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Two types of boundedly rational monopolists are considered, when they are unable to determine the profit maximizing output levels. In the first case, the monopolist knows the price function and in the second case it can access only past output and price values. In applying gradient dynamics, the marginal profit is either known or approximated by finite differences based on two past profit data. Stability conditions are derived first with discrete time scales, which are also applied in a special case. Two models of continuous-time dynamics are then introduced. The first is a natural modification of the discrete model, and the other includes an inertia coefficient with the derivative. In each case, a delay differential equation is obtained with two delays. Stability conditions are derived and the stability-switching curves are constructed and illustrated.

This study is devoted to consider the dynamic properties of a Lotka–Volterra competition model with two discrete delays. First, it reviews stability conditions of the no-delay model as a benchmark. Second, it examines the delay model and analytically establishes the stability switching curve on which stability is switched to instability and vice versa. Finally, numerical simulations are performed to confirm the theoretical results such as Hopf bifurcations and, further, demonstrate a wide variety of dynamics ranging from simple limit cycle to the existence of multi-stability and the birth of chaotic dynamics.

Repeated asymmetric contest games are examined under conditions which guarantee the existence and uniqueness of pure Nash equilibrium (Hirai and Szidarovszky 2013). Conditions are derived for the local asymptotical stability of the equilibrium under continuous and discrete dynamics with gradient adjustments. In both cases, a crucial assumption is the nonexistence of a dominant player at the equilibrium level. In the case of continuous time scales, this is sufficient for stability, and in the discrete case, the speeds of adjustments have to be sufficiently small. As special cases, symmetric and semi-symmetric games are analyzed.

It is well-known that Goodwin’s nonlinear delay accelerator model can generate diverse oscillations (i.e., smooth and sawtooth oscillations). It is, however, less-known what conditions are needed for these dynamics to emerge. In this study, using a piecewise linear investment function, we solve the governing delay differential equation and obtain the explicit forms of the time trajectories. In doing so, we detect conditions for persistent oscillations and also conditions for the birth of such cyclic dynamics.

This paper reexamines the Lotka-Volterra competition model with two delays. The steady state is shown to be locally asymptotically stable without delay. If the two delays are identical, then the model becomes a one-delay system. The critical value of the delay is determined when stability might be lost. If the delays are different, then the stability switching curves are analytically defined and numerically verified. It is demonstrated that the unstable two-delay system may exhibit periodic behavior, multistability, quasi period-doubling cascade and even complicated dynamics depending on model parameters.

A three-delay duopoly is considered where the firms have identical implementation delays with different information delays. The equilibrium is locally asymptotically stable without delays however this stability is lost with increasing values of the delays. The stability properties of the equilibrium depend on the common implementation delay of the firms and on the sum of the two information delays. The stability switching curves are first analytically characterized and illustrated, and then the direction of the stability switching is determined at each point of the curves. The possibility of multiple pure imaginary eigenvalues is also discussed when the directions of the stability switches cannot be determined. Simulation examples illustrate the theoretical results.

The dynamic behavior of n-firm oligopolies is examined without product differentiation and with linear price and cost functions. Continuous time scales are assumed with best response dynamics, in which case the equilibrium is asymptotically stable without delays. The firms are assumed to face both implementation and information delays. If the delays are equal, then the model is a single delay case, and the equilibrium is oscillatory stable if the delay is small, at the threshold stability is lost by Hopf bifurcation with cyclic behavior, and for larger delays, the trajectories show expanding cycles. In the case of the non-equal delays, the stability switching curves are constructed and the directions of stability switches are determined. In the case of growth rate dynamics, the local behavior of the trajectories is similar to that of the best response dynamics. Simulation studies verify and illustrate the theoretical findings.

Dynamic asymmetric contest games are examined under the assumption that the assessed value of the prize by each agent depends on the total effort of all agents, and each agent has only delayed information about the efforts of the competitors. Assuming gradient dynamics with continuous time scales, first the resulting one-delay model is investigated. Then, assuming additional delayed information about the agents’ own efforts, a two-delay model is constructed and analyzed. In both cases, first the characteristic equation is derived in the general case, and then two special cases are considered. First, symmetric agents are assumed and then general duopolies are examined. Conditions are derived for the local stability of the equilibrium including stability thresholds and stability switching curves.

A growth model with a special production function augmented with physical capital and human capital is modified and used to explain the birth of cyclical dynamics. Crucial feature of the model is the assumption that there are a gestation delay and a maturation delay in constructing physical capital and human capital, respectively. Dynamics is described by a continuous time system of delay differential equations. Stability switching curves and directions of stability switches are analytically derived where stability of the model might be lost and regained. Its shape is numerically verified and it is confirmed that the two-delay model can generate a wide variety of dynamics from simple dynamics to complex dynamics.

This paper describes a heuristic-based approach for transforming conditioned-based data (CBD) signatures into other signatures. Those signatures are highly correlated to changes in value (dP) of a parameter of interest (P 0 ) that degrades to a level of damage at which a component and its assembly no longer functions within operation specifications: functional failure occurs. CBD signatures can be expressed in terms of feature data (FD) plus noise where the magnitude of FD changes to create a signature in response to degradation. That FD-based signature can be expressed as ratio of a current measurement (FD i ) to a nominal value (FD 0 ) to form a fault-to-failure progression (FFP) signature. Further, a present measurement can be expressed as a nominal value times a degradation function of a change (dP i ) in a parameter that has a nominal value P 0 in the absence of degradation. An FD-based signature is transformed into a fault-to-failure progression (FFP) signature by dividing a present FD by the nominal FD value and solving in terms of (dP i /P 0 ) to produce a degradation-progression signature {DPS i } which is a linearized version of an FFP signature. A DPS signature is further transformable into a functional failure signature, {FFS}, by dividing a DPS signature by a defined failure level. An FFS is amenable to processing as input to prediction algorithms because (1) its characteristic curve approaches an ideal straight-line transfer curve as noise is ameliorated and/or mitigated; (2) its data has zero or negative values in the absence of degradation; (3) its data has positive values below 100 when there is degradation below a defined level of functional failure; and (4) its data has values at or above 100 when the level of degradation is at or above a level defined as functional failure. Even in the presence of noise and other effects, such as feedback, and even when the rate of degradation is nonlinear, an FSS is still has a very close to linear transfer curve. Seven sets of models have been developed to use for transformation of CBD signatures into FFS data. The models were developed based on the characteristic shapes of degradation signatures and confirmed by physics-of-failure (PoF) analysis and failure-mode effects analysis (FMEA). This paper presents a methodology for selecting and using those models to transform CBD-based signature data into FFS data for use as input to a prediction framework of a PHM system.

An extended n-firm oligopoly with product differentiation is considered. It is assumed that the government selects an emission standard for the industry and based on the output and technology of each firm it selects a maximum allowed amount of emission for each firm. If the actual amount is higher than the allowed maximum, then the firm has to pay a constant multiple of the excess to the government; otherwise it is rewarded similarly based on the saved emission amount. The existence of the unique interior equilibrium is first proved, and then the effect of the level of penalty or reward and that of the emission standard on the industry output and therefore on the total emission level is also examined. Time delay is introduced into the penalties the firms have to pay and into the rewards the firms receive. In analyzing the stability of the equilibrium both discrete and continuous time scales are considered. For mathematical simplicity the case of symmetric firms is analyzed. In the discrete case the various values of the delay length are examined. The equilibrium is stable if either the total industry output is sufficiently large or the common speed of adjustment of the firms is sufficiently small. In the continuous case, either the equilibrium is always stable or stability occurs if the delay is sufficiently small and at the critical value Hopf bifurcation occurs.

We call the intercept of the price function with the vertical axis the maximum price and the slope of the price function the marginal price. In this paper it is assumed that a monopolistic firm has full information about the marginal price and its own cost function but is uncertain on the maximum price. However, by repeated interaction with the market, the obtained price observations give a basis for an adaptive learning process of the maximum price. It is also assumed that the price observations have fixed delays, so the learning process can be described by a delayed differential equation. In the cases of one or two delays, the asymptotic behavior of the resulting dynamic process is examined, stability conditions are derived. Three main results are demonstrated in the two delay learning processes. First, it is possible to stabilize the equilibrium which is unstable in the one delay model. Second, complex dynamics involving chaos, which is impossible in the one delay model, can emerge. Third, alternations of stability and instability (i.e., stability switches) occur repeatedly.

Two time delays are assumed in a boundedly rational monopoly. The characteristic equation is derived for the general case, and a complete stability analysis is conducted both analytically and numerically in two special cases. In the first case, wherein the continuously distributed time delays have different weights, it is shown that a monopoly equilibrium is destabilized to generate a limit cycle via Hopf bifurcation. In the second case in which one delay is continuously distributed and the other is fixed, it is demonstrated that the stability of the monopoly equilibrium can change finite number of times and eventually becomes periodically or aperiodically unstable. It is of interest to notice that the two delay dynamics can be qualitatively different from the one delay dynamics.

Dynamics of Goodwin’s accelerator business cycle model is reconsidered. The model is characterized by a nonlinear accelerator and an investment time delay. The role of the nonlinearity for the birth of persistent oscillations is fully discussed in the existing literature. On the other hand, not much of the role of the delay has yet been revealed. The purpose of this paper is to show that the delay really matters. In the original framework of Goodwin [6] , it is first demonstrated that there is a threshold value of the delay: limit cycles arise for smaller values than the threshold and so do sawtooth oscillations for larger values. In the extended framework in which a consumption or saving delay, in addition to the investment delay, is introduced, three main results are demonstrated under assumption of the identical length of investment and consumption delays. The dynamics with consumption delay is basically the same as that of the single delay model. Second, in the case of saving delay, the steady state can coexist with the stable and unstable limit cycles in the stable case. Third, in the unstable case, there is an interval of delay in which the limit cycle or the sawtooth oscillation emerges depending on the choice of the constant initial function.

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.

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.

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.

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.

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.

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.

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.

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.

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.

We investigate love dynamics of two individuals in a delay love affair model in which both are assumed to be cautious, the most natural romantic style. The local stability analysis proves first that the steady state is fairly stable when there are no delays and second that solving the characteristic equation generates a set of positive delays for which the steady state loses stability. Through numerical analysis, we confirm the following three main results: (1) cyclic oscillations of love feeling emerge via Hopf bifurcation; (2) multiple delays cause the double edge effect implying that alternation of stability and instability repeatedly occurs; (3) complicated dynamics involving chaotic oscillations emerges and then merges to a limit cycle as the length of one delay increases with fixed values of the other delay.

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.

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.

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.

The efficient use of the Ordered Weighted Averaging (OWA) operator in decision making problems depends on the choice of the order weights. Using fuzzy quantifiers is one of the most popular methods to obtain them. In this study, a new method will be introduced for determining the order weights from the quantifiers, which is especially useful in the case of unimodal quantifiers. The new method is generic and has better computational efficiency in comparison to the previously applied methods. In addition, a new measure for sensitivity analysis on the outputs of OWA operator will be introduced. The theoretical results will be illustrated by a Ph.D. student selection problem discussed earlier in the literature.

All realistic Multi-Criteria Decision Making (MCDM) problems face various kinds of uncertainty. Since the evaluations of alternatives with respect to the criteria are uncertain they will be assumed to have stochastic nature. To obtain the uncertain optimism degree of the decision maker fuzzy linguistic quantifiers will be used. Then a new approach for fuzzy-stochastic modeling of MCDM problems will be introduced by merging the stochastic and fuzzy approaches into the OWA operator. The results of the new approach, entitled FSOWA, give the expected value and the variance of the combined goodness measure for each alternative. Robust decision depends on the combined goodness measures of alternatives and also on the variations of these measures under uncertainty. In order to combine these two characteristics a composite goodness measure will be defined. The theoretical results will be illustrated in a watershed management problem. By using this measure will give more sensitive decisions to the stakeholders whose optimism degrees are different than that of the decision maker. FSOWA can be used for robust decision making on the competitive alternatives under uncertainty.

All realistic Multi Criteria Decision Making (MCDM) problems in water resources management face various kinds of uncertainty. In this study the evaluations of the alternatives with respect to the criteria will be assumed to be stochastic. Fuzzy linguistic quantifiers will be used to obtain the uncertain optimism degree of the Decision Maker (DM). A new approach for stochastic-fuzzy modeling of MCDM problems will be then introduced by merging the stochastic and fuzzy approaches into the Ordered Weighted Averaging (OWA) operator. The results of the new approach, entitled SFOWA, give the expected value and the variance of the combined goodness measure for each alternative, which are essential for robust decision making. In order to combine these two characteristics, a composite goodness measure will be defined. By using this measure the model will give more sensitive decisions to the stakeholders whose optimism degrees are different than that of the decision maker. The methodology will be illustrated by using a water resources management problem in the Central Tisza River in Hungary. Finally, SFOWA will be compared to other methods known from the literature to show its suitability for MCDM problems under uncertainty.

In an optimization problem , we have a single decision maker, his feasible decision alternative set, and an objective function depending on the selected alternative. In game theoretical models, we have several decision makers who are called the players , each of them has a feasible alternative set, which is called the player’s strategy set, and each player has an objective function what is called the player’s payoff function. The payoff of each player depends on the strategy selections of all players , so the outcome depends on his own decision as well as on the decisions of the other players . Let N be the number of players , \(S_k\) the strategy set of player \(k (k=1, 2, \ldots , N)\) and it is assumed that the payoff function \(\phi _k\) of player k is defined on \(S_1 \times S_2 \times \cdots \times S_N\) and is real valued. That is, \(\phi _k : S_1 \times S_2 \times \cdots \times S_N \mapsto \mathbb {R}\). So if \(s_1, s_2, \ldots , s_N\) are the strategy selections of the players , \(s_k \in S_k (k=1, 2, \ldots , N)\), then the payoff of player k is \(\phi _k(s_1, s_2, \ldots , s_N)\). The game can be denoted as \(\varGamma (N; S_1, S_2, \ldots , S_N; \phi _1, \phi _2, \ldots , \phi _N)\) which is usually called the normal form representation of the game.

Let N denote the number of players . It is usually assumed that the set of all feasible strategies of each player has at least two elements. If \(S_k\) is the strategy set of player k, then its payoff function \(\phi _k\) is defined on the set of all simultaneous strategies , which is denoted by \(S=S_1 \times S_2 \times \dots \times S_N\), and \(\phi _k(\underline{s})\), for all \(\underline{s} \in S\) is a real number. The normal form of the game is given as \(G=\{ N; S_1, S_2, \dots , S_N; \phi _1, \dots , \phi _N \}\). The game is continuous, if all sets \(S_k\) are connected and all payoff functions \(\phi _k\) are piecewise continuous.

In the cases of noncooperative games the players cannot or do not want to make binding agreements, so they select strategies independently of each other, and receive the corresponding payoffs. The Nash equilibrium does not need agreement between the players, since at an equilibrium situation the interest of each player is to keep the equilibrium strategy, otherwise his payoff decreases. Each player considers his own selfish interest without any consideration to the other players. A simple example illustrates, the players can be able to increase their payoffs by cooperation.

In the previous chapter, conditions were given for the existence of equilibria in N-person games. The Nikaido-Isoda theorem was based on the Kakutani fixed point theorem, which guarantees the existence of at least one fixed point without providing computational method for finding the fixed points. In this chapter, a general method will be introduced and applied to some special game classes.

The equilibrium problem of N-person games has strong relation to other important mathematical problem areas such as optimization and fixed point problems. In this chapter we will discuss this issue.