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.     

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.     

On Harrod's first refining principle

Harrod introduced a refinement to crude Utilitarianism with the aim of reconciling it with common sense ethics. It is shown (a) that this refinement (later known as Rule Utilitarianism) does not maximise utility (b) the principle which truly maximizes utility, marginal private benefit equals marginal social cost, requires that a number of forbidden acts like lying be performed. Hence Harrod's claim that his refined Utilitarianism is the foundation of moral institutions cannot be sustained. Some more modern forms of Utilitarianism are reinterpreted in this paper as utility maximizing decision rules. While they produce more utility than Harrod's rule, they require breaking the moral rules some of the time, just like the marginal rule mentioned above. However, Harrod's rule is useful in warning the members of a group, considered as a single moral agent, of the externalities that lie beyond the immediate consequences of the collective action.     

Decision-making under ignorance with implications for social choice

A new investigation is launched into the problem of decision-making in the face of complete ignorance, and linked to the problem of social choice. In the first section the author introduces a set of properties which might characterize a criterion for decision-making under complete ignorance. Two of these properties are novel: independence of non-discriminating states, and weak pessimism. The second section provides a new characterization of the so-called principle of insufficient reason. In the third part, lexicographic maximin and maximax criteria are characterized. Finally, the author's results are linked to the problem of social choice.     

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.     

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.     

Rational choice and economic behavior

This paper discusses several concepts that can be used to provide a foundation for a unified, theory of rational, economic behavior. First, decision-making is defined to be a process that takes place with reference to both subjective and objective time, that distinguishes between plans and actions, between information and states and that explicitly incorporates the collection and processing of information. This conception of decision making is then related to several important aspects of behavioral economics, the dependence of values on experience, the use of behavioral rules, the occurrence of multiple goals and environmental feedback.Our conclusions are (1) the non-transitivity of observed or revealed preferences is a characteristic of learning and hence is to be expected of rational decision-makers; (2) the learning of values through experience suggests the sensibleness of short time horizons and the making of choices according to flexible utility; (3) certain rules of thumb used to allow for risk are closely related to principles of Safety-First and can also be based directly on the hypothesis that the feeling of risk (the probability of disaster) is identified with extreme departures from recently executed decisions. (4) The maximization of a hierarchy of goals, or of a lexicographical utility function, is closely related to the search for feasibility and the practice of satisficing. (5) When the dim perception of environmental feedback and the effect of learning on values are acknowledged the intertemporal optimality of planned decision trajectories is seen to be a characteristic of subjective not objective time. This explains why decision making is so often best characterized by rolling plans. In short, we find that economic man - like any other - is an existential being whose plans are based on hopes and fears and whose every act involves a leap of faith.This paper is based on a talk presented at the Conference, New Beginnings in Economics, Akron, Ohio, March 15, 1969. Work on this paper was supported by a grant from the National Science Foundation.     

On Playing Fair: Professor Binmore on Game Theory and the Social Contract

This paper critically reviews Ken Binmores non- utilitarian and game theoretic solution to the Arrow problem. Binmores solution belongs to the same family as Rawls maximin criterion and requires the use of Nash bargaining theory, empathetic preferences, and results in evolutionary game theory. Harsanyi has earlier presented a solution that relies on utilitarianism, which requires some exogenous valuation criterion and is therefore incompatible with liberalism. Binmores rigorous demonstration of the maximin principle for the first time presents a real alternative to a utilitarian solution.     

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.     

Fairness,Public Good,and Emotional Aspects of Punishment Behavior

We report an experiment on two treatments of an ultimatum minigame. In one treatment, responders reactions are hidden to proposers. We observe high rejection rates reflecting responders intrinsic resistance to unfairness. In the second treatment, proposers are informed, allowing for dynamic effects over eight rounds of play. The higher rejection rates can be attributed to responders provision of a public good: Punishment creates a group reputation for being tough and effectively educate proposers. Since rejection rates with informed proposers drop to the level of the treatment with non-informed proposers, the hypothesis of responders enjoyment of overt punishment is not supported.     

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     

Mean variance preferences and the heat equation

Chipman (1979) proves that for an expected utility maximizer choosing from a domain of normal distributions with mean and variance ² the induced preference functionV(, ) satisfies a differential equation known as the heat equation. The purpose of this note is to provide a generalization and simple proof of this result which does not depend on the normality assumption.     

Testing Descriptive Utility Theories: Violations of Stochastic Dominance and Cumulative Independence

Choices between gambles show systematic violations of stochastic dominance. For example, most people choose ($6, .05; $91, .03; $99, .92) over ($6, .02; $8, .03; $99, .95), violating dominance. Choices also violate two cumulative independence conditions: (1) If S = (z, r; x, p; y, q) R = (z, r; x, p; y, q) then S = (x, r; y, p + q) R = (x, r + p; y, q). (2) If S = (x, p; y, q; z, r) R = (x, p; y, q; z, r) then S = (x, p + q; y, r) R = (x, p; y, q + r), where 0 < z < x < x < y < y < y < z.Violations contradict any utility theory satisfying transivity, outcome monotonicity, coalescing, and comonotonic independence. Because rank-and sign-dependent utility theories, including cumulative prospect theory (CPT), satisfy these properties, they cannot explain these results.However, the configural weight model of Birnbaum and McIntosh (1996) predicted the observed violations of stochastic dominance, cumulative independence, and branch independence. This model assumes the utility of a gamble is a weighted average of outcomes\' utilities, where each configural weight is a function of the rank order of the outcome\'s value among distinct values and that outcome\'s probability. The configural weight, TAX model with the same number of parameters as CPT fit the data of most individuals better than the model of CPT.     

Rational preference: Decision theory as a theory of practical rationality

In general, the technical apparatus of decision theory is well developed. It has loads of theorems, and they can be proved from axioms. Many of the theorems are interesting, and useful both from a philosophical and a practical perspective. But decision theory does not have a well agreed upon interpretation. Its technical terms, in particular, utility and preference do not have a single clear and uncontroversial meaning.How to interpret these terms depends, of course, on what purposes in pursuit of which one wants to put decision theory to use. One might want to use it as a model of economic decision-making, in order to predict the behavior of corporations or of the stock market. In that case, it might be useful to interpret the technical term utility as meaning money profit. Decision theory would then be an empirical theory. I want to look into the question of what utility could mean, if we want decision theory to function as a theory of practical rationality. I want to know whether it makes good sense to think of practical rationality as fully or even partly accounted for by decision theory. I shall lay my cards on the table: I hope it does make good sense to think of it that way. For, I think, if Humeans are right about practical rationality, then decision theory must play a very large part in their account. And I think Humeanism has very strong attractions.     

Constructs and empirical basis in theories of economic behavior

Theories of economic behavior often use as-if-languages: for example, analytical sentences or definitions are used as if they were synthetic and factual-normative theoretical constructs are used as if they were empirical concepts. Such as-if-languages impede the acquisition of knowledge and are apt to encourage the wrong assessment of actual research strategies. The author's criticism is first leveled at revealed-preference theory. In this theory observed behavior is often understood in an empirical sense although it is a pure theoretical construct. Another example can be found in von Mises' representations on marketing behavior: here theoretical valuations are used to achieve a spurious streamlining of reality. Result: Scientists should not ogle with reality if they have nothing to say about it.     

Analysis,modeling, and the management of international negotiations

Negotiation management, a framework for practical implementation of computer-based support to negotiators and their staffs, is defined and described in this paper. In particular, three integrated tools and their underlying methodologies, are illustrated. The collaborative use of computers and information techniques in complex negotiation is framed in terms of a discovery and design paradigm in which parties engage in dialogue, learn, and develop trust that can support the negotiation itself through a process of joint analysis and modeling.     

On the foundations of mean-variance analyses

Let (, ) and (, ) be mean-standard deviation pairs of two probability distributions on the real line. Mean-variance analyses presume that the preferred distribution depends solely on these pairs, with primary preference given to larger mean and smaller variance. This presumption, in conjunction with the assumption that one distribution is better than a second distribution if the mass of the first is completely to the right of the mass of the second, implies that (, ) is preferred to (, ) if and only if either > or ( = and < ), provided that the set of distributions is sufficiently rich. The latter provision fails if the outcomes of all distributions lie in a finite interval, but then it is still possible to arrive at more liberal dominance conclusions between (, ) and (, ).This research was supported by the Office of Naval Research.     

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.     

Liberals,information and Webster's principles

Critiques two social choice principles employed by Webster's analysis of using information to resolve Sen's paradox of the Impossibility of a Paretian Liberal.     

Expert rule versus majority rule under partial information

We study the uncertain dichotomous choice model. Under some assumptions on the distribution of expertise of the various panel members, the probability of the expert rule to be the optimal one is compared to that of the majority rule to be optimal. It turns out that for the former probability exceeds the latter by far, especially as the panel size becomes large.