Inferring a linear ordering over a power set

An observer attempts to infer the unobserved ranking of two ideal objects, A and B, from observed rankings in which these objects are `accompanied' by `noise' components, C and D. In the first ranking, A is accompanied by C and B is accompanied by D, while in the second ranking, A is accompanied by D and B is accompanied by C. In both rankings, noisy-A is ranked above noisy-B. The observer infers that ideal-A is ranked above ideal-B. This commonly used inference rule is formalized for the case in which A,B,C,D are sets. Let X be a finite set and let be a linear ordering on 2^X. The following condition is imposed on . For every quadruple (A,B,C,D)Y, where Y is some domain in (2^X)⁴, if and , then . The implications and interpretation of this condition for various domains Y are discussed.     

Is There a “Pessimistic” Bias in Individual Beliefs? Evidence from a Simple Survey

It is an important issue for economic and finance applications to determine whether individuals exhibit a behavioral bias toward pessimism in their beliefs, in a lottery or more generally in an investment opportunities framework. In this paper, we analyze the answers of a sample of 1,540 individuals to the following question “Imagine that a coin will be flipped 10 times. Each time, if heads, you win $10\texttt{C}\!\!\!\rule[2.3pt]{.4em}{.3pt}\!\!\rule[3.3pt]{.4em}{.3pt}It is an important issue for economic and finance applications to determine whether individuals exhibit a behavioral bias toward pessimism in their beliefs, in a lottery or more generally in an investment opportunities framework. In this paper, we analyze the answers of a sample of 1,540 individuals to the following question “Imagine that a coin will be flipped 10 times. Each time, if heads, you win . How many times do you think that you will win?” The average answer is surprisingly about 3.9 which is below the average 5, and we interpret this as a pessimistic bias. We find that women are more “pessimistic” than men, as are old people relative to young. We also analyze how our notion of pessimism is related to more general notions of pessimism previously introduced in psychology.     

On the intensity of downside risk aversion

The degree of downside risk aversion (or equivalently prudence) is so far usually measured by . We propose here another measure, , which has specific and interesting local and global properties. Some of these properties are to a wide extent similar to those of the classical measure of absolute risk aversion, which is not always the case for . It also appears that the two measures are not mutually exclusive. Instead, they seem to be rather complementary as shown through an economic application dealing with a simple general equilibrium model of savings.

A Note on the Computation of the Mean Random Consistency Index of the Analytic Hierarchy Process (Ahp)

In this paper, we use Saaty's Eigenvector Method and the Power Method as well as =1, 2, , 9, 1/2, 1/3, , 1/9} and -={1,2, ,9,1, 1/2, ,1/9} as the sets from which the pairwise comparison judgments are assigned at random to examine the variation in the values determined for the mean random consistency index. By extensive simulation analysis, we found that both methods produce the same values for the mean random consistency random index. Also, we found that the reason for producing two different sets of values is the use of vs. - and not the selection of the Power Method vs. Saaty's Eigenvector Method.     

A formulation of the determinism hypothesis

The author tries to formulate what a determinist believes to be true. The formulation is based on some concepts defined in a systems-theoretical manner, mainly on the concept of an experiment over the sets A ^m (a set of m-tuples of input values) and B ⁿ (a set of n-tuples of output values) in the time interval (t ₁, ..., t _k ) (symbolically E[t ₁,..., t _k , A ^m , B ⁿ ]), on the concept of a behavior of the system S ^m,n (=(A ^m , B ⁿ )) on the basis of the experiment E[t ₁, ..., t _k , A ^m , B ⁿ ] and, indeed, on the concept of deterministic behavior .... The resulting formulation of the deterministic hypothesis shows that this hypothesis expresses a belief that we always could find some hidden parameters.     

Why I am not an objective Bayesian; some reflections prompted by Rosenkrantz

Summary The objective Bayesian program has as its fundamental tenet (in addition to the three Bayesian postulates) the requirement that, from a given knowledge base a particular probability function is uniquely appropriate. This amounts to fixing initial probabilities, based on relatively little information, because Bayes' theorem (conditionalization) then determines the posterior probabilities when the belief state is altered by enlarging the knowledge base. Moreover, in order to reconstruct orthodox statistical procedures within a Bayesian framework, only privileged ignorance probability functions will work.To serve all these ends objective Bayesianism seeks additional principles for specifying ignorance and partial information probabilities. H. Jeffreys' method of invariance (or Jaynes' modification thereof) is used to solve the former problem, and E. Jaynes' rule of maximizing entropy (subject to invariance for continuous distributions) has recently been thought to solve the latter. I have argued that neither policy is acceptable to a Bayesian since each is inconsistent with conditionalization. Invariance fails to give a consistent representation to the state of ignorance professed. The difficulties here parallel familiar weaknesses in the old Laplacean principle of insufficient reason. Maximizing entropy is unsatisfactory because the partial information it works with fails to capture the effect of uncertainty about related nuisance factors. The result is a probability function that represents a state richer in empirical content than the belief state targeted for representation. Alternatively, by conditionalizing on information about a nuisance parameter one may move from a distribution of lower to higher entropy, despite the obvious increase in information available.Each of these two complaints appear to me to be symptoms of the program's inability to formulate rules for picking privileged probability distributions that serve to represent ignorance or near ignorance. Certainly the methods advocated by Jeffreys, Jaynes and Rosenkrantz are mathematically convenient idealizations wherein specified distributions are elevated to the roles of ignorance and partial information distributions. But the cost that goes with the idealization is a violation of conditionalization, and if that is the ante that we must put up to back objective Bayesianism then I propose we look for a different candidate to earn our support.³¹     

Towards a more precise decision framework

The terms negative utility of gambling and risk aversion conflate three things:

Notes on the so-called incompleteness problem and on the proposed alternative concept of rational behavior

In reply to McClennen, the paper argues that his criticism is based on a mistaken assumption about the meaning of rationality postulates, to be called the Implication Principle. Once we realize that the Implication Principle has no validity, McClennen's criticisms of what he calls the Reductio Argument and what he calls the Incentive Argument fall to the ground. The rest of the paper criticizes the rationality concept McClennen proposes in lieu of that used by orthodox game theory. It is argued that McClennen's concept is inconsistent with the behavior of real-life intelligent egoists; it is incompatible with the way payoffs are defined in game theory; and it would be highly dangerous as a practical guide to human behavior.The author is indebted to the National Science Foundation for financial support trough Grant GS-3222, administered through the Center for Research in Management Science, University of California, Berkeley.     

Experiments with cooperative 2 × 2 games

Nash's solution of a two-person cooperative game prescribes a coordinated mixed strategy solution involving Pareto-optimal outcomes of the game. Testing this normative solution experimentally presents problems in as much as rather detailed explanations must be given to the subjects of the meaning of threat strategy, strategy mixture, expected payoff, etc. To the extent that it is desired to test the solution using naive subjects, the problem arises of imparting to them a minimal level of understanding about the issue involved in the game without actually suggesting the solution.Experiments were performed to test the properties of the solution of a cooperative two-person game as these are embodied in three of Nash's four axioms: Symmetry, Pareto-optimality, and Invariance with respect to positive linear transformations. Of these, the last was definitely discorroborated, suggesting that interpersonal comparison of utilities plays an important part in the negotiations.Some evidence was also found for a conjecture generated by previous experiments, namely that an externally imposed threat (penalty for non-cooperation) tends to bring the players closer together than the threats generated by the subjects themselves in the process of negotiation.     

Objective knowledge out of ignorance: Popper on body,mind, and the third world

In the fifties, Popper defended an interactionistic version of body-mind dualism. It distinguished between the world of physical bodies and states and the world of mental states. Later he added a third world of objective thought contents. He claims the assumption that there is the third world is a necessary presupposition of problem-solving in general and of his philosophy of science in particular. The present article contains separate reasonings to the effect that this presupposition is neither necessary nor even possible. It is further argued that postulating the existence of entities makes sense only relative to a criterion of ontological commitment, which Popper does not mention and obviously does not have, and that in addition it presupposes a theory, which is tentatively accepted as true and which according to the criterion implies the existence of the entities. But as yet there is no testable theory involving terms like mind, intention etc., which made the notion that itself or its terms are essentially different from what is already known in the empirical sciences at least plausible. Therefore the body-mind controversy is still pointless. Popper's stand on it seems to be but a reflex of his anti-behavioristic and anti-psychologistic attitude.     

Can we rationally learn to coordinate?

In this paper we examine the issue whether individual rationality considerations are sufficient to guarantee that individuals will learn to coordinate. This question is central in any discussion of whether social phenomena (read: conventions) can be explained in terms of a purely individualistic approach. We argue that the positive answers to this general question that have been obtained in some recent work require assumptions which incorporate some convention. This conclusion may be seen as supporting the viewpoint of institutional individualism in contrast to psychological individualism.     

Games,goals, and bounded rationality

A generalization of the standard n-person game is presented, with flexible information requirements suitable for players constrained by certain types of bounded rationality. Strategies (complete contingency plans) are replaced by policies, i.e., endmean pairs of goals and controls (partial contingency plans), which results in naturally disconnected player choice sets. Well-known existence theorems for pure strategy Nash equilibrium and bargaining solutions are generalized to policy games by modifying connectedness (convexity) requirements.     

Risk seeking with diminishing marginal utility in a non-expected utility model

The present work takes place in the framework of a non-expected utility model under risk: the RDEU theory (Rank Dependent Expected Utility, first initiated by Quiggin under the denomination of Anticipated Utility), where the decision maker's behavior is characterized by two functionsu andf. Our first result gives a condition under which the functionu characterizes the decision maker's attitude towards wealth. Then, defining a decision maker as risk averter (respectively risk seeker) when he always prefers to any random variable its expected value (weak definition of risk aversion), the second result states that a decision maker who has an increasing marginal utility of wealth (a convex functionu) can be risk averse, if his functionf issufficiently below his functionu, hence if he is sufficientlypessimistic. Obviously, he can also be risk seeking with a diminishing marginal utility of wealth. This result is noteworthy because with a stronger definition of risk aversion/risk seeking, based on mean-preserving spreads, Chew, Karni, and Safra have shown that the only way to be risk averse (in their sense) in RDEU theory is to have, simultaneously, a concave functionu and a convex functionf.     

Simulation methodology

Operational researchers, management scientists, and industrial engineers have been asked by Russell Ackoff to become systems scientists, yet he stated that Systems Science is not a science. (TIMS Interfaces, 2 (4), 41). A. C. Fabergé (Science 184, 1330) notes that the original intent of operational researchers was that they be scientists, trained to observe. Hugh J. Miser (Operations Research 22, 903), views operations research as a science, noting that its progress indeed is of a cyclic nature.The present paper delineates explicitly the attributes of simulation methodology. Simulation is shown to be both an art and a science; its methodology, properly used, is founded both on confirmed (validated) observation and scrutinised (verified) art work.The paper delineates the existing procedures by which computer-directed models can be cyclically scrutinised and confirmed and therefore deemed credible. The complexities of the phenomena observed by social scientists are amenable to human understanding by properly applied simulation; the methodology of the scientist of systems (the systemic scientist).

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Résumé Russell Ackoff propose à ceux qui s'occupent de recherches opérationnelle, industrielle, et de gestion, d'agir en systems scientists, et pourtant il affirme que systems science n'est pas une science (TIMS Interfaces 2 (4), 41). A. C. Fabergé (Science 184, 1330) remarque, qu'à l'origine, le but de ceux qui s'occupaient de recherche opérationnelle était d'agir en hommes de science instruits à observer. Hugh J. Miser (Operational Research 22, 903) considère la recherche opérationnelle comme science, notant que ses progrès sont en effet de nature cyclique.La présente étude délimite explicitement les attributs de la méthode de la simulation. Il est démontré que la simulation est à la fois un art et une science; sa méthode, lorsqu'utilisée correctement, repose sur l'observation validée et le modèle vérifié.L'étude délimite les moyens actuels dont nous disposons pour vérifier et valider cycliquement les modèles bâtis à l'aide d'ordinateurs, établissant ainsi leur crédibilité. La nature complexe des phénomènes étudiés par les sciences sociales peut être comprise à l'aide de la simulation: la méthode dont se servent les hommes de science qui étudient les systèmes (les scientistes systémiques).

     

Coherent decision analysis with inseparable probabilities and utilities

This article explores the extent to which a decision maker's probabilities can be measured separately from his/her utilities by observing his/her acceptance of small monetary gambles. Only a partial separation is achieved: the acceptable gambles are partitioned into a set of belief gambles, which reveals probabilities distorted by marginal utilities for money, and a set of preference gambles, which reveals utilities reciprocally distorted by marginal utilities for money. However, the information in these gambles still enables us to solve the decision maker's problem: his/her utility-maximizing decision is the one that avoids arbitrage (i.e., incoherence or Dutch books).     

The Optimality of the Expert and Majority Rules Under Exponentially Distributed Competence

We study the uncertain dichotomous choice model. In this model a set of decision makers is required to select one of two alternatives, say support or reject a certain proposal. Applications of this model are relevant to many areas, such as political science, economics, business and management. The purpose of this paper is to estimate and compare the probabilities that different decision rules may be optimal. We consider the expert rule, the majority rule and a few in-between rules. The information on the decisional skills is incomplete, and these skills arise from an exponential distribution. It turns out that the probability that the expert rule is optimal far exceeds the probability that the majority rule is optimal, especially as the number of the decision makers becomes large.     

How can an expert system help in choosing the optimal decision?

We introduce a rationality principle for a preference relation on an arbitrary set of lotteries. Such a principle is a necessary and sufficient condition for the existence of an expected utility agreeing with . The same principle also guarantees a rational extension of the preference relation to any larger set of lotteries. When the extended relation is unique with respect to the alternatives under consideration, the decision maker does not need a numerical evaluation in order to make a choice. Such a rationality condition needs little information in order to be applied, and its verification amounts to solving a linear system.The present research is supported by the Research Contract of CNR (Research National Council) 1989 and 1990 Decision Models under uncertainty and risk, for expert systems with incomplete and revisable information.     

Lattices,bargaining and group decisions

This essay aims at constructing an abstract mathematical system which, when interpreted, serves to portray group-choices among alternatives that need not be quantifiable. The system in question is a complete distributive lattice, on which a class of non-negative real-valued homomorphisms is defined. Reinforced with appropriate axioms, this class becomes a convex distributive lattice. If this lattice is equipped with a suitable measure, and if the mentioned class of homomorphisms is equipped with a metric, then the class and its convex sets are seen to possess certain characteristic properties. The main result (Theorem 6) follows from a combination of these results and a famous result due to Choquet.The mathematical scheme is then interpreted in the subject-language of choice among alternatives. It is shown, by means of an example, that the system furnishes all the ingredients for describing multi-group choices. Whether or not the same ingredients are also adequate for a behavioural theory of multi-group choices is an issue that will not be gone into. However, the example effectively illustrates how a process of bargaining can be described with the aid of the mathematical scheme.In the second example, a class of bargaining situations is modelled in the symbolism of linear programming with several objective functions combined with unknown weights; the cost vectors in such formulations are identified with homomorphisms, and the main theorem of this essay is applied.     

On Seidenfeld‘s Criticism of Sophisticated Violations of the Independence Axiom 1

An agent who violates independence can avoid dynamic inconsistency in sequential choice if he is sophisticated enough to make use of backward induction in planning. However, Seidenfeld has demonstrated that such a sophisticated agent with dependent preferences is bound to violate the principle of dynamic substitution, according to which admissibility of a plan is preserved under substitution of indifferent options at various choice nodes in the decision tree. Since Seidenfeld considers dynamic substitution to be a coherence condition on dynamic choice, he concludes that sophistication cannot save a violator of independence from incoherence. In response to McClennens objection that relying on dynamic substitution when independence is at stake must be question-begging, Seidenfeld undertakes to prove that dynamic substitution follows from the principle of backward induction alone, provided we assume that the agents admissible choices from different sets of feasible plans are all based on a fixed underlying preference ordering of plans. This paper shows that Seidenfelds proof fails: depending on the interpretation, it is either invalid or based on an unacceptable assumption.     

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.