GETTING MORE OUT OF MATHEMATICS THAN WE PUT IN

Mathematical concepts have often proved to have more in them than in their original definition. Some of this "latent information," originally ignored by mathematicians and physicists alike, has proved to be the key to very deep and surprising properties of the physical universe. The views of Penrose, Plato, and Descartes will be discussed. Penrose's 1098 page book, The Road to Reality, will be used as a prop in a topological experiment.

BACKWARD INDUCTION AND COMMON STRONG BELIEF OF RATIONALITY

In 1995, Aumman showed that with appropriate definitions, common knowledge of rationality (CKR) in games of perfect information (PI) implies that the backward induction (BI) outcome is reached. His work was criticized because it assumes rationality also at unreached vertices of the game tree. Here we avoid the criticism by replacing "common knowledge" by "strong belief" where a player strongly believes a proposition p if he believes p at every v that is logically consistent with p. The relation to previous work of Battigalli and Siniscalchi is discussed.

2007 Nobel Laureates + Multi-Agent Initiatives with Type Dependent Externalities

  We model situations in which a principal provides incentives to a group of agents to participate in a project (such as a social event or a commercial activity). Agents' benefits from participation depend on the identity of other participating agents. We assume bilateral externalities and characterize the optimal incentive mechanism. Using a graph-theoretic approach we show that the optimal mechanism provides a ranking of incentives for the agents, which can be described as arising from a virtual popularity tournament among the agents.

Space and Conscious Awareness

Being consciously aware of our environment is something we take for granted.     We don't need to study philosophy to have the undeniable subjective feeling of being aware of objects and events occurring around us. However, investigating patients who suffered brain damage, and therefore lost awareness of aspects of their environment, makes one acutely aware of the fact that being aware is not so trivial.

FMRI AND GAME THEORY

Until recently, economists have always been content to treat the human brain as a "black box" and suggest mathematical equations which simplify what the brain is doing. This "rational choice" approach has been enormously successful. However, there is not yet a one theory that fully describes human behavior.Recent advances in neuroscience have made it possible to observe detailed processes in the brain better than ever before.

How Old is Our Planet and How Can WeTell?

Abstract: The concept of "Deep Time", and measuring extremely long time, are two of geology's major contributions to our culture. Whereas the recognition of relative ages of events and establishing stratigraphic sequences are part of the foundations of geology, the "absolute" measurement of ages of rocks with reasonable accuracy is an achievement of the last half-century. Such measurement is based on radioactive clocks, in which a parent nuclide decays with a known decay constant to a daughter nuclide. Both "decay clocks" and "accumulation clocks " will be discussed.

Monotonicity and Implementability

Consider a model with finite number of goods, and with buyers with private values and quasi-linear utility functions. A domain of valuation functions for a buyer is a monotonicity domain if every finite-valued monotone randomized allocation rule defined on it is implementable, in the sense that there exists a randomized truth-telling direct mechanism that implements this allocation rule. We prove that a domain of valuations of dimension at least 2 is a monotonicity domain if and only if its closure is convex.

Implementation and Mechanism Design with Evidence (joint work with Bart Lipman)

The standard theory of implementation and mechanism design is limitedin two ways: First it is assumed that the social choice function depends onlyon the pro.le of preferences of the agents over the set of social alternatives.Second, it is assumed that agents cannot support their statements with hardevidence. The general goal of this research is to extend the theory by study-ing a more general model which is not limited in these two ways.

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