Game logic and its applications I

This paper provides a logic framework for investigations of game theoretical problems. We adopt an infinitary extension of classical predicate logic as the base logic of the framework. The reason for an infinitary extension is to express the common knowledge concept explicitly. Depending upon the choice of axioms on the knowledge operators, there is a hierarchy of logics. The limit case is an infinitary predicate extension of modal propositional logic KD4, and is of special interest in applications. In Part I, we develop the basic framework, and show some applications: an epistemic axiomatization of Nash equilibrium and formal undecidability on the playability of a game. To show the formal undecidability, we use a term existence theorem, which will be proved in Part II.The authors thank Hiroakira Ono for helpful discussions and encouragements from the early stage of this research project, and Philippe Mongin, Mitio Takano and a referee of this journal for comments on earlier versions of this paper. The first and second authors are partially supported, respectively, by Tokyo Center of Economic Research and Grant-in-Aids for Scientific Research 04640215, Ministry of Education, Science and Culture.Presented by H. Ono     

Game Logic and its Applications II

This paper provides a Genzten style formulation of the game logic framework GLm (0 m ), and proves the cut-elimination theorem for GLm. As its application, we prove the term existence theorem for GL used in Part I.     

John Nash: Three Phases in the Career of a Beautiful Mind

This article is the Helen Flanders Dunbar Lecture presented at Columbia-Presbyterian Hospital in New York City in 2002, the centennial year of her birth. It focuses on three periods in the evolution of Nash’s mental illness, the predelusional, delusional, and postdelusional periods, and provides a psychoanalytic interpretation of each. Donald Capps is Professor of Pastoral Psychology at Princeton Theological Seminary. His books include Men, Religion, and Melancholia (1997), Living Stories: Pastoral Counseling in Congregational Context (1998), Social Phobia: Alleviating Anxiety in an Age of Self-Promotion (1999), Jesus: A Psychological Biography (2000), Giving Counsel: A Minister’s Guidebook (2001), Men and Their Religion: Honor, Hope, and Humor (2002), and A Time to Laugh: The Religion of Humor (2005). He has served as President of the Society for the Scientific Study of Religion and has an honorary doctorate in theology from the University of Uppsala, Sweden. Correspondence to Donald Capps, joan.blyth@ptsem.edu     

社会两难行为理性特点的实验分析

本研究运用定量和定性的方法探讨了动态适应性学习条件下两难行为的理性特点及其影响因素。72名杭州本科生参加了本实验,结果发现:1、在有限次重复的两难博弈实验中,人们的两难行为不符合纳什博弈理论,进一步的定性分析指出,这种两难行为体现出“社会理性”的特征;2、在高奖罚的条件下,人们更容易做出接近“纳什均衡点”决策;3、群体规模和反馈方式对人们的对策行为影响不显著,但是,进一步的定性分析部分支持了群体规模的社会理性解释。     

The “Ages” of Ogden Nash

This essay looks at the way Ogden Nash, America's Genius of Light Verse, saw his life and times in the 30's, 40's, and 50's. It focuses on a group of poems in which the author mentions his age at the time of writing and explores their insights. They move from young adulthood to retirement and thus become Nash's equivalent of Shakespeare's Ages of Humanity. The author notes how his study of Nash has provided insights in living and illustrates this through poems of his own. In following this particular journey through the stages of life, we might well discover ways of mining the gold in our own experience.     

A one-person doxastic characterization of Nash strategies

Within a formal epistemic model for simultaneous-move games, we present the following conditions: (1) belief in the opponents’ rationality (BOR), stating that a player believes that every opponent chooses an optimal strategy, (2) self-referential beliefs (SRB), stating that a player believes that his opponents hold correct beliefs about his own beliefs, (3) projective beliefs (PB), stating that i believes that j’s belief about k’s choice is the same as i’s belief about k’s choice, and (4) conditionally independent beliefs (CIB), stating that a player believes that opponents’ types choose their strategies independently. We show that, if a player satisfies BOR, SRB and CIB, and believes that every opponent satisfies BOR, SRB, PB and CIB, then he will choose a Nash strategy (that is, a strategy that is optimal in some Nash equilibrium). We thus provide a sufficient collection of one-person conditions for Nash strategy choice. We also show that none of these seven conditions can be dropped.     

议价博弈决策的权利认知效应

在议价博弈中,决策者可能存在因对自己及其他参与者拥有权利的认识程度不同,从而影响其策略选择及决策行为的现象。为验证该“权利认知效应”的存在,用实验法考察了240名不同年龄被试在两种议价博弈任务——最后通牒和纳什议价中的决策表现,结果发现：（1）6到12岁儿童和大学生中均存在权利认知效应;（2）随着年龄增长,受权利认知影响的人数呈现逐渐减少的趋势;（3）在议价博弈中,儿童对分配权的认识程度高于对否决权的认知。     

The Evolution of John Nash's Male Melancholia: From Honor to Hope to Humor

This article is a response to the psychoanalytic study of mathematical genius John Nash by Donald Capps, and I apply Capps's own theory of male melancholia to John Nash. Capps interpreted Nash's life by dividing it into three phases: 1) predelusional; 2) delusional; and 3) postdelusional. I correlate Capps's three forms of male religiousness (i.e., honor, hope, and humor) with these three phases, respectively. The explanatory value of this interpretation is that it locates Capps's psychobiography of Nash within Capps's larger writings, thus providing an understanding of Nash as a deeply (if unconventionally) religious man.     

John Nash: The Sufferings of Those Who Loved Him

This article, as a sequel to three previous papers on John Nash by Capps (2003, 2004a, & 2004b), focuses on the other side of Nashs beautiful mind by examining the sufferings of those who loved him. Nashs mistress and wife, appear to have suffered the most by Nashs narcissistic and paranoid schizophrenic tendencies due to his noncommittal and disrespectful attitudes. Nashs exploitive and experiential ways of relating to his close male friends (special friendships) seemed to have caused much pain and agony in at least two of his special friends. In addition, Nashs two sons also suffered from his lack of affection and responsibility. Thus it is crucial for people in mental health professionals as well as in pastoral care and counseling not only to care for the mentally ill but also for those who are around him or her.     

John Nash's Predelusional Phase: A Case of Acute Identity Confusion

This article uses Erik H. Erikson's concept of acute identity confusion to provide a psychoanalytic explanation for why John Nash, the mathematical genius, experienced a mental breakdown at age thirty. Particular emphasis is given to the problem of intimacy, a central feature of identity confusion, especially as this manifested itself in bisexual confusion. Special note is taken of the traditional psychoanalytic emphasis on the regressive pull in cases of preschizophrenic illness, together with Erikson's own emphasis on commitment pressures and consequent psychosocial and psychosexual foreclosures.