A partial information non-zero sum differential game of backward stochastic differential equations with applications

This paper is concerned with a new kind of non-zero sum differential game of backward stochastic differential equations (BSDEs). It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motion. We establish a necessary condition in the form of maximum principle with Pontryagin’s type for open-loop Nash equilibrium point of this type of partial information game, and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a partial information linear-quadratic (LQ) game and a partial information financial problem.     

Slowing deforestation pace through subsidies: a differential game

This paper deals with the issue of deforestation, one of the main global environmental problems. We consider two players having different utilities for forest conservation. The first one (call it North) represents a set of countries having as an objective to maximize the size of tropical forest at a certain time horizon. The (South) second player's revenue function involves a trade-off between forest exploitation and agricultural activities. We study two scenarios. In the first one, the assumption is a laisser-faire policy, where the South solves an optimal control problem over a finite horizon. Revenue and optimal forest exploitation obtained represent a benchmark for the second scenario, where the North offers subsidies to the South to reduce the deforestation rate. The two scenarios are analyzed and compared in terms of strategies, outcomes and forest conservation.     

Path-vector contracting: Profit maximization and risk management

We consider an Internet Service Provider’s (ISP’s) problem of providing end-to-end (e2e) services with bandwidth guarantees, using a path-vector based approach. In this approach, an ISP uses its edge-to-edge (g2g) single-domain contracts and vector of contracts purchased from neighboring ISPs as the building blocks to construct, or participate in constructing, an end-to-end “contract path”. We develop a spot-pricing framework for the e2e bandwidth guaranteed services utilizing this path contracting strategy, by formulating it as a stochastic optimization problem with the objective of maximizing expected profit subject to risk constraints. In particular, we present time-invariant path contracting strategies that offer high expected profit at low risks, and can be implemented in a fully distributed manner. Simulation analysis is employed to evaluate the contracting and pricing framework under different network and market conditions. An admission control policy based on the path contracting strategy is developed and its performance is analyzed using simulations.     

A differential game related to terrorism: min-max zero-sum two persons differential game

In this work, we are going to study a differential game related to terrorism: min-max two persons differential game, the question and discuss the qualitative of how best to prosecute the “war on terror” leads to strategic interaction in an inter-temporal setting. We consider a min-max differential game between a governments and a terrorist organization. To obtain the optimal strategy of this problem, we study the analytic form of min-max two persons differential game and a necessary conditions of this problem. Furthermore, we study a saddle point of a min-max differential game. Finally, we make a comparison between the game of the government and the terrorist organization. In the final, I hope from among this study introduce the optimal control and state trajectory to the governments to counter-terrorist.

     

Non-zero sum differential graphical game: cluster synchronisation for multi-agents with partially unknown dynamics

This paper investigates the cluster synchronisation problem for multi-agent non-zero sum differential game with partially unknown dynamics. The objective is to design a controller to achieve the cluster synchronisation and to ensure local optimality of the performance index. With the definition of cluster tracking error and the concept of Nash equilibrium in the multi-agent system (MAS), the previous problem can be transformed into the problem of solving the coupled Hamilton–Jacobi–Bellman (HJB) equations. To solve these HJB equations, a data-based policy iteration algorithm is proposed with an actor–critic neural network (NN) structure in the case of the MAS with partially unknown dynamics; the weights of NNs are updated with the system data rather than the complete knowledge of system dynamics and the residual errors are minimised using the least-square approach. A simulation example is provided to verify the effectiveness of the proposed approach.     

Deforestation and foreign transfers: a Stackelberg differential game approach

In this paper, our concern is with deforestation as a global environmental issue. Foreign transfers from developed or Northern countries to developing or Southern countries have been proposed to deal with that issue. We use a Stackelberg differential game approach where the two players are: a donor community (the leader) and a recipient country (the follower). We study the impact of different specifications for the transfer function, related to the different information structures the players can consider when playing the Stackelberg game. The different scenarios are compared both from the environmental and economic points of view.     

The Nash solution of an advertising differential game: Generalization of a model by Leitmann and Schmitendorf

In this paper the differential game between two profit maximizing firms considered in [15] is extended to the case of general (concave) effectiveness functions of advertising. For this model, a phase portrait analysis of Nash equilibrium solutions is carried out providing qualitative insights into the structure of optimal advertising rates. Applying this analysis to the Leitmann-Schmitendorf advertising model, we obtain more detailed information on the structure of the optimal controls. Moreover, Nash-optimal solutions are derived for infinite duration of the game.     

Designing interactive systems through a game lens: An ethnographic approach

This research has the aim to find new meaningful elements, in the video game world, that could inspire the design of novel gamified systems. Starting from the players’ point of view, I looked at the field of the Massively Multi Player Online Role-Playing Games as a source of inspiration, conducting an ethnographic study in World of Warcraft. Thus, drawing on the findings gathered in my empirical work, I identified 9 recommendations to suggest new directions for the gamification design of interactive systems. Some of these recommendations are devised to suit the applications that pivot on user social participation. Others are aimed at imagining new forms of online communities. Others address those interactive systems that aim at changing user behaviors. These recommendations, by suggesting to support the development of intrinsic motivations, proposing new and diversified game elements and recommending to look at systemic design strategies, aim at addressing the limits of the current gamification techniques. At the high grade of abstraction they are left, they are meant to be applied to different fields.     

A differential game of retailer promotions

The paper identifies equilibrium marketing strategies over time in a marketing channel of distribution. A number of retailers promote locally a manufacturer's brand in order to increase their sales. However, sustained and substantial retailer promotions will damage the image of the manufacturer's brand. The manufacturer advertises nationally to improve the brand image. Demand at the retail outlets increase both by the local promotions and by the strength of brand image. First, we identify a noncooperative equilibrium of a differential game played with Markovian strategies. Next, we study a cooperative game where the players make coordinated marketing decisions and address the question whether the manufacturer can design an incentive strategy such that the retailers will stick to their parts of the agreed solution.     

Multi-agent team cooperation: A game theory approach

The main goal of this work is to design a team of agents that can accomplish consensus over a common value for the agents’ output using cooperative game theory approach. A semi-decentralized optimal control strategy that was recently introduced by the authors is utilized that is based on minimization of individual cost using local information. Cooperative game theory is then used to ensure team cooperation by considering a combination of individual cost as a team cost function. Minimization of this cost function results in a set of Pareto-efficient solutions. Among the Pareto-efficient solutions the Nash-bargaining solution is chosen. The Nash-bargaining solution is obtained by maximizing the product of the difference between the costs achieved through the optimal control strategy and the one obtained through the Pareto-efficient solution. The latter solution results in a lower cost for each agent at the expense of requiring full information set. To avoid this drawback some constraints are added to the structure of the controller that is suggested for the entire team using the linear matrix inequality (LMI) formulation of the minimization problem. Consequently, although the controller is designed to minimize a unique team cost function, it only uses the available information set for each agent. A comparison between the average cost that is obtained by using the above two methods is conducted to illustrate the performance capabilities of our proposed solutions.     

A differential game of international pollution control

Pollution is an inevitably by-product of production and damages the environment. The trade-off between production as a good and pollution as a bad over time can be analysed in the framework of a control model, which yields a path of emission charges that force the producers to behave in associally optimal way. Pollution that crosses national borders calls for international coordination of emission charges. In order to estimate the benefits of coordination one must use a realistic non-cooperative equilibrium concept. A natural extension of the control model implies unrealistic assumptions on information and commitment, and under-estimates the damage to the environment of not coordinating emission charges. In this note the more realistic subgame-perfect non-cooperative equilibrium is derived, which reinforces the case for international agreement og on pollution control.     

A mean-variance approach to advertising media selection

We propose a simple model of advertising media selection taking into account the uncertainty of the audiences reached effectively. The model is based upon the mean-variance point of view but the formulation differs from the now classical model used in portfolio theory.Here the goal is to minimize the variability of the total desirable audience. A practical solution method is proposed and applied to some examples.     

User opinion classification in social media: A global consistency maximization approach

Social media is a major platform for opinion sharing. In order to better understand and exploit opinions on social media, we aim to classify users with opposite opinions on a topic for decision support. Rather than mining text content, we introduce a link-based classification model, named global consistency maximization (GCM) that partitions a social network into two classes of users with opposite opinions. Experiments on a Twitter data set show that: (1) our global approach achieves higher accuracy than two baseline approaches and (2) link-based classifiers are more robust to small training samples if selected properly.     

A nonzero-sum differential game with curious solution properties

An example of a nonzero-sum differential game is given that illustrates some of the difficulties that might be encountered in using various solution concepts in differential game theory.     

A differential game in three dimensions: The aerial dogfight scenario

In this article, an aerial dogfight in three dimensions between two aircraft was modeled, with each aircraft having combined qualitative objectives of capture with avoidance. A spherical polar-coordinate system was devised to describe the system. From a standard point-mass model of aircraft dynamics for three-dimensional flight, a kinematic model was derived, with its corresponding controls for each aircraft. This kinematic model was then used, assuming bang-bang control functions for each aircraft, in order to determine a map of the game. A technique of trajectory dissection was introduced, whereby the airplane loci were decomposed in terms of regions of zero-level and saturation-level control values, and controllable and winning regions were determined for each aircraft using a Liapunov-function approach, the winning regions being calculated using the Getz-Leitmann theorem. A map of the game was constructed, and the barrier was found to be nonvoid. The concept of the posthumous mutual-kill strategy was introduced. Editor: M. Ardema     

A differential game among sectors in a macroeconomy

The purpose of this paper is primarily to model the Danish economy as a differential game among the sectors, and to solve this game using an algorithm (A) published in full detail elsewhere.This numerical algorithm (A) is first described briefly. It is used for the approximation of open-loop Nash-Cournot equilibrium controls in a differential game of fixed duration and initial state; it is based upon a hierarchical decomposition of the differential game into optimal control problems, with a fictive referee for the Nash-Cournot playing rule. Each constrained optimal control problem is solved by means of the generalized reduced gradient using constraint co-ordination.The algorithm has been applied to a nonlinear dynamic sectoral model of the Danish economy, which is described in detail. The control functions are: investments, labour, write-offs, marginal tax rates in each sector. The state variables are: foreign debt, state budget excess. Some results for the 1947–1952 period are given and they show that the open-loop Nash equilibrium controls obtained are somehow closer to the actual historical controls than those yielded by maximizing classical welfare criterions.     

A neural networks approach to minority game

The minority game (MG) comes from the so-called “El Farol bar” problem by W.B. Arthur. The underlying idea is competition for limited resources and it can be applied to different fields such as: stock markets, alternative roads between two locations and in general problems in which the players in the “minority” win. Players in this game use a window of the global history for making their decisions, we propose a neural networks approach with learning algorithms in order to determine players strategies. We use three different algorithms to generate the sequence of minority decisions and consider the prediction power of a neural network that uses the Hebbian algorithm. The case of sequences randomly generated is also studied. Research supported by Local Project 2004–2006 (EX 40%) Università di Foggia. A. Sfrecola is a researcher financially supported by Dipartimento di Scienze Economiche, Matematiche e Statistiche, Università degli Studi di Foggia, Foggia, Italy.