Another impossibility result for normal form games

It is shown that the axioms Cubitt and Sugden (1994; Economic J. 104: 798) impose on a theory of rationally justifiable play (TRJP) do not prevent the possibility that two players necessarily disagree concerning the probability they ascribe to the choice of a third player. This appears to indicate that those axioms are not sufficient for defining a `reasonable' TRJP. In addition, for the case in which a player's beliefs are statistically independent, conditions for a TRJP are suggested under which the existence of a quasi-strict equilibrium is sufficient, but the existence of a consistent n-pair is not, for defining a TRJP meeting those requirements.     

Endogenous correlated equilibria in noncooperative games

Most of the results of modern game theory presuppose that the choices rational agents make in noncooperative games are probabilistically independent. In this paper I argue that there is noa priori reason for rational agents to assume probabilistic independence. I introduce a solution concept for noncooperative games called anendogenous correlated equilibrium, which generalizes the Nash equilibrium concept by dropping probabilistic independence. I contrast the endogenous correlated equilibrium with the correlated equilibrium defined by Aumann (1974, 1987). I conclude that in general the endogenous correlated equilibrium concept is a more appropriate solution concept for noncooperative game theory than the less general Nash equilibrium concept. I close by discussing the relationship between endogenous correlated equilibrium and the game solution concept calledrationalizability introduced by Bernheim (1984) and Pearce (1984).     

Folk theorems for the observable implications of repeated games

The fact that infinitely repeated games have many different equilibrium outcomes is known as the Folk Theorem. Previous versions of the Folk Theorem have characterized only the payoffs of the game. This paper shows that over a finite portion of an infinitely repeated game, the concept of perfect equilibrium imposes virtually no restrictions on observable behavior. The Prisoner's Dilemma is presented as an example and discussed in detail.I would like to thank an anonymous referee, Sushil Bikhchandani, David Hirshleifer, David Levine, Thomas Voss, and participants in the UCLA Game Theory Seminar for helpful comments.     

A recursive core for partition function form games

We present a well-defined generalisation of the core to coalitional games with externalities, where the value of a deviation is given by an endogenous response, the solution (if nonempty: the core) of the residual game.     

A test of the characteristic function and the Harsanyi function in N-person normal form sidepayment games

This paper reports the results of a laboratory experiment investigating sidepayment games represented in normal form. Attempts to predict payoff allocations via the application of solution concepts (such as the Shapley value or the nucleolus) encounter a problem in games of this form, because the game must first be transformed into some other form. Commonly, this other form is a set function defined over coalitions, such as the von Neumann-Morgenstern characteristic function. Because there are numerous possible transformations, the question arises as to which one provides the most accurate basis for prediction of payoffs.The laboratory experiment tested three such transformations - the mixed strategy characteristic function, the pure strategy characteristic function, and the Harsanyi threat function. Payoff predictions from two solution concepts (Shapley value, nucleolus) were computed on the basis of each of these transformations, making a total of six theories under test.Results of the study show, in general, that payoff predictions based on the Harsanyi threat function and on the mixed strategy characteristic function were more accurate than those based on the pure strategy characteristic function. The most accurate theories were the Shapley value computed from the Harsanyi function, the nucleolus computed from the Harsanyi function, and the Shapley value computed from the mixed strategy characteristic function. Less accurate were the nucleolus computed from the mixed strategy characteristic function and both the nucleolus and the Shapley value computed from the pure strategy characteristic function.This research was supported by grants SOC-7726932 and SES-8319322 from the National Science Foundation. The authors express appreciation to Yat-Tuck See, Jyh-Jen Horng Shiau, and Raymond Wong for assistance in computer programming, and to Jennifer Brandt, Young C. Choi, David C. Dettman, Laurel Dettman, Stephen B. Geisheker, Irving J. Ginsberg, Mike P. Griffin, Kimberly Ihm, Todd Isaacson, Christy Kinney, Mary Kohl, Sue Pope, Tammy Schmieden, Jill Schwarze, Susan Winter, and Kenneth Yuen for assistance in data collection and analysis.     

Self enforceable paths in extensive form games

This paper explores the idea of forward induction for extensive games. It interprets this idea as a general behavioral principle the technical details of which have to be worked out in each specific case. Because of its cooperative ingredient, this approach should be contrasted with the usual approaches of non-cooperative game theory which are rooted in individual rationality.     

Focal points in pure coordination games: An experimental investigation

This paper reports an experimental investigation of the hypothesis that in coordination games, players draw on shared concepts of salience to identify focal points on which they can coordinate. The experiment involves games in which equilibria can be distinguished from one another only in terms of the way strategies are labelled. The games are designed to test a number of specific hypotheses about the determinants of salience. These hypotheses are generally confirmed by the results of the experiment.     

Some notes on Church's thesis and the theory of games

This paper considers games in normal form played by Turing Machines. The machines are fed as input all the relevent information and then are required to play the game. Some impossibility results are derived for this set-up. In particular, it is shown that no Turing Machine exists which will always play the correct strategy given its opponent's choice. Such a result also generalizes to the case in which attention is restricted to economically optimizing machines only. The paper also develops a model of knowledge. This allows the main results of the paper to be interpreted as stemming out of the impossibility of always deciding whether a player is rational or not in some appropriate sense.     

On the logic of common belief and common knowledge

The paper surveys the currently available axiomatizations of common belief (CB) and common knowledge (CK) by means of modal propositional logics. (Throughout, knowledge — whether individual or common — is defined as true belief.) Section 1 introduces the formal method of axiomatization followed by epistemic logicians, especially the syntax-semantics distinction, and the notion of a soundness and completeness theorem. Section 2 explains the syntactical concepts, while briefly discussing their motivations. Two standard semantic constructions, Kripke structures and neighbourhood structures, are introduced in Sections 3 and 4, respectively. It is recalled that Aumann's partitional model of CK is a particular case of a definition in terms of Kripke structures. The paper also restates the well-known fact that Kripke structures can be regarded as particular cases of neighbourhood structures. Section 3 reviews the soundness and completeness theorems proved w.r.t. the former structures by Fagin, Halpern, Moses and Vardi, as well as related results by Lismont. Section 4 reviews the corresponding theorems derived w.r.t. the latter structures by Lismont and Mongin. A general conclusion of the paper is that the axiomatization of CB does not require as strong systems of individual belief as was originally thought — onlymonotonicity has thusfar proved indispensable. Section 5 explains another consequence of general relevance: despite the infinitary nature of CB, the axiom systems of this paper admit of effective decision procedures, i.e., they aredecidable in the logician's sense.     

Social Representations: The Beautiful Invention

Psychoanalysis: Its Image and Its Public is a perfect illustration of Tarde's claim that ‘beautiful’ should be reserved for ideas that lead to a discovery of more ideas and to an invention that we can judge as fruitful for the future. The article examines the influence of the book in geographical, historical and scientific contexts and traces the development and diffusion of the theory of social representations throughout four periods. The article highlights the difference between the first edition in 1961, and the second one in 1976, and it examines the development of Moscovici's ideas over time. The first edition of Psychoanalysis, based on the criticism of sociology of knowledge, underlines common sense as a new object of study, attempting to capture complex social phenomena. It shows common sense as a legitimate object of study that interweaves social, cultural and individual dimensions in the constitution of social reality. The 1976 edition emphasises that language and communication and its role in the theory of social knowledge. The book has had manifold impact on new areas of study. An overview of international research in social representations shows how this work has inspired scholars all over the world, has lead to creating Schools of research and to innovative proposals not only in terms of domains that have been studied but also in terms of theoretical models and methodological concerns.     

A Banzhaf share function for cooperative games in coalition structure

A cooperative game with transferable utility–or simply a TU-game– describes a situation in which players can obtain certain payoffs by cooperation. A value function for these games assigns to every TU-game a distribution of payoffs over the players. Well-known solutions for TU-games are the Shapley and the Banzhaf value. An alternative type of solution is the concept of share function, which assigns to every player in a TU-game its share in the worth of the grand coalition. In this paper we consider TU-games in which the players are organized into a coalition structure being a finite partition of the set of players. The Shapley value has been generalized by Owen to TU-games in coalition structure. We redefine this value function as a share function and show that this solution satisfies the multiplication property that the share of a player in some coalition is equal to the product of the Shapley share of the coalition in a game between the coalitions and the Shapley share of the player in a game between the players within the coalition. Analogously we introduce a Banzhaf coalition structure share function. Application of these share functions to simple majority games show some appealing properties.     

Axiomatization of a class of share functions for n-person games

The Shapley value is the unique value defined on the class of cooperative games in characteristic function form which satisfies certain intuitively reasonable axioms. Alternatively, the Banzhaf value is the unique value satisfying a different set of axioms. The main drawback of the latter value is that it does not satisfy the efficiency axiom, so that the sum of the values assigned to the players does not need to be equal to the worth of the grand coalition. By definition, the normalized Banzhaf value satisfies the efficiency axiom, but not the usual axiom of additivity.In this paper we generalize the axiom of additivity by introducing a positive real valued function on the class of cooperative games in characteristic function form. The so-called axiom of -additivity generalizes the classical axiom of additivity by putting the weight (v) on the value of the gamev . We show that any additive function determines a unique share function satisfying the axioms of efficient shares, null player property, symmetry and -additivity on the subclass of games on which is positive and which contains all positively scaled unanimity games. The axiom of efficient shares means that the sum of the values equals one. Hence the share function gives the shares of the players in the worth of the grand coalition. The corresponding value function is obtained by multiplying the shares with the worth of the grand coalition. By defining the function appropiately we get the share functions corresponding to the Shapley value and the Banzhaf value. So, for both values we have that the corresponding share functions belong to this class of share functions. Moreover, it shows that our approach provides an axiomatization of the normalized Banzhaf value. We also discuss some other choices of the function and the corresponding share functions. Furthermore we consider the axiomatization on the subclass of monotone simple games.     

The incoherence of agreeing to disagree

The agreeing-to-disagree theorem of Aumann and the no-expected-gain-from-trade theorem of Milgrom and Stokey are reformulated under an operational definition of Bayesian rationality. Common knowledge of beliefs and preferences is achieved through transactions in a contingent claims market, and mutual expectations of Bayesian rationality are defined by the condition of joint coherence,i.e., the collective avoidance of arbitrage opportunities. The existence of a common prior distribution and the impossibility of agreeing to disagree follow from the joint coherence requirement, but the prior must be interpreted as a risk-neutral distribution: a product of probabilities and marginal utilities for money. The failure of heterogenous information to create disagreements or incentives to trade is shown to be an artifact of overlooking the potential role of trade in constructing the initial state of common knowledge.     

Knowing what we know about knowledge in social work: The search for a comprehensive model of knowledge production

Influenced by its historical beginnings, the production and modelling of knowledge in social work has become a small but diverse field. Ten key models exploring the nature of knowledge and its production in social work are reviewed against significant ideas from the interdisciplinary knowledge production literature beyond social work. In so doing, the place of holistic approaches to modelling the knowledge production process within social work and the human services is highlighted. It is argued that despite the number of models that attempt to describe the knowledge production process, there is scope for a more comprehensive, holistic, complex approach to modelling knowledge production in social work and the human services.     

Preferences for information on probabilities versus prizes: The role of risk-taking attitudes

Many real-world decisions entail choices between information on either probabilities or payoffs (i.e., prizes). Simplified versions of such decisions are examined to gain insight into preferences for different types of information as a function of risk-attitudes. General and simple decision rules are derived for cases where the utility function is concave (or convex) over the relevant payoff interval.The article further describes several experiments to test business students' intuitions concerning these optimal decision rules. In general, risk-taking attitudes did not correlate significantly with subjects' preferences for information, in violation of theorems regarding mean-preserving spreads of risk. Other tests, e.g., narrowing certain probability ranges, also resulted in preferences contrary to expected utility (EU) theory.     

Prospect theory for continuous distributions

We extend the original form of prospect theory by Kahneman and Tversky from finite lotteries to arbitrary probability distributions, using an approximation method based on weak-⋆ convergence. The resulting formula is computationally easier than the corresponding formula for cumulative prospect theory and makes it possible to use prospect theory in future applications in economics and finance. Moreover, we suggest a method how to incorporate a crucial step of the “editing phase” into prospect theory and to remove in this way the discontinuity of the original model.

The Knowledge Spectrum: A Framework for Teaching Knowledge and its Use in Social Work Practice

For students, educators play a pivotal role in identifying andexplaining the knowledge that can guide social work practice.Yet there is an absence of educational tools that can assistthe educator with this objective. The knowledge spectrum frameworkis an innovative tool that was developed to explain the possibilitiesof what can inform social work practice. As a tool, it can beutilized to show that knowledge use in practice is a dynamicprocess, illustrating that knowledge can be created, modifiedand discarded by practitioners. The framework can also be utilizedto capture the complexity of knowledge used by recognizing boththe tacit (unconscious) and explicit (conscious) domains, therebyidentifying different knowledge use possibilities. Strengtheningstudents’ and/or practitioners’ ability to recognizeand identify the basis of their professional behaviour is criticalfor clear, knowledge-guided practice.     

Training Supervisors in Two Cultures: Toward a Model for Codifying Practice Wisdom and Local Knowledge

This article describes parallel learning issues that emerged in supervision training and research at a school for social work in the United States and at a university in Northern Thailand. It presents a conceptual model for understanding and addressing these shared training issues and explores the model's cross-cultural relevance and utility.