Learning from others: conditioning versus averaging

How should we revise our beliefs in response to the expressed probabilistic opinions of experts on some proposition when these experts are in disagreement? In this paper I examine the suggestion that in such circumstances we should adopt a linear average of the experts’ opinions and consider whether such a belief revision policy is compatible with Bayesian conditionalisation. By looking at situations in which full or partial deference to the expressed opinions of others is required by Bayesianism I show that only in trivial circumstances are the requirements imposed by linear averaging compatible with it.     

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.     

On the evaluation of solution concepts

It is proposed that solution concepts for games should be evaluated in a way that is analogous to the way a logic is evaluated by a model theory for the language. A solution concept defines a set of strategy profiles, as a logic defines a set of theorems. A model theoretic analysis for a game defines a class of models, which are abstract representations of particular plays of the game. Given an appropriate definition of a model, one can show that various solution concepts are characterized by intuitively natural classes of models in the same sense that the set of theorems of a logic is characterized by a class of models of the language. Sketches of characterization results of this kind are given for rationalizability, Nash equilibrium, and for a refinement of rationalizability —strong rationalizability — that has some features of an equilibrium concept. It is shown that strong rationalizability is equivalent to Nash equilibrium in perfect information games. Extensions of the model theoretic framework that represent belief revision and that permit the characterization of other solution concepts are explored informally.     

Population monotonic path schemes for simple games

A path scheme for a game is composed of a path, i.e., a sequence of coalitions that is formed during the coalition formation process and a scheme, i.e., a payoff vector for each coalition in the path. A path scheme is called population monotonic if a player’s payoff does not decrease as the path coalition grows. In this study, we focus on Shapley path schemes of simple games in which for every path coalition the Shapley value of the associated subgame provides the allocation at hand. Obviously, each Shapley path scheme of a game is population monotonic if and only if the Shapley allocation scheme of the game is population monotonic in the sense of Sprumont (Games Econ Behav 2:378–394, 1990). We prove that a simple game allows for population monotonic Shapley path schemes if and only if the game is balanced. Moreover, the Shapley path scheme of a specific path is population monotonic if and only if the first winning coalition that is formed along the path contains every minimal winning coalition. We also show that each Shapley path scheme of a simple game is population monotonic if and only if the set of veto players of the game is a winning coalition. Extensions of these results to other efficient probabilistic values are discussed.     

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.     

Rawls and Bentham reconciled

The paper deals with the characterization of a class of social welfare orderings. The social evaluation functions which represent these orderings are separable in the components of the ordered utility vector. The characterization is based on the Strong Pareto Property, Co-cardinality, Continuity and a new Independence Property. Since this class encompasses the utilitarian rule and since there are members of this family which almost coincide with the rules of rank dictatorship this family bridges the gap between pure utilitarianism and rank dictatorship.     

Law of demand and stochastic choice

Theory and Decision - We consider random choice rules that, by satisfying a weak form of Luce’s choice axiom, embody a form probabilistic rationality. We show that for this important class of...     

Eliciting beliefs

We develop an algorithm that can be used to approximate a decisionmaker’s beliefs for a class of preference structures that includes, among others, α-maximin expected utility preferences, Choquet expected utility preferences, and, more generally, constant additive preferences. For both exact and statistical approximation, we demonstrate convergence in an appropriate sense to the true belief structure.     

Market uncertainty and the process of belief formation

Subjective or personal belief formation has been up to now grounded on axioms (e.g., the axioms of Savage) that can be described asintrospection axioms, for they bear only on judgments formed by the individual within himself (e.g., judgments on the ordering of the “acts” in Savage's contribution), excluding every reference to other, possibly external, sources of opinion formation. This current view, linked to the traditional concept ofinstrumental economic rationality, implicitly assumes cognitive capacities that are most of the time out of the reach of any individual. Therefore this view is little or improperly used in the practical world of economics and/or management. An alternative view of belief formation on a market is offered in this article, grounded on a much more modest demand of cognitive capacities from the individual and on a different concept of economic rationality known ascognitive rationality. The model introduces to this end the global concept of market psychology as well as the concept of regular revision of the sets of opinions. Belief formation then appears as a process of deliberation between the individual and the market, rationality emerging within this process as deeply intertwined with the way the individual revises his opinions during the process. The rational treatment of beliefs in view of reaching a decision can consequently be carefully distinguished from their rational formation process.     

Expected utility with ambiguous probabilities and ‘irrational’ parameters

Savage's expected utility theory has been criticized for various reasons. Two of these are: (1) Subjective probabilities may be more or less ambiguous; this may affect the preference order. Savage's theory rules out any such effect. (2) Savage's theory rules out any influence of certain parameters of a probability distribution on the preference order. Such parameters are fractiles or pairs of fractiles, e.g., which have gained widespread attention as risk measures.This paper generalizes expected utility theory to include effects of ambiguity and those parameters on the preference order. Basic to this generalization is the classic paper by Herstein/Milnor.     

因病信教农民的宗教心理及其演变:试论把握“信念”概念对理解中国农村宗教实践的启示

本文通过对苏北望村基督徒因病信教现象的考察,关注信教农民的宗教心理及其演变。用"信念"来把握农民"信"的起点心理,它蕴含了个体面临生命困境时寻求宗教帮助并对之赋以期许的自然心态和家庭伦理。"信念"的宗教建构、"信念"的情感嵌入以及"信仰"要求三个宗教操演环节促使个体心理的由俗入圣。基于原有病型及其疗效基础上的个体与宗教间的互动,信徒发生了信念坍塌、信念过渡和信仰升级三种类型的心理分化。"信念"操演与分化构成了农民宗教心理演变的机理,也形塑了农民的信仰方式。文章进一步探讨"信念"对于理解中国农村宗教实践中的个体宗教心理、宗教皈信、宗教复兴等议题所具有的启发。     

On Stalnaker's Notion of Strong Rationalizability and Nash Equilibrium in Perfect Information Games

Counterexamples to two results by Stalnaker (Theory and Decision, 1994) are given and a corrected version of one of the two results is proved. Stalnaker's proposed results are: (1) if at the true state of an epistemic model of a perfect information game there is common belief in the rationality of every player and common belief that no player has false beliefs (he calls this joint condition strong rationalizability), then the true (or actual) strategy profile is path equivalent to a Nash equilibrium; (2) in a normal-form game a strategy profile is strongly rationalizable if and only if it belongs to C , the set of profiles that survive the iterative deletion of inferior profiles.     

Feddersen and Pesendorfer meet Ellsberg

The Condorcet Jury Theorem formalises the “wisdom of crowds”: binary decisions made by majority vote are asymptotically correct as the number of voters tends to infinity. This classical result assumes like-minded, expected utility maximising voters who all share a common prior belief about the right decision. Ellis (Theor Econo 11(3): 865–895, 2016) shows that when voters have ambiguous prior beliefs—a (closed, convex) set of priors—and follow maxmin expected utility (MEU), such wisdom requires that voters’ beliefs satisfy a “disjoint posteriors” condition: different private signals lead to posterior sets with disjoint interiors. Both the original theorem and Ellis’s generalisation assume symmetric penalties for wrong decisions. If, as in the jury context, errors attract asymmetric penalties then it is natural to consider voting rules that raise the hurdle for the decision carrying the heavier penalty for error (such as conviction in jury trials). In a classical model, Feddersen and Pesendorfer (Am Politi Sci Rev 92(1):23–35, 1998) have shown that, paradoxically, raising this hurdle may actually increase the likelihood of the more serious error. In particular, crowds are not wise under the unanimity rule: the probability of the more serious error does not vanish as the crowd size tends to infinity. We show that this “Jury Paradox” persists in the presence of ambiguity, whether or not juror beliefs satisfy Ellis’s “disjoint posteriors” condition. We also characterise the strictly mixed equilibria of this model and study their properties. Such equilibria cannot exist in the absence of ambiguity but may exist for arbitrarily large jury size when ambiguity is present. In addition to uninformative strictly mixed equilibria, analogous to those exhibited by Ellis (Theor Econo 11(3): 865–895, 2016), there may also exist strictly mixed equilibria which are informative about voter signals.

     

The essential ranking of decision rules in small panels of experts

For small panels of experts (e.g., boards of managers, courts, specialized committees), n 5, this paper provides an algorithm for ranking the seven efficient and commonly used weighted majority rules by their respective performance. These rules are terned efficient since they constitute the set of potentially optimal decision rules in uncertain, symmetric, pairwise choice situations. The main contribution of this study is the discovery of an essential ordering of six of these rules which entails that the set of possible ranking of the seven rules is almost single peaked.The essential ordering significantly reduces the number of possible rankings of the rules, and thus, simplifies the development of the ranking algorithm. The essential ordering has important applications when the available information on the experts' decisional skills is incomplete.     

A Special Case of Penrose’s Limit Theorem When Abstention is Allowed

In general, analyses of voting power are performed through the notion of a simple voting game (SVG) in which every voter can choose between two options: ‘yes’ or ‘no’. Felsenthal and Machover [Felsenthal, D.S. and Machover, M. (1997), International Journal of Game Theory 26, 335–351.] introduced the concept of ternary voting games (TVGs) which recognizes abstention alongside. They derive appropriate generalizations of the Shapley–Shubik and Banzhaf indices in TVGs. Braham and Steffen [Braham, M. and Steffen, F. (2002), in Holler, et al. (eds.), Power and Fairness, Jahrbuch für Neue Politische Ökonomie 20, Mohr Siebeck, pp. 333–348.] argued that the decision-making structure of a TVG may not be justified. They propose a sequential structure in which voters first decide between participation and abstention and then between ‘yes’ or ‘no’. The purpose of this paper is two-fold. First, we compare the two approaches and show how the probabilistic interpretation of power provides a unifying characterization of analogues of the Banzhaf (Bz) measure. Second, using the probabilistic approach we shall prove a special case of Penrose’s Limit Theorem (PLT). This theorem deals with an asymptotic property in weighted voting games with an increasing number of voters. It says that under certain conditions the ratio between the voting power of any two voters (according to various measures of voting power) approaches the ratio between their weights. We show that PLT holds in TVGs for analogues of Bz measures, irrespective of the particular nature of abstention.     

Capacity Updating Rules and Rational Belief Change

Choquet expected utility substitutes capacities for subjective probabilities to explain uncertainty aversion and related phenomena. This paper studies capacities as models of belief. The notions of inner and outer acceptance context are defined. These are shown to be the natural acceptance contexts when belief expansion is described by naïve Bayesian and Dempster–Shafer updating of capacities respectively. We also show that Eichberger and Kelsey's (1999b) use of Dempster–Shafer updating as a model of belief revision may lead to violations of the AGM axioms for rational belief change.     

Measuring revisions to subjective expectations

This paper develops a new metric to measure revisions to subjective expectations and proposes a survey design strategy that enables the estimation of the metric. As an application, I analyze how women update their ex pectations about the effectiveness of contraception methods. The women interviewed exhibit substantial heterogeneity in the way they revise their expectations with receipt of the same information. When relating the heterogeneity in the updating process to observable characteristics, I find that schooling, having a regular sexual partner and knowledge and use of birth control methods have a large impact on the revision process about the effectiveness of contraceptives.

Information-seeking during acquaintanceship: Effects of level of social understanding and personal relevance

Two studies examined how individual differences in social understanding influence children's information-seeking orientations during an initial meeting with a peer, in which they do or do not anticipate future interaction. Study 1 involved a relatively controlled, laboratory setting in which 7-9 year old children were presented a list of questions from which to choose what to ask the peer. The findings indicate that children who view a person's behavior in terms of stable dispositional characteristics (high SDC), express greater interest in gathering trait-related information about an unfamiliar peer than children who do not perceive people in terms of stable traits (low SDC), but only when they expect to play games with that peer in a future interaction. Study 2 involved a more naturalistic observation of the actual questions 7-9 year old children ask during an initial meeting with an unfamiliar peer. The findings showed that high SDC children asked more questions of the peer when they expected interaction than when they did not, whereas low SDC children did not differ significantly across conditions. Taken together, the two studies indicate that the future, predictability orientation of high SDC children leads to a more extensive information search about a peer when they expect interaction with that peer involving instrumental choices.     

Decreasing higher-order absolute risk aversion and higher-degree stochastic dominance

Fishburn and Vickson (Stochastic dominance: an approach to decision-making under risk, Lexington Books, D.C. Heath and Company, Lexington, pp. 39–113, 1978) showed that, when applied to random alternatives with an equal mean, 3rd-degree and decreasing absolute risk aversion stochastic dominances represent equivalent rules. The present paper generalizes this result to higher degrees. Specifically, higher-degree stochastic dominance rules and common preference by all decision makers with decreasing higher-order absolute risk aversion are shown to coincide under appropriate constraints on the respective moments of the random variables to be compared.     

Long-Term Behavior in the Theory of Moves

This paper proposes a revised Theory of Moves (TOM) to analyze matrix games between two players when payoffs are given as ordinals. The games are analyzed when a given player i must make the first move, when there is a finite limit n on the total number of moves, and when the game starts at a given initial state S. Games end when either both players pass in succession or else a total of n moves have been made. Studies are made of the influence of i, n, and S on the outcomes. It is proved that these outcomes ultimately become periodic in n and therefore exhibit long-term predictable behavior. Efficient algorithms are given to calculate these ultimate outcomes by computer. Frequently the ultimate outcomes are independent of i, S, and n when n is sufficiently large; in this situation this common ultimate outcome is proved to be Pareto-optimal. The use of ultimate outcomes gives rise to a concept of stable points, which are analogous to Nash equilibria but consider long-term effects. If the initial state is a stable point, then no player has an incentive to move from that state, under the assumption that any initial move could be followed by a long series of moves and countermoves. The concept may be broadened to that of a stable set. It is proved that every game has a minimal stable set, and any two distinct minimal stable sets are disjoint. Comparisons are made with the results of standard TOM.