Abstract: We provide a model for reasoning about knowledge anti probabil-ity together. We a.llow explicit mention of probabilities in formulas, so that our language has formulas tha.t essentia.lly say "a.ccording to agent i, formula. (p holds with probability a.t least o~. " The language is powerfid enough to allow reason-ing a~bout higher-order probabilities, as well as allowing explicit comparisons of the probabilities an agent places on distinct events. We present a general framework for interpreting such formulas, a.nd consider various properties that might hold of the in-terrelationship between agents ' subjective probability spaces at different states. We provide a. complete a.xiomatiza.tion for rea.soning about knowledge a.nd probability, prove a. small model property, and obtain decision procedures. We then consider the effects of adding common knowledge and a. probabilistic va.ria.nt of common knowledge to the language.

Abstract. We study information elicitation mechanisms in which a principal agent attempts to elicit the private information of other agents using a carefully selected payment scheme based on proper scoring rules. Scoring rules, like many other mechanisms set in a probabilistic environment, assume that all participating agents share some common belief about the underlying probability of events. In real-life situations however, the underlying distributions are not known precisely, and small differences in beliefs of agents about these distributions may alter their behavior under the prescribed mechanism. We propose designing elicitation mechanisms in a manner that will be robust to small changes in belief. We show how to algorithmically design such mechanisms in polynomial time using tools of stochastic programming and convex programming, and discuss implementation issues for multiagent scenarios. 1

An (A) expected (strong) bubble is said to exist if it is mutual knowledge that the price of the asset is higher than the expected (possible) dividend. By requiring common knowledge instead of mutual knowledge, the new concept of common expected bubble (common strong bubble) is developed. In a simple …nite horizon model with asymmetric information and short sale constraints, which follows Allen, Morris and Postlewaite (1993), it is showed that two results hold true for any …nite number of agents: First, common strong bubbles never exist in any rational expectations equilibrium; Second, it is possible to have a bubble, which is both a strong bubble and a common expected bubble, in a rational expectations equilibrium, even with common knowledge of trades. Furthermore, the …rst result crucially depends on the implicit assumption of perfect memory, hence an example of common strong bubbles can be constructed in case that agents are forgetful. Based on these results, this paper, as well as Conlon (2004), provides a partial answer to what properties rational bubbles can have and cannot have in a rational expectations equilibrium.

A broad nontechnical coverage of many of the developments in game theory since the 1950s is given together with some comments on important open problems and where some of the developments may take place. The nearly 90 references given serve only as a minimal guide to the many thousands of books and articles that have been written. The purpose here is to present a broad brush picture of the many areas of study and application that have come into being. The use of deep techniques flourishes best when it stays in touch with application. There is a vital symbiotic relationship between good theory and practice. The breakneck speed of development of game theory calls for an appreciation of both the many realities of conflict, coordination and cooperation and the abstract investigation of all of them.