis, Herbert. 2000. *Game Theory Evolving*.

Kuhn, Steven. 2000. "Prisoner’s Dilemma", in Stanford Encyclopedia
of Philosophy. (plato.stanford.edu/entries/prisoner-dilemma/#Bib)

Luce, R. Duncan and Raiffa, Howard. 1957.* Games
and Decisions: Introduction and Critical Survey*.

Myerson, Roger. 1991. *Game Theory: Analysis of Conflict*.

Nash, John. 1950a. 'The Bargaining Problem.' *Econometrica*
18: 155-162.

Nash, John. 1950b. 'Equilibrium points in -person games.'
Proceedings of the

Nash, John. 1951. 'Non-Cooperative Games.' *Annals of Mathematics* 54:
286-295.

Nash, John. 1953. 'Two-Person Cooperative Games.' *Econometrica* 21:
128-140.

Nash, John. 2002. *The
Essential John Nash*, ed. Harold Kuhn and Sylvia Nasar.

Osborne, Martin J. and Rubinstein, Ariel. 1994. *A Course in Game Theory*.

Osborne, Martin. 2004. *An Introduction to Game Theory*.

Ross, Kenneth. 1980. *Elementary Analysis: The Theory of Calculus*. Springer.

Stromberg, Karl. 1981. *Introduction
to Classical Real Analysis*.

Von Neumann, John and Morgenstern, Oskar. 1944 (2004). *Theory of Games and
Economic Behavior*, ed. Ariel Rubinstein.

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Fudenberg and Levine (1998) have published a comprehensive work on dynamical updating models of equilibrium selection. Fudenberg and Levine is a superlative reference, but even their book does not cover all the relevant dynamical models because the field has grown and continues to grow so rapidly. Their presentation is superb, though some readers will find their work mathematically formidable.

Skyrms (1990) takes an in-depth look at issues connected to the equilibrium
selection problem from a philosopher’s perspective. Surprisingly, so far this fine work’s impact
has been greater in the social sciences than in philosophy.

Maynard Smith and Price (1973) and Maynard Smith (1982) introduced the idea
of equilibrium selection via the replicator dynamics and established
evolutionary game theory as an important new field. Weibull (1995) gives a
mathematically sophisticated treatment of evolutionary game theory, including
in particular replicator dynamics and some of its variants. Skyrms (1996)
introduces a much more general evolutionary game theory that allows for the
possibility that encounters between agents are
correlated by their types. Skyrms' presentation is informal and highly
readable. The formal theory suggested by Skyrms’ informal presentation has yet
to be developed. Weibull and Skyrms both limit their treatments to one-shot
games. Alexander (2003) gives a fine
overview of the philosophical issues raised by evolutionary game theory.

Young (1998) and Vanderschraaf (2001) present accounts of equilibrium selection based upon generalizations of the inductive learning rule known as fictitious play. Both books can be studied by relative newcomers to game theory.

In a new work, Young (2004) overviews and compares a number of dynamical
models of equilibrium selection, including fictitious play and the replicator
dynamics.

**References**

Alexander, Jason. “Evolutionary
Game Theory” in Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/game-evolutionary/

Fudenberg, Drew and Levine, David. 1998. *The
Theory of Learning in Games*.

Maynard Smith, John. 1982. *Evolution and the Theory of Games*.

Maynard Smith, John, and Price, G. R. 1973. 'The logic of animal conflict'. Nature 146: 15-18.

Skyrms, Brian. 1990. *The
Dynamics of Rational Deliberation*.

Skyrms, Brian. 1996. *Evolution of the Social Contract*.

Weibull, Jorgen W. (1995). *Evolutionary Game Theory*.

Vanderschraaf, Peter. (2001) *Learning and Coordination: Inductive
Deliberation, Equilibrium and Convention*.

Young, H. Peyton. (1998) *Individual Strategy and Social Structure: An
Evolutionary Theory of Institutions*.

Young, H. Peyton. 2004. *Strategic
Learning and Its Limits*.

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Vanderschraaf and Sillari (2005) give a survey of alternate accounts of
common knowledge and applications of common knowledge in philosophy and the
social sciences. One can find a shorter
introduction to the notion of common knowledge in Aumann’s classic 1976 esssay.
Aumann's presentation is elegant and he introduces one of the most interesting
applications of common knowledge, the celebrated "No Disagreement
Theorem". Geanakoplos (1992) gives another excellent and in-depth overview
article on common knowledge. Geanakoplos' discussion is primarily conceptual,
and he discusses few applications. Brandenburger (1992) surveys many of the
known results connecting mutual and common knowledge to solution concepts in
game theory in a rich and highly readable article.

Lewis introduced the first explicit analysis of common knowledge and applied
it in his analysis of convention (1969). Lewis’ analysis of common knowledge is
uncharacteristically opaque. Cubitt and
Sugden (2003), Sillari (2005) and Vanderschraaf (1998) attempt to reconstruct
Lewis’ analysis of common knowledge formally, and their reconstructions
conflict to a certain extent. Schiffer (1972) and Aumann (1976) independently
developed different definitions of common knowledge. Aumann (1976) gave the first mathematically
rigorous formulation of common knowledge using set theory. In his presentation,
Schiffer used the knowledge operators of epistemic logic to illustrate common
knowledge between two agents. Bacharach (1989) and Bicchieri (1993) adopted
Schiffer's idea of using knowledge operators to analyze common knowledge, and
developed a logic of common knowledge that includes
soundness and completeness theorems. Schiffer's hierarchical definition has
become the best-known definition of common knowledge in philosophy. Barwise
(1988) introduced a fixed point analysis of common knowledge he argues is
inspired by Lewis’ original analysis. Aumann's,
Barwise’s and Lewis’ definitions all imply Schiffer's definition. Gilbert (1989)
presented yet another account of common knowledge she dubs the “smooth-reasoner”
account. Gilbert’s analysis is motivated
in part by her objections to Lewis' and Aumann's accounts of common knowledge.
Nevertheless, Lewis', Schiffer's and Aumann's analyses remain widely accepted.

Chwe (2001) presents a penetrating and delightful discussion of the role
common knowledge plays in understanding social processes. This work is filled with fascinating examples
and is brilliantly written.

**References**

Aumann, Robert.: 1976. "Agreeing
to Disagree", *Annals of Statistics* 4, 1236-9.

Bacharach, Michael. 1989. "Mutual Knowledge and Human Reason", mimeo.

Barwise, Jon. 1988. "Three Views of Common
Knowledge", in *Proceedings of the Second Conference on Theoretical
Aspects of Reasoning About Knowledge*, ed. M.Y.
Yardi.

Bicchieri, Cristina. 1993.* Rationality and Coordination*.

Brandenburger, Adam. 1992. "Knowledge and Equilibrium in Games", *Journal
of Economic Perspectives* 6:83-101.

Chwe, Michael. 2001. *Rational
Ritual: Culture, Coordination and Common Knowledge*.

Cubitt, Robin and Sugden, Robert. 2003.
"Common Knowledge, Salience and Convention: A Reconstruction of David
Lewis' Game Theory", *Economics and Philosophy* 19: 175-210.

Geanakoplos, John. 1992. "Common Knowledge",* Journal of
Economic Perspectives* 6:53-82.

Gilbert, Margaret. 1989. *On** Social Facts*.

Lewis, David. 1969. *Convention: A Philosophical Study*.

Schiffer, Stephen. 1972. *Meaning*.

Sillari, Giacomo. 2005. "A Logical Framework for Convention", *Synthese*
147(2): 379-400.

Vanderschraaf, Peter. 1998. "Knowledge, Equilibrium and
Convention", *Erkenntnis* 49: 337-369.

Vanderschraaf, Peter and Sillari, Giacomo. 2005. “Common
Knowledge”, in Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/common-knowledge/#Rel

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Verbeek and Morris (2004) give a readable and up to date overview of the uses, and possible misuses, of game theory in moral and political philosophy.

Two pioneering works that apply game theory to moral and political
philosophy game theory both first appeared in 1986. In *Morals
By Agreement*, David Gauthier developed a social
contract as the product of rational choice using game theory. One of Gauthier's central claims is that
orthodox decision theory requires reform in light of certain problems in game
theory. Gauthier presented a theory of constrained maximization, which he argued
solves the problem of cooperation in social dilemma problems like the
Prisoners' Dilemma. Gauthier's arguments have generated some heated controversy
in philosophy, and at times he is not the best expositor of his own theory. For
a fine, though ultimately critical, overview of constrained maximization, see
Bicchieri (1993).

Bob Sugden's *Economics of Rights, Cooperation and Welfare *(1986)
considers how elements of a social contract might evolve using game theory.
This book is a neglected classic, highly readable, light on the math and yet
filled with fascinating material. Sugden’s book was out of print for years, but
in 2004 was reissued with an new concluding chapter
where Sugden discusses the relevant developments of the past twenty years. I regard Gauthier’s and Sugden’s works as the
classics of the rational choice and evolutionary game theoretic approaches to
analyzing the social contract.

The opening chapter of Barry (1989) presents a beautiful and informal discussion of the Nash bargaining problem and its potential uses in analyzing problems of distributive justice.

Young (1994) applies a number of different game theoretic concepts to
specific problems of resource division.
Young’s examples are fascinating and his discussion in the main body of
the text will appeal to both specialists and nonspecialists alike. Young gives complete presentations of the
underlying mathematics in appendices.
One will not find a more illuminating overview of axiomatic bargaining
theory in any other work.

Evolutionary game theory is now coming into its own as an analytical tool
for the social philosopher. Axelrod (1984) published an early study of
evolutionary game theory applied to the problem of explaining cooperation in
the repeated Prisoners' Dilemma. Axelrod's book is stimulating and highly
readable. Like many pioneering studies, Axelrod's study has come under severe
attack and many of his original claims have been shown to be suspect. Still,
Axelrod remains a good starting point for those who would like to get
acquainted with evolutionary game theory and see it applied to a problem in
moral philosophy.

Along with Sugden (1986), Binmore (1994, 1998, 2005)
and Brian Skyrms' *Evolution of the Social Contract*
(1996) and *The Stag Hunt and The
Evolution of Social Structure* (2004) are important works in this growing
field. In *Evolution of the Social Contract Skyrms*
investigates the consequences of relaxing the assumptions of traditional
evolutionary game theory when one analyzes problems of justice such as the
bargaining problem and the ultimatum game.
In *The Stag Hunt and The Evolution of Social Structure* Skyrms explores how
local interaction structures might emerge using the coordination problem known
as stag hunt as a motivating problem. Skyrms'
presentation in these books is informal, and summarizes the first findings of
several research programs that generalize traditional evolutionary game theory
and its applications in explaining social contracts in new and exciting ways.
Readers will be left hungering for more after studying Skyrms, but they will
have to wait and see how these research programs progress --- or join in the
fray!

Binmore (1994, 1998, 2005) presents a social
contract theory using evolutionary game theory that is a serious rival to the
well-known rational choice contractarian theories of philosophers such as
Harsanyi and Rawls. Binmore gives a succinct and provocative statement of his
theory of the social contract in his new *Natural
Justice*. While Binmore describes his
theory as an evolutionary theory, he combines elements of both the evolutionary
and rational choice approaches to game theory.
Binmore bases his *Natural Justice*
on his much more comprehensive treatment in his earlier two volume work *Game Theory and the Social Contract*. This earlier work is filled with applications
of game theory to moral and political philosophy that specialists will love,
even if they do not always agree with Binmore’s specific claims. Indeed, I found that Binmore’s splendid
bibliographies in *Game Theory and the
Social Contract alone* were worth the price
of these volumes. However, the
presentation in *Game Theory and the
Social Contract* is exceedingly complex, and will give nonspecialists
trouble if they do not first study *Natural
Justice*.

In her new *The Grammar of Society*
(2006), Bicchieri uses game theory in her presentation of a novel account of
social norms that challenges several of the fundamental methodological
assumptions of the social sciences. Bicchieri
argues that the stress social scientists place upon rational deliberation
obscures the fact that many successful choices, and in particular many
successfully coordinated activities, occur even though the individuals make
their choices without much deliberation.
Bicchieri explores in depth the more automatic components of
coordination. She proposes a heuristic
account of coordination that complements the more traditional deliberational
account. According to the heuristic
account, individuals conform with a social norm as an automatic response to
cues in their situation that focus their attention on this particular
norm. A social norm is analyzed as a
rule for choosing in a mixed-motive game, such as the Prisoners' Dilemma, that
members of a population prefer to follow on condition that they expect
sufficiently many in the population to follow the rule. Bicchieri applies this account of social
norms and heuristic selection of norms to a number of important problems in the
social sciences, including bargaining, the Prisoners' Dilemma and suboptimal
norms based upon pluralistic ignorance.

**References**

Axelrod, Robert. 1984. *The Evolution of Cooperation*.

Barry, Brian. 1989. *Theories of Justice*.

Bicchieri, Cristina. 1993. *Rationality and Coordination*.

Bicchieri, Cristina. 2006. *The
Grammar of Society*.

Binmore, Ken. 1994.* Game Theory and the Social Contract Volume I: Playing
Fair*.

Binmore, Ken. 1998.* Game Theory and the Social Contract Volume II: Just
Playing.*

Binmore, Ken. 2005. *Natural
Justice*.

Gauthier, David. 1986. *Morals By Agreement*.

Hampton, Jean. 1986. *Hobbes and the Social Contract Tradition*.

Kavka, Gregory. 1986. *Hobbesian Moral and Political Theory*.

Skyrms, Brian. 1996. *Evolution of the Social Contract*.

Sugden, Robert. 1986 (2004). *The
Economics of Rights, Co-operation and Welfare*.

Verbeek, Bruno and Morris, Christopher. 2004.
“Game Theory and Ethics,” in Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/game-ethics/

Young, H. Peyton. 1994. *Equity
in Theory and Practice*.

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To date, there is no one set of computer software that covers many of the
applications of game theory in a manner comparable to the way systems like SAS,
S-Plus and Minitab cover wide areas of statistics. The most well-developed game
theory software package is **Gambit**, a software library developed
primarily by Richard McKelvey, Andrew McLennan and Theodore Turocy. Gambit is
available free of charge and can be downloaded at http://econweb.tamu.edu/gambit/ . Gambit contains both an interactive module and
a language for developers. One may download either or both to one’s own system.
Gambit will run on under MS Windows, Linux and Solaris. The authors of Gambit
have also made their C++ source code freely available. With the Gambit
interactive module, one can construct games in the strategic and the extensive
forms and compute their Nash equilibria. The Gambit interactive module is a
great achievement, given how computationally difficult it is to compute Nash
equilibria. However, the periodic releases of Gambit sometimes contain bugs,
and strategic form games with more than two players are quite hard to construct
and analyze in Gambit.

Erika Nickrenz, Adela Pena and Sara Sant'Ambrogio form the greatest
classical chamber ensemble in the world (my humble opinion). They are known
internationally as the Eroica Trio and
I discuss their performances in some of my classes and my written work as
examples of complex systems of conventions in action!

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