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We consider a network Nash equilibrium problem where players are represented as nodes of the graph of social relationships and each player’s action is influenced by the actions of her social contacts. Within this model, we investigate the problem of ranking the importance of different players. To this end, we first consider some classic topological measures, commonly used to assess the importance of nodes of a graph, and combine them using the entropy weighting method to obtain a single topological measure. We then consider a different measure of importance, based on the social welfare function, computed at the Nash equilibrium, for the original network, as well as for all the networks obtained from the initial one by removing a single player. The two measures are then illustrated and compared with the help of small size test problems

We investigate a class of parametric network games which encompasses both the cases of strategic complements and strategic substitutes. In the case of a bounded strategy space, we derive a representation formula for the unique Nash equilibrium. We also prove a comparison result between the Nash equilibrium and the social optimum and then compute the price of anarchy for some simple test problems

We investigate a class of Nash equilibrium problems on networks where both the utility function and the strategy space of each player are parametrized by means of continuous functions of a vector parameter. We provide a variational inequality reformulation of the Nash equilibrium problem which turns out to be much simpler than the original problem. Moreover, under additional assumptions, the continuity of the solution with respect to the parameter is proved.

We consider a class of finite-dimensional variational inequalities where both the operator and the constraint set can depend on a parameter. Under suitable assumptions, we provide new estimates for the Lipschitz constant of the solution, which considerably improve previous ones. We then consider the problem of computing the mean value of the solution with respect to the parameter and, to this end, adapt an algorithm devised to approximate a Lipschitz function whose analytic expression is unknown, but can be evaluated in arbitrarily chosen sample points. Finally, we apply our results to a class of Nash equilibrium problems, and generalized Nash equilibrium problems on networks.

We propose a new algorithm for solving a class of linear-quadratic network games with strategic complements and bounded strategies. The algorithm is based on the sequential solution of linear systems of equations and we prove that it finds the exact Nash equilibrium of the game after a finite number of iterations. The new algorithm is then applied to a social network model of juvenile delinquency which has been investigated recently where we also consider random perturbations of some data. Experimental results show the efficiency of the algorithm in solving large scale problems.

In this note we extend a previously proposed model of international human migration by introducing the possibility that some of the destination countries agree to establish a common upper bound on the migratory flows they are willing to accept jointly. In this framework, we propose a new equilibrium definition and prove its equivalence to a suitably defined variational inequality. Some numerical examples show that the flow distribution under joint regulations can differ from those corre-sponding to a situation where each government autonomously establishes migration bounds.

In this paper, we consider policy interventions for international migrant flows and quantify their ramifications. In particular, we further develop a recent equilibrium model of international human migration in which some of the destination countries form coalitions to establish a common upper bound on the migratory flows that they agree to accept jointly. We also consider here a scenario where some countries can leave or join an initial coalition and investigate the problem of finding the coalitions that maximize the overall social welfare. Moreover, we compare the social welfare at equilibrium with the one that a supranational organization might suggest in an ideal scenario. This research adds to the literature on the development of mathematical models to address pressing issues associated with problems of human migration with insights for policy and decision-makers.

We consider a class of games played on networks in which the utility 4 functions consist of both deterministic and random terms. In order to find the Nash 5 equilibrium of the game we formulate the problem as a variational inequality in a 6 probabilisticLebesguespacewhichissolvednumericallytoprovideapproximations 7 for the mean value of the random equilibrium. We also numerically compare the 8 solution thus obtained, with the solution computed by solving the deterministic 9 variational inequality derived by taking the expectation of the pseudo-gradient of 10 the game with respect to the random parameters.

We consider a congested traffic network where users behave according to the Wardrop equilibrium principle, but the data are uncertain and only known through their probability distributions. Within this framework, we propose a stochastic equilibrium model to analyse the network performance, which allows for nonlinear cost functions. The effectiveness of our approach is shown through numerical experiments on medium-size networks.

We investigate a class of network games with strategic complements and congestion effects, by using the variational inequality approach. Our contribution is twofold. We first express the boundary components of the Nash equilibrium by means of the Katz-Bonacich centrality measure. Then, we propose a new ranking of the network nodes based on the social welfare at equilibrium and compare this solution-based ranking with some classical topological ranking methods.

We consider a game-theory model of congestion control in communica- tion networks, where each player is a user who wishes to maximize his/her flow over a path in the network. We allow for stochastic fluctuations of the cost function of each player, which consists of two parts: a pricing and a utility term. The solution concept we look for is the mean value of the (unique) variational Nash equilibrium of the game. Furthermore, we assume that it is possible to invest a certain amount of money to improve the network by enhancing the capacity of its links and, because of limited financial resources, an optimal choice of the links to improve has to be made. We model the investment problem as a nonlinear knapsack problem with gen- eralized Nash equilibrium constraints in probabilistic Lebesgue spaces and solve it numerically for some examples.

In this paper we propose an approximation procedure for a class of monotone variational inequalities in probabilistic Lebesgue spaces. The implementation of the functional approximation in Lp, with p > 2, leads to a finite dimensional variational inequality whose structure is different from the one obtained in the case p = 2, already treated in the literature. The proposed computational scheme is applied to the random traffic equilibrium problem with polynomial cost functions.

The paper deals with a traffic network with random demands in which some of the roads need maintenance jobs. Due to budget constraints, a central authority has to choose which of them are to be maintained in order to decrease as much as possible the average total travel time spent by all the users, assuming that the network flows are distributed according to the Wardrop equilibrium principle. This optimal road maintenance problem is modeled as an integer nonlinear program, where the objective function evaluation is based on the solution of a stochastic variational inequality. We propose a regularization and approximation procedure for its computation and prove its convergence. Finally, the results of preliminary numerical experiments on some test networks are reported.

We investigate a generalized Nash equilibrium problem where players are modeled as nodes of a network and the utility function of each player depends on his/her own action as well as on the actions of his/her neighbors in the network. In the case of a quadratic reference model with shared constraints we are able to derive the variational solution of the game as a series expansion which involves the powers of the adjacency matrix, thus extending a previous result. Our analysis is illustrated by means of some numerical examples.

We investigate a class of network games with strategic complements and bounded strategy sets by using the variational inequality approach. In the case where the Nash equilibrium of the game has some boundary components, we derive a formula which connects the equilibrium to the Katz-Bonacich centrality measure, thus generalizing the classical result for the interior solution case. Furthermore, we prove that any component of the Nash equilibrium is less than or equal to the corresponding component of the social optimal solution and numerically study the price of anarchy for a small size test problem.

We consider a popular model of congestion control in communication networks within the theory of generalized Nash equilibrium problems with shared constraints, where each player is a user who has to send his/her flow over a path in the network. The cost function of each player consists of two parts: a pricing and a utility term. Within this framework we assume that the network system manager can invest a given amount of money to improve the network by enhancing the capacity of its links and, because of limited financial resources, has to make a choice as to which of the links to improve. This choice is made with the help of a performance function which is computed for each set of improvements under consideration and has the property that, once the equilibrium has been reached, maximizes the aggregate utility and minimizes the sum of delays at the links. We model this problem as a nonlinear knapsack problem with generalized Nash equilibrium constraints and show some preliminary numerical experiments.

We use the theory of stochastic variational inequalities to develop a network equilibrium model of the whole supply chain of electricity markets: power generation, supply, transmission, and consumption. In particular, we take into account the case where the market demand functions are not exactly known but are affected by some kind of uncertainty. A discretization and truncation procedure is used to numerically solve the stochastic variational inequality model. Monotonicity properties of the operator are investigated and the affine case is analysed in detail. Finally, numerical experiments show the impact of different probability densities of the random variables on the approximated solutions and the scalability of the proposed numerical method for real-world sized problems.

We consider a traffic network in which some of the roads need maintenance jobs. Due to budget constraints not all of the roads can be maintained and a central authority has to choose which of them are to be improved.We propose a nonlinear programming model where this choice is made according to its effects on the relative variation of the total cost, assuming that users behave according toWardrop equilibrium principle. To assess the network improvement after maintenance we use the Bureau of Public Road link cost functions.