Conditioning Capacities and Choquet Integrals: The Role of Comonotony

Choquet integrals and capacities play a crucial role in modern decision theory. Comonotony is a central concept for these theories because the main property of a Choquet integral is its additivity for comonotone functions. We consider a Choquet integral representation of preferences showing uncertainty aversion (pessimism) and propose axioms on time consistency which yield a candidate for conditional Choquet integrals. An other axiom characterizes the role of comonotony in the use of information. We obtain two conditioning rules for capacities which amount to the well-known Bayes' and Dempster–Schafer's updating rules. We are allowed to interpret both of them as a lack of confidence in information in a dynamic extension of pessimism. This revised version was published online in June 2006 with corrections to the Cover Date.     

Cognitive rationality and alternative belief measures

In this article, we first examine the various criticisms of the probabilistic model. Then we introduce capacities in order to show that if a probability measure corresponds to anesthetizing the belief of the agent's knowledge, it is then possible to suggest another type of rationality—namely, being able to describe a wise and a rash behavior when facing risk—and therefore another model of belief under uncertainty. While trying to specify various alternative measures, possibility, necessity, and measures resulting from a triangular norm or from a triangular conorm, we finally try to define the field of application of the probabilistic model as well as a sign of the rationality choice: constraint of mass-unity for traditional rationality, and constraint of duality for the one we present.This article is a revised version of a paper presented during the FUR V Congress, Duke University, North Carolina, USA, June 10–13, 1990. I am indebted to Bertrand Munier, Alain Chateauneuf, Olivier Favereau, Jacques-François Thisse, Bernard Walliser, and Kip Viscusi for helpful comments on earlier versions of this article.     

Modus Ponens and Modus Tollens for Conditional Probabilities,and Updating on Uncertain Evidence

There are narrowest bounds for P(h) when P(e)  =  y and P(h/e)  =  x, which bounds collapse to x as y goes to 1. A theorem for these bounds – Bounds for Probable Modus Ponens – entails a principle for updating on possibly uncertain evidence subject to these bounds that is a generalization of the principle for updating by conditioning on certain evidence. This way of updating on possibly uncertain evidence is appropriate when updating by ‘probability kinematics’ or ‘Jeffrey-conditioning’ is, and apparently in countless other cases as well. A more complicated theorem due to Karl Wagner – Bounds for Probable Modus Tollens – registers narrowest bounds for P(∼h) when P(∼e) =  y and P(e/h)  =  x. This theorem serves another principle for updating on possibly uncertain evidence that might be termed ‘contraditioning’, though it is for a way of updating that seems in practice to be frequently not appropriate. It is definitely not a way of putting down a theory – for example, a random-chance theory of the apparent fine-tuning for life of the parameters of standard physics – merely on the ground that the theory made extremely unlikely conditions of which we are now nearly certain. These theorems for bounds and updating are addressed to standard conditional probabilities defined as ratios of probabilities. Adaptations for Hosiasson-Lindenbaum ‘free-standing’ conditional probabilities are provided. The extended on-line version of this article (URL: ) includes appendices and expansions of several notes. Appendix A contains demonstrations and confirmations of elements of those adaptations. Appendix B discusses and elaborates analogues of modus ponens and modus tollens for probabilities and conditional probabilities found in Elliott Sober’s “Intelligent Design and Probability Reasoning.” Appendix C adds to observations made below regarding relations of Probability Kinematics and updating subject to Bounds for Probable Modus Ponens.      

What accounts for China''s trade balance dynamics?

This paper proposes a structural VAR model which extends the frameworks of Hoffmaister and Roldós [Hoffmaister, A. W., & Roldós, J. E. (2001). The sources of macroeconomic fluctuations in developing countries: Brazil and Korea. Journal of Macroeconomics, 23, 213–239] and Prasad [Prasad, E. S. (1999). International trade and the business cycle. Economic Journal, 109, 588–606]. The model is then used to analyse the sources of China's trade balance fluctuations in the period of 1985–2000. Efforts are made to distinguish the forces which underlie the long-run trend in trade balance from those with transitory impacts. The effects of four types of shock are examined—the foreign supply shock, the domestic supply shock, the relative demand shock and the nominal shock. Among other findings, two emerge as important. First, the movements in China's trade balance are largely the result of real shocks. Second, the Renminbi is undervalued, yet changes in the exchange rate bear little on the trade balance. Therefore, monetary measures would not suffice to redress China's trade ‘imbalance’.     

A Model of Minimal Probabilistic Belief Revision

In the literature there are at least two models for probabilistic belief revision: Bayesian updating and imaging [Lewis, D. K. (1973), Counterfactuals, Blackwell, Oxford; Gärdenfors, P. (1988), Knowledge in flux: modeling the dynamics of epistemic states, MIT Press, Cambridge, MA]. In this paper we focus on imaging rules that can be described by the following procedure: (1) Identify every state with some real valued vector of characteristics, and accordingly identify every probabilistic belief with an expected vector of characteristics; (2) For every initial belief and every piece of information, choose the revised belief which is compatible with this information and for which the expected vector of characteristics has minimal Euclidean distance to the expected vector of characteristics of the initial belief. This class of rules thus satisfies an intuitive notion of minimal belief revision. The main result in this paper is to provide an axiomatic characterization of this class of imaging rules.     

Government size and economic growth in Taiwan: A threshold regression approach

The Armey curve developed by [Armey, R. (1995). The freedom revolution. Washington, DC: Rognery Publishing Co.] and [Vedder, R. K., & Gallaway, L. E. (1998). Government size and economic growth. Joint Economic Committee] demonstrates that there is a non-linear relationship between government size and economic growth. In order to search for the threshold effects, this paper employs [Hansen, B. E. (2000). Sample splitting and threshold estimation. Econometrica, 68(3), 575–603] threshold regression model to test whether the Armey curve exists in Taiwan, allowing for endogenous government size thresholds. We apply the two-sector production function developed by [Ram, R. (1986). Government size and economic growth: A new framework and some evidence from cross-section and time-series data. American Economic Review, 76(1), 191–203] to construct the threshold regression model. Three classifications of government size are tested in sequence as threshold variables. The result indicates that all three classifications of government size have a threshold effect and that a non-linear relationship of the Armey curve exists in Taiwan.     

Executive function and theory of mind as predictors of socially withdrawn behavior in institutionalized children

False‐belief understanding and executive functions are two main sociocognitive abilities reliably linked to child social competence. Although institution‐reared children are especially at risk for behavioral problems and cognitive delays, the role that executive function and false‐belief understanding might play in the social withdrawal of institutionalized children has not been examined. The current study used two‐wave data to investigate the concurrent and longitudinal relations of social withdrawal with executive function and false‐belief understanding in institutionalized children; it also allowed investigation of the directionality between executive function and false‐belief understanding. Data were collected from 66 Turkish children (T1 M = 57.83 months, SD = 9.20; T2 M = 69.58 months, SD = 8.45) residing in institutions, at two time points, approximately 1 year apart. We measured false‐belief understanding and executive function via individual assessments, and social withdrawal via care provider reports at both time points. Results showed that both executive function and false‐belief understanding increased between T1 and T2, while social withdrawal did not show a significant change. Path analysis revealed that when T1 age and language were controlled, T1 executive function predicted T2 executive function, and in turn, T2 executive function predicted lessened social withdrawal at T2. In addition, T1 executive function predicted T2 false‐belief understanding. T1 false‐belief understanding was not related to T2 false‐belief understanding, executive function, or social withdrawal. Findings suggested that executive function is an important predictor of social withdrawal in high‐risk populations.     

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.     

A Rule For Updating Ambiguous Beliefs

When preferences are such that there is no unique additive prior, the issue of which updating rule to use is of extreme importance. This paper presents an axiomatization of the rule which requires updating of all the priors by Bayes rule. The decision maker has conditional preferences over acts. It is assumed that preferences over acts conditional on event E happening, do not depend on lotteries received on E ^c, obey axioms which lead to maxmin expected utility representation with multiple priors, and have common induced preferences over lotteries. The paper shows that when all priors give positive probability to an event E, a certain coherence property between conditional and unconditional preferences is satisfied if and only if the set of subjective probability measures considered by the agent given E is obtained by updating all subjective prior probability measures using Bayes rule.     

A test of the Core,Bargaining Set,Kernel and Shapley models in n-person quota games with one weak player

n-person (n – 1)-quota-games where the quotas are positive for certain n – 1 (strong) players and negative for the remaining (weak) player, are discussed. Normative solutions predicted by the Core,the Kernel, the Bargaining Set, the Competitive Bargaining Set, and by the Shapley Value are presented and exemplified.Each of twelve groups of subjects participated in a four-person and a five-person (n – 1)-quota games with one weak player. The weak player was always excluded from the ratified coalition. The division of payoffs among the strong players was more egalitarian than the Kernel solution but less egalitarian than the Shapley value. The Core and the Bargaining Sets were fully supported for the two strongest players, but less supported for the other players. Analyses of the bargaining process confirmed a dynamic interpretation of the Bargaining Set Theory.This research was performed while the author was at the University of North Carolina. The research was partially supported by a PHS Research Grant No. MH-10006 from the National Institute of Mental Health. The author thanks Professor Amnon Rapoport for helpful advice in the design of this study.     

Rational beliefs in extensive games

Given an extensive game, with every node x and every player i a subset k _i (x) of the set of terminal nodes is associated, and is given the interpretation of player i's knowledge (or information) at node x. A belief of player i is a function that associates with every node x an element of the set K _i (x). A belief system is an n-tuple of beliefs, one for each player. A belief system is rational if it satisfies some natural consistency properties. The main result of the paper is that the notion of rational belief system gives rise to a refinement of the notion of subgame-perfect equilibrium.     

Theory of mind,gender, gains in friendships versus peer acceptance and anxious solitude from middle childhood through early adolescence

This investigation proposes that theory of mind (ToM) may be related more strongly to change in friendships than peer acceptance in late middle childhood through early adolescence, and examines the relation between ToM and anxious solitude. Fourth grade ToM was tested as a predictor of change in reciprocated friendships, peer acceptance, and anxious solitude from 4th to 7th grade, and, conversely, reciprocated friendships, peer acceptance, and anxious solitude in 3^rd grade were tested as predictors of 4th grade ToM. Gender moderation of these relations was evaluated. Participants were 688 American public-school children (51.5% girls), 193 of whom completed a ToM questionnaire in 4th grade. In 3rd–7th grade children and their peers reported reciprocated friendship, and peers reported peer acceptance and anxious solitude annually. A multi-group (gender split) autoregressive cross-lagged panel analysis modeled relations between ToM and reciprocated friendship, peer acceptance, and anxious solitude over time. Consistent with hypotheses, girls’ more advanced 4th grade ToM predicted incremental gains in their number of friendships two years later, but not their peer acceptance. In contrast, boys’ more advanced 4th grade ToM did not predict change in their number of friendships or peer acceptance over time. Gender differences in the relation between ToM and friendship are discussed in the context of gender-specific peer relations patterns in late middle childhood and early adolescence. Additionally, more advanced 4th grade ToM predicted gains in anxious solitude in middle school for both genders. This somewhat surprising result is discussed in relation to ToM assessment and peer relations in anxious solitary children.     

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.     

Ramsey's Foundations Extended to Desirabilities

In his Truth and Probability (1926), Frank Ramsey provides foundations for measures of degrees of belief in propositions and preferences for worlds. Nonquantitative conditions on preferences for worlds, and gambles for worlds and certain near-worlds, are formulated which he says insure that a subject's preferences for worlds are represented by numbers, world values. Numbers, for his degrees of belief in propositions, probabilities, are then defined in terms of his world values. Ramsey does not also propose definitions of desirabilities for propositions, though he is in a position to do this. Given his measures for probabilities of propositions and values of worlds, he can frame natural definitions for both evidential and causal desirabilities that would measure respectively the welcomeness of propositions as items of news, and as facts. His theory is neutral with respect to the evidential/causal division. In the present paper, as Ramsey's foundations are explained, several problems and limitations are noted. Their distinctive virtue – their evidential/causal neutrality – is demonstrated. Comparisons are made with other foundational schemes, and a perspective is recommended from which nonquantitative foundations are not the be all for quantitative theories of ideal preferences and credences.     

Product evaluation bias under minimal group situations

The present study was designed to identify the source of product evaluation bias. Three experiments were conducted to see if the evaluation bias occur in the minimal group situation defined by [Tajifel, H., Billig, M., Bundy, R., & Flament, C. (1971). Social categorization and intergroup behavior. European Journal of Social Psychology, 1, 149–178], and to determine the factors responsible for making evaluation bias. Relative measure of product evaluation played a major role in producing an evaluation bias in minimal group situation. Similarly, relative measure of distribution was a critical factor affecting distribution bias. The possible boundary condition of [Tajifel, H., Billig, M., Bundy, R., & Flament, C. (1971). Social categorization and intergroup behavior. European Journal of Social Psychology, 1, 149–178] minimal group situation in evaluation bias were also discussed.     

The effects of free trade areas on non-members: Modelling Kemp–Vanek admissibility

A multi-sector multi-country Numerical General Equilibrium model is used to endogenously determine the trade taxes that FTA members need to charge on non-member trade so that after the FTA is formed, FTA-member trade volumes with non-members remain at their pre-FTA level. We apply the notion of Kemp–Vanek admissibility in McMillan [McMillan, J. (1993), Does regional integration foster open trade? Economic theory and GATT's Article XXIV” In Anderson, K. & R. Blackhurst (Eds.), Regional Integration and the Global Trading System. London: Harvester-Wheatsheaf] to construct an FTA which does not make non-members of the FTA worse off, ensuring that the FTA is necessarily a building block and not a stumbling block to global free trade. We implement the path-independent welfare decomposition in Harrison et al. [Harrison, G., Rutherford, T. & Wooton, I. (1993). An alternative welfare decomposition for customs unions. Canadian Journal of Economics, 26(4), 961–68]. Results of such experiments are relevant to the current debate under the WTO's Doha Round of trade negotiations over GATT (1994) Article XXIV which evaluates the consistency of FTAs with the WTO.     

Second-order probabilities and belief functions

A second-order probability Q(P) may be understood as the probability that the true probability of something has the value P. True may be interpreted as the value that would be assigned if certain information were available, including information from reflection, calculation, other people, or ordinary evidence. A rule for combining evidence from two independent sources may be derived, if each source i provides a function Q _i (P). Belief functions of the sort proposed by Shafer (1976) also provide a formula for combining independent evidence, Dempster's rule, and a way of representing ignorance of the sort that makes us unsure about the value of P. Dempster's rule is shown to be at best a special case of the rule derived in connection with second-order probabilities. Belief functions thus represent a restriction of a full Bayesian analysis.     

The Instruction set of Questionnaires can Affect the Structure of the Data: Application to Self-Rated State Anxiety

The present study tested the assumption that self-ratings, such as those used for measuring state anxiety, do not measure a one-dimensional transcendent entity but involve decisions based on a multi-dimensional judgment. Two groups of subjects were presented with a balanced nine-item state anxiety questionnaire. Each group received a different set of instructions (a standard set and an altered instruction set suggesting unidimensionality of the questions in the questionnaire). It was hypothesized that this change in instructions would impact the structure of the data. The impact of the instructions was detectable at the level of the observed correlation between the negative and positive composites, Cohen's (1988) q=–0.27$. A multigroup confirmatory factor analysis showed that positive and negative wording factors correlated more strongly when unidimensionality was suggested than when standard instructions were used, q=–0.54. Theoretical and practical implications are discussed.