Narrow Results

The properties of Cournot mixed oligopoly consisting of one public firm and one or more than one private firms have mostly been analyzed for simple cases on the basis of numerical calculations of the equilibrium values for a linear market demand function and linear or quadratic cost functions. In this chapter, after proving the existence of a unique equilibrium in Cournot mixed oligopoly under general conditions on the market demand and each firm’s cost function, we derive conditions ensuring the existence of a unique Nash equilibrium for the mixed oligopoly where one public firm and at least one of the private firms are active in a general model of Cournot mixed oligopoly with one public firm and several private firms.

We develop an equilibrium model for market impact of trades when investors with private signals execute via a trading desk. Fat tails in the signal distribution lead to a power law for price impact, while the impact is logarithmic for lighter tails. Moreover, the tail distribution of the equilibrium trade volume obeys a power law. The spread decreases with the degree of noise trading and increases with the number of insiders. In case of a monopolistic insider, the last slice traded against the limit order book is priced at the fundamental value of the asset reminiscent of the Kyle models in continuous time. However, competition among insiders leads to aggressive trading, hence vanishing profit in the limit. The model also predicts that the order book flattens as the amount of noise trading increases converging to a model with proportional transactions costs with non-vanishing spread.

In this paper, I review our current understanding of the applicability of hydrodynamics to modeling the quark–gluon plasma (QGP), focusing on the question of hydrodynamization/ thermalization of the QGP and the anisotropic hydrodynamics (aHydro) far-from-equilibrium hydrodynamic framework. I discuss the existence of far-from-equilibrium hydrodynamic attractors and methods for determining attractors within different hydrodynamical frameworks. I also discuss the determination of attractors from exact solutions to the Boltzmann equation in relaxation time approximation and effective kinetic field theory applied to quantum chromo-dynamics. I then present comparisons of the kinetic attractors with the attractors obtained in standard second-viscous hydrodynamics frameworks and aHydro. I demonstrate that, due to the resummation of terms to all orders in the inverse Reynolds number, the aHydro framework can describe both the weak- and strong-interaction limits. I then review the phenomenological application of aHydro to relativistic heavy-ion collisions using both quasiparticle aHydro and second-order viscous aHydro. The phenomenological results indicate that aHydro provides a controlled extension of dissipative relativistic hydrodynamics to the early-time far-from-equilibrium stage of heavy-ion collisions. This allows one to better describe the data and to extract the temperature dependence of transport coefficients at much higher temperatures than linearized second-order viscous hydrodynamics.

The discovery of a priori bounds for semilinear parabolic equations with superlinear nonlinearity enables the use of the parabolic semiflow in proving existence results for equilibria, as an alternative to minimax methods from the calculus of variations. We present an overview of results that use this approach, discuss its shortcomings and advantages, and outline possible future research directions.

We consider jamming in wireless networks in the framework of zero-sum games with linearized Shannon capacity utility function. The base station has to distribute the power fairly among the users in the presence of a jammer. The jammer in turn tries to distribute its power among the channels to produce as much harm as possible. This game can also be viewed as a minimax problem against the nature. We show that the game has the unique equilibrium and investigate its properties and also we developed an efficient algorithm which allows to find the optimal strategies in finite number of steps.

Many multi-cellular organisms exhibit remarkably similar patterns of aging and mortality. Because this phenomenon appears to arise from the complex interaction of many genes, it has been a challenge to explain it quantitatively as a response to natural selection. We survey attempts by the author and his collaborators to build a framework for understanding how mutation, selection and recombination acting on many genes combine to shape the distribution of genotypes in a large population. A genotype drawn at random from the population at a given time is described by a Poisson random measure on the space of loci and its distribution is characterized by the associated intensity measure. The intensity measures evolve according to a continuous-time measure-valued dynamical system. We present general results on the existence and uniqueness of this dynamical system and how it arises as a limit of discrete generation systems. We also discuss existence of equilibria.

In a recent article [4], we have developed a market model where an insider trades using future information on the value of the underlying, through these trades it creates an effect on the drift of the underlying model. We find points of partial equilibria. That is, when the filtration is fixed the chosen portfolio is optimal, leads to finite utility of the insider and prices are semimartingales in their own filtration. In this article, we treat two explicit examples in detail. The first is an insider which has a medium term effect on the price. The second is an insider which has a long term effect on the price with memory effects. These examples were quoted in [4] but no details were given.

Sufficient conditions for Nash equilibrium in an n-person game are given in terms of what the players know and believe — about the game, and about each other's rationality, actions, knowledge, and beliefs. Mixed strategies are treated not as conscious randomizations, but as conjectures, on the part of other players, as to what a player will do. Common knowledge plays a smaller role in characterizing Nash equilibrium than had been supposed. When n = 2, mutual knowledge of the payoff functions, of rationality, and of the conjectures implies that the conjectures form a Nash equilibrium. When n ≥ 3 and there is a common prior, mutual knowledge of the payoff functions and of rationality, and common knowledge of the conjectures, imply that the conjectures form a Nash equilibrium. Examples show the results to be tight.

Utility indifference pricing is an effective method for investors to construct a strategy in an incomplete market. In fact, if an investor can trade a random endowment under the criteria shown by utility indifference pricing, they can devise financial contracts that are optimized according to their preferences. However, because it does not have the direct implication of equilibrium, the value of the random endowment given by indifference pricing is not necessarily the same as the market price. In this study, we attempt to derive the equilibrium of random endowment under the framework of indifference pricing. However, letting the utility function be of exponential type means that any trade involving random endowment will not appear in equilibrium. Thus, we show that non-zero trade in equilibrium appears by introducing uncertainty in a model, which is one of the sources of market incompleteness.