E8112: Topics in Mathematical Economics

Logistical Information

Classes: Tues. and Thurs., 11:15 am. 12:30 pm., Spring 2010 (1st half), 4-168.

Instructor: David Rahman (dmr@umn.edu, econ.umn.edu/dmr).

Office hours: Wednesday, 1:00 2:30 pm. or by appointment, 4-147 Hanson Hall.

Prerequisites: First-year graduate economics, basic real analysis.

Course Description

The purpose of this course is to provide students with tools and topics that will help

them find ideas for their dissertations at the frontier of research in economic theory.

As regards tools, this course will use duality as a methodological thread to connect the

material presented. As for topics, the course will begin with well-known applications

of duality in game theory, followed by increasingly recent work in contract theory and

mechanism design, to end with open questions.

Assessment

Final exam: Take-home distributed on the last day of class.

Homework: Two problem sets distributed in class on weeks 3 and 5.

Grading: A students overall grade will be a weighted average of homework and final.

Reading

The course will rely mainly on articles and lecture notes for its content. The following

textbooks are recommended reading.

Gale, D., The Theory of Linear Economic Models, University of Chicago Press, 1960.

Myerson, R., Game Theory: Analysis of Conflict, Harvard University Press, 1991.

Rockafellar, R. T., Convex Analysis, Princeton University Press, 1970.

Topkis, D., Supermodularity and Complementarity, Princeton University Press, 1998.

Vohra, R. V., Advanced Mathematical Economics, Routledge, 2005.

The following textbooks are tangentially related and may be useful in the future.

Korte, B., and J. Vygen, Combinatorial Optimization, Springer, 2000.

Murota, K., Discrete Convex Analysis, SIAM, 2003.

1

mailto:dmr@umn.eduhttp://www.econ.umn.edu/~dmr/

Tentative Outline

Below is an outline of the course content with articles assigned for reading.

1. Linear Programming. The Theorem of the Alternative, weak and strong

duality, the Minimax Theorem for two-person zero-sum games.

Read : Vohra (2005, Charpter 4); Rockafellar (1970, 5, 16 and 24).

2. Correlated Equilibrium. Definition, examples, a revelation principle, proof

of existence.

Read : Aumann (1974, 1987); Myerson (1997); Hart and Schmeidler (1989).

Supplement : Nau and McCardle (1990); Nau et al. (2003); Hart (2005).

3. Mechanism Design I. Rochets Theorem, surplus-extracting mechanisms.

Dominant-strategy versus Bayesian incentives.

Read : Rochet (1987); Cremer and McLean (1988); dAspremont et al.

(2004); Neeman (2004).

Supplement : Cremer and McLean (1985); Heifetz and Neeman (2006);

Kosenok and Severinov (2008).

4. Contract Theory I. Literature review, basic problems, budget balance, nearly-

efficient partnerships.

Read : Alchian and Demsetz (1972); Holmstrom (1982); Legros and Matthews

(1993).

Supplement : Legros and Matsushima (1991); Strausz (1997).

5. Contract Theory II. Detection and enforcement, attribution, exact versus

virtual enforcement.

Read : Rahman (2008); Rahman and Obara (2010).

Supplement : dAspremont and Gerard-Varet (1998).

6. Mechanism Design II. Standard one-dimensional problems.

Read : Mirrlees (1971); Myerson (1981).

Supplement : Mirrlees (1999).

2

7. Mechanism Design III. VCG mechanisms, Roberts Theorem, Revenue Equiv-

alence.

Read : Mas-Colell et al. (1995, Chapter 23); Lavi et al. (2007); Heydenreich

et al. (2009).

8. Mechanism Design IV. Robust mechanisms, monotonicity, efficiency.

Read : Bergemann and Morris (2005); Bikhchandani et al. (2006); Mon-

derer (2007); Jehiel and Moldovanu (2001); Jehiel et al. (2006, 2007).

References

Alchian, A. and H. Demsetz (1972): Production, Information Costs, and Eco-

nomic Organization, American Economic Review, 62, 777795.

Aumann, R. (1974): Subjectivity and Correlation in Randomized Strategies,

Journal of Mathematical Economics, 1, 6796.

(1987): Correlated Equilibrium as an Expression of Bayesian Rationality,

Econometrica, 55, 118.

Bergemann, D. and S. Morris (2005): Robust Mechanism Design, Economet-

rica, 73.

Bikhchandani, S., S. Chatterji, A. Sen, R. Lavi, A. Mualem, and

N. Nisan (2006): Weak Monotonicity Characterizes Deterministic Dominant-

Strategy Implementation, Econometrica, 74, 11091132.

Cremer, J. and R. McLean (1985): Optimal Selling Strategies under Uncer-

tainty for a Discriminating Monopolist when Demands are Interdependent, Econo-

metrica, 53, 345361.

(1988): Full Extraction of the Surplus in Bayesian and Dominant Strategy

Auctions, Econometrica, 56, 12471257.

dAspremont, C., J. Cremer, and L.-A. Gerard-Varet (2004): Balanced

Bayesian Mechanisms, Journal of Economic Theory, 115, 385396.

dAspremont, C. and L.-A. Gerard-Varet (1998): Linear Inequality Methods

to Enforce Partnerships under Uncertainty: An Overview, Games and Economic

Behavior, 25, 311336.

3

Hart, S. (2005): Adaptive Heuristics, Econometrica, 73, 14011430.

Hart, S. and D. Schmeidler (1989): Existence of Correlated Equilibria, Math-

ematics of Operations Research, 14, 1825.

Heifetz, A. and Z. Neeman (2006): On the Generic (Im)Possibility of Full

Surplus Extraction in Mechanism Design, Econometrica, 74, 213233.

Heydenreich, B., R. Muller, M. Uetz, and R. Vohra (2009): Characteri-

zation of Revenue Equivalence, Econometrica, 77, 307316.

Holmstrom, B. (1982): Moral Hazard in Teams, Bell Journal of Economics, 13,

324340.

Jehiel, P. and B. Moldovanu (2001): Efficient Design with Interdependent

Valuations, Econometrica, 69, 12371259.

Jehiel, P., M. M. ter Vehn, and B. Moldovanu (2007): Ex Post Implemen-

tation and Preference Aggregation via Potentials, Economic Theory, 37, 469490.

Jehiel, P., M. M. ter Vehn, B. Moldovanu, and W. R. Zame (2006): The

Limits of Ex Post Implementation, Econometrica, 74, 585610.

Kosenok, G. and S. Severinov (2008): Individually Rational, Balanced-Budget

Bayesian Mechanisms and the Allocation of Surplus, Journal of Economic Theory,

140, 126261.

Lavi, R., A. Mualem, and N. Nisan (2007): Two Simplified Proofs of Roberts

Theorem, Mimeo.

Legros, P. and H. Matsushima (1991): Efficiency in Partnerships, Journal of

Economic Theory, 55, 296322.

Legros, P. and S. Matthews (1993): Efficient and Nearly Efficient Partner-

ships, Review of Economic Studies, 60, 599611.

Mas-Colell, A., M. D. Whinston, and J. R. Green (1995): Microeconomic

Theory, Oxford University Press.

Mirrlees, J. A. (1971): An Exploration in the Theory of Optimum Income Tax-

ation, Review of Economic Studies, 38, 175208.

(1999): The Theory of Moral Hazard and Unobservable Behavior: Part I,

Review of Economic Studies, 66, 321.

4

Monderer, D. (2007): Monotonicity and Implementability, Mimeo.

Myerson, R. (1981): Optimal Auction Design, Mathematics of Operations Re-

search, 6, 5873.

(1997): Dual Reduction and Elementary Games, Games and Economic

Behavior, 21, 183202.

Nau, R. F., S. Gomez-Canovas, and P. Hansen (2003): On the Geometry of

Nash Equilibria and Correlated Equilibria, International Journal of Game Theory,

32, 443453.

Nau, R. F. and K. F. McCardle (1990): Coherent Behavior in Noncooperative

Games, Journal of Economic Theory, 50, 424444.

Neeman, Z. (2004): The Relevance of Private Information in Mechanism Design,

Journal of Economic Theory, 117, 5577.

Rahman, D. (2008): But Who Will Monitor the Monitor? Mimeo.

Rahman, D. and I. Obara (2010): Mediated Partnerships, Econometrica, to

appear.

Rochet, J. C. (1987): A Necessary and Sufficient Condition for Rationalizability

in a Quasi-Linear Context, Journal of Mathematical Economics, 16, 191200.

Rockafellar, R. T. (1970): Convex Analysis, Princeton, New Jersey: Princeton

University Press.

Strausz, R. (1997): Delegation of Monitoring in a Principal-Agent Relationship,

Review of Economic Studies, 64, 337357.

Vohra, R. V. (2005): Advanced Mathematical Economics, Routledge.

5

References