Compiled by Peter Vanderschraaf

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.

Φ Convention
Φ Game Theory
Φ Equilibrium Selection
Φ Common Knowledge
Φ Game Theory in Moral and Political Philosophy
Φ Computer Software
Φ 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.


Gilbert, Margaret. 1989, On Social Facts. Princeton: Princeton University Press.

Hume, David. 1740 (2000). A Treatise of Human Nature. ed. David Fate Norton and Mary J. Norton. Oxford: Clarendon Press.

Hume, David. 1777 (1998). An Enquiry Concerning the Principles of Morals, ed. Tom Beauchamp. Oxford: Clarendon Press.

Lewis, David. 1969 (2002). Convention: A Philosophical Study. Cambridge, Massachusetts: Harvard University Press.

Schelling, Thomas. 1960. The Strategy of Conflict. Cambridge, Massachusetts: Harvard University Press.

Skyrms, B. 1990. The Dynamics of Rational Deliberation. Cambridge, Massachusetts: Harvard University Press.

Sugden, Robert. 1986. The Economics of Rights, Co-operation and Welfare. Oxford: Basil Blackwell, Inc.

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

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

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Game Theory

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 60th 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.)


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

Aumann, Robert. 1989. "Game Theory", in The New Palgrave: Game Theory, ed. John Eatwell, Murray Milgate and Peter Newton. New York and London: MacMillan Press Ltd., pp. 1-53.

Bicchieri, Cristina. 1993. Rationality and Coordination. Cambridge: Cambridge University Press.

Binmore, Ken. 1992. Fun and Games. Lexington, Massachusetts: D. C. Heath and Company.

Border, Kim.  1989.  Fixed Point Theorems with Applications to Economics and Game Theory.  Cambridge: Cambridge University Press.

Fudenberg, Drew and Tirole, Jean. 1991. Game Theory. Cambridge, Massachusetts: Harvard University Press.

Gintis, Herbert. 2000. Game Theory Evolving. Princeton: Princeton University Press.

Kuhn, Steven. 2000. "Prisoner’s Dilemma", in Stanford Encyclopedia of Philosophy. (

Luce, R. Duncan and Raiffa, Howard. 1957. Games and Decisions: Introduction and Critical Survey. New York: John Wiley and Sons.

Myerson, Roger. 1991. Game Theory: Analysis of Conflict. Cambridge, Massachusetts: Harvard University Press.

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

Nash, John. 1950b. 'Equilibrium points in -person games.' Proceedings of the National Academy of Sciences of the United States 36: 48-49.

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.  Cambridge: Cambridge University Press.

Osborne, Martin. 2004.  An Introduction to Game Theory.  Oxford: Oxford University Press.

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

Stromberg, Karl.  1981.  Introduction to Classical Real Analysis.  Belmont, California:  Wadsworth.

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

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Equilibrium Selection

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.


Alexander, Jason.  “Evolutionary Game Theory” in Stanford Encyclopedia of Philosophy.

Fudenberg, Drew and Levine, David. 1998. The Theory of Learning in Games. Cambridge, Massachusetts: MIT Press.

Maynard Smith, John. 1982. Evolution and the Theory of Games. Cambridge: Cambridge University Press.

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.  Cambridge, Massachusetts:  Harvard University Press.

Skyrms, Brian. 1996. Evolution of the Social Contract. Cambridge: Cambridge University Press.

Weibull, Jorgen W. (1995). Evolutionary Game Theory. Cambridge, Massachusetts and London: MIT Press.

Vanderschraaf, Peter. (2001) Learning and Coordination: Inductive Deliberation, Equilibrium and Convention. New York: Routledge.

Young, H. Peyton. (1998) Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton, New Jersey: Princeton University Press.

Young, H. Peyton.  2004.  Strategic Learning and Its Limits.  Oxford:  Oxford University Press.

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Common Knowledge

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.


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. San Francisco: Morgan Kaufman, pp. 365-379.

Bicchieri, Cristina. 1993. Rationality and Coordination. Cambridge: Cambridge University Press.

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.  Princeton: Princeton University Press.

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. Princeton: Princeton University Press.

Lewis, David. 1969. Convention: A Philosophical Study. Cambridge, Massachusetts: Harvard University Press.

Schiffer, Stephen. 1972. Meaning. Oxford: Oxford University Press.

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.

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Game Theory in Moral and Political Philosophy

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.

Hampton's (1986) and Kavka's (1986) books give brilliant and complementary analyses of Hobbesian social contract theory. Both Kavka and Hampton make extensive use of elementary concepts from game theory. Both are highly readable and require no prior knowledge of game theory. Hampton and Kavka are both lasting contributions to contemporary political philosophy, and are great examples of how game theory can be applied in social philosophy. Ironically, while Hume's informal game theoretic insights are far more sophisticated that those of Hobbes, no one has yet produced a work on Humean social philosophy incorporating game theory that approaches the quality of Hampton's and Kavka's books.

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.


Axelrod, Robert. 1984. The Evolution of Cooperation. New York: Basic Books, Inc.

Barry, Brian. 1989. Theories of Justice. Berkeley: University of California Press.

Bicchieri, Cristina. 1993. Rationality and Coordination. Cambridge: Cambridge University Press.

Bicchieri, Cristina.  2006.  The Grammar of Society.  Cambridge: Cambridge University Press.

Binmore, Ken. 1994. Game Theory and the Social Contract Volume I: Playing Fair. Cambridge, Massachusetts: MIT Press.

Binmore, Ken. 1998. Game Theory and the Social Contract Volume II: Just Playing. Cambridge, Massachusetts: MIT Press.

Binmore, Ken.  2005.  Natural Justice.  Oxford: Oxford University Press.

Gauthier, David. 1986. Morals By Agreement. Oxford: Clarendon Press.

Hampton, Jean. 1986. Hobbes and the Social Contract Tradition. Cambridge: Cambridge University Press.

Kavka, Gregory. 1986. Hobbesian Moral and Political Theory. Princeton: Princeton University Press.

Skyrms, Brian. 1996. Evolution of the Social Contract. Cambridge: Cambridge University Press.

Sugden, Robert. 1986 (2004). The Economics of Rights, Co-operation and Welfare. Oxford: Basil Blackwell, Inc.

Verbeek, Bruno and Morris, Christopher.  2004.  “Game Theory and Ethics,” in Stanford Encyclopedia of Philosophy.

Young, H. Peyton.  1994.  Equity in Theory and Practice.  Princeton: Princeton University Press.

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Computer Software

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 .  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.

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Classical Music

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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