An invited contribution to a symposium on Information and Knowledge in Economics,  to appear in Econ Journal Watch. Topics discussed include the distinction between information and knowledge; awareness; logical omniscience; the cost of calculation; semantic and syntactic models of knowledge, and the equivalence between them; and common knowledge of the model. Finally, some aspects of the symposium contributions of Ken Binmore, Jim Friedman, and Eric Rasmusen are discussed.

Effectivity functions for finitely many players and alternatives are considered. It is shown that every monotonic and superadditive effectivity function can be augmented with equal chance lotteries to a finite lottery model - i.e., an effectivity function that preserves the original effectivity in terms of supports of lotteries - which has a Nash consistent representation. In other words, there exists a finite game form which represents the lottery model and which has a Nash equilibrium for any profile of utility functions, where lotteries are evaluated by their expected utility. No additional condition on the original effectivity function is needed.

These two papers (one in English and one in Hebrew) describe the unique educational experience of the Neve Shalom school, which is a fully integrated Jewish-Arab school within a system where Jews and Arabs regularly study in separate schools and languages. They are similar but have different emphases since the audiences are expected to be different in terms of background knowledge and familiarity with facts and situations. The paper studies the way the school handles issues of culture, language, religion and national identity. It then analyzes the school and makes suggestions concerning both the improvement of the effectiveness of the school itself and what can be learned from its experience to the educational system in Israel as a whole.

In this paper we study the social preferences obtained from monotone neutral social welfare functions for random individual preferences. It turns out that there are two extreme types of behavior. On one side, there are social welfare functions, such as the majority rule, that lead to stochastic stability of the outcome in terms of perturbations of individual preferences. We identify and study a class of social welfare functions that demonstrate an extremely different type of behavior which is a completely chaotic: they lead to a uniform probability distribution on all possible social preference relations and, for every''>0, if a small fraction'' of individuals change their preferences (randomly) the correlation between the resulting social preferences and the original ones tends to zero as the number of individuals in the society increases. This class includes natural multi-level majority rules.

Recent advancement in genetics testing for late-onset diseases raises fundamental decision dilemmas. The first study surveyed people's willingness to undergo predictive testing to find out about their own predisposition for certain incurable, late-onset diseases. The second study investigated the respondents' willingness to be tested as a function of the base rate of the disease, test diagnosticity, and the availability of treatment for the disease. In addition we surveyed (in the first study) people's willingness to disclose to others personal information about their genetic predisposition. The findings show that people often prefer not to know, as if they are choosing "protective ignorance". Respondents' verbal justifications of their choices were also analyzed. Respondents offered emotional, cognitive-instrumental, and strategic reasons for their preferences. The findings are compared with other issues in behavioral decision theory, including attitudes towards uncertainty and desire for control. The implications of the findings for policies and legislation on genetic tests are also considered.

We study the computational power and limitations of iterative combinatorial auctions. Most existing iterative combinatorial auctions are based on repeatedly suggesting prices for bundles of items, and querying the bidders for their "demand under these prices. We prove several results regarding such auctions that use a polynomial number of demand queries: (1) that such auctions can simulate several other natural types of queries; (2) that such auctions can solve linear programming relaxations of winner determination problems; (3) that they can approximate the optimal allocation as well as generally possible using polynomial communication or computation, while weaker types of queries can not do so. We also initiate the study of how can the prices of bundles be represented when they are not linear, and show that the "default representation has severe limitations.

We embark on a systematic analysis of the power and limitations of iterative ascending-price combinatorial auctions. We prove a large number of results showing the boundaries of what can be achieved by different types of ascending auctions: item prices vs. bundle prices, anonymous prices vs. personalized prices, deterministic vs. non-deterministic, ascending vs. descending, preference elicitation vs. full elicitation, adaptive vs. non-adaptive, and single trajectory vs. multi trajectory. Two of our main results show that neither ascending item-price auctions nor ascending anonymous bundle-price auctions can determine the optimal allocation among general valuations. This justifies the use of personalized bundle prices in iterative combinatorial auctions like the FCC spectrum auctions.

Seeking advice is a basic practice in making real life decisions. Until recently, however, little attention has been given to it in either empirical studies or theories of decision making. The studies reported here investigate the influence of advice on judgment and the consequences of advice use for judgment accuracy. Respondents were asked to provide final judgments on the basis of their initial opinions and advice presented to them. The respondents' weighting policies were inferred. Analysis of the these policies show that (a) the respondents tended to place a higher weight on their own opinion than on the advisor's opinion (the self/other effect); (b) more knowledgeable individuals discounted the advice more; (c) the weight of advice decreased as its distance from the initial opinion increased; and (d) the use of advice improved accuracy significantly, though not optimally. A theoretical framework is introduced which draws in part on insights from the study of attitude change to explain the influence of advice. Finally the usefulness of advice for improving judgment accuracy is considered.

Casual observation suggests that athletics records tend to cluster over time. After prolonged periods without new records, a record breaking performance spurs other athletes to increase effort and thereby repeatedly set new standards. Subsequently, record breaking subsides and the pattern repeats itself. The clustering hypothesis is tested for the mile run, the marathon, the world hour record and long jump. For all four disciplines, the hypothesis of non-clustering is rejected at the 4 level or below. A theoretical rationale for this phenomenon is provided through a model of social learning under limited awareness. The agents are assumed to be unaware of the true limits to performance and to take the current record as the upper bound. The observation of a record breaking achievement spurs the agents to try harder and thus temporarily increase the probability of new records. Subsequently, record breaking trails off and the process is repeated.

In many situations, the decision maker observes items in sequence and needs to determine whether or not to retain a particular item immediately after it is observed. Any decision rule creates a set of items that are selected. We consider situations where the available information is the rank of a present observation relative to its predecessors. Certain "natural" selection rules are investigated. Theoretical and Monte Carlo results are presented pertaining to the evolution of the number of items selected, measures of their quality and the time it would take to amass a group of a given size. A comparison between rules is made, and guidelines to the choice of good procedures are offered.

We consider stability properties of equilibria in stochastic evolutionary dynamics. In particular, we study the stability of mixed equilibria in strategic form games. In these games, when the populations are small, all strategies may be stable. We prove that when the populations are large, the unique stable outcome of best-reply dynamics in 2 x 2 games with a unique Nash equilibrium that is completely mixed is the mixed equilibrium. The proof of this result is based on estimating transition times in Markov chains.

'We will try to explore, primarily from the complexity-theoretic point of view, limitations of error-correction and fault-tolerant quantum computation. We consider stochastic models of quantum computation on n qubits subject to noise operators that are obtained as products of tiny noise operators acting on a small number of qubits. We conjecture that for realistic random noise operators of this kind there will be substantial dependencies between the noise on individual qubits and, in addition, the dependence structure of the noise acting on individual qubits will necessarily depend (systematically) on the dependence structure of the qubits themselves. The main hypothesis of this paper is that these properties of noise are sufficient to reduce quantum computation to probabilistic classical computation. Some potentially relevant mathematical issues and problems will be described. Our line of thought appears to be related to that of physicists Alicki, Horodecki, Horodecki and Horodecki [AHHH].'

A correlated equilibrium of a strategic form n-person game is called tight if all the incentive constraints are satisfied as equalities. The game is called tight if all of its correlated equilibria are tight. This work shows that the set of tight games has positive measure.

We design two computationally-efficient incentive-compatible mechanisms for combinatorial auctions with general bidder preferences. Both mechanisms are randomized, and are incentive-compatible in the universal sense. This is in contrast to recent previous work that only addresses the weaker notion of incentive compatibility in expectation. The first mechanism obtains an O('ˆ\v sm)-approximation of the optimal social welfare for arbitrary bidder valuations – this is the best approximation possible in polynomial time. The second one obtains an O(log2 m)-approximation for a subclass of bidder valuations that includes all submodular bidders. This improves over the best previously obtained incentive-compatible mechanism for this class which only provides an O('ˆ\v sm)-approximation.

This paper reports an attempt to teach game theory in order to increase students' motivation to learn mathematics. A course in game theory was created in order to introduce students to new mathematical content presented in a different way.

Modern game theory was born in 1928, when John von Neumann published his Minimax Theorem. This theorem ascribes to all two-person zero-sum games a value-what rational players may expect-and optimal strategies-how they should play to achieve that expectation. Seventy-seven years later, strategic game theory has not gotten beyond that initial point, insofar as the basic questions of value and optimal strategies are concerned. Equilibrium theories do not tell players how to play and what to expect; even when there is a unique Nash equilibrium, it it is not at all clear that the players "should" play this equilibrium, nor that they should expect its payoff. Here, we return to square one: abandon all ideas of equilibrium and simply ask, how should rational players play, and what should they expect. We provide answers to both questions, for all n-person games in strategic form.

{A version of a secretary problem is considered: Let Xj

Although the fact that the eye is moving constantly has been known for a long time, the role of fixational eye movements (FEM) is still in dispute. Whatever their role, it is structurally clear that, since the eye is a ball, the size of these movements diminishes for locations closer to the poles. Here we propose a new perspective on the role of FEM from which we derive a prediction for a three-way interaction of a stimulus' orientation, location, and spatial frequency. Measuring time-to-disappearance for gratings located in the periphery we find that, as predicted, gratings located to the left and right of fixation fade faster when horizontal than when vertical in low spatial frequencies and faster when vertical than when horizontal in high spatial frequencies. The opposite is true for gratings located above and below fixation.

Consider an election between two candidates in which the voters' choices are random and independent and the probability of a voter choosing the first candidate is p > 1/2. Condorcet's Jury Theorem which he derived from the weak law of large numbers asserts that if the number of voters tends to infinity then the probability that the first candidate will be elected tends to one. The notion of influence of a voter or its voting power is relevant for extensions of the weak law of large numbers for voting rules which are more general than simple majority. In this paper we point out two different ways to extend the classical notions of voting power and influences to arbitrary probability distributions. The extension relevant to us is the "effect" of a voter, which is a weighted version of the correlation between the voter's vote and the election's outcomes. We prove an extension of the weak law of large numbers to weighted majority games when all individual effects are small and show that this result does not apply to any voting rule which is not based on weighted majority.