This paper discusses the difference between Harsanyi and Bayesian equilibria for games of incomplete information played onuncountable belief spaces. A conjecture belonging to ergodic theory is presented. If the conjecture were valid then there would exist a game played on an uncountable belief space with a common prior for which there are Bayesian equilibria but no Harsanyi equilibrium.

In this paper, we show that test makers and test takers have a strong and systematic tendency for hiding correct answers – or, respectively, for seeking them – in middle positions. In single, isolated questions, both prefer middle positions over extreme ones in a ratio of up to 3 or 4 to 1. Because test makers routinely, deliberately and excessively balance the answer key of operational tests, middle bias almost, though not quite, disappears in those keys. Examinees taking real tests also produce answer sequences that are more balanced than their single question tendencies, but to a lesser extent than the correct key. In a typical 4-choice test, about 55% of erroneous answers (which are the only answers whose position is determined by the test taker, not the test maker) are in the two central positions. We show that this bias is large enough to have real psychometric consequences, as questions with middle correct answers are easier and – what's more important – less discriminating than questions with extreme correct answers, a fact some of whose implications we explore.

In interactive contexts such as games and economies, it is important to take account not only of what the players believe about substantive matters (such as payoffs), but also of what they believe about the beliefs of other players. Two different but equivalent ways of dealing with this matter, the semantic and the syntactic, are set forth. Canonical and universal semantic systems are then defined and constructed, and the concepts of common knowledge and common priors formulated and characterized. The last two sections discuss relations with Bayesian games of incomplete information and their applications, and with interactive epistemology - the theory of multi-agent knowledge and belief as formulated in mathematical logic.

Intergroup conflicts are characterized by conflicts of interests within the competing groups as well. The intragroup conflict stems from a basic fact: while all group members are better off if they all cooperate in competing against the outgroup, each individual group member is better off defecting. The Intergroup Prisoner's Dilemma (IPD) game is proposed as a theoretical framework for combining the intragroup and intergroup levels of conflict. This framework is used to examine major issues concerning individual and group behavior in intergroup conflict. These include: the effect of real intergroup conflict on intragroup cooperation; the motivational basis of cooperation; the distinction between non-cooperative groups, unitary groups, and individuals; and alternative routes to conflict resolution.

The purpose of this paper is to examine the extent to which the concepts of individual and collective choice unsed in economic theory desribe "predictable" or "learnable" behavior. Given a set X of N alternatives, a choice function c is a mapping which assigns to nonempty subsets S of X an element c(S) of S. A rational choice function is one for which there is a linear ordering on the alternatives such that c(S) is the maximal element of S according to that ordering. Using the basic concept of PAC-learnability from statistical learning theory we define a class of choice functions on a ground set of N elements as learnable if it is possible to predict, with small amount of error, the chosen element from a set A after viewing a "few examples." Here, "few" means a polynomial number in N. Learnability is quite a strict condition on a class of choice functions. The main point we discuss in this regard are: The class of rational choice function can be learned quickly and efficiently. Various natural classes of choice functions, which represent indidivual choices and strategic choices of several interacting agents, are learnable. The class of rational choice functions has superior learnability properties in comparison to other classes. We make the conjecture that classes of choice functions that represent a genuine aggregation of individual choices in a large society are never learnable. We also ask to what extent learnability can replace or reinforce the rationality hypothesis in some economic situations.

Carroll's apparently impeccable solution to one of his probability problems is shown to answer another problem that is based on reasonable assumptions. His original assumptions, however, are self-contradictory, hence entailing paradoxical results.

With respect to the S5 multi-agent epistemic logic, we define a cell to be a minimal subset of knowledge structures known in common semantically by all the agents. A cell has finite fanout if at every knowledge strcuture every agent considers only a finite number of other knowledge structures to be possible. A set of formulas in common knowledge is finitely generated if the common knowledge of some finite subset implies the common knowledge of the whole set. For every finitely generated set of formulas held in common knowledge at some knowledge structure either this set determines uniquely a finite cell or there are uncountable many cells of finite fanout (and also uncountably many cells of uncountable size) at which exactly this set of formulas is known in common. The situation is very different, however, for sets of formulas held in common knolwedge that are not finitely generated - if there are uncountably many corresponding cells then either none of these cells or all of them could have finite fanout.

We study the incentives of privately informed traders who have access to two forms of trade: direct negotiations with a small number of buyers and sellers (or decentralized trade), and centralized markets with a relatively large number of buyers and sellers. We show that "weak" trader types (that is, buyers with a high willingness to pay and sellers with low costs) will prefer to trade through centralized markets. This leads to a complete unraveling of direct negotiations, so that ultimately, all "serious" buyers and sellers opt for trading through the centralized market. Once this happens, no trader can pro''¯tably trade through direct negotiations.

The concept of an effectivity function is adopted as a formal model of a constitution. A game form models the actions available and permissible to individuals in a society. As a representation of the constitution such a game form should endow each group in society with the same power as it has under the constitution. Another desirable property is Nash consistency of the game form: Whatever the individual preferences, the resulting game should be minimally stable in the sense of possessing a Nash equilibrium. A first main result of the paper is a characterization of all effectivity functions that have a Nash consistent representation for the case without special structure on the sent of alternatives (social states). Next, a similar result is derived for the case where the set of alternatives is a compact metric space and the effectivity function is topological. As a sepcial case, veto functions are considered. Further results concern Pareto optimality of Nash equilibrium outcomes.

With the S5 multi-agent epistemic logic we consider the canonical maps from Krpke structures to knowledge structures. We define a cell to be a minimal subset of knowledge structures known in common semantically by the agents. A cell has finite fanout if at every point every agent considers only a finite number of other points to be possible. We define a cell to be surjective if every Kripke structure that maps to it does so surjectively. All cells with finite fanout are surjective, but the converse does not hold. To construct a counter-example we need topological insights concerning the relationship between the logic and its semantic models. The difference between syntactic and semantic common knowledge is central to this construction.

A situation in which the regularity in nature can be utilized while competition is to be avoided is modeled by the Parasite game. In this game regular behavior could enhance guessing nature but strategic randomization is required to avoid being outguessed. In an experiment, 60 pairs of participants (partner design) played many rounds of the Parasite game. The treatements differed in nature's probabilities and whether or not these probabilities were announced in advance or oculd only be experienced over time. Before playing, the working memory (WM) of participants was measured. Data analyses test the correspondence of participants' behavior to game-theoretic benchmarks and the effect of participants' WM on their behavior.

In this paper, we analyze the investment patterns of a large number of clients of a major Israeli brokerage house during 1994. We compare the behavior of clients making independent investment decisions to that of investors whose accounts were managed by brokerage professionals. Our main objective is to investigate whether the disposition effect (i.e., the tendency to sell winners quicker than losers), demonstrated in the US only for individual investors, also holds for professional investors. This analysis is important, as accepted financial theory predicts that prices are determined mainly by decisions made by professionals. We show that both professional and independent investors exhibit the disposition effect, although the effect is stronger for independent investors. The second objective of our study is the comparison of trade frequency, volume and profitability between independent and professionally managed accounts. We believe that these comparisons not only provide insights of their own, but also help to put the differences in the disposition effect in a wider perspective. We demonstrate that professionally managed accounts were more diversified and that round trips were both less correlated with the market and slightly more profitable than those of independent accounts.

The positive core is a nonempty extension of the core of transferable utility games. If the core is nonempty, then it coincides with the core. It shares many properties with the core. Six well-known axioms which are employed in some axiomatizations of the core, the prenucleolus, or the positive prekernel, and one new intuitive axiom, characterize the positive core on any infinite universe of players. This new axiom requires that the solution of a game, whenever it is nonempty, contains an element which is invariant under any symmetry of the game.

Large flowers often contain larger nectar rewards, and receive more pollinator visits, than small flowers. We studied possible behavioral mechanisms underlying the formation of flower size preferences in bumblebees, using a two-phase laboratory experiment. Experimentally naive Bombus terrestris (L.) foraged on artificial flowers that bore either a big (3.8cm diameter) or a small (2.7cm diameter) display of a uniform color. Only flowers of one display size contained nectar rewards. We changed the display color and the locations of big and small flowers in the second experiment phase. We recorded the bees' choices in both phases. Almost one half of the bees (41) made their first visit to a small flower. The bees learned to associate display size with food reward, and chose rewarding flowers with >85 accuracy by the end of each experimental phase. Some learning occurred within the bees' first three flower visits. Learning of the size-reward association was equally good for big and small displays in the first experimental phase, but better for small displays in the second phase. Formation of size-reward associations followed a similar course in both phases. This suggests that the bees did not apply their experience from the first learning phase to the new situation of the second phase. Rather, they treated each phase of the experiment as an independent learning task. Our results suggest that associative learning is involved in the formation of preferences for large displays by bees. Moreover, bees that had learned to prefer large displays in one foraging situation may not transfer this preference to a novel situation that is sufficiently different. We propose that this feature of the bees' behavior can select for honest advertising in flowers.

When people make moral or legal judgments in isolation, they produce a pattern of outcomes that they would themselves reject, if only they could see that pattern as a whole. A major reason is that human thinking is category-bound. When people see a case in isolation, they spontaneously compare it to other cases that are mainly drawn from the same category of harms. When people are required to compare cases that involve different kinds of harms, judgments that appear sensible when the problems are considered separately often appear incoherent and arbitrary in the broader context. Another major source of incoherence is what we call the translation problem: The translation of moral judgments into the relevant metrics of dollars and years is not grounded in either principle or intuition, and produces large differences among people.. The incoherence produced by category-bound thinking is illustrated by an experimental study of punitive damages and contingent valuation. We also show how category-bound thinking and the translation problem combine to produce anomalies in administrative penalties. The underlying phenomena have large implications for many topics in law, including jury behavior, the valuation of public goods, punitive damages, criminal sentencing, and civil fines. We consider institutional reforms that might overcome the problem of predictably incoherent judgments. Connections are also drawn to several issues in legal theory, including valuation of life, incommensurability, and the aspiration to global coherence in adjudication.

Let X_i be non-negative, independent random variables with finite expectation, and X*_n=maxX_1,...,X_n. The value EX*_n is what can be obtained by a "prophet". A "mortal" on the other hand, may use k>=1 stopping rules t_1,...,t_k, yielding a return of E[max_i=1,...,k X_t_i]. For n>=k the optimal return is V^n_k(X_1,...,X_n)=supE[max_i=1,...,k X_t_i] where the supremum is over all stopping rules t_1,...t_k such that P(t_i

When modelling large economies by nonatomic measure spaces of agents, one defines "coalitions" as measurable - not arbitrary - sets of agents. Here we suggest a rationale for this restriction: "Real" economies have finitely many agents. In them, coalitions are associated with various measures, like total endowment, which play a vital role in the analysis. So in the model, too, one should be able to associate similar measures with coalitions; this means that they must be "measurable." Thus, though in the finite case a coalition is simply an arbitrary set of players, the appropriate generalization to the infinite case is not an arbitrary but a measurable set.

The paper presents a notion of rationalizing choice functions that violate the Independence of Irrelevant Alternatives  axiom. A collection of linear orderings is said to provide a rationalization by multiple rationales for a choice function if the choice from any choice set can be rationalized by one of the orderings. We characterize a tight upper bound on the minimal number of orderings that is required to rationalize arbitrary choice functions, and calculate the minimal number for several specific choice procedures.

The present chapter brings together parts of the theory of polynomial equalities and inequalities used in the theory of stochastic games. The theory can be considered as a theory of polynomial equalities and inequalities over the field of real numbers or the field of real algebraic numbers or more generally over an arbitrary real closed field.

By means of an example it is shown that the prenucleolus is not the only minimal solution that satisfies nonemptiness, Pareto optimality, covariance, the equal treatment property and the reduced game property, even if universe of players is infinite. This example also disproves a conjecture of Gurvich et al. Moreover, we prove that the prenucleolus is axiomatized by nonemptiness, covariance, the equal treatment property, and the reconfirmation property, provided the universe of players is infinite.