University of Wisconsin – Madison Department of Economics

Summary: The first essay extends the belief-based approach for the repeated prisoners’ dilemma with asymmetric private monitoring to approximate full efficiency and to obtain certain folk theorems. The second essay introduces a general probabilistic endowment and inheritance rule for the house allocation problem. The third essay studies the effect of heterogeneous discounting on Pareto frontiers and feasible payoff sets for general two-player finitely repeated games. References: Larry Samuelson (Major Professor) – [email protected] Johanna Hertel – [email protected] William H. Sandholm –[email protected] Papers: “Belief-Based Strategies in the Repeated Prisoners’ Dilemma with Asymmetric Private Monitoring,” (Job Market Paper) “The Pareto Frontier of a Finitely Repeated Game with Unequal Discounting”, Economics Letters, Forthcoming. “The Role of Heterogeneous Discounting in General Two-Player Finitely Repeated Games,” UW -Madison, December, 2004. “The Probabilistic Endowment and Inheritance Rule for the House Allocation Problem,” UW Madison, April, 2005. “Mechanism Design with Positive Externalities: An Example,” (with Tanapong Potipiti) UW Madison, January, 2006. Teaching Experience: Teaching Assistant (Principle of Microeconomics – Econ 101, Intermediate Microeconomics – Econ 301, Mathematical Economics – Econ 703). Fields of Interest: Microeconomic Theory, Industrial Organization, Econometric Theory. Three Essays in Game Theory Dissertation Abstract Bo Chen Department of Economics University of Wisconsin Madison Chapter 1. Belief-Based Strategies in the Repeated Prisoners’Dilemma with Asymmetric Private Monitoring This paper extends the "belief-based" approach to the repeated prisoners’dilemma with asymmetric private monitoring technologies. We …rst found that previous type of construction in Sekiguchi (1997) and Bhaskar and Obara (2002) may not be su¢ cient to accommodate all asymmetric private monitoring scenarios in that the previous belief-based strategy pro…les may fail to be Nash equilibria when the private monitoring structure is asymmetric enough. We then modify the previous belief-based strategies by letting the player with smaller observation errors always randomize between "C" and "D" along the cooperative path of the play. It is shown that full e¢ ciency can be approximated using a modi…ed belief-based strategy pro…le, provided that observation errors are small and a public randomization device is available for the players. We then apply the techniques to obtain certain forms of folk theorems for the repeated prisoners’dilemma with asymmetric private monitoring structures. Finally, we construct an example to show that the modi…ed "belief-based" strategies can be potentially generalized to more general two-player repeated games with almost perfect private monitoring structures. Chapter 2. The Probabilistic Endowment and Inheritance Rule for the House Allocation Problem This paper is concerned with random assignments for the house allocation problem introduced by Shapley and Scarf (1974) when monetary compensation is not allowed. Motivated by TA assignment processes in an academic department and the hierarchical exchange rule in Pápai (2000), we introduce a general "probabilistic endowment and inheritance (PEI)" rule where the primary set of endowments (houses) is randomly assigned to a subset of agents and the property rights of the endowments are also transferred randomly as the sequential trade process proceeds. When the initial set of endowments is …nite, we …nd that the PEI rule is non-bossy, strategy-proof and ex post e¢ cient. We also show that the PEI rule is a general stochastic mechanism for the house allocation problem (all the existing deterministic and random mechanisms in the literature are special cases of the PEI rule). In particular, we establish an equivalence result between an anonymous PEI rule and the random serial dictatorship mechanism in Abdulkadiro1⁄4 glu and Sönmez (1998). Chapter 3. The Role of Heterogeneous Discounting in General Two-Player Finitely Repeated Games The …rst part of this paper characterizes the Pareto frontier of the average payo¤ set for a general two-player …nitely repeated game with heterogeneous discounting. We …nd that the Pareto frontier shifts outward as the time horizon of the repeated game increases. We further extends the result to show that for any two-player …nitely repeated game with a nondegenerate stage game, the feasible payo¤ set of the T th-repeated game is a strict subset of that of the (T + 1)-repeated game. In the second part of the paper, we present an example to show that heterogeneous discounting may also enable players to obtain lower equilibrium payo¤s than the e¤ective minmax values (Wen (1994)), or there exists some nontrivial relationship between the dimensionality condition and unequal discounting for general in…nitely repeated games. The main idea is that we can use unequal discounting to design some nonstationary punishments that have di¤erent e¤ects on di¤erent players. Lower equilibrium payo¤s can thus be sustained as players have heterogeneous discounting.