Theory of Repeated Games | 繰り返しゲームの理論
Special Lectures in Economics | 経済学特論
Syllabus in English | 講義シラバス（日本語）
1st semester, Thursdays 2nd, Room 演1
Office Hours: Wednesdays 12:50 - 13:50
Last update | 最終更新日: July 17, 2015
Announcement | お知らせ
[2015/4/29] --- There will be NO lecture on June 4.
[2015/4/23] --- There will be NO lecture on April 30. The next lecture is May 7.
[2015/3/30] --- If you plan to attend this course (even if you are just sitting), please fill THIS FORM on-line! <= Important!
[2015/3/30] --- The first lecture will be on April 9 (Thu). Lectures will be given in English, while your questions/comments as well as presentations can be Japanese.
Course Objective and Content | 講義概要・目的
We study the theory of repeated games. First, lectures based on the textbook/survey articles are given. Then, we read essential papers on this subject.
The following is a tentative plan of the course. Half of the lectures are based on your presentations. The detailed topics are provided in the Course Schedule section.
Central Results
Applications
Related Issues
Variations on the Game
Grading Policy | 成績評価
Based on the presentation and final report.
Auditing students SHOULD also make presentation but need NOT submit the final report.
Presentation can be given in Japanese or English, but SLIDES (you are encouraged to use them) must be typed in ENGLISH. I plan to invite your presentation in the lectures that are colored blue on course schedule below.
Final report can be written in Japanese or English. Suggested length is 3 - 6 pages. Your report should be either
1) an original paper/idea on repeated games,
2) a referee report on a paper (which is not directly covered in class), or
3) a survey on a specific issue of repeated games. <= New!
Main Textbook | 講義テキスト
Main Textbook: [MS] Mailath and Samuelson, Repeated Games and Reputations: Long-run Relationships, 2006. Amazon
The relevant parts will be distributed in class, so you need NOT purchase this book.
Survey Articles | サーベイ論文
Kandori, M. (2002). Introduction to repeated games with private monitoring. Journal of Economic Theory, 102(1), 1-15.
Kandori, M. (2008). Repeated games. New Palgrave Dictionary of Economics, 2nd edition, Palgrave Macmillan.
Mertens, Sorin and Zamir, Repeated Games, 2015. Amazon
Pearce, D. G. (1992). Repeated games: Cooperation and rationality. In Advances in Economic Theory: Sixth World Congress (Vol. 1, pp. 132-74).
Samuelson, L. (2006). The economics of relationships. Econometric Society Monographs, 41, 136.
松島斉. (1994). 過去, 現在, 未来: 繰り返しゲームと経済学. 現代の経済理論』, 東京大学出版会, 57, 101.
松島斉. (2001). 繰り返しゲームの新展開: 私的モニタリングによる暗黙の協調.
Related Survey | 関連サーベイ
Fudenberg, D. (1992). Explaining cooperation and commitment in repeated games. In Laffont, J.-J.(Hrsg): Advances in economic theory—Sixth World Congress (Vol. 1, p. 89).
Harrington, J. E. (2008a). Detecting cartels. Handbook of antitrust economics, 213, 245.
MacLeod, W. B. (2007). Reputations, relationships, and contract enforcement. Journal of economic literature, 595-628.
Mertens, J. F. (2002). Stochastic games. Handbook of game theory with economic applications, 3, 1809-1832.
Sannikov, Y. (2012). Contracts: The theory of dynamic principal-agent relationships and the continuous-time approach. In 10th World Congress of the Econometric Society.
Sobel, J. (2002). Can we trust social capital?. Journal of economic literature, 139-154.
Vieille, N. (2002). Stochastic games: Recent results. Handbook of game theory with economic applications, 3, 1833-1850.
You are expected to read all double-starred (**) articles and encouraged to read starred(*) articles below.
Course Schedule - Detailed (tentative) | 講義日程詳細版（予定）
Lecture 1 Introduction Apr. 9
Stage Game, History, Discounting
Finite vs. Infinite Horizon
Subgame Perfect Equilibrium
Trigger Strategy
Folk Theorem
Tit-for-Tat
References
**Kandori (2008)
*Pearce (1992)
*MS: Ch1
CENTRAL RESULTS Lecture Slides
Lecture 2 Equilibrium Strategy Apr. 16
Unimprovability
One-shot Deviation
Optimal Penal Code
Simple Strategy
Stick and Carrot
References
**Pearce (1992)
*MS: Ch2.1-2.4, Ch2.6
**Abreu (1988)
Abreu (1986)
Rubinstein (1979)
Lecture 3 Cooperation in Equilibrium Apr. 23
Trigger Strategies
Folk Theorem
Minimax Value
Feasible Payoffs
Individual Rationality
Mutual Minimax
References
**MS: Ch3.1-3.5
*Pearce (1992)
**Fudenberg and Maskin (1986)
Abreu, Dutta and Smith (1994)
Benoit and Krishna (1985)
Wen (1994)
Lectures 4 Cancelled
Lectures 5 Imperfect Public Monitoring May 7
Bounded/Imperfect Recall
Public Monitoring
Private Monitoring
Dynamic Programing
Signal
Modified Trigger Strategies
References
*Pearce (1992)
*MS: Ch7.1, 7.2
*Green and Porter (1984)
Abreu, Pearce and Stacchetti (1986)
Fudenberg and Levine (1994)
Lectures 6 General Folk Theorem May 14
Public Randomization
Dynamic Programing
Self-Generation
Bang-Bang
Public Perfect Equilibrium
Folk Theorem
References
*Pearce (1992)
*Ch7.3, 7.5, 9.1-9.4
**Abreu, Pearce and Stacchetti (1990)
**Fudenberg, Levine and Maskin (1994)
*Matsushima (1989)
*Radner, Myerson and Maskin (1986)
Fudenberg and Levine (1991)
The Following Topics Will Be Covered through Students' Presentation
APPLICATIONS
Lecture 7 Tacit Collusion May 21
Bernheim and Whinston (1990) --- YOSHIDA, Shohei
Athey and Bagwell (2001) --- KINOSHITA, Takuya
References
MS: Ch5.3, Ch6.1, Ch11.2
Aoyagi (2003)
Athey and Bagwell (2008)
Bos and Harrington (2010)
Jullien and Rey (2007)
Motta and Polo (2003)
Rotemberg and Saloner (1986)
Skrzypacz and Hopenhayn (2004)
Lecture 8 Relational Contracts May 28
Levin (2003) --- TAKASAGO, Takao
Baker, Gibbons and Murphy (2002) --- ODA, Shohei
References
MS: Ch11.4
Chassang (2010)
MacLeod (2003)
MacLeod and Malcomson (1989)
Lecture 9 Will be Cancelled
RELATED ISSUES
Lecture 10 Asynchronous Choice June 11
Maskin and Tirole (1988a) --- SHIOZAWA, Kohei
Maskin and Tirole (2001) --- OOHATA, Ryota
References
MS: Ch5.5-5.7
Lagunoff and Matsui (1997)
Maskin and Tirole (1988b)
Maskin and Tirole (1987)
Lecture 11 Information and Bounded Rationality June 18
Abreu and Rubinstein (1988) --- QUENTIN, Carayon
Short lecture on related papers.
References
Abreu, Milgrom and Pearce (1991)
Aumann and Sorin (1989)
Banks and Sundaram (1990)
Miller (1996)
Neyman (1985)
Normann and Wallace (2012)
Lecture 12 Communication June 25
Athey and Bagwell (2001) --- KINOSHITA, Takuya (2nd)
Kandori and Matsushima (1998) --- Kh.Bolorchuluun
References
Compte (1998)
Aoyagi (2003)
Aoyagi (2007)
Athey, Bagwell and Sanchirico (2004)
Harrington and Skrzypacz (2011)
Obara (2009)
Ben-Porath and Kahneman (1996)
VARIATIONS ON THE GAMES
Lecture 13 Overlapping Generations July 2
Aoyagi (2003) --- <= from Communication
Kandori (1992a) --- SHIOZAWA, Kohei
References
Aoyagi (2007)
Bhaskar (1998)
Smith (1992)
Lagunoff and Matsui (2004)
Lecture 14 Random Matching July 9
Kandori (1992b) --- KINOSHITA, Takuya
Ghosh and Ray (1996) --- BLANCHARD, Tom
References
MS: Ch5.1, 5.2
Ellison, G. (1994)
Fujiwara-Greve and Okuno-Fujiwara (2009)
Greif, Milgrom and Weingast (1994)
Matsushima (1990)
Okuno-Fujiwara and Postlewaite (1995)
Takahashi (2010)
Lecture 15 Macroeconomics July 16
Alesina (1987) --- TAKASAGO, Takao
Rogoff (1987) --- ODA, Shohei
References
Athey, Atkeson and Kehoe (2001)
Barro and Gordon (1983)
Tabellini (1987)
OTHER POSSIBLE TOPICS
Renegotiation
*Farrell and Maskin (1989)
Benoit and Krishna (1993)
Revision Game
*Kamada and Kandori (2009)
Calcagno, Kamada, Lovo and Sugaya (2014)
Experiments
*Bó (2005) ---
Andreoni and Miller (1993)
Aoyagi and Fréchette (2009)
Bó and Fréchette (2011)
Cooper, DeJong, Forsythe and Ross (1996)
Duffy and Ochs (2009)
Private Monitoring
*Sekiguchi (1997)
Bhaskar and Obara (2002)
Ely and Välimäki (2002)
Kandori (2002)
Matsushima (2004)
Sugaya (2011)
Yamamoto (2007)
Stochastic Games
*Dutta (1995)
Dutta and Madhavan (1997)
Fudenberg and Yamamoto (2011)
Hörner, Sugaya, Takahashi and Vieille (2011)
Building Trust
Watson (1999)
Watson (2002)
Sobel (2006)
Reputation
*Fudenberg and Levine (1989)
*Kreps, Milgrom, Roberts and Wilson (1982)
Fudenberg and Levine (1992)
Ely and Välimäki (2003)
Hörner, J. (2002)
Continuous Time Model
*Sannikov (2007)
*Sannikov (2008)
Fudenberg and Levine (2007)
Papers on Repeated Games | 繰り返しゲームに関する論文
Abreu, D. (1986). Extremal equilibria of oligopolistic supergames. Journal of Economic Theory, 39(1), 191-225.
Abreu, D. (1988). On the theory of infinitely repeated games with discounting. Econometrica: Journal of the Econometric Society, 383-396.
Abreu, D., Dutta, P. K., & Smith, L. (1994). The folk theorem for repeated games: a NEU condition. Econometrica: Journal of the Econometric Society, 939-948.
Abreu, D., Milgrom, P., & Pearce, D. (1991). Information and timing in repeated partnerships. Econometrica: Journal of the Econometric Society, 1713-1733.
Abreu, D., Pearce, D., & Stacchetti, E. (1986). Optimal cartel equilibria with imperfect monitoring. Journal of Economic Theory, 39(1), 251-269.
Abreu, D., Pearce, D., & Stacchetti, E. (1990). Toward a theory of discounted repeated games with imperfect monitoring. Econometrica: Journal of the Econometric Society, 1041-1063.
Abreu, D., & Rubinstein, A. (1988). The structure of Nash equilibrium in repeated games with finite automata. Econometrica: Journal of the Econometric Society, 1259-1281.
Alesina, A. (1987). Macroeconomic policy in a two-party system as a repeated game. The Quarterly Journal of Economics, 651-678.
Andreoni, J., & Miller, J. H. (1993). Rational cooperation in the finitely repeated prisoner's dilemma: Experimental evidence. The economic journal, 570-585.
Aoyagi, M. (2003). Bid rotation and collusion in repeated auctions. Journal of Economic Theory, 112(1), 79-105.
Aoyagi, M. (2007). Efficient collusion in repeated auctions with communication. Journal of Economic Theory, 134(1), 61-92.
Aoyagi, M., & Fréchette, G. (2009). Collusion as public monitoring becomes noisy: Experimental evidence. Journal of Economic theory, 144(3), 1135-1165.
Athey, S., Atkeson, A., & Kehoe, P. J. (2001). On the optimality of transparent monetary policy. Federal Reserve Bank of Minneapolis, Research Department.
Athey, S., & Bagwell, K. (2001). Optimal collusion with private information. RAND Journal of Economics, 32(3), 428-465.
Athey, S., & Bagwell, K. (2008). Collusion with persistent cost shocks. Econometrica, 76(3), 493-540.
Athey, S., Bagwell, K., & Sanchirico, C. (2004). Collusion and price rigidity. The Review of Economic Studies, 71(2), 317-349.
Aumann, R. J., & Sorin, S. (1989). Cooperation and bounded recall. Games and Economic Behavior, 1(1), 5-39.
Baker, G., Gibbons, R., & Murphy, K. J. (2002). Relational Contracts and the Theory of the Firm. Quarterly journal of economics, 39-84.
Banks, J. S., & Sundaram, R. K. (1990). Repeated games, finite automata, and complexity. Games and Economic Behavior, 2(2), 97-117.
Barro, R. J., & Gordon, D. B. (1983). Rules, discretion and reputation in a model of monetary policy. Journal of monetary economics, 12(1), 101-121.
Benoit, J. P., & Krishna, V. (1985). Finitely Repeated Games. Econometrica: Journal of the Econometric Society, 905-922.
Benoit, J. P., & Krishna, V. (1993). Renegotiation in finitely repeated games. Econometrica: Journal of the Econometric Society, 303-323.
Ben-Porath, E., & Kahneman, M. (1996). Communication in repeated games with private monitoring. Journal of Economic Theory, 70(2), 281-297.
Bernheim, B. D., & Whinston, M. D. (1990). Multimarket contact and collusive behavior. The RAND Journal of Economics, 1-26.
Bhaskar, V. (1998). Informational constraints and the overlapping generations model: Folk and anti-folk theorems. The Review of Economic Studies, 65(1), 135-149.
Bhaskar, V., & Obara, I. (2002). Belief-based equilibria in the repeated prisoners' dilemma with private monitoring. Journal of Economic Theory, 102(1), 40-69.
Binmore, K. G., & Samuelson, L. (1992). Evolutionary stability in repeated games played by finite automata. Journal of economic theory, 57(2), 278-305.
Bó, P. D. (2005). Cooperation under the shadow of the future: experimental evidence from infinitely repeated games. American Economic Review, 1591-1604.
Bó, P. D., & Fréchette, G. R. (2011). The evolution of cooperation in infinitely repeated games: Experimental evidence. The American Economic Review, 101(1), 411-429.
Bos, I., & Harrington Jr, J. E. (2010). Endogenous cartel formation with heterogeneous firms. The RAND Journal of Economics, 41(1), 92-117.
Calcagno, R., Kamada, Y., Lovo, S., & Sugaya, T. (2014). Asynchronicity and coordination in common and opposing interest games. Theoretical Economics, 9(2), 409-434.
Chassang, S. (2010). Building routines: Learning, cooperation, and the dynamics of incomplete relational contracts. The American Economic Review,100(1), 448-465.
Compte, O. (1998). Communication in repeated games with imperfect private monitoring. Econometrica, 597-626.
Cooper, R., DeJong, D. V., Forsythe, R., & Ross, T. W. (1996). Cooperation without reputation: experimental evidence from prisoner's dilemma games. Games and Economic Behavior, 12(2), 187-218.
Duffy, J., & Ochs, J. (2009). Cooperative behavior and the frequency of social interaction. Games and Economic Behavior, 66(2), 785-812.
Dutta, P. K. (1995). A folk theorem for stochastic games. Journal of Economic Theory, 66(1), 1-32.
Dutta, P. K., & Madhavan, A. (1997). Competition and collusion in dealer markets. The Journal of Finance, 52(1), 245-276.
Ellison, G. (1994). Cooperation in the prisoner's dilemma with anonymous random matching. The Review of Economic Studies, 61(3), 567-588.
Ely, J. C., & Välimäki, J. (2002). A robust folk theorem for the prisoner's dilemma. Journal of Economic Theory, 102(1), 84-105.
Ely, J. C., & Välimäki, J. (2003). Bad reputation. The Quarterly Journal of Economics, 785-814.
Farrell, J., & Maskin, E. (1989). Renegotiation in repeated games. Games and economic behavior, 1(4), 327-360.
Fudenberg, D., & Levine, D. K. (1989). Reputation and equilibrium selection in games with a patient player. Econometrica: Journal of the Econometric Society, 759-778.
Fudenberg, D., & Levine, D. K. (1991). An approximate folk theorem with imperfect private information. Journal of Economic Theory, 54(1), 26-47.
Fudenberg, D., & Levine, D. K. (1992). Maintaining a reputation when strategies are imperfectly observed. The Review of Economic Studies, 59(3), 561-579.
Fudenberg, D., & Levine, D. K. (1994). Efficiency and Observability with Long-Run and Short-Run Players. Journal of Economic Theory, 62(1), 103-135.
Fudenberg, D., & Levine, D. K. (2007). Continuous time limits of repeated games with imperfect public monitoring. Review of Economic Dynamics, 10(2), 173-192.
Fudenberg, D., Levine, D., & Maskin, E. (1994). The folk theorem with imperfect public information. Econometrica: Journal of the Econometric Society, 997-1039.
Fudenberg, D., & Maskin, E. (1986). The folk theorem in repeated games with discounting or with incomplete information. Econometrica: Journal of the Econometric Society, 533-554.
Fudenberg, D., & Yamamoto, Y. (2011). The folk theorem for irreducible stochastic games with imperfect public monitoring. Journal of Economic Theory, 146(4), 1664-1683.
Fujiwara-Greve, T., & Okuno-Fujiwara, M. (2009). Voluntarily separable repeated prisoner's dilemma. The Review of Economic Studies, 76(3), 993-1021.
Ghosh, P., & Ray, D. (1996). Cooperation in community interaction without information flows. The Review of Economic Studies, 63(3), 491-519.
Green, E. J., & Porter, R. H. (1984). Noncooperative collusion under imperfect price information. Econometrica: Journal of the Econometric Society, 87-100.
Greif, A., Milgrom, P., & Weingast, B. R. (1994). Coordination, commitment, and enforcement: The case of the merchant guild. Journal of political economy, 745-776.
Harrington, J. E. (2008b). Optimal Corporate Leniency Programs. The Journal of Industrial Economics, 56(2), 215-246.
Harrington, J. E., & Skrzypacz, A. (2011). Private monitoring and communication in cartels: Explaining recent collusive practices. The American Economic Review, 101(6), 2425-2449.
Hörner, J. (2002). Reputation and competition. The American Economic Review, 92(3), 644-663.
Hörner, J., Sugaya, T., Takahashi, S., & Vieille, N. (2011). Recursive methods in discounted stochastic games: An algorithm for δ→ 1 and a folk theorem. Econometrica, 79(4), 1277-1318.
Jullien, B., & Rey, P. (2007). Resale price maintenance and collusion. The RAND Journal of Economics, 38(4), 983-1001.
Kalai, E., & Stanford, W. (1988). Finite rationality and interpersonal complexity in repeated games. Econometrica: Journal of the Econometric Society, 397-410.
Kamada, Y., & Kandori, M. (2009). Revision games. Harvard University, 2009.
Kandori, M. (1992a). Repeated games played by overlapping generations of players. The Review of Economic Studies, 59(1), 81-92.
Kandori, M. (1992b). Social norms and community enforcement. The Review of Economic Studies, 59(1), 63-80.
Kandori, M. (1992c). The use of information in repeated games with imperfect monitoring. The Review of Economic Studies, 59(3), 581-593.
Kandori, M., & Matsushima, H. (1998). Private observation, communication and collusion. Econometrica, 627-652.
Kandori, M., & Obara, I. (2006). Efficiency in repeated games revisited: The role of private strategies. Econometrica, 74(2), 499-519.
Kreps, D. M., Milgrom, P., Roberts, J., & Wilson, R. (1982). Rational cooperation in the finitely repeated prisoners' dilemma. Journal of Economic theory, 27(2), 245-252.
Lagunoff, R., & Matsui, A. (1997). Asynchronous choice in repeated coordination games. Econometrica: Journal of the Econometric Society, 1467-1477.
Lagunoff, R., & Matsui, A. (2004). Organizations and overlapping generations games: Memory, communication, and altruism. Review of Economic Design, 8(4), 383-411.
Levin, J. (2003). Relational incentive contracts. The American Economic Review, 93(3), 835-857.
Matsushima, H. (1989). Efficiency in repeated games with imperfect monitoring. Journal of Economic Theory, 48(2), 428-442.
MacLeod, B. W. (2003). Optimal contracting with subjective evaluation. The American Economic Review, 93(1), 216-240.
MacLeod, W. B., & Malcomson, J. M. (1989). Implicit contracts, incentive compatibility, and involuntary unemployment. Econometrica: Journal of the Econometric Society, 447-480.
Maskin, E., & Tirole, J. (1987). A theory of dynamic oligopoly, III: Cournot competition. European Economic Review, 31(4), 947-968.
Maskin, E., & Tirole, J. (1988a). A theory of dynamic oligopoly, I: Overview and quantity competition with large fixed costs. Econometrica: Journal of the Econometric Society, 549-569.
Maskin, E., & Tirole, J. (1988b). A theory of dynamic oligopoly, II: Price competition, kinked demand curves, and Edgeworth cycles. Econometrica: Journal of the Econometric Society, 571-599.
Maskin, E., & Tirole, J. (2001). Markov perfect equilibrium: I. Observable actions. Journal of Economic Theory, 100(2), 191-219.
Matsushima, H. (1989). Efficiency in repeated games with imperfect monitoring. Journal of Economic Theory, 48(2), 428-442.
Matsushima, H. (1990). Long-term partnership in a repeated prisoner's dilemma with random matching. Economics Letters, 34(3), 245-248.
Matsushima, H. (2004). Repeated games with private monitoring: Two players. Econometrica, 72(3), 823-852.
Miller, J. H. (1996). The coevolution of automata in the repeated prisoner's dilemma. Journal of Economic Behavior & Organization, 29(1), 87-112.
Motta, M., & Polo, M. (2003). Leniency programs and cartel prosecution. International journal of industrial organization, 21(3), 347-379.
Neyman, A. (1985). Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma. Economics letters, 19(3), 227-229.
Normann, H. T., & Wallace, B. (2012). The impact of the termination rule on cooperation in a prisoner’s dilemma experiment. International Journal of Game Theory, 41(3), 707-718.
Obara, I. (2009). Folk theorem with communication. Journal of Economic Theory, 144(1), 120-134.
Okuno-Fujiwara, M., & Postlewaite, A. (1995). Social norms and random matching games. Games and Economic behavior, 9(1), 79-109.
Radner, R., Myerson, R., & Maskin, E. (1986). An example of a repeated partnership game with discounting and with uniformly inefficient equilibria. The Review of Economic Studies, 53(1), 59-69.
Rogoff, K. (1987). Reputational constraints on monetary policy. In Carnegie-Rochester conference series on public policy (Vol. 26, pp. 141-181). North-Holland.
Rotemberg, J. J., & Saloner, G. (1986). A supergame-theoretic model of price wars during booms. The American Economic Review, 390-407.
Rubinstein, A. (1979). Equilibrium in supergames with the overtaking criterion. Journal of Economic Theory, 21(1), 1-9.
Salant, D. J. (1991). A repeated game with finitely lived overlapping generations of players. Games and Economic Behavior, 3(2), 244-259.
Sannikov, Y. (2007). Games with imperfectly observable actions in continuous time. Econometrica, 75(5), 1285-1329.
Sannikov, Y. (2008). A continuous-time version of the principal-agent problem. The Review of Economic Studies, 75(3), 957-984.
Sekiguchi, T. (1997). Efficiency in repeated prisoner's dilemma with private monitoring. Journal of Economic Theory, 76(2), 345-361.
Skrzypacz, A., & Hopenhayn, H. (2004). Tacit collusion in repeated auctions. Journal of Economic Theory, 114(1), 153-169.
Smith, L. (1992). Folk theorems in overlapping generations games. Games and Economic Behavior, 4(3), 426-449.
Sobel, J. (2006). For better or forever: formal versus informal enforcement. Journal of Labor Economics, 24(2), 271-297.
Sugaya, T. (2011). Folk theorem in repeated games with private monitoring. Economic Theory Center Working Paper, (011-2011).
Tabellini, G. (1987, May). Reputational constraints on monetary policy a comment. In Carnegie-Rochester conference series on public policy (Vol. 26, pp. 183-190). North-Holland.
Takahashi, S. (2010). Community enforcement when players observe partners' past play. Journal of Economic Theory, 145(1), 42-62.
Watson, J. (1999). Starting small and renegotiation. Journal of economic Theory, 85(1), 52-90.
Watson, J. (2002). Starting small and commitment. Games and Economic Behavior, 38(1), 176-199.
Wen, Q. (1994). The" folk theorem" for repeated games with complete information. Econometrica: Journal of the Econometric Society, 949-954.
Yamamoto, Y. (2007). Efficiency results in N player games with imperfect private monitoring. Journal of Economic Theory, 135(1), 382-413.
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