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.

One way to express bounded rationality of a player in a game theoretic models is by specifying a set of feasible strategies for that player. In dynamic game models with finite automata and bounded recall strategies, for example, feasibility of strategies is determined via certain complexity measures: the number of states of automata and the length of recall. Typically in these models, a fixed finite bound on the complexity is imposed resulting in finite sets of feasible strategies. As a consequence, the number of distinct feasible strategies in any subgame is finite. Also, the number of distinct strategies induced in the first T stages is bounded by a constant that is independent of T. In this paper, we initiate an investigation into a notion of feasibility that reflects varying degree of bounded rationality over time. Such concept must entail properties of a strategy, or a set of strategies, that depend on time. Specifically, we associate to each subset ¨i of the full (theoretically possible) strategy set a function yi from the set of positive integers to itself. The value ˆi(t) represents the number of strategies in ¨i that are distinguishable in the first t stages. The set ¨i may contain infinitely many strategies, but it can differ from the fully rational case in the way yi grows reflecting a broad implication of bounded rationality that may be alleviated, or intensified, over time. We examine how the growth rate of yi affects equilibrium outcomes of repeated games. In particular, we derive an upper bound on the individually rational payoff of repeated games where player 1, with a feasible strategy set ¨1, plays against a fully rational player 2. We will show that the derived bound is tight in that a specific, and simple, set ¨1 exists that achieves the upper bound. As a special case, we study repeated games with non-stationary bounded recall strategies where the length of recall is allowed to vary in the course of the game. We will show that a player with bounded recall can guarantee the minimax payoff of the stage game even against a player with full recall so long as he can remember, at stage t, at least K log(t) stages back for some constant K >0. Thus, in order to guarantee the minimax payoff, it suffices to remember only a vanishing fraction of the past. A version of the folk theorem is provided for this class of games.

While traditional mechanism design typically assumes isomorphism between the agents' type- and action spaces, in many situations the agents face strict restrictions on their action space due to, e.g., technical, behavioral or regulatory reasons. We devise a general framework for the study of mechanism design in single-parameter environments with restricted action spaces. Our contribution is threefold. First, we characterize sufficient conditions under which the information-theoretically optimal social-choice rule can be implemented in dominant strategies, and prove that any multilinear social-choice rule is dominant-strategy implementable with no additional cost. Second, we identify necessary conditions for the optimality of action-bounded mechanisms, and fully characterize the optimal mechanisms and strategies in games with two players and two alternatives. Finally, we prove that for any multilinear social-choice rule, the optimal mechanism with k actions incurs an expected loss of O(1/k2) compared to the optimal mechanisms with unrestricted action spaces. Our results apply to various economic and computational settings, and we demonstrate their applicability to signaling games, public-good models and routing in networks.

The paper starts from the fact that Israel is described as a 'Jewish and Democratic' state. It opens with a rejection of some preliminary charges that Israel cannot be both Jewish and democratic or that maintaining its Jewish particularity is in principle unjustified. The main part of the paper analyzes various issues, such as the right to participate in elections, immigration, and education, to show what arrangements may be required by a wish to maintain the Jewishness of the state while, at the same time, respecting the human rights of all its residents.

The paper analyses the development of the constitutional process in Israel since 1950, and especially since the 1992 basic laws. It argues that this process should be viewed within a frameworks distinguishing between three stages of constitution-making: the initial enactment of a constitution, amendments of the constitution, and application and interpretation of the constitution. The distinction between stages has institutional implications. Constitution-making should be primarily done by constituent assemblies. Regular legislatures are a second choice. The process should seek broad consensus, and involve big compromises between segments of the public. Amendments should also be undertaken by legislatures with broad consensus, but they can be more local, and their ratification procedures may be less demanding. Application and interpretation should be done in an ongoing way by all branches of government. Courts are authoritative interpreters but they do not necessarily have the final word on the constitution. When we study the Israeli process we see that does not conform to this model at all. It reflects judicial involvement in all stages, including the initial making of the constitution. There is thus a 'legitimacy deficit' in the constitutional process, which may perpetuate the current instability in the constitutional situation.