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In this paper we study an oligopoly market where profit-maximizing firms and socially concerned firms compete in quantities. Confronting remarks by Milton Friedman and Gary Becker, we are using an evolutionary setting to investigate the endogenous choice of the proper objective of business firms and the influence of product differentiation on the long run survival of firms which pursue non-profit motives. We find that firms which consider a combination of profit and consumer welfare can indeed have larger market shares and profits than their profit-maximizing rivals. One insight is that it might pay off for shareholders to consider stakeholder welfare, but that the level of social concern should not be too high. Based on a strategy׳s profitability, we consider asynchronous evolutionary updating with firms selecting Nash quantities or choosing best replies to the expected market quantity. Here we observe that the consumers׳ willingness to pay a price premium for products is crucial for the long run survival of socially concerned firms. Depending on the degrees of product differentiation and social concern, long run outcomes consist either of both types of firms or only one type of firm. If the firms׳ propensity to switch between a social or a profit-maximizing strategy is sufficiently large, steady states are unstable and even complicated dynamics can occur.

In this paper, we formulate and analyse a long-term multi-objective dynamic model for controlling invasive species. This optimization framework is then applied to the case of buffelgrass control in the Arizona desert. The proposed model simultaneously optimizes three objectives corresponding to three different valued and threatened resources including saguaros (a native cactus species), buildings and vegetation. The model is used to decide the optimal allocation of labour to these resources to control the population of the species in a multi-period planning horizon. The computational method to solve this problem is based on multi-objective integer programming.

This paper aims to show that delay matters in continuous- and discrete-time framework. It constructs a simple dynamic model of a boundedly rational monopoly. First the existence of the unique equilibrium state is proved under general price and cost function forms. Conditions are derived for its local asymptotical stability with both continuous and discrete time scales. The global dynamic behavior of the systems is then numerically examined, demonstrating that the continuous system is globally asymptotically stable without delay and in the presence of delay if the delay is sufficiently small. Then stability of the continuous system is lost via Hopf bifurcation. In the discrete case without delay, the steady state is locally asymptotically stable if the speed of adjustment is small enough, then stability is lost via period-doubling bifurcation. If the delay is one or two steps, then stability loss occurs via Neimark–Sacker bifurcation.

Oligopolies with product differentiation are examined with both quantity-adjusting and price-adjusting firms. Conditions are given for the positivity of the equilibrium outputs as well as for that of the equilibrium prices. The stability of the positive equilibria is then investigated; the positivity and stability conditions are also compared. The dependence of these conditions on the number of firms, their qualifications and the degree of relation between the goods is discussed. The analysis is given by assuming both discrete- and continuous-time scales.

Whether it be one security expert covering more systems or reducing total man-hours, there has always been a push to do more with less. Intuitively, we realize different systems need different levels of security. To aid in this effort, we develop multiresolution attacker/defender games by combining two game theoretic approaches: resource assignment and optimal response. We use the resource assignment game to determine the level of detail necessary to build the game needed to respond optimally to attacks. To aid in the selection of a resource assignment game and an optimal response game, we present considerations and survey numerous works. Further resource savings are possible when the optimal response games share features. Even though effort sharing between systems ought to be addressed during the resource-allocation game, we present both a linear effort sharing model and a method for solving post hoc. An illustrative example demonstrates the potential savings from our technique.

Dynamic monopolies are investigated with discrete and continuous time scales by assuming general forms of the price and cost functions. The existence of the unique profit maximizing output level is proved. The discrete model is then constructed with gradient adjustment. It is shown that the steady state is locally asymptotically stable if the speed of adjustment is small enough and it goes to chaos through period-doubling cascade as the speed becomes larger. The non-negativity condition that prevents time trajectories from being negative is derived. The discrete model is converted into the continuous model augmented with time delay and inertia. It is then demonstrated that stability can be switched to instability and complex dynamics emerges as the length of the delay increases and that instability can be switched to stability as the inertia coefficient becomes larger. Therefore the delay has the destabilizing effect while the inertia has the stabilizing effect.

Mathematical models for conflict resolution are very important in integrated water resources and environmental management. This study proposes a new methodology to resolve conflicts among different water users and water suppliers while considering environmental requirements and the system’s constraints. A two-level leader–follower model is applied to maximize the net benefit with the Iran Water Resources Management Company as the leader and agricultural, domestic, and industrial users as followers subject to the system’s constraints. As a comparison, the Nash bargaining solution is also used to find a solution when simultaneous moves are assumed by the participants. The suggested method is then applied to the real case of the Zarrinehrud River basin that is one of the areas facing water shortages in Iran. For the actual optimization, Genetic Algorithm is used in order to avoid local optimum. As the contribution of this study, the results show that benefits for the leader in the leader–follower model increased in comparison with the Nash bargaining solutions.

Optimal resource allocation is first found in defending possible targets against random terrorist attacks subject to budget constraint. The mathematical model is a nonconvex optimization problem which can be transformed into a convex problem by introducing new decision variables, so standard methods can be used for its solution. Without budget constraint the simplified model can be solved by a very simple algorithm which requires the solution of a single variable monotone equation.

Buffelgrass (Pennisetum ciliare) is a fire-prone, African bunchgrass spreading rapidly across the southern Arizona desert. This article introduces a model that simulates buffelgrass spread over a gridded landscape over time to evaluate strategies to control this invasive species. Weed-carrying capacity, treatment costs, and damages vary across grid cells. Damage from buffelgrass depends on its density and proximity to valued resources. Damages include negative effects on native species (through spatial competition) and increased fire risk to land and buildings. We evaluate recommended “rule of thumb” control strategies in terms of their ability to prevent weed establishment in newly infested areas and to reduce damage indices over time. Two such strategies—potential damage weighting and consecutive year treatment—used in combination, provided significant improvements in long-term control over no control and over a strategy of minimizing current damages in each year. Results suggest specific recommendations for deploying rapid-response teams to prevent establishment in new areas. The long-run population size and spatial distribution of buffelgrass is sensitive to the priority given to protecting different resources. Land managers with different priorities may pursue quite different control strategies, posing a challenge for coordinating control across jurisdictions.

An analytical approach is employed to investigate the transient and steady-state stresses in an isotropic, homogeneous half-space subjected to moving concentrated loads with subsonic speeds. Applying the Stokes–Helmholtz resolution to the Navier’s equation of motion for the half-space results in a system of wavetype partial differential equations. Based on the new moving coordinate system, a modified system of partial differential equations is obtained. Applying a concurrent two-sided and one-sided Laplace transformation, this system is modified to a system of ordinary differential equations, the solutions of which are obtained with respect to boundary conditions. The transformed transient stresses can be inverted by the Cagniard–de Hoop method. Special properties of Laplace transformation yield the steady-state stresses through an analytical approach. Numerical examples are presented to illustrate the methodology. Final results revealed the importance of considering the stresses related to the initial stages of the loading.

The recent developments of advanced intrusion detection systems in the cyber security field provide opportunities to proactively protect the computer network systems and minimize the impacts of attackers on network operations. This paper is intended to assist the network defender find its best actions to defend against multistage attacks. The possible sequences of interactions between the attackers and the network defender are modeled as a two-player non-zero-sum non-cooperative dynamic multistage game with incomplete information. The players are assumed to be rational. They take turns in making decisions by considering previous and possible future interactions with the opponent and use Bayesian analysis after each interaction to update their knowledge about the opponents. We propose a Dynamic game tree-based Fictitious Play DFP approach to describe the repeated interactive decisions of the players. Each player finds its best moves at its decision nodes of the game tree by using multi-objective analysis. All possibilities are considered with their uncertain future interactions, which are based on learning of the opponent's decision process including risk attitude and objectives. Instead of searching the entire game tree, appropriate future time horizons are dynamically determined for both players. In the DFP approach, the defender keeps tracking the opponent's actions, predicts the probabilities of future possible attacks, and then chooses its best moves. Thus, a new defense algorithm, called Response by DFP RDFP, is developed. Numerical experiments show that this approach significantly reduces the damage caused by multistage attacks and it is also more efficient than other related algorithms. Copyright © 2013 John Wiley & Sons, Ltd.

A continuously distributed time lag is considered in a monopoly where the time window of past data is bounded from below and its length is fixed. The dynamic behavior of the resulting system is described by a special delayed differential equation with infinite spectrum. The location of the stability switch is determined and a simple rule is developed to determine which ones lead to the loss of stability or the regaining of stability. A simple computer example illustrates the theoretical findings. The distribution delay with finite time lag may be considered as an intermediate case between a finite fixed delay and an infinitely distributed delay, as is usually proposed in the existing literature.

This paper discusses the delay dynamics of monopoly with discrete timescales. It is assumed that a monopoly has delayed and limited information on demand. It is also assumed that the firm wants to react to an average of past data instead of reacting to sudden market changes and this average is used to determine the marginal profit. In the case of one-step delay, the output of the previous time period is selected. In the cases of two and three delays where data at one, two, and three previous time periods are available, it is shown that the steady state undergoes to complex dynamics through either a period-doubling or a Neimark–Sacker bifurcation, depending on the specified values of the parameters. Numerical examples illustrate the theoretical results. Finally, the case of geometric delay is also analyzed to show the birth of the period-doubling bifurcation.

We develop a long-term dynamic model for controlling invasive species using the theory of cooperative games. The model is applied to control of invasive buffelgrass in the Arizona desert, which directly competes with indigenous species and can increase wildfire risk. Interest groups care about damages to three threatened resources: saguaro, cactus, riparian vegetation, and buildings. The model optimally allocates labor and a budget to protect these resources by controlling the buffelgrass population over a multi-period planning horizon. The solution is based on computing the Shapley values for the interest groups. A homeowner strategy of creating defensible space around structures to protect against wildfire affords less protection to the other resources. A similar result holds for protection of saguaros, which are also spatially concentrated. Under the optimal solution, groups caring about spatially-dispersed, riparian vegetation would compensate homeowners and groups caring about saguaros for a reallocation of resources toward greater protection of dispersed vegetation. Results highlight the importance of the spatial configuration of players and the resources they wish to protect in invasive species control problems.

Single-product oligopolies without product differentiation are examined with linear production, production adjustment, flexible workforce and investment costs. The price function is assumed to be hyperbolic which makes the nonlinearity of the model much stronger than in the case of linear price function examined earlier in the literature. The best responses of the firms are determined which are not monotonic in contrast to the linear case. The set of all steady states is then characterized and in the case of a duopoly it is illustrated. The asymptotical behavior of the steady states is examined by using simulation. We analyze the effects of such costs on the industry dynamics and compare them to the prediction by the well known model with hyperbolic price function and no product adjustment and investments costs.

Management of surface and ground water resources in drought-stricken regions with increasing water consumption poses a major challenge to water resource managers and decision makers. The si...

A modified Cournot oligopoly is introduced and examined in which the firms can treat their wastes up to a certain amount, and if the amount of waste is even higher then an outside facility is used with a given fixed cost and higher unit cost. The resulting payoff functions become discontinuous. The best response functions of firms are nonincreasing and also might be discontinuous. The best response functions can be modified to depend on the total industry output, which also might be discontinuous as well as multiple valued. The existence of at least one equilibrium is proved and numerical examples show the possibility of a unique equilibrium as well as that of multiple equilibria.

The intersection of the price function with the vertical axis is called the maximum price and the slope of the price function is called the marginal price. It is assumed that a monopoly has full information about the marginal price and its own cost function but is uncertain about the maximum price. Based on repeated price observations an adaptive learning process can be developed for the maximum price. If the price observations have fixed delays, then the learning process can be described by a delayed differential equation. In the cases of one or two delays, the asymptotic behavior of the associated dynamic process is examined. Stability conditions are derived and the occurrence of Hopf bifurcation is shown at the critical values. The nonlinear learning process can generate complex dynamics in the case of local instability when the delay is sufficiently long.

A new model and solution methods are introduced to find the optimal time of preventive replacement of a working object. Most models consider only the replacement costs and the distribution of the time to failure. Very few models take the random termination time of ongoing projects into account, when project interruption results in additional cost. Traditionally, the object is replaced upon failure. Otherwise it is replaced at the earlier (replace first) or later (replace last) occurrence of either the termination of the project or the scheduled replacement time. The long-term optimum models were formulated in [1] and the solution methodology was illustrated with examples. This paper presents the single-cycle counterpart of these models which are extended to include repair costs of repairable failures, generated revenues and salvage value of the unit in a one-cycle cost per unit time minimization model. The models are further extended to the case when the unit repeats identical projects in time and the terminating time of the ongoing project or that of a given number of projects is considered in the replace next model, which is also formulated. Conditions are given for the existence of finite optima, and examples illustrate the models and solutions. A comparison of the alternative models discussed in this paper has an interesting conclusion. Consider the derivatives h1(T), h2(T) and h3(T) of the objective functions of the classical, "replace first", "replace last" models, respectively. They all have an identical first term, which is constant if all distributions are exponential and strictly increasing if at least one distribution is Weibull, which is the case if the irreparable failure is Weibull. In addition to strictly increasing, it is zero at T = 0 and tends to infinity if T_*** The second term of h2(t) is always larger than the second term of h3(t), so the solution of h2(t) = 0 is larger than the solution of h3(t) = 0. The second term of h1 (t) is between those of h2 (t) and h3 (t), so the solution of the first order condition of the classical model is between the solutions of the other two first order conditions. This has a sense. If "replacement last" model is used, then the replacement is often done after time T, when failure occurs or working cycle terminates, so the actual replacement is often performed after time T. However, in the case of "replacement first" the actual replacement often occurs before time T, when either working cycle terminates or unit fails. That is, in using "replacement last" policy the value of T underestimates the actual replacement time, while in the case of "replacement first" the value of T overestimates it. If the classical model is used, then replacement is done at time T unless failure occurs, no project termination time is considered. The classical model, "replacement first" and "replacement last" models have a simple closed form objective function, so they can be solved by simple computer methods. The most frequently used procedures are discussed in many textbooks on computer methods, for example in [2]. In the general case the "replacement next" model needs simulation to find its optimum, closed form representation of its objective function is only possible in special cases.

This chapter studies the dynamics of the Kaldor–Kalecki model of national income and capital stock. The investment function is assumed to have not only a Kaldorian characteristics, namely, a S-shaped form but also a Kaleckian characteristics, that is, a gestation delay between “investment decision” and “investment implementation.” We divide the analysis into two parts. In the first part, we assume that the time period under consideration is short enough so that the capital stock is not affected by the flow of investment and then examine the delay effect on dynamics of national income. In the second part, taking the capital accumulation into account, we draw attention to how the delay affects cyclic dynamics observed in the nondelay Kaldor–Kalecki model. It is demonstrated that the investment delay quantitatively affects the dynamic behavior but not qualitatively.

A modified version of single-product discrete Cournot oligopolies is introduced, where the additional costs of decreasing or increasing the output level from the previous time period as well as increasing the output level from the already built up capacity limit are included. In this way the costs of laying off or hiring new workers and making investments for increasing capacity are essential parts of the model. The best response functions of the firms are analytically determined, which are non-differentiable even if the price and the production cost functions are linear. The set of all equilibria are characterized by a system of linear inequalities, which is illustrated in the case of duopoly. The asymptotic properties of the equilibria are examined by using computer simulation. Finally, it is shown how some specific models can be derived from the general approach.

Sequential sampling process models provide promising approaches to characterizing human judgment and decision making. In this paper, we study one class of sequential sampling process models referred to as Decision Field Theory (DFT). These models not only provide qualitative explanations of empirical findings, but also provide quantitative predictions of real time performance. We analyze major DFT models in the literature, and demonstrate that models which are seemingly similar can behave very differently, depending upon the cognitive basis for model parameters. Based on this analysis, we propose a mathematical framework which attempts to unify DFT models.

The purpose of this paper is to study the dynamics of a monopolistic firm in a continuous-time framework. The firm is assumed to be boundedly rational and to experience time delays in obtaining and implementing information on output. The dynamic adjustment process is based on the gradient of the expected profit. The paper is divided into three parts: we examine delay effects on dynamics caused by one-time delay and two-time delays in the first two parts. Global dynamics and analytical results on local dynamics are numerically confirmed in the third part. Four main results are demonstrated. First, the stability switch from stability to instability occurs only once in the case of a single delay. Second, the alternation of stability and instability can continue if two time delays are involved. Third, the occurence of Hopf bifurcation is analytically shown if stability is lost. Finally, in a bifurcation process, there are a period-doubling cascade to chaos and a period-halving cascade to the equilibrium point in the case of two time delays if the difference between the two delays is large.

In this paper, we construct a three-country model with two governments and two firms and consider dynamic behavior of the sequential subsidy game in which the governments determine their optimal trade policies and then the firms determine their optimal outputs. We first show the existence of an optimal trade policy under realistic conditions. In the case of symmetric firms, the governments adopt periodic mixed trade policy (i.e., one government gives subsidy and the other levies tax in one period and then the governments interchange their policies in the next period) if the adjustment is naive, and the governments adopt a stable mixed policy if adaptive. In the case of asymmetric firms, a firm receives subsidy if its cost is lower and pays tax if higher. If the Cournot output point under the optimal subsidy is locally stable, then its dynamics can be periodic which is synchronized with the periodic trade policy. If it is locally unstable, then complex dynamics involving chaos emerges regardless of the cost difference.

A dynamic model of controlling invasive weeds is first developed which is a large scale, nonlinear 0-1 integer programming problem. This model is then applied for the case of control of the invasive grass, Pennisetum ciliare (buffelgrass), in the Arizona desert. The large size of the problem makes the application of direct optimization methods impossible, instead the most frequently suggested strategies were analyzed and their consequences compared. The model is more advanced and complex than those examined in earlier studies.

Linear differential duopolies are constructed with continuous time scales, constant coefficients and two types of information delays: fixed and continuously distributed time delays. System dynamics are considered with delays in the diagonal terms. By analyzing the associated characteristic equations, it is found that the stability is lost when the lengths of delays cross some critical values. Then it is shown that the destabilizing effect caused by the fixed delays is stronger than the destabilizing effect of the distributed delays having exponentially-declining weighting function. It is further demonstrated that the strength of the destabilizing effect is reversed if the distributed delay has a bell-shaped weighting function.

The Herfindahl–Hirschman Index is one of the most commonly used indicators to detect anticompetitive behaviour in industries. In fact, an increase in the value of the index is usually interpreted as an indicator of actions which may lessen competition or even create a monopoly. In this article we show that this is not always the case, since in certain instances it is impossible to detect cooperation. We also show an example when competition even has a decreasing effect on the value of the index.

This paper develops a continuous-time neoclassical growth model with time delay. Despite of its simple structure, the resulting dynamic system shows emergence of erratic fluctuations in the capital accumulation process when the production function is unimodal and the delay in production is explicitly considered. It complements the seminal paper of Day (1982) in which a discrete-time neoclassical growth model displayed chaotic behavior for some configurations of the propensity to save, the growth rate of labor and the capital depreciation rate. Our analysis has at least two implications. First, nonlinearities and delay matter for a birth of aperiodic fluctuations of national product in the continuous-time model. Second, comparing the effects caused by two different time delays, fixed and continuously distributed time delays, reveals that the continuous-time model with the former can generate complex fluctuations while the one with the latter generates simple fluctuations when the shape parameter of the density function is small and complex fluctuations when large.

AnN-firm production game known as oligopoly will be examined with isoelastic price function and linear cost under al Cournot competition. After the best responses of the firms are determined, a dynamic system with adaptive expectations is introduced. It is first shown that the local asymptotic behavior of the system is identical with that of the adaptive adjustment process in which the firms cautiously determine their outputs. Dynamic analysis is confined to two special cases, one in whichNis divided into two groups and the other in whichNis divided into three groups. Then stability conditions will be derived and the global behavior of the equilibria will be illustrated including chaos control. Lastly the two- and three-group models are compared with two-firm (duopoly) and three-firm (triopoly) models to shed light on roles of the number of the firms.

Empirical evidence, and theoretical results have shown that, in an industry, higher concentration index indicates higher price-cost margin. In order to detect collusive behaviour the antitrust authorities often monitor the Herfindahl-Hirschman Index. We consider N-firm oligopolies where a group of firms partially cooperate with each other, and monitor the Herfindahl-Hirschman Index as well. After suspecting that the authorities might notice the violation of antitrust regulations, they stop their cooperation. The group will not cooperate again until the Index moves back to the legal domain. This flip-flop dynamical model is formulated, the equilibria are determined, and the asymptotic properties of the system are examined.

In considering economic dynamics, it has been known that time delays are inherent in economic phenomena and could be crucial sources for oscillatory behavior. The main aim of this study is to shed light on what effects the delays can generate. To this end, different models of Cournot duopoly with different delays are built in a continuous time framework and their local and global dynamics are analytically and numerically examined. Three major findings are obtained. First, the stability switching conditions are analytically constructed. Second, it is numerically demonstrated that different lengths of the delays are sources for the birth of simple and complicated dynamics. Third, the delay for collecting information on the competitors' output alone does not affect stability.

The main purpose of this paper is to consider effects caused by time delays on stability of continuous models when the dynamic system is piecewise differentiable. To this end, we construct a continuous version of the discrete Hicksian trade cycle model and introduce continuously distributed time delays. It is demonstrated that time delays stabilize an otherwise unstable system, and a piecewisely connected limit cycle is generated around an unstable stationary state.

This paper will demonstrate a strong relationship between the Compromise Programming (CP) method and the Ordered Weighted Averaging (OWA) operator. We will mathematically prove that the application of the CP method reflects the pessimistic view of the Decision Maker (DM), if the distance from the ideal point is minimized. It will however present an optimistic DM, if the distance from the nadir is maximized. The OWA operator may cover all of the optimistic, neutral and pessimistic views. A practical relationship will be developed between the power value of the CP method and the optimism degree of the DM based on fuzzy quantifiers. The theoretical results will be illustrated by a real case study of a water resources management problem.

Heterogeneous duopolies with product differentiation and isoelastic price functions are examined, in which one firm is quantity setter and the other is price setter. The reaction functions and the Cournot–Bertrand (CB) equilibrium are first determined. It is shown that the best response dynamics with continuous time scales and without time delays, is always locally asymptotically stable. This stability can be, however, lost in the presence of time delays. Both fixed and continuously distributed time delays are examined, stability conditions derived and the stability regions determined and illustrated. The results are compared to Cournot–Cournot (CC) and Bertrand–Bertrand (BB) dynamics. It turns out that continuously distributed lags have a smaller destabilizing effect on the equilibria than fixed lags, and both homogeneous (CC and BB) competitions are more stable than the heterogeneous competitions.

A general framework of partial cooperation and shareholding interlocks in oligopolies is first introduced, and then the best responses of the firms are determined. The monotonic dependence of the equilibrium industry output on the coopeation levels of the firms is proved. Conditions are given for the local asymptotic stability of the equilibrium which require sufficiently small speed of adjustment. Antitrust thresholds are then introduced into the model which may result in the loss of equilibrium or in the presense of multiple equilibria. The dynamic behavior of the associate dynamic models with adpative output adjustments also becomes more complex: period-2 cycles may emerge and coexist with stationary states.

Mexican Valley is one of the Mexico’s most critical areas of water supply, the groundwater resources are overexploited at a rate of 100% or more, generating ground sinking up to 0.4 cm/year in some areas. Water shortage in this area is already at an alarmingly critical level. The situation is expected to worsen due to increasing domestic, industrial and agricultural water demands. The limited water resources, the competing users and the combination of several water sources with different qualities require the development of an adequate water distribution system. The objective of this work is to generate a water distribution model as a three-person linear game in which the users are the players, the supplied amounts from five sources are the strategies, and the total water supplies are the payoffs. Since the available water supplies are lower than the total demands from the users, a tradeoff has to be determined. The nonsymmetric Nash bargaining method is used, which requires the solution of a special optimization problem with nonlinear objective function and linear constraints. For all water distribution scenarios there is no water distribution strategy that satisfies the domestic demand with the current system. Therefore investments and further developments are needed in combination with more efficient water usage by the three sectors in the near future to secure the satisfaction of domestic users. A market driven water pricing policy also would give an incentive to the users for more efficient usage of water.

The interactions between attackers and network administrator are modeled as a non-cooperative non-zero-sum dynamic game with incomplete information, which considers the uncertainty and the special properties of multi-stage attacks. The model is a Fictitious Play approach along a special game tree when the attacker is the leader and the administrator is the follower. Multi-objective optimization methodology is used to predict the attacker’s best actions at each decision node. The administrator also keeps tracking the attacker’s actions and updates his knowledge on the attacker’s behavior and objectives after each detected attack, and uses it to update the prediction of the attacker’s future actions. Instead of searching the entire game tree, appropriate time horizons are dynamically determined to reduce the size of the game tree, leading to a new, fast, adaptive learning algorithm. Numerical experiments show that our algorithm has a significant reduction in the damage of the network and it is also more efficient than other existing algorithms.

This study reconsiders a duopoly model with advertisement introduced earlier by Ahmed, Agiza and Hassan [1999]. It demonstrates three major findings. The first is that the model can be destabilized via either flip bifurcation or Hopf bifurcation. The second is that a half-pitchfork bifurcation of the output occurs when the advertisement dynamics is periodic and the nonlinearity of the output dynamics becomes stronger. Finally the third is that the existence of attractor and the coexistence of attracting sets are the main features of the model when it is locally unstable.

The effects of government subsidies are examined in a spatial duopoly with a conventional and an electronic retailer. The Nash equilibrium is first determined and then the optimal cost subsidy rates are computed that maximize social welfare. The stability of the equilibrium is also investigated and it is shown that in the case of delayed information there is the possibility of cyclic behavior.

Multi criteria decision making (MCDM) problems are usually under uncertainty. One of these uncertain parameters is the decision maker (DM)’s degree of optimism, which has an important effect on the results. Fuzzy linguistic quantifiers are used to obtain the assessments of this parameter from DM and then, because of its uncertainty it is assumed to have stochastic nature. A new approach, entitled FSROWA, is introduced to combine the Fuzzy and Stochastic features into a Revised OWA operator. If the DM is not aware of the risk in decision, then the decision objective is to maximize the expected combined goodness measure. If the DM cares only about the risk, then minimizing the variance of the combined goodness measure is his/her objective. The results of the FSROWA provide the expected value and the variance of the combined goodness measure for each alternative. In order to combine these two attributes a composite goodness measure is introduced. This measure gives more robust decision outcomes. The theoretical results are illustrated in a watershed management problem.

A multiobjective optimization model is developed for the Mexican Valley to find optimal water distribution scenarios for three water users: agriculture, industry and domestic. Surface, ground, treated and imported water supplies are considered. The model with three objectives and 15 constraints was solved by the distance based method with distance l1.

N -firm single-product oligopolies will be examined without product differentiation and with production adjustment costs. The best response functions of the firms will be first determined in terms of the outputs of the rest of the industry and the firm's own outputs of the previous time period. This function is always decreasing, not necessarily continuous, and might have two different values. Then the best response will be determined, when the output of the rest of the industry is replaced by the output of the entire industry. We will show that this function might have infinitely many values leading to infinitely many equilibria. A complete description of the equilibrium set will also be given. The theoretical findings will be illustrated with the case of linear price and linear cost functions.

A common mistake in modeling systems is mixing elements of different levels in the same model. This paper explains levels, shows how levels can be obtained with decomposition techniques and gives many examples of levels in models, architectures and frameworks. Examples are given that show the confusion that results when levels are mixed. The paper shows that complex models are usually composed of many aspects. Each aspect has levels and the levels of one aspect are different from the levels of another aspect. The model must be constructed using elements of the same level for each aspect.

The n -person Battle of Sexes game is first introduced. The game’s properties are discussed and payoff functions are modeled under the main assumption that players’ payoffs are based on whether they like their choices or not and also on how many other players have the same choices. Linear payoff functions are assumed and the existence of the Nash equilibrium is examined. The dependence of the equilibrium on model parameters is also analyzed.