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Topics in Foundations of Game Theory Topics in Foundations of Game Theory Module I. Introduction to Axiomatic Decision Theory A typical agent in economics maximizes some utility function. The underlying jus- tification is, as long as the agent has a stable preference and makes choices consistently according to it, then it is as if he maximizes this utility function. This “as if” approach plays the critical role in economics research. In this module we look at the axiomatic foundation of single-person decision theory. This line of research makes it precise what this “as if” approach means, what are the underlying behavioral assumptions behind a utility function, what are the implications of observed behavioral anomalies in light of the axioms, and so on. We’ll use Kreps’ book “notes on the theory of choice” for basic material. The course will combine lectures, students presentations and discussions. No prerequisite. The class meets every Wednesday evening 6-8:30pm, Allen 306. Module plan. 1. (Oct. 18) Lecture one: representation of preference relations; revealed preference approach Readings: Kreps Ch. 1-3 2. (Oct. 25) Lecture two: vNM model, the mixture space Readings: Kreps Ch. 4-5, (optional: ch 6) 3. (Nov. 1) Lecture three: Allais paradox, the state space approach to uncertainty and subjective probabilities; Student presentation: AA model. Readings: Kreps Ch. 7; Anscombe and Aumann (1963); 4. (Nov. 8) Lecture four: Savage Readings: Kreps Ch. 8-9 5. *(Nov. 15) Lecture five: “bounded rationality” models – temptation and self- control, preference for flexibility, ambiguity, maximin, interactive decision theory (games), (macro: robust control, Epstein and Zin) 6. *(Nov. 29) Lecture six: new trend – behavioral anomalies, neuroeconomics Readings: (all articles are available online, try google!) Kahneman and Tversky ed., (2000) choices, values, and frames 1 Rabin (1998), psychology and economics Gul and Pesendorfer (2005), the case for mindless economists Rubinstein (2006), Comments on Behavioral Economics * subject to changes. 2 Topics in Foundations of Game Theory Module II. Information, Knowledge, Belief and Rationality These are the key concepts in modeling players who can reason. In this module we study the formal modeling of these concepts, and use them to investigate issues such as the foundations of popular solution concepts in games, when do rational agents trade, etc. This is a research-oriented course meant to engage everyone in asking questions about things often taken for granted and thinking about new ideas. There is no textbook. We’ll read selected book chapters and papers, and since there are many, many interesting works in the frontier, will try to adjust to what interests the class most. The course will combine lectures, students presentations and discussions. Familiarity with basic game theory is a plus (it can be taken simultaneously with first-year game theory course). The class meets every Wednesday evening 4:25-6:55pm at Allen 226. Module plan: 1. (Jan 10) Lecture one: the standard state space models and their conceptual issues; Readings: Aumann (1999a, 1999b), Geanakoplos (1992), Optional readings: Bacharach (1985); (for unawareness) Dekel, Lipman and Rusti- chini (1998a), Li (2006a); (for syntactic approach) Fagin, Halpern, Moses and Vardi (1995). 2. (Jan 17) Lecture two: no trade theorems; Readings: Aumann (1976), Rubinstein and Wolinsky (1990); Optional readings: Milgrom and Stokey (1982), Geanakoplos (1989); Homework: Read Aumann (1987, 1998) and Gul (1998), prepare for discussion next week in class. 3. (Jan 24) Lecture three: the CPA. Readings: Brandenburger and Dekel (1993), Morris (1995), Lipman (2003); Optional readings: Mertens and Zamir (1985), Samet (1998a, 1998b), Heifetz (2003), Bergemann and Morris (2005), Ely and Peski (2006); 4. (Jan 31) Lecture four: approximate common knowledge; Readings: Rubinstein (1989), Monderer and Samet (1989), Morris (1999); 3 Optional reading: Neeman (1996a), Morris (2002), Morris, Postlewaite and Shin (1995a); (applications in game theory) Morris, Rob and Shin (1995b), Kajii and Morris (1997); (applications on no-trade type results) Neeman (1996b); 5. (Feb 7) Lecture five: global games; Readings: Morris and Shin (2003), Atkeson (2000); Optional reading: Morris and Shin (1998), Carlsson and van Damme (1993), Morris and Shin (2000); 6. (Feb 14, 21) Lecture six/seven: epistemic foundation for games Readings: Dekel and Gul (1997), Brandenburger (2006), Bernheim (1984), Batti- galli and Siniscalchi (1999); Optional readings: Aumann and Brandenburger (1995), Ben-Porath (1997), Bran- denburger (2006), Brandenburger and Dekel (1987), Brandenburger, Friedenburg and Keisler (2006), Myerson (1986b), Blume, Brandenburger and Dekel (1991), Brandenburger, Friedenburg and Keisler (2006b), Reny (1993), Pearce (1984). References Atkeson, Andrew, “Discussion of Morris and Shin’s “Rethinking Multiple Equilibria in Macroeconomic Modelling,” NBER Macroeconomics Annual, 2000, 15. Aumann, Robert J., “Agreeing to Disagree,” Annals of Statistics, 1976, 76 (4), 1236– 1239. , “Correlated Equilibrium as an Expression of Bayesian Rationality,” Econometrica, 1987, 55 (1), 1–18. , “Common Priors: A Reply to Gul,” Econometrica, 1998, 66 (4), 929–938. , “Interactive Epistemology I: Knowledge,” International Journal of Game Theory, 1999, 28, 263–300. , “Interactive Epistemology II: Probability,” International Journal of Game Theory, 1999, 28, 300–314. and Adam Brandenburger, “Epistemic Conditions for Nash Equilibrium,” Econometrica, 1995, 63 (5), 1161–1180. 4 Bacharach, Michael, “Some Extensions of a Claim of Aumann in an Axiomatic Model of Knowledge,” Journal of Economic Theory, 1985, 37, 167–90. Battigalli, Pierpaolo and Marciano Siniscalchi, “Hierarchies of Conditional Beliefs and Interactive Epistemology in Dynamic Games,” Journal of Economic Theory, 1999, 88, 188–230. Ben-Porath, Elchanan, “Rationality, Nash Equilibrium and Backwards Induction in Perfect-Information Games,” Review of Economic Studies, 1997, 64, 23–46. Bergemann, Dirk and Stephen Morris, “Robust Mechanism Design,” Econometrica, 2005, 73, 1521–1534. Bernheim, B. Douglas, “Rationalizable Strategic Behavior,” Econometrica, 1984, 52, 1007–1028. Blume, Lawrence, Adam Brandenburger, and Eddie Dekel, “Lexicographic Prob- abilities and Choice Under Uncertainty,” Econometricca, 1991, 59, 61–79. Brandenburger, Adam, “The Power of Paradox: Some Recent Developments in In- teractive Epistemology,” 2006. forthcoming, International Journal of Game Theory. , Amanda Friedenberg, and H. Jerome Keisler, “Admissibility in Games,” 2006. memeo. , , and , “Notes on Relationship between Strong Belief and Assumption,” 2006b. memeo. and Eddie Dekel, “Rationalizability and Correlated Equilibria,” Econometrica, 1987, 55 (6), 1391–1402. and , “Hierarchies of Beliefs and Common Knowledge,” Journal of Economic Theory, 1993, 59, 189–198. Carlsson, Hans and Eric van Damme, “Global Games and Equilibrium Selection,” Econometrica, 1993, 61 (5), 989–1018. Dekel, Eddie and Faruk Gul, “Rationality and Knowledge in Game Theory,” in David M. Kreps and Kenneth F. Wallis, eds., Advances in Economic Theory, Seventh World Congress, Cambridge: Cambridge University Press, 1997. , Barton L. Lipman, and Aldo Rustichini, “Standard State-Space Models Preclude Unawareness,” Econometrica, 1998a, 66 (1), 159–173. Ely, Jeffrey and Marcin Peski, “Hierarchies of Belief and Interim Rationalizability,” Theoretical Economics, 2006, 1, 19–65. 5 Fagin, Ronald, Joseph Y. Halpern, Yoram Moses, and Moshe Y. Vardi, Rea- soning About Knowledge, The MIT Press, 1995. Feinberg, Yossi, “Characterizing Common Priors in the Form of Posteriors,” Journal of Economic Theory, 2000, 91, 127–179. Geanakoplos, John, “Game Theory without Partitions, and Applications to Specula- tion and Consensus,” 1989. Cowles Foundation Discussion Paper No. 914. , “Common Knowledge,” Journal of Economic Perspectives, 1992, 6 (4), 53–82. Gul, Faruk, “A Comment on Aumann’s Bayesian View,” Econometrica, 1998, 98 (4), 923–928. Halpern, Joseph Y., “Characterizing the Common Prior Assumption,” Journal of Economic Theory, 2002, 106, 316–355. Heifetz, Aviad, “The Positive Foundation of the Common Prior Assumption,” 2003. Forthcoming, Games and Economic Behavior. Kajii, Atsushi and Stephen Morris, “The Robustness of Equilibria to Incomplete Information,” Econometrica, 1997, 65 (6), 1283–1309. Li, Jing, “Information Structures with Unawareness,” 2006a. Working paper, University of Pennsylvania. Lipman, Barton L., “Finite Order Implications of Common Priors,” Econometrica, 2003, 71, 1255–1267. , “Finite Order Implications of Common Priors in Infinite Models,” 2005. Working paper. Mertens, Jean-Francois and Shmuel Zamir, “Formulation of Bayesian Analysis for Games with Incomplete Information,” International Journal of Game Theory, 1985, 14 (1), 1–29. Milgrom, Paul and Nancy Stokey, “Information, Trade and Common Knowledge,” Journal of Economic Theory, 1982, 26, 17–27. Monderer, Dov and Dov Samet, “Approximating Common Knowledge with Common Beliefs,” Games and Economic Behavior, 1989, 1, 170–190. Morris, Stephen, “The Common Prior Assumption in Economic Theory,” Economics and Philosophy, 1995, 11, 227–253. 6 , “Approximate Common Knowledge Revisited,” International Journal of Game Theory, 1999, 28, 433–445. , “Coordination, Communication, and Common Knowledge: A Retrospective on the Electronic-Mail Game,” Oxford Review of Economic Policy, 2002, 18 (4), 385–408. and Hyun Song Shin, “Unique Equilibrium in a Model of Self-Fulfilling Attacks,” American Economic Review, 1998,
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