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Within the framework of stochastic two-person nonzero-sum games, we deal with two commonly used models in engineering and economics—namely, the LQG (Linear-Quadratic-Gaussian) and the duopoly problems. We investigate how variations in information available to either player affect the equilibrium Nash strategies for these two models, whose existence and uniqueness have been proven in the paper. We show that for the LQG model better information for either player results in lower average Nash costs for both players; whereas for the duopoly model better information for one player helps him alone to achieve a higher average Nash profit, and it hurts the other player in the sense that his average Nash profit decreases. We further relate these properties of the Nash solutions for these two games to some of the distinct features of zero-sum games and team problems.

This paper investigates some convergence properties of Fiacco and McCormick's SUMT algorithm. A convex programming problem is given for which the SUMT algorithm generates a unique unconstrained minimizing trajectory having an infinite number of accumulation points, each a global minimizer to the original convex programming problem.

In this paper, we considern-person weighted majority games with major and minor players. We prove under certain conditions that if the minor players split into equal “mini-minor” players the nucleolus' components which correspond to the major players do not change.

A simple two-person exchange model is considered first as a cooperative game without side payments, then as a non-cooperative game. It is shown that the Nash equilibrium solution of the latter agrees with the von Neumann-Morgenstern and core solutions of the former. This is made possible by the Pareto optimality of the Nash equilibrium, which is the main theoretical result presented.

A class of differentiated products is completely described by a vector of objectively measured characteristics. Observed product prices and the specific amounts of characteristics associated with each good define a set of implicit or "hedonic" prices. A theory of hedonic prices is formulated as a problem in the economics of spatial equilibrium in which the entire set of implicit prices guides both consumer and producer locational decisions in characteristics space. Buyer and seller choices, as well as the meaning and nature of market equilibrium, are analyzed. Empirical implications for hedonic price regressions and index number construction are pointed out.

The string-to-string correction problem is to determine the distance between two strings as measured by the minimum cost sequence of “edit operations” needed to change the one string into the other. The edit operations investigated allow changing one symbol of a string into another single symbol, deleting one symbol from a string, or inserting a single symbol into a string. An algorithm is presented which solves this problem in time proportional to the product of the lengths of the two strings. Possible applications are to the problems of automatic spelling correction and determining the longest subsequence of characters common to two strings.

The use of quantile criteria in two-person matrix games is explored: maximizing the confidence of a fixed payoff level or maximizing the payoff level for a given confidence level.

A general net profit model is developed consisting of the difference between the gross profit and the total inventory costs. This model considers that the demand is a function of the selling price, that the selling price depends on the pricing policy and on the unit cost which in its turn varies according to the order quantity. It is shown that the correct treatment of inventory problems is that which considers its interaction with problems of pricing since the obtainable net profit can be increased over the amounts achieved when inventory problems were regarded as independent subsystems. Further it has been found that there exists a critical demand elasticity such that the preferable pricing policy is different for prevailing elasticities above this value than for elasticities below it.

Under discriminatory pricing, the firm's marginal revenue is equated with its marginal cost at each market point in the space economy. This pricing practice is more profitable than the f. o.b. pricing schedule which involves equality of aggregate marginal revenue and marginal cost. From this departure point, the present paper determines the effect on the theory of spatial price discrimination of homogeneous and heterogeneous local demand schedules and different degrees of competition over economic space. By relaxing the customary assumptions of homogeneity and assuming varying levels of competition, better explanations of spatial pricing policies obtain. A case study is also presented to indicate the importance of the alternative assumptions made in spatial price discriminating theory. It is shown that different demand schedules and varying levels of competition tend to promote discriminatory prices over economic space.

In this paper the shortest confidence interval for the ratio of two variances is derived when the populations are assumed to be normal. A table, which will help in computing the shortest confidence interval, is also provided.

This paper considers the price adjustment process in a market which retains the characteristics of a perfectly competitive market except that individual firms are price-setters. Buyers, unaware initially of what prices which firms are charging, indulge in search by contacting a sample of firms and buy (according to a demand curve) from the lowest-price firm encountered. Firms set prices to maximise profits over their perceived (or estimated) demand curve, and update their estimated demand curve in accordance with the observed change in demand between successive time periods. It is shown that the price distribution converges to a degenerate distribution centred on the monopoly price.

This paper presents an algorithm for the global maximization of a convex function subject to linear inequality constraints. It is computationally finite and is designed to converge rapidly on problems in which there are few local optima or the global optimum is significantly better than most of the other local optima.

Some properties of “Davidon”, or variable metric, methods are studied from the viewpoint of convex analysis; they depend on the convexity of the function to be minimized rather than on its being approximately quadratic. An algorithm is presented which generalizes the variable metric method, and its convergence is shown for a large class of convex functions.

Iran has nationalized water and is faced with the problems of pricing it. A proposal for pricing irrigation water is presented in which water is managed by nonprofit water authorities. An irrigation project is divided into zones, and a single price is proposed for each zone. Operation and maintenance costs are set as a lower limit on price, and a certain fraction of ability to pay is set as an upper limit. Zones are ranked by assigned reimbursable costs, and a system is proposed that would produce water revenues in aggregate equal to total project reimbursable costs. For those zones where water is priced above reimbursable cost the price is set at that level where the price minus reimbursable costs, as a percentage of the difference between the upper limit and reimbursable cost, is the same for all zones. For those zones where the price is below reimbursable cost, the recommended price is at the upper limit.

Among other things a global version of the fundamental theorem of welfare economics is proved. One starts with a pure exchange economy with fixed total resources where hypotheses of differentiability, convexity, and monotonicity are made on the utility functions. Let ʌ be the set of price equilibria where the initial allocation coincides with the final one. Then the map which assigns to such a price equilibrium, the corresponding allocation is a diffeomorphism (a complete correspondence) between ʌ and the set of Pareto Optima.

In this paper a variant of the Scarf–Hansen fixed-point algorithm is applied to approximate a Nash equilibrium point in an N-person game. A numerical example is given to illustrate the application of the algorithm. A procedure is suggested to generate several Nash equilibrium points in the case where there is more than one. A numerical example is given to illustrate the procedure.

Here we study the structure of Nash equilibrium points forN-person games. For two-person games we observe that exchangeability and convexity of the set of equilibrium points are synonymous. This is shown to be false even for three-person games. For completely mixed games we get the necessary inequality constraints on the number of pure strategies for the players. Whereas the equilibrium point is unique for completely mixed two-person games, we show that it is not true for three-person completely mixed game without some side conditions such as convexity on the equilibrium set. It is a curious fact that for the special three-person completely mixed game with two pure strategies for each player, the equilibrium point is unique; the proof of this involves some combinatorial arguments.

This paper analyzes behavior of a cooperative of raw material producers. The cooperative sells a production input to producers, provides a "free" service to members, and bargains with processors for raw material price. One analysis assumes the cooperative's objective is maximization of the raw material price received by members. Another assumes the objective is maximization of quantity marketed through the cooperative. The cooperative has three instruments to manipulate to attain its objective. First-order maximization conditions for the two objectives are quite different from each other and from "marginal cost equals marginal revenue" conditions.

THE traditional and still widely accepted theory of the firm is based on the assumption of profit maximization. This assumption has, however, been extensively criticized. In particular, it is argued that the businessman will maximize his utility but not his profits. This criticism may have some plausibility in the case of big corporations where ownership is separated from control. For the case of an owner-managed firm, I wish to show that utility maximization leads to profit maximization, provided 'profit' is appropriately defined. By the appropriate definition of profit, I mean, in particular, that 'profit' is not only net of the outlays on hired or purchased inputs (including interest on capital), but also net of the imputed wages to the owner-manager qua manager. It is obvious that the maximization of profit gross of this imputed wages makes no sense as it implies that, as long as the marginal return is positive, the owner-manager will work 24 hours a day and 365 days a year. The definition of profit as being net of the owner-manager's imputed wages was in fact used by Scitovsky (I943) when he attempted to show that a utility-maximizing owner-manager would maximize profit only if he has a peculiar indifference map between income and leisure. I (Ng, I969) have already shown that profit maximization does not imply this peculiar indifference map. However, both Scitovsky and I did not take account of the possibility of hiring or selling managerial services by the owner-manager. In this paper, I wish to generalize my argument to include such possibilities and also to supplement the argument by a general mathematical treatment.'

Systems selling is a concept that many firms have adopted in recent years to guide their product and marketing strategy. In this paper by Lars-Gunnar Mattsson, economic consequences of systems selling for the industrial goods seller are analyzed. Four different types of revenue consequences are identified. It is argued that the seller should let its market analysis be influenced by which of these four consequences are consistent with its objectives in choosing a systems selling approach. It is shown that systems selling increases product differentiation and barriers to entry into a market, thus increasing the system selling firm’s profit opportunities. Cost consequences are analyzed in terms of size of investments needed and heterogeneity of customers’ needs, and the paper concludes with a discussion of the factors influencing the relative advantage for systems selling in the international market.

The optimum price switching problem consists in determining simultaneously the optimum prices as well as the optimum decision times for switching from price to price. This paper proposes a dynamic formulation for the pricing problem faced by a firm, and provides neee9aary conditions for the optimality of both prices and price switches. Subsequently, the paper discusses the economics of price selection and the economics of decision times.

The major purpose of this study is to analyze the distinction between Pareto's " ophelimity " and " utility " theories, a distinction which has been overlooked in the literature on Pareto. This paper will also serve to bring out the reasons why the distinction was of importance to Pareto himself. Essentially, the distinction between Pareto's ophelimity and utility theories has a methodological foundation, specifically, his views on scope. The paper is organized into three main parts. The first part presents Pareto's views on scope. The second part focuses attention on the equilibrium of systems defined in terms of scope. Finally, Pareto's ophelimity and utility theories are used as illustrations of Paretian welfare theory applied to two different systems.

When future production is a random variable, the equilibrium price for a commodity will be that which yields the maximum expected discounted utility of the consumption quantities and the supply of storage. The equilibrium consumption schedule is obtained with the dynamic programming techniques of inventory theory.

A rational price system is one of the most important, complex, and unsolved problems of contemporary economic science. The positions of defenders of different conceptions of price formation differ mainly by their approach to the method of determining the share of profit in the price of each product. At present, there exist four basic conceptions of price formation, solving this problem by one of the following methods: the share of profit is defined in proportion to wages; in proportion to enterprise cost of production; in proportion to the total value of fixed and working capital used in the production of a given product; the price system is based on a calculation of shadow prices.

General convergence theorems for Newton method applied to convex operators for partially ordered topological linear spaces

Four kinds of word strings were heard behind masking noise in a speech-perception experiment. There were ordinary sentences (strings), semantically anomalous “sentences” (strings) and scrambled versions of each, called scrambled ordinary and scrambled anomalous strings. Words were systematically erased from each string such that string length varied from one to (all) five principal words, making a 4-string types × 5-string lengths design. All strings were rated by other S s on scales of meaningfulness or grammaticality. Some results were: (a) ordinary sentences were heard better through noise and rated as more grammatical and as more meaningful; (b) for all three measures the ordinary-string scores diminished relatively less or not at all as a function of increasing string length; and (c) the anomalous strings were rated as more grammatical than the scrambled ordinary strings, but the scrambled ordinary strings were rated more meaningful and were heard relatively better.

Profit-making organizations have moved from the single goal of profit maximization to ill-defined goals including corporate citizenship and responsibility to society. The single goal of maximum pro...

The Price System in Socialist Economy. After analyzing the short-comings of the current price system in vigor in Poland and in most of the Socialist economies, the author presents various measures for optimal price-fixing. He first demonstrates how methods of planned price-fixing, based on production costs, evolve in a vicious circle with prices depending on costs, and costs on prices, causing discrepancies in the price structure and making periodic adjustments necessary. Likewise, the market price is not a sound indicator of the optimal volume of supply and demand, nor of the level of satisfaction of social needs. Consequently, measurable, objective standards must be established in order to determine optimum social needs within the given economic situation. The author proposes that a uniform degrés of satisfaction of the demand for all available goods and services could be used as the standard of optimization. This standard would be achieved by balancing prices in such a way as to eliminate exaggerated profits or losses; that is, when an average profit, expressed in terms of a mean profit rate, is guaranteed. The last part of the article is devoted to different means of establishing the sought for equilibrium between the balanced market price and the production price or vice versa, to obtain the mean profit rate., Après avoir analysé les défauts du système des prix actuellement en vigueur en Pologne et dans la plupart des économies socialistes, l'auteur propose différentes formules de fixation des prix. Il montre d'abord que les méthodes de fixation planifiée des prix fondées sur les coûts de production aboutissent à un cercle vicieux où les prix dépendent des coûts et les coûts des prix. Il en résulte des distorsions importantes de la structure des prix et la nécessité de procéder à des ajustements périodiques. De même, le prix d'équilibre du marché ne peut à lui seul renseigner sur la dimension optimale de l'offre et de la demande et sur le degré de satisfaction des besoins sociaux. Par conséquent, il convient de déterminer des critères objectifs et mesurables permettant de fixer de manière optimale les besoins sociaux dans le cadre d'une situation économique donnée. L'auteur propose de retenir comme critère d'optimalisation, le degré uniforme de satisfaction de la demande pour tous les biens et services accessibles. Le niveau sera obtenu quand le prix d'équilibre formé sur le marché n'assure ni un bénéfice excessif, ni un déficit, c'est-à-dire qu'il garantit un profit moyen exprimé par le taux de profit moyen. La fin de l'article est consacrée aux diverses méthodes permettent d'égaliser le prix d'équilibre du marché et le prix de production ou inversement, égalisation qui selon l'auteur assurera le taux de profit moyen recherché., Popkiewicz Józef. Le système des prix en économie socialiste. In: Revue de l'Est, vol. 4, 1973, n°3. La Pologne. pp. 49-63.

A simulated model of an auction market is developed showing the relationship between the variation in valuations, the price variation and the number of independent bidders in the market. Average prices paid in a market with two or three bidders are less than average valuations. Average prices are progressively greater than average valuations as the number of bidders increases beyond four. Some applications of this model in the Australian wool market are discussed.

Here, we present various new forms of the Bernstein-Vazirani algorithm beyond qubit systems. First, we review the Bernstein-Vazirani algorithm for determining a bit string. Second, we discuss the generalized Bernstein-Vazirani algorithm for determining a natural number string. The result is the most notable generalization. Thirdly, we discuss the generalized Bernstein-Vazirani algorithm for determining an integer string. Finally, we discuss the generalized Bernstein-Vazirani algorithm for determining a complex number string. The speed of determining the strings is shown to outperform the best classical case by a factor of the number of the systems in every cases. Additionally, we propose a method for calculating many different matrices simultaneously. The speed of solving the problem is shown to outperform the classical case by a factor of the number of the elements of them. We hope our discussions will give a first step to the quantum simulation problem.

Herein, a ring is not required to have an identity. All rings are associative but not necessarily commutative. However, we specialize to the commutative case for some of our results. The paper is concerned primarily with rings having the property that all unbounded ideals or all unbounded homomorphic images are isomorphic to the ring. We say that a ring R is bounded if nR = 0 for some positive integer n; alternately, R, with or without 1, is said to have finite characteristic. Unbounded rings having the property that all proper subrings are bounded were characterized in [8].

A description of the characteristics of a poly phase alternating current revolving-field generator provided with a single closed-circuit field winding and commutator. Effect of shifting brushes at various inductive and non-inductive loads, represented graphically.

Does unit-price information alter purchase decisions of consumers in supermarkets? Will the change to a unit-pricing system create burdensome costs for retailers? The authors report the results of unit-pricing experiments and assess the benefits of unit pricing.

PROFIT maximization over time is the goal of economic production regardless of whether the production process is of fixed or variable length. The classical criteria for profit maximization are consistent with maximization over time when the production process is of fixed length. Variable length production processes, as encountered in livestock feeding, require somewhat different criteria from those given by the classical theory of the firm. The length of a process can affect both production costs and revenues (e.g., length of feeding of livestock). When there is a time lag between the end of one variable length process and the beginning of another, profit maximization is achieved by extending the process so long as the extension results in additions to revenue greater than or equal to additions to cost.

The general equilibrium of the national economy is an important problem in socialist construction. Some time ago, a great many discussions concerning this problem were conducted by China's economic circles. In this article, some of my personal approaches to such problems as the nature of general equilibrium, its content, its focus, and the principle of planning general equilibrium, etc., will be presented.

The comparative price and differential price of a commodity are the components of a commodity price system. The price relationship among different commodities forms the comparative price of a commodity; the price discrepancies of the same kind of commodity during various phases of production and circulation form the differential price of a commodity. We offer, in this article, our shallow view on the problem of the differentialprice of a commodity.

We prove a theorem on the existence of an invariant subspaces of special type for maximal J-dissipative and maximal J-symmetric operators.

Commutating capacitor voltage and current equations are derived for the commutation period of the main and auxiliary SCRs. The circuit turn-off time is computed as a function of x, where x is the ratio of the maximum load current to be commutated to the peak discharge current of the commutating capacitor. A suitable choice of x is suggested. In addition to the worst case design for L1, L2, and C, a nomograph for the selection of L1, L2, and C is given as a function of x.

Neighboring optimum feedback guidance to motivate min-distance lookup parameter determined by minimizing metric function of perturbed state and reference trajectory

We present algorithms which use only $O(\sqrt n )$ nonscalar multiplications (i.e. multiplications involving “x” on both sides) to evaluate polynomials of degree n, and proofs that at least $\sqrt ...

It is difficult to measure the price of life insurance protection because the policy contract is complex and because the necessary data frequently are not readily available. Thus, the market for individual life insurance is characterized by price ignorance and a lack of effective price competition. Many policyholders are paying more than necessary for their life insurance protection, in the sense that some companies are charging much higher prices than others for essentially the same coverage. The appropriate device to make life insurance price competition more effective is a rigorous system of price disclosure that would permit careful buyers to make reasonably informed purchase decisions.

The theory of effective rate of protection (ERP) has been developed in recent years in an attempt to seek a concept G,? protection which, in the presence of traded inputs, would be able to perform analytically the role that nominal tariffs played in the “older’“, traditional theory which was premised on a model which excluded traGed inputs. Thus, in the Zraditional model, with two traded goods produced with standard restrictions on the production function? by two primary factors in given endowment, and the small-country assumption, a tariff on a good would lead to: (i) a rise in the (gross) output of the protected good; (ii) a rise in the nominal value of its output; (iii) a rise in the use of each primary factor therein; (iv) a rise in the real value-added therein (which coincides with output, when real value-addeC is defined as deflated by the price of “own output”); and (;lr) a rise in the nominal value-added therein (which coincides of course with the nominal value r Thanks are due to the National Science Foundation for supporting the research reported in this paper. We have had the benefit of correspondence and/or mutual discussions over the last year with Chulsoon Khang and, in particular, Michael Bruno, whose paper (1973) in this Symposium complements ours admirably. The careful comments of John Chipman have also led to many improvements. Above all, we are greatly indebted to Yasuo Uekawa whose extremely careful reading has resulted in the removal of errors from earlier drafts. 2 These should be linear homogene NIS, concave, and factor-intensities should difFer in equilibrium.

Problem 1.-How can these players operate simultaneously in both the dimensions of rationality and irrationality, of probability and magic? These cab driver-crapshooters usually bet according to known probability ("rational" aspect of the game), yet they have many magical practices in their betting and shooting that make little sense to an outside observer (the "irrational" aspects of the game). Once one understands the players' basic belief system, their system of cause and effect, one then sees that their magical practices are also "rational," that is, the strategies these players usse to maximize their own control over the dice when they are shooting and to minimize the control of other shooters are logically consistent within their belief system. Without understanding their belief system, we do not understand their behavior. Problem 2.- What is the origin of their magical behavior and belief system? An attempt is made to reconcile theories of Malinowski and Kroeber concerning the origin of magic whith ...

In this paper authors relate a method for solving a polynomial using an automatic computer. This method is one which developed further Lehmer Method, finds the all roots of a polynomial by knowing follows. 1) When a given polunomial has no root on the unit circle, the number or roots inside it. 2) When a given polynomial has a root or roots on the unit circle, the argument of all roots on it.

The repeated prisoner's dilemma game is converted into a differential game by assuming that the players, instead of making decisions individually for each repetition of the prisoner's dilemma game, make decisions on the ratio of cooperative and noncooperative games that they wish to play over the next few moves, and that the actual plays are then determined using this ratio and a randomizing procedure. Although it sounds like a significant departure, this assumption is probably not too different from reality.

This work is concerned primarily with presenting techniques for constructing bimatrix games with predetermined equilibrium points. Some partial results on Nash-solvability are also given.

The mathematical programming problem discussed is the convergence of a certain popular type of gradient procedure for maximizing a function under inequality constraints. An example shows that convergence to a solution need not always occur, and a theorem shows that under certain circumstances the gradient method does converge.

An equilibrium point in game theory is a situation where a unilateral change by only one of the players does not yield an increase of his profit or “payoff” function.In this paper it will be shown that an equilibrium point exists if: the payoff functions are continuous in all the variables and are concave in the variables belonging to each relevant player, the region in which the game is played is closed, bounded and convex and satisfies a weak continuity condition.The continuity condition is automatically satisfied if the region is a convex polyhedron. It is not assumed that the region in which the game is played is the product space of the subregions of each individual player. Therefore, the existence of an equilibrium point is shown for the general case of the so-called “mixed constraints”.

A game with quadratic pay-off, whose dynamics are described by evolution equations in Hilbert space, is considered. One player, trying to maximize the pay-off, chooses his control function for the game at the outset and we first investigate how the opposing player, who knows this choice, will use his controls to minimize the payoff. The controls of the minimizing player may vary throughout a Hilbert space, whilst the controls of the maximizing player are possibly restricted to a subset of a Hilbert space. An unexpected linear simplification shows that the maximizing player should try, roughly speaking, to get as far away as possible from the free motion of the minimizing player, and with this information a maximum principle is obtained.

A general approach to the decision problem confronting the individual player in a noncooperative game is outlined and applied to one player's choice of bid in a competitive sealed auction for a valuable object. That player's Bayes-optimal bid is characterized under both general and specific assumptions as to the subjective probability distributions expressing his judgments. Under the specific assumptions, analytically tractable results (in terms of Gamma fractiles) are presented. Equilibrium and maximin bids are presented for comparative purposes.