Performance sampling models of duration dependence in employee turnover and firm exit predict that hazard rates will initially be low, gradually rise to a maximum, and then fall. As we note in this paper, however, several empirical duration distributions have bimodal hazard rates. This paper shows that such bimodal hazard rates can be derived from existing models of performance sampling by small changes in the assumptions. In particular, bimodal hazard rates emerge if the mean or the variance of performances changes over time, which would occur if employees or firms face more challenging tasks over time. Using data on turnover in law firms, we show that the hazard rate predicted by these models fit data better than existing models.

Behavior in competitive situations requires decision makers to evaluate their own as well as their competitors' positions. Using data from a realistic competitive risk-taking setting, Jeopardy's Tournament of Champions (TOC), we test whether players choose the strategic best response when making their betting decisions. Analyses show that the percentage of players choosing the strategic best response is very low, a surprising finding because the TOC is contested by the best and most experienced players of the game. We conjecture that performance aspiration and survival targets that guide risk-taking behavior in competitive situations may lead players to select inferior competitive strategies.

We examine incentive-compatible mechanisms for fair financing and efficient selection of a public budget (or public good). A mechanism selects the level of the public budget and imposes taxes on individuals. Individuals' preferences are quasilinear. Fairness is expressed as weak monotonicity (called scale monotonicity) of the tax imposed on an individual as a function of his benefit from an increased level of the public budget. Efficiency is expressed as selection of a Pareto-optimal level of the public budget. The budget deficit is the difference between the public budget and the total amount of taxes collected from the individuals. We show that any efficient scale-monotonic and incentive-compatible mechanism may generate a budget deficit. Moreover, it is impossible to collect taxes that always cover a fixed small fraction of the total cost.

An overview of the landmark contributions of Robert J. Aumann, winner of the 2005 Nobel Memorial Prize in Economic Sciences.

On the occasion of Robert J. Aumann's being awarded the 2005 Nobel Prize in Economics, this paper gives a personal view on some of Aumann's contributions, and primarily on his approach to foundational issues in game theory, economics, and science as a whole. It is based on numerous discussions and e-mail exchanges we had in the 1990's, dealing with various scientific and political matters, including our long debate on the ``Bible Code' controversy.

The Shapley value is an a priori evaluation of the prospects of a player in a multi-person game. Introduced by Lloyd S. Shapley in 1953, it has become a central solution concept in cooperative game theory. The Shapley value has been applied to economic, political, and other models.

In many tournaments investments are made over time and conducting a review only once at the end, or also at points midway through, is a strategic decision of the tournament designer. If the latter is chosen, then a rule according to which the results of the different reviews are aggregated into a ranking must also be determined. This paper takes a first step in the direction of answering how such rules should be optimally designed. A characterization of the optimal aggregation rule is provided for a two-agent two-stage tournament. In particular, we show that treating the two reviews symmetrically may result in an equilibrium effort level that is inferior to the one in which only a final review is conducted. However, treating the two reviews lexicographically by first looking at the final review, and then using the midterm review only as a tie-breaking rule, strictly dominates the option of conducting a final review only. The optimal mechanism falls somewhere in between these two extreme mechanisms. It is shown that the more effective the first-stage effort is in determining the final review's outcome, the smaller is the weight that should be assigned to the midterm review in determining the agents' ranking.

The vertical inflorescences of several plant species are terminated by colorful bracts, which attract insect pollinators. The bracts contrast in color with the leaves below them, and are oriented perpendicular to the flowers on the inflorescence. We conducted laboratory experiments to determine the effects of color contrast and perpendicular orientation on the feeding choices of bumblebees. We first trained bees to feeders with color-contrasting perpendicular displays, composed of a horizontal and a vertical display component. We subsequently recorded the bees' choices among feeders that displayed only one of these cues. The bees preferred perpendicular displays that resembled the training model in the color of the horizontal component. None of them chose a color-contrasting display that was not perpendicular. We then evaluated the effects of the horizontal vs. vertical components of perpendicular displays on the bees' choices. After training bees to color-contrasting perpendicular displays, we allowed them to choose between displays that had either the same horizontal or the same vertical component as the training model. Foragers mostly oriented to the horizontal displays to which they had been trained. Our results suggest that (a) bumblebees can learn to associate three-dimensional perpendicular color-contrasting displays with food rewards; (b) these displays are processed hierarchically, with orientation dominating color contrast; (c) The horizontal component of perpendicular displays dominates the vertical component. We discuss possible implications of our findings for the evolution of flower signals based on extra-floral bracts.

Nobel Lecture.

Who is Robert Aumann? Is he an economist or a mathematician? A rational scientist or a deeply religious man? A deep thinker or an easygoing person? These seemingly disparate qualities can all be found in Aumann; all are essential facets of his personality. A pure mathematician who is a renowned economist, he has been a central figure in developing game theory and establishing its key role in modern economics. He has shaped the field through his fundamental and pioneering work, work that is conceptually profound, and much of it mathematically deep. He has greatly influenced and inspired many people: his students, collaborators, colleagues, and anyone who has been excited by reading his papers or listening to his talks. Aumann promotes a unified view of rational behavior, in many different disciplines: chiefly economics, but also political science, biology, computer science, and more. To mention just a few of the areas of Aumann's groundbreaking work: perfect competition, repeated games, correlated equilibrium, interactive knowledge and rationality, and coalitions and cooperation. But Aumann is not just a theoretical scholar, closed in his ivory tower. He is interested in real-life phenomena and issues, to which he applies insights from his research. He is a devoutly religious man; and he is one of the founding fathers-and a central and most active member-of the multidisciplinary Center for the Study of Rationality at the Hebrew University in Jerusalem. Aumann enjoys skiing, mountain climbing, and cooking-no less than working out a complex economic question or proving a deep theorem. He is a family man, a very warm and gracious person-of an extremely subtle and sharp mind. This interview catches a few glimpses of Robert Aumann's fascinating world. It was held in Jerusalem on three consecutive days in September of 2004. I hope the reader will learn from it and enjoy it as much as we two did.

Let A be a finite set of m alternatives, let N be a finite set of n players and let RN be a profile of linear preference orderings on A of the players. Let uN be a profile of utility functions for RN. We define the NTU game VuN that corresponds to simple majority voting, and investigate its Aumann-Davis-Maschler and Mas-Colell bargaining sets. The first bargaining set is nonempty for m 3 and it may be empty for m ¥ 4. However, in a simple probabilistic model, for fixed m, the probability that the Aumann-Davis-Maschler bargaining set is nonempty tends to one if n tends to infinity. The Mas-Colell bargaining set is nonempty for m 5 and it may be empty for m ¥ 6. Furthermore, it may be empty even if we insist that n be odd, provided that m is sufficiently large. Nevertheless, we show that the Mas-Colell bargaining set of any simple majority voting game derived from the k-th replication of RN is nonempty, provided that k ¥ n + 2.

This paper undertakes a careful examination of the concept of conditional probability and its use. The ideas are then applied to resolve a conceptual puzzle related to Savage's "Sure-Thing Principle."

Consciousness is the last great frontier of science. Here we discuss what it is, how it differs fundamentally from other scientific phenomena, what adaptive function it serves, and the difficulties in trying to explain how it works. The emphasis is on the adaptive function.

We analyze an independent private values model where a number of objects are sold in sequential first- and second-price auctions. Bidders have unit demand and their valuation for an object is decreasing in the rank number of the auction in which it is sold. We derive efficient equilibria if prices are announced after each auction or if no information is given to bidders. We show that the sequence of prices constitutes a supermartingale. Even if we correct for the decrease in valuations for objects sold in later auctions we find that average prices are declining.

This paper investigates the effectiveness of shaming penalties. It establishes that there may be an inverse relation between the rate of shaming penalties and their deterrent effects - the more people are shamed the less deterring shaming penalties become. This conclusion is based on a search model in which the costs of searching for law-abiding partners increase with the rate of shaming, and may lead to lower expected sanction for offenders. The inverse relation between the rate of shaming penalties and their effectiveness is later used to show that increasing the probability of detection, increasing the magnitude of shaming penalties or reducing the number of wrongful acquittals does not necessarily increase the deterrent effects of shaming penalties (and may, in fact, decrease these effects).

A 4-person quota game is analyzed and discussed, in which players find it beneficial to pay others, in order to encourage favorable coalition structure.

This work follows "Evolutionary dynamics and backward induction" (Hart [2000]) in the study of dynamic models consisting of selection and mutation, when the mutation rate is low and the populations are large. Under the assumption that there is a single backward induction (or subgame perfect) equilibrium of a perfect information game, Hart [2000] proved that this point is the only stable state. In this work, we examine the case where there are multiple backward induction equilibria.

The existence of a value and optimal strategies is proved for the class of two-person repeated games where the state follows a Markov chain independently of players' actions and at the beginning of each stage only player one is informed about the state. The results apply to the case of standard signaling where players' stage actions are observable, as well as to the model with general signals provided that player one has a nonrevealing repeated game strategy. The proofs reduce the analysis of these repeated games to that of classical repeated games with incomplete information on one side.