Pareto-optimal Nash equilibrium in capacity allocation game for self-managed networks

In this paper we introduce a capacity allocation game which models the problem of maximizing network utility from the perspective of distributed noncooperative agents. Motivated by the idea of self-managed networks, in the developed framework the decision-making entities are associated with individual transmission links, deciding on the way they split capacity among concurrent flows. An efficient decentralized algorithm is given for computing a strongly Pareto-optimal strategies, constituting a pure Nash equilibrium. Subsequently, we discuss the properties of the introduced game related to the Price of Anarchy and Price of Stability. The paper is concluded with an experimental study.     

Nash strategies for M-person differential games with mixed information structures

This paper is concerned with a class of M-person linear-quadratic nonzero-sum differential games in which a subset of the players have access to closed-loop (CL) information and the rest to open-loop (OL) information. The state equation contains an additive random perturbation term, inclusion of which has been shown to be necessary in order to obtain, a unique globally optimal Nash equilibrium solution. For each player with CL information, the optimal strategy is a linear function of the current and the initial states, and for each player with OL information, the optimal strategy is a linear function of the initial state.     

Deterministic calibration and Nash equilibrium

We provide a natural learning process in which the joint frequency of (time-averaged) empirical play converges into the set of convex combinations of Nash equilibria. Furthermore, the actual distribution of players' actions is close to some (approximate) Nash equilibria on most rounds (on all but a vanishing fraction of the rounds). In this process, all players rationally choose their actions using a public prediction made by a deterministic, weakly calibrated algorithm. For this to be possible, we show that such a deterministic (weakly) calibrated learning algorithm exists.     

Sufficient conditions for existence of a fixed point in stochasticreward net-based iterative models

Stochastic Petri net models of large systems that are solved by generating the underlying Markov chain pose the problem of largeness of the state-space of the Markov chain. Hierarchical and iterative models of systems have been used extensively to solve this problem. A problem with models which use fixed-point iteration is the theoretical proof of the existence, uniqueness and convergence of the fixed-point equations, which still remains an “art”. In this paper, we establish conditions, in terms of the net structure and the characteristics of the iterated variables, under which existence of a solution is guaranteed when fixed-point iteration is used in stochastic Petri nets. We use these conditions to establish the existence of a fixed point for a model of a priority scheduling system, at which tasks may arrive according to a Poisson process or due to spawning or conditional branching of other tasks in the system     

Sufficient conditions for the existence and asymptotic behaviour of solution to a quasi-linear elliptic problem

In this article, we combine the existing regularity theory, perturbation method and the lower and upper solutions method to study the existence and asymptotic behaviour of positive solution to a boundary value problem for the p-Laplacian operator. More exactly, we study the existence and asymptotic behaviour of the positive solution to a quasi-linear elliptic problem of the form?Δ _p u=λ a(x)g(u) in D′(Ω), u>0 in Ω, lim _{x→? Ω} u(x)=0. Under some conditions on a and g, we show that there is a non-negative number Λ₀ such that for all λ∈(0, Λ₀], the problem has a solution u _λ in the sense of distribution, which is bounded from above by some positive numbers μ(λ). Such estimates and the asymptotic behaviour are important in computer programs when we know an algorithm for determining the solution.     

Nash equilibrium in differential games and the quasi-strategy formalism

We consider a two-player nonzero-sum differential game in the case where players use nonanticipative strategies. We define the Nash equilibrium in this case and obtain a characterization of Nash equilibrium strategies. We show that a Nash equilibrium solution can be approximately realized by control-with-guide strategies.     

Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions

This paper investigates the existence and uniqueness of solutions for an impulsive mixed boundary value problem of nonlinear differential equations of fractional order α∈(1,2]. Our results are based on some standard fixed point theorems. Some examples are presented to illustrate the main results.     

A Nash equilibrium solution in an oligopoly market: The search for Nash equilibrium solutions with replicator equations derived from the gradient dynamics of a simplex algorithm

The present analysis applies continuous time replicator dynamics to the analysis of oligopoly markets. In the present paper, we discuss continuous game problems in which decision-making variables for each player are bounded on a simplex by equalities and non-negative constraints. Several types of problems are considered under conditions of normalized constraints and non-negative constraints. These problems can be classified into two types based on their constraints. For one type, the simplex constraint applies to the variables for each player independently, such as in a product allocation problem. For the other type, the simplex constraint applies to interference among all players, creating a market share problem. In the present paper, we consider a game problem under the constraints of allocation of product and market share simultaneously. We assume that a Nash equilibrium solution can be applied and derive the gradient system dynamics that attain the Nash equilibrium solution without violating the simplex constraints. Models assume that three or more firms exist in a market. Firms behave to maximize their profits, as defined by the difference between their sales and cost functions with conjectural variations. The effectiveness of the derived dynamics is demonstrated using simple data. The present approach facilitates understanding the process of attaining equilibrium in an oligopoly market.     

Task graph pre-scheduling, using Nash equilibrium in game theory

Prescheduling algorithms are targeted at restructuring of task graphs for optimal scheduling. Task graph scheduling is a NP-complete problem. This article offers a prescheduling algorithm for tasks to be executed on the networks of homogeneous processors. The proposed algorithm merges tasks to minimize their earliest start time while reducing the overall completion time. To this end, considering each task as a player attempting to reduce its earliest time as much as possible, we have applied the idea of Nash equilibrium in game theory to determine the most appropriate merging. Also, considering each level of a task graph as a player, seeking for distinct parallel processors to execute each of its independent tasks in parallel with the others, the idea of Nash equilibrium in game theory can be applied to determine the appropriate number of processors in a way that the overall idle time of the processors is minimized and the throughput is maximized. The communication delay will be explicitly considered in the comparisons. Our experiments with a number of known benchmarks task graphs and also two well-known problems of linear algebra, LU decomposition and Gauss–Jordan elimination, demonstrate the distinguished scheduling results provided by applying our algorithm. In our study, we consider ten scheduling algorithms: min–min, chaining, A ^?, genetic algorithms, simulated annealing, tabu search, HLFET, ISH, DSH with task duplication, and our proposed algorithm (PSGT).     

Discovering theorems in game theory: Two-person games with unique pure Nash equilibrium payoffs

In this paper we provide a logical framework for two-person finite games in strategic form, and use it to design a computer program for discovering some classes of games that have unique pure Nash equilibrium payoffs. The classes of games that we consider are those that can be expressed by a conjunction of two binary clauses, and our program re-discovered Kats and Thisse?s class of weakly unilaterally competitive two-person games, and came up with several other classes of games that have unique pure Nash equilibrium payoffs. It also came up with new classes of strict games that have unique pure Nash equilibria, where a game is strict if for both player different profiles have different payoffs.     

On global existence of solutions to coupled matrix Riccatiequations in closed-loop Nash games

Presents comparison and global existence results for solutions of coupled matrix Riccati differential equations appearing in closed loop Nash games and in mixed H₂/H_∞-type problems. Convergence of solutions is established for the diagonal case, solutions of the corresponding algebraic equations are discussed using numerical examples     

Hybrid coevolutionary programming for Nash equilibrium search in games with local optima

The conventional local optimization path and coevolutionary processes are studied when "local Nash equilibrium (NE) traps" exist. Conventional NE search algorithms in games with local optima can misidentify NE by following a local optimization path. We prove that any iterative NE search algorithms based on local optimization cannot differentiate real NE and "local NE traps". Coevolutionary programming, a parallel and global search algorithm, is applied to overcome this problem. In order to enhance the poor convergence of simple coevolutionary programming, hybrid coevolutionary programming is suggested. The conventional NE algorithms, simple coevolutionary programming, and hybrid coevolutionary algorithms are tested through a simple numerical example and transmission-constrained electricity market examples.     

Sufficient conditions for parameter convergence in linearizable systems

The major drawback of the adaptive compensation design in the previous paper (2002) in terms of parameter identification is that the criterion for parameter convergence cannot be checked in advance. The purpose of this paper aims at circumventing such difficulties by providing a priori verifiable sufficient conditions that guarantee the asymptotic stability of the parametric equilibrium.     

Sufficient conditions for deceptive and easy binary functions

This paper finds sufficient conditions for fully or partially deceptive binary functions by calculating schema average fitness values. Deception conditions are first derived for functions of unitation (functions that depend only on the number of 1s in the string) and then extended for any binary function. The analysis is also extended to find a set of sufficient conditions for fully easy binary functions. It is found that the computational effort required to investigate full or partial deception in a problem of sizel using these sufficient conditions isO(2 ^l ) and using all necessary conditions of deception isO(4 ^l ). This calculation suggests that these sufficient conditions can be used to quickly test deception in a function. Furthermore, it is found that these conditions may also be systematically used to design a fully deceptive function by performing onlyO(l ²) comparisons and to design a partially deceptive function to orderk by performing onlyO(kl) comparisons. The analysis shows that in the class of functions of unitation satisfying these conditions of deception, an order-k partially deceptive function is also partially deceptive to any lower order. Finally, these sufficient conditions are used to investigate deception in a number of currently-used deceptive problems.     

Uniqueness and efficiency of Nash equilibrium in a family of randomly generated repeated games

This paper describes the results of an analysis of the Nash equilibrium in randomly generated repeated games. We study two families of games: symmetric bimatrix games G(A, B) with B = A^⊤ and nonsymmetric bimatrix games (the first includes the classical games of prisoner dilemma, battle of the sexes, and chickens). We use pure strategies, implemented by automata of size two, and different strategy domination criteria. We observe that, in this environment, the uniqueness and efficiency of equilibria outcomes is the typical result.     

Sufficient conditions for connectivity maintenance and rendezvous in leader-follower networks

In this paper we derive a set of constraints that are sufficient to guarantee maintained connectivity in a leader-follower multi-agent network with a proximity based communication topology. In the scenario we consider, only the leaders are aware of the global mission, which is to converge at a known destination point. Thus, the followers need to stay in contact with the group of leaders in order to reach the goal. In the paper we show that we can maintain the initial network structure, and thereby connectivity, by setting up bounds on the ratio of leaders-to-followers and on the magnitude of the goal attraction force experienced by the leaders. The results are first established for an initially complete communication graph and then extended to an incomplete graph. The results are illustrated by computer simulations.     

Static team problems--Part I: Sufficient conditions and the exponential cost criterion

The stationary conditions of Radner are shown under relaxed hypotheses to be sufficient to establish the global optimality of candidate control laws for static team problems with convex cost. This extension of Radner's theorem establishes the optimality of affine laws for the exponential of a quadratic performance index with jointly Gaussian state and observation variables. The class of performance indexes of team problems for which the optimal solution is known is thereby enlarged.     

Sufficient conditions on stability of interval matrices:connections and new results

The stability properties of interval matrices are discussed. Sufficient conditions are given for these properties based upon Gershgorin's theorem and its extension, and they are shown to improve previous results. The derivation helps clarify relations between certain previously available results and shows they can be derived using Gershgorin's theorem. Furthermore, the problem of determining the stability of interval matrices is related to that of characterizing certain nonsingular M-matrices     

Feedback control on Nash equilibrium for discrete-time stochastic systems with Markovian jumps: Finite-horizon case

In this paper, we consider the feedback control on nonzero-sum linear quadratic (LQ) differential games in finite horizon for discrete-time stochastic systems with Markovian jump parameters and multiplicative noise. Four-coupled generalized difference Riccati equations (GDREs) are obtained, which are essential to find the optimal Nash equilibrium strategies and the optimal cost values of the LQ differential games. Furthermore, an iterative algorithm is given to solve the four-coupled GDREs. Finally, a suboptimal solution of the LQ differential games is proposed based on a convex optimization approach and a simplification of the suboptimal solution is given. Simulation examples are presented to illustrate the effectiveness of the iterative algorithm and the suboptimal solution.