On Dynamic Investigations of Cournot Duopoly Game: When Firms Want to Maximize Their Relative Profits

This paper studies a Cournot duopoly game in which firms produce homogeneous goods and adopt a bounded rationality rule for updating productions. The firms are characterized by an isoelastic demand that is derived from a simple quadratic utility function with linear total costs. The two competing firms in this game seek the optimal quantities of their production by maximizing their relative profits. The model describing the game’s evolution is a two-dimensional nonlinear discrete map and has only one equilibrium point, which is a Nash point. The stability of this point is discussed and it is found that it loses its stability by two different ways, through flip and Neimark–Sacker bifurcations. Because of the asymmetric structure of the map due to different parameters, we show by means of global analysis and numerical simulation that the nonlinear, noninvertible map describing the game’s evolution can give rise to many important coexisting stable attractors (multistability). Analytically, some investigations are performed and prove the existence of areas known in literature with lobes.

Fracture Properties of Nash Point Limestone and Implications for Fracturing of Layered Carbonate Sequences

Carbonate reservoirs accommodate a significant proportion of global hydrocarbon reserves. However they are often tight and permeability is therefore usually dependent on either flow through existing fractures or through those produced by hydraulic stimulation. Hence, understanding how fracture networks develop in carbonate reservoir rocks is key to efficient and effective production. However, despite their prevalence as reservoir rocks, there is a paucity of data on key fracture properties of carbonate rocks, particularly in more than one orientation. Here, therefore we report measurements of both the tensile strength and fracture toughness of Nash Point limestone in the three principal fracture orientations to determine what effect any mechanical anisotropy might have on fracture propagation. We find Nash Point limestone to be essentially isotropic in terms of both its microstructure and its fracture properties. When comparing the fracture toughness of Nash Point limestone with that of others limestones, we find that fracture toughness decreases with increasing porosity, although this dependency is not as strong as found in other porous rocks. Finally, as many so-called carbonate reservoirs actually comprise layered sequences, we extend our analysis to consider the layered sequence of limestones and shales at Nash Point. We find that the fracture toughness of Nash Point limestone is higher than Nash Point shale but that the fracture energy is lower. We therefore discuss how the implications of fracturing through multi-layered sequences could be explored in future work.

Local and global analysis of a nonlinear duopoly game with heterogeneous firms

In this paper, we introduce a nonlinear duopoly game whose players are heterogeneous and their inverse demand functions are derived from a more general isoelastic demand. The game is modeled by a discrete time dynamic system whose Nash equilibrium point is unique. The conditions of local stability of Nash point are calculated. It becomes unstable via two types of bifurcations: flip and Neimark–Sacker. Some local and global numerical investigations are performed to show the dynamic behavior of game’s system. We show that the system is noninvertible and belongs to $Z_{2}-Z_{0}$ Z 2 − Z 0 type. We also show some multistability aspects of the system including basins of attraction and regions known as lobes.

Fluid-driven tensile fracture and fracture toughness in Nash Point Shale at elevated pressure.

<p>Fluid-driven fracturing is a key process in enhancing production in both the hydrocarbon and geothermal energy extraction industries. However, whilst a large number of studies have now developed laboratory methods to simulate the process in a range of settings, and across a number of different rock types, data relating the fundamental material parameters (such as fracture toughness) to the overall rock mechanics response as a function of parameters such as confining and pore pressure remain limited. Here we report a new analysis to recover fracture toughness across a range of effective pressures from hydraulic fracturing experiments that use a modified thick-walled cylinder sample mounted in a conventional triaxial deformation apparatus. We use samples that are 90mm in length and 40mm diameter, with a central, axially drilled borehole 12.6 mm in diameter. An array of 16 ports in the engineered, nitrile, sample jacket allows us to record radial strain (4 channels), acoustic emission output (11 channels) and borehole fluid pressure (1 channel) continuously throughout each test. The sample material was Nash Point shale (NPS) from the south coast of Wales, UK, with samples cored both normal and parallel to bedding in order to investigate the effect of anisotropy. Earlier, ambient pressure fracture toughness tests using the Semi-Circular Bend sample geometry had indicated significant anisotropy, values of 0.24 – 0.30 MPa.m<sup>1/2 </sup>in the Short-Transverse (ST) orientation, and 0.71 - 0.73 MPa.m<sup>1/2 </sup>in the Divider (DIV) orientation.</p><p>Here, we present results from a suite of 9 experiments, 6 with samples cored parallel to bedding (ST fracture orientation) and 3 with samples cored normal to bedding (DIV fracture orientation). We find that the fluid injection pressure required to fracture our annular shell samples is significantly higher for DIV samples than for ST samples, and increases significantly with increasing confining pressure in both orientations; ranging from 10 to 36 MPa for ST samples and 30 to 58 MPa for DIV samples as confining pressure is increased from 2.2 to 20.5 MPa. We note that the fluid injection pressure undergoes a number of oscillations between fracture nucleation and the fracture reaching the sample boundary. Such oscillations are more common in ST samples than in DIV samples, and in experiments at lower confining pressures. We use the magnitude of each pressure oscillation to estimate the associated increment of fracture extension via the proportion of AE energy generated relative to the total energy accumulated when the fracture reaches the sample boundary. This analysis produces fracture toughness values ranging from 0.36 to 2.76 MPa.m<sup>1/2</sup> (ST orientation) and 2.98 to 4.05 MPa.m<sup>1/2 </sup>(DIV orientation) as confining pressure was increased from 2.2 to 20.5 MPa. We further find that the increase in fracture toughness increases essentially linearly with increasing effective pressure, and this trend appears to be independent of orientation and the material anisotropy.</p>

Nash versus coarse correlation

We run a laboratory experiment to test the concept of coarse correlated equilibrium (Moulin and Vial in Int J Game Theory 7:201–221, 1978), with a two-person game with unique pure Nash equilibrium which is also the solution of iterative elimination of strictly dominated strategies. The subjects are asked to commit to a device that randomly picks one of three symmetric outcomes (including the Nash point) with higher ex-ante expected payoff than the Nash equilibrium payoff. We find that the subjects do not accept this lottery (which is a coarse correlated equilibrium); instead, they choose to play the game and coordinate on the Nash equilibrium. However, given an individual choice between a lottery with equal probabilities of the same outcomes and the sure payoff as in the Nash point, the lottery is chosen by the subjects. This result is robust against a few variations. We explain our result as selecting risk-dominance over payoff dominance in equilibrium.

Investigations of Nonlinear Triopoly Models with Different Mechanisms

This paper studies the dynamic characteristics of triopoly models that are constructed based on a 3-dimensional Cobb–Douglas utility function. The paper presents two parts. The first part introduces a competition among three rational firms on which their prices are isoelastic functions. The competition is described by a 3-dimensional discrete dynamical system. We examine the impact of rationality on the system’s steady state point. Studying the stability/instability of this point, which is Nash equilibrium and is unique in those models, is illustrated. Numerically, we give some global analysis of Nash point and its stability. The second part deals with heterogeneous scenarios. It consists of two different models. In the first model, we assume that one competitor adopts the local monopolistic approximation mechanism (LMA) while the other opponents are rational. The second model assumes two heterogeneous players with LMA mechanism against one rational firm. Studies show that the stability of NE point of those models is not guaranteed. Furthermore, simulation shows that when firms behave rational with symmetric costs, the stability of NE point is achievable.