The Pennsylvania State University The Graduate School Department of Economics ESSAYS ON NETWORKS AND HIERARCHIES A Dissertation in Economics by Myeonghwan Cho °c 2008 Myeonghwan Cho Submitted in Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy August 2008 The dissertation of Myeonghwan Cho was reviewed and approved¤ by the following: Kalyan Chatterjee Distinguished Professor of Economics and Management Science Dissertation Advisor, Chair of Committee Edward J. Green Professor of Economics James S. Jordan Professor of Economics Anthony M. Kwasnica Associate Professor of Business Economics Robert C. Marshall Professor of Economics Head of the Department of Economics ¤ Signatures are on ¯le in the Graduate School. Abstract This dissertation consists of four chapters. Chapter 1 studies a repeated prisoner's dilemma game in a network in which each agent interacts with his neighbors and cannot observe the actions of other agents who are not directly connected to him. If there is global information processing through a public ran- domization and global communication, it is not di±cult to construct a sequential equilibrium which supports cooperation and satis¯es a re¯nement, called stability, which requires that cooperation resumes after any history. Here, we allow agents to locally communicate with their neighbors and show that it is possible to construct such an equilibrium without global information processing. The role of local communication is to resolve the discrepancy of agents' expectations on their neighbor's future actions. Chapter 2 considers the same environment as does Chapter 1, except that agents are not allowed to communicate with their neighbors but instead can observe a public randomiza- tion. By introducing a public randomization, we also can construct a sequential equilibrium for su±ciently patient agents, which supports cooperation and in which cooperation even- tually resumes after any history. In addition, if there is a small possibility of mistakes, the equilibrium we construct can result in a more e±cient outcome than the trigger strategy equilibrium, even though they produce the same payo®s in the limit. Chapter 3 studies a situation in which agents can make a binding agreement both on the amount of local public goods and on the structure of networks through which they share the bene¯ts of public goods. An agent enjoys the bene¯t of public goods produced by others who are (directly or indirectly) connected to him. There is a cost to maintain a link as well as to produce a public good. Since agents can choose the amount of public goods, the value of a link is endogenously determined. In the analysis, we ¯rst consider two di®erent models of sequential bargaining games through which a contract on allocations is established. In the ¯rst model, we allow agents to propose a pure allocation and show that there is no symmetric stationary perfect equilibrium for su±ciently patient agents. In the iii second model, agents are allowed to propose a distribution on allocations. As a result, we ¯nd a symmetric stationary perfect equilibrium in which probabilistic choices are made on an equivalent class of allocations. Next, we characterize core allocations, which consist of a minimally connected network and an e®ort pro¯le in which at most one agent does not produce the maximum amount of public good. Chapter 4 studies the problem of organizational design in a setting where workers di®er in their abilities and each worker is required to be monitored to prevent shirking. The need to monitor workers generates a hierarchical structure in an organization, which can be represented as a rooted tree. The value of an organization is determined by the wage cost, the total number of productive tasks carried out, and the length of chain of supervision in the organizational hierarchy. In this setting, we construct and characterize an e±cient structure of hierarchy which maximizes the value of the organization. In addition, we show that a hierarchy with the optimal size has a form of balanced rooted tree. iv Table of Contents List of Figures vii List of Tables viii Acknowledgments ix 1 Cooperation in the Repeated Prisoner's Dilemma Game with Local In- teraction and Local Communication 1 1.1 Introduction .................................... 1 1.2 The Model .................................... 5 1.3 Strategy σ¤ .................................... 8 1.4 Belief System ¹ .................................. 13 1.5 Sequential Equilibrium .............................. 15 1.6 Discussion ..................................... 17 1.7 Conclusion .................................... 20 Appendix A: Proof of Lemma 1.3 ........................... 22 Appendix B: Actions in period s ¸ t ......................... 25 Appendix C: Proof of Proposition 1.1 ........................ 28 2 Public Randomization in the Repeated Prisoner's Dilemma Game with Local Interaction 34 2.1 Introduction .................................... 34 2.2 The Model .................................... 36 2.3 Sequential Equilibrium .............................. 40 2.4 Stability and Robustness of σp ......................... 45 2.5 Concluding Remarks ............................... 51 Appendix: Proof of Proposition 2.1 .......................... 52 v 3 Endogenous Formation of Networks for Local Public Goods 57 3.1 Introduction .................................... 57 3.2 Models ....................................... 60 3.3 Sequential Bargaining Games .......................... 62 3.3.1 Non-Existence of Symmetric Stationary Perfect Equilibrium with Pure Proposals ................................. 65 3.3.2 Existence of Symmetric Stationary Perfect Equilibrium with Mixed Proposals ................................. 68 3.4 Allocations in the Core .............................. 74 3.5 Concluding Remarks ............................... 80 Appendix A: Proofs in Section 3.3.1 ......................... 81 Appendix B: Proofs in Section 3.3.2 ......................... 83 Appendix C: Proofs in Section 3.4 .......................... 86 4 E±cient Structure of Organization with Heterogeneous Workers 90 4.1 Introduction .................................... 90 4.2 Model ....................................... 93 ¤ 4.3 E±cient Structure Ga .............................. 97 4.4 Optimal Size of Organization . 104 4.5 Characterization of Structure Rule . 107 4.6 Conclusion .................................... 109 Appendix: Proof of Proposition 4.6 . 110 References 113 vi List of Figures 1.1 σ¤-path conditioning on history h2 ....................... 12 1.2 histories ht and h^t ................................ 14 1.3σ ^-path conditioning on h3 ............................ 18 ¤ 3 ¤ 3 1.4 ®N (σ ; h ) and ®N (σ ; h~ ) ........................... 19 ¤ 2 1.5 ®N (σ ; h ) .................................... 20 1.6 Assumption of small l .............................. 21 1.7 Assumption of small g .............................. 21 3.1 (G; x) and (G0; x0) ................................ 76 0 0 3.2 (x1; x2) satisfying NBi = 0 ........................... 77 3.3 Examples of (G; x) for Proposition 3.6 ..................... 79 4.1 Counting up and down rank systems ...................... 96 ¤ 4.2 Example of Ga .................................. 97 0 4.3 Example of Ga and Ga for Lemma 4.1 ..................... 99 00 0 4.4 Example of Ga, Ga, and Ga for Lemma 4.2 . 100 4.5 Counting down rank system . 103 4.6 Example for the proof of Proposition 4.5 . 107 4.7 Structure rule' ¹ ................................. 109 4.8 Example for the proof of Lemma 4.6 . 112 vii List of Tables 1.1 Payo®s in prisoner's dilemma game with communication ........... 6 2.1 Payo® of prisoner's dilemma game ....................... 37 viii Acknowledgments I am deeply indebted to Professor Kalyan Chatterjee for his invaluable advice and encour- agement during the process of developing this research. Without his guidance and support, I would not have reached this point. I also would like to thank Professor Edward Green, Professor James Jordan, Professor Vijay Krishna, Professor Anthony Kwasnica, Profes- sor Marek Pycia, Professor Neil Wallace, and participants at the 18th Stony Brook Game Theory conference and PET07 conference for their helpful comments. In particular, I am grateful to Professor John Riew for his encouragement. The ¯nancial support of the Robert C. Daniels Graduate Fellowship, the Will E. Mason Memorial Endowment Award, and the David W. and Carolyn P. Grow Graduate Fellowship are gratefully acknowledged. Finally and personally, I would like to thank my parents for their continued love and support throughout my entire life and Sun Young for being with me through the last year at Penn State. ix Chapter 1 Cooperation in the Repeated Prisoner's Dilemma Game with Local Interaction and Local Communication 1.1 Introduction From the results called the folk theorem, it is well known that any outcome that Pareto dominates a Nash equilibrium (or, an individually rational outcome) in a stage game can be supported as an equilibrium in in¯nitely repeated games for su±ciently patient agents. The earliest works on this theorem, such as Friedman (1971) and Fudenberg and Maskin (1986), require that each agent perfectly monitors the actions chosen by all the other agents. After those works, many studies focus on situations in which monitoring is not perfect. For example, Kandori (1992b) and Ellison (1994) study repeated games with a random matching process, where agents are matched randomly in each period and each agent