This file contains the proof of Remark 1 of "Voluntary Partnerships and Cooperation: An Experimental Study" and the discussion regarding the VSRPD with the first round.

Voluntarily Separable Repeated Prisoner's Dilemma (Fujiwara-Greve and Okuno-Fujiwara, 2009) has many kinds of equilibria. Focusing on monomorphic and bimorphic equilibria, we show that a bimorphic equilibrium consisting of cooperators and defectors is most efficient, under a mild payoff condition. This is a striking contrast to ordinary repeated Prisoner's Dilemma, where the symmetric efficient payoff is achieved by the symmetric C-trigger equilibrium. Our result indicates that behavioral diversity can be beneficial for the society, when players are free to escape from personalized punishments.

Abstract This chapter discusses the relevant parts of the contributions included in this volume. A summary ofThe East Asian Miraclepublished by the World Bank in 1993 is presented. The potential role of government in resolving coordination failures; how government policy can complement decentralized coordination; the use of contingent rents versus the use of direct policies; a framework for analyzing the roles of the private sector and government in coordinating economic activity; the political-economy structures of East Asian economies; and the motivations of government and the bureaucracy are discussed.

Voluntarily Separable Repeated Prisoner's Dilemma (Fujiwara-Greve and Okuno-Fujiwara, 2009) has many kinds of equilibria. Focusing on monomorphic and bimorphic equilibria, we show that a bimorphic equilibrium consisting of cooperators and defectors is most efficient, under a mild payoff condition. This is a striking contrast to ordinary repeated Prisoner's Dilemma, where the symmetric efficient payoff is achieved by the symmetric C-trigger equilibrium. Our result indicates that behavioral diversity can be beneficial for the society, when players are free to escape from personalized punishments. Takako Fujiwara-Greve TCER and Keio University Dept. of Economics 2-15-45 Mita, Minatoku, Tokyo 108-8345 JAPAN takakofg@econ.keio.ac.jp Masahiro Okuno-Fujiwara TCER and Musashino University Dept. of Political Science and Economics 3-3-3 Ariake, Kotoku, Tokyo 135-8181 JAPAN okuno.fujiwara@gmail.com Nobue Suzuki TCER and Komazawa University Faculty of Economics 1-23-1 Komazawa, Setagayaku, Tokyo 154-8525 JAPAN nsuzuki@komazawa-u.ac.jp Efficiency may Improve when Defectors Exist∗ Takako Fujiwara-Greve† Dept. of Economics Keio University Masahiro Okuno-Fujiwara‡ Dept. of Political Science and Economics Musashino University Nobue Suzuki§ Faculty of Economics Komazawa University October 16, 2013 Abstract: Voluntarily Separable Repeated Prisoner’s Dilemma (Fujiwara-Greve and Okuno-Fujiwara, 2009) has many kinds of equilibria. Focusing on monomorphic and bimorphic equilibria, we show that a bimorphic equilibrium consisting of cooperators and defectors is most efficient, under a mild payoff condition. This is a striking contrast to ordinary repeated Prisoner’s Dilemma, where the symmetric efficient payoff is achieved by the symmetric C-trigger equilibrium. Our result indicates that behavioral diversity can be beneficial for the society, when players are free to escape from personalized punishments.

In the literature of voluntarily repeated Prisoner's Dilemma, the focus is on how long-term cooperation is established, when newly matched partners cannot know the past actions of each other. In this paper we investigate how non-cooperative and cooperative players co-exist. In many incomplete information versions of a similar model, inherently non-cooperative players are assumed to exist in the society, but their long-run fitness has not been analyzed. In reality and in experiments, we also observe that some people are cooperative, while others never cooperate. We show that a bimorphic equilibrium of the most cooperative strategy and the most myopic strategy exists for sufficiently high survival rate of players, and that it is evolutionarily stable under uncoordinated mutations. For lower survival rates, adding initial periods of defection makes similar bimorphic equilibria. Both types of equilibria confirm persistence of defectors. Length: 48 pages

We consider the voluntarily separable repeated Prisoner's Dilemma model in which players randomly meet and form pairs to repeatedly play Prisoner's Dilemma only by mutual agreement. While the literature has dealt with the case of no information flow across partnerships, we consider the case in which players can issue a "reference letter" to verify at least that the partnership entered the cooperation phase. We show that such reference letters can be voluntarily provided by the partners even at some cost, and that the sheer existence of a letter shortens the trust-building periods and thus improves social efficiency.

We extend the voluntarily separable repeated Prisoner's Dilemma (Fujiwara-Greve and Okuno-Fujiwara, 2009) to continuous actions. We show that there is a (constrained) ecient bimorphic equilibrium which is robust under evolutionary pressure. It consists of a cooperative strategy and a myopic defection strategy so that our model provides a foundation to incomplete information models as well.

We extend the voluntarily separable repeated Prisoner's Dilemma (Fujiwara-Greve and Okuno-Fujiwara, 2009) to continuous actions. We show that there is a (constrained) ecient bimorphic equilibrium which is robust under evolutionary pressure. It consists of a cooperative strategy and a myopic defection strategy so that our model provides a foundation to incomplete information models as well.

In Fujiwara-Greve and Okuno-Fujiwara (2009), an evolutionary stability concept was defined by allowing mutations of any strategy. However, in human societies, not all strategies are likely to be tried out when a player considers what happens in the future. In this paper we introduce the "shared belief" of potential continuation strategies, generated and passed on in a society, and mutations are restricted only among best responses against the shared belief. We show that a myopic strategy becomes a part of a bimorphic equilibrium under a shared belief and contributes to a higher payoff than ordinary neutrally stable distributions'.

Ordinary repeated games do not apply to real societies where one can cheat and escape from partners. We formulate a model of endogenous relationships that a player can unilaterally end and start with a randomly assigned new partner with no information flow. Focusing on two-person, two-action Prisoner's Dilemma, we show that the endogenous duration of partnerships generates a significantly different evolutionary stability structure from ordinary random matching games. Monomorphic equilibria require initial trust building, while a polymorphic equilibrium includes earlier cooperators than any strategy in monomorphic equilibria and is thus more efficient. This is due to the non-linearity of average payoffs.

We consider voluntarily separable repeated Prisoner's Dilemma in which a pair of players meet randomly and repeatedly play Prisoner's Dilemma only by mutual agreement. Fujiwara-Greve and Okuno-Fujiwara (2007) consider the case that once a partnership is dissolved there is no information flow to other partner- ships. We consider the case that players can issue a reference letter to the partner if they entered cooperation periods, but the content of a letter is not verifiable. We show that the sheer existence of a letter shortens the trust-building periods of new matches and thus improves eciency in equilibrium.

Hirofumi Uzawa is one of the giants of modern economic theory. Hiro is probably best known to the readers of Macroeconomic Dynamics (MD) for his seminal articles on two-sector economic growth. The two-sector technology is more general than the one-sector technology: it allows a production possibility frontier that is strictly concave to the origin as opposed to being necessarily flat. This generality allows richer and more complex dynamics. This makes it especially useful for the analysis of economic fluctuations. The two-sector model is perfect for dynamic international trade.Hiro is also well known to macroeconomists for his seminal contribution to endogenous growth. In his article in the 1965 IER, productivity permanently increases as the result of permanent accumulation of human capital. Uzawa was thus a first mover in the new growth theory. The symbol H (for Human Capital, or for Hiro?) is today everywhere in models of economic dynamics.On his own and through his many students and mentees, Hiro has been the major inspiration for the modern theory of optimal economic growth. He taught a generation of pure and applied economists how to apply Pontryagin's maximum principle in economic dynamics. It seems that Uzawa introduced—or at least pushed the use of—phase diagrams in economic dynamics. Where would we be without this essential tool?Most readers of MD are likely to think first of Uzawa's contributions to macro, but Hiro is equally well known for his superb works on mathematical economics, general equilibrium, and demand theory. Hiro's mathematics is elegant and often very deep. Like the quality mathematician that he is, he does not apply technique for technique's sake.Hiro has made fundamental contributions to nonlinear programming. For the convex (but not necessarily smooth) case, he employed Slater's condition to obtain Kuhn–Tucker multipliers that satisfy the saddlepoint property necessary for an optimum. For the smooth (but not necessarily convex) case, Arrow, Hurwicz, and Uzawa introduced the current version of the constraint qualification, which ensures that optimality implies the existence of Kuhn–Tucker multipliers satisfying the saddlepoint property.Hiro's paper “Walras's Existence Theorem and Brouwer's Fixed Point Theorem” in the Economic Studies Quarterly (1962) is a hidden gem on general equilibrium. This paper can be seen as foreshadowing Sonnenschein's result on excess demand functions. Hiro clarified old, important questions about recovering preference maps from demand functions. Hiro was probably the first to convincingly show—in the context of tatonnement adjustment—the important distinction between local stability and global stability in economic dynamics.We have given here only a glimpse into the very large body of beautiful, influential Uzawa papers. Hiro's splendid bibliography is given at the end of the interview. Some of the work that Hiro has pursued energetically has yet to be widely recognized. One thinks, for example, of the Penrose Effect, Hiro's modeling of the organizational costs incurred in adding capital or making other changes in the way a firm does business.Hiro has had many successful students and mentees. Your MD interviewers are lucky to have been among those whom Hiro has influenced profoundly. A very incomplete list of the others would also include Dave Cass, Steve Goldman, Harl Ryder, Hajime Oniki, Bob Lucas, George Akerlof, Joe Stiglitz, Miguel Sidrauski, Morris Teubal, Assaf Razin, Guillermo Calvo, Bill Ethier, and Lenny Mirman.Hiro is widely recognized and even revered in Japan. He was elected to the very selective Japan Academy in 1989 at a remarkably young age. He was named “A Person of Cultural Merit” in 1983 and elected to the Order of Culture in 1997. Hiro has received significant international recognition. He was President of the Econometric Society. He is a Fellow of the Econometric Society, Member of the American Academy of Arts and Sciences, Foreign Honorary Member of the American Economic Association, and Foreign Associate of the U.S. National Academy of Sciences.This interview took place nearly 10 years ago. We apologize to the readers and to Professor Uzawa for the delay in getting the transcript to the editor. The interview was held at the Research Center on Global Warming of the Development Bank of Japan, at which Hiro plays an important role. Four of us—Uzawa, the two interviewers, and Yumiko Baba, who was then a post-doc in economics at the University of Tokyo, there to operate the tape recorder—were collected at the Meiji Gakuin University in central Tokyo and whisked away in a large black automobile to Hiro's home court at the Bank. Hiro is an imposing figure: tall and erect with a very long, pointed white beard. His eyes are very active. He strokes his beard in a soothing manner. It is not difficult to be in awe of him. The interview took an even more formal tack because there were two in the room with the nickname “Hiro.” It was hence efficient to use last names at times.The interviewers had agreed to try to steer Uzawa toward a discussion of his well-known basic technical contributions and away from his less well-known and more political contributions. In the end, we failed to steer Hiro onto any course other than his own. This is mostly as it should be. In this interview, you will hear about some of the technical contributions for which Hiro is widely known. You will also hear about what motivated him to enter economics, his strong social concerns and strong political views, the turbulence of the war years and the postwar years, and his recent work and interests. A few of the paragraphs at the end of the interview were added to bring the record up to date. What comes through is a picture of Hirofumi Uzawa, a truly distinguished scholar and a person dedicated to human betterment.Hiro talked in his usual warm, friendly voice. He peppered the interview with his strong opinions about other major economists, often with lively anecdotes. Of course, Hiro's opinions are his own, not those of the interviewers or the editors. We hope that the readers will get as much out of this conversation with Hiro as we did.

Unlike the ordinary repeated games, in the real world, people can run away after cheating. In this paper we construct a social game, in which players can repeat Prisoners' Dilemma only if both players agree to continue the partnership. We investigate how a social sanction prevents moral hazard in such a voluntary relationship. We have three conclusions. First, it is possible to enforce voluntary long-term cooperation by trust-building. Second, the trust-building periods can be shortened under diverse strategy distributions. Third, if there is a reference letter system which conveys information that a partnership ended by an unavoidable cause, then the trust-building periods can be shortened as well.

In today's science-driven industries, such as the semiconductor industry, firms are increasingly engaged in across-firm research and development projects in the form of a research consortium or a strategic alliance. Those collaboration processes, however, have complex aspects due to the competing relationship of the firms in product markets and will not be successful unless the participating firms have enough incentives to reveal their private information and to exert sufficient efforts. The paper attempts to explore the conditions under which firms have enough incentives to reveal their information and/or to expend collaborative efforts. Three existing economic models are examined for this purpose. It is argued that those incentives depend upon the nature of competition in the product markets, information structure, and the way that each firm's private information affects this competition. The models examined in the paper suggest that some mechanism is necessary to evaluate private technical information of each firm and to convey it to the other firms without distortion. This conclusion coincides with the observed fact that a neutral third-party plays an indispensable role in a successful research consortium.