Lina Mallozzi

An application of optimization theory to the study of equilibria for games: a survey

This contribution is a survey about potential games and their applications. In a potential game the information that is sufficient to determine Nash equilibria can be summarized in a single function on the strategy space: the potential function. We show that the potential function enable the application of optimization theory to the study of equilibria. Potential games and their generalizations are presented. Two special classes of games, namely team games and separable games, turn out to be potential games. Several properties satisfied by potential games are discussed and examples from concrete situations as congestion games, global emission games and facility location games are illustrated.

Equilibrium strategies via GA to stackelberg games under multiple follower's best reply

In this paper, we study a two‐person game between one leader and one follower, called the Stackelberg game. The leader player enounces a decision before the others, and the follower takes into account this decision and solves an optimization problem that may have multiple solutions. Then, the leader optimizes his objective by assuming a given followers reaction depending on his behavior. We consider in this paper a hierarchical equilibrium solution for a two‐level game, particularly the strong Stackelberg solutions that corresponds to an optimistic leaders point of view and we give a numerical procedure based on a genetic algorithm (GA) evolution process to compute them. The use of a multimodal genetic algorithm allows us to approach the possible multiple solutions to the lower level problem. The algorithm convergence is illustrated by means of some test cases.

Cooperative games in facility location situations with regional fixed costs

A continuous single-facility location problem, where the fixed cost (or installation cost) depends on the region where the new facility is located, is studied by mean of cooperative Game Theory tools. Core solutions are proposed for the total cost allocation problem. Sufficient conditions in order to have a nonempty core are given, then the Weber problem with regional fixed costs is studied.

A Hierarchical MultiModal Hybrid Stackelberg–Nash GA for a Leader with Multiple Followers Game

In this paper a numerical procedure based on a genetic algorithm (GA) evolution process is given to compute a Stackelberg solution for a hierarchical n + 1-person game. There is a leader player who enounces a decision before the others, and the rest of players (followers) take into account this decision and solve a Nash equilibrium problem. So there is a two-level game between the leader and the followers, called Stackelberg–Nash problem. The idea of the Stackelberg-GA is to bring together genetic algorithms and Stackelberg strategy in order to process a genetic algorithm to build the Stackelberg strategy. In the lower level, the followers make their decisions simultaneously at each step of the evolutionary process, playing a so called Nash game between themselves. The use of a multimodal genetic algorithm allows to find multiple Stackelberg strategies at the upper level. In this model the uniqueness of the Nash equilibrium at the lower-level problem has been supposed. The algorithm convergence is illustrated by means of several test cases.

Sequential Production Situations and Potentials

This paper studies an important type of production problems in which production takes place in several stages. These problems are modeled as sequential production games, a specific class of extensive form games with imperfect information. These games are related to potential games. Despite the presence of imperfect information, it is shown that pure strategy subgame perfect Nash equilibria do exist, thus allowing for easily adopt able recommendations to firms seeking the advice of game theorists.

Optimal transport and a bilevel location-allocation problem

In this paper a two-stage optimization model is studied to find the optimal location of new facilities and the optimal partition of the consumers (location-allocation problem). The social planner minimizes the social costs, i.e. the fixed costs plus the waiting time costs, taking into account that the citizens are partitioned in the region according to minimizing the capacity costs plus the distribution costs in the service regions. By using optimal transport tools, existence results of solutions to the location-allocation problem are presented together with some examples.

Multi-Leader Multi-Follower Model with Aggregative Uncertainty

We study a non-cooperative game with aggregative structure, namely when the payoffs depend on the strategies of the opponent players through an aggregator function. We assume that a subset of players behave as leaders in a Stackelberg model. The leaders, as well the followers, act non-cooperatively between themselves and solve a Nash equilibrium problem. We assume an exogenous uncertainty affecting the aggregator and we obtain existence results for the stochastic resulting game. Some examples are illustrated.

Game Theoretical Tools for Wing Design

In the general setting of modeling for system design it is assumed that all decision-makers cooperate in order to choose the optimal set of the variable design. Sometimes there are conflicts between the different tasks, so that the design process is studied as a multi-player game. In this work we deal with a preliminary application of the design of a wing by studying its optimal configuration by considering some of the standard parameters of the plant design. The choice of the parameters value is done by optimizing two tasks: the weight and the drag. This two-objective optimization problem is approached by a cooperative model, just minimizing the sum of the weight and the drag, as well by a non-cooperative model by means of the Nash equilibrium concept. Both situations will be studied and some test cases will be presented.

A Planar Location-Allocation Problem with Waiting Time Costs

We study a location-allocation problem where the social planner has to locate some new facilities minimizing the social costs, i.e. the fixed costs plus the waiting time costs, taking into account that the citizens are partitioned in the region according to minimizing the capacity acquisition costs plus the distribution costs in the service regions. In order to find the optimal location of the new facilities and the optimal partition of the consumers, we consider a two-stage optimization model. Theoretical and computational aspects of the location-allocation problem are discussed for a planar region and illustrated with examples.

A Game Theoretical Model for Experiment Design Optimization

In this paper we present a noncooperative game theoretical model for the well known problem of experimental design. A virtual player decides the design variables of an experiment and all the players solve a Nash equilibrium problem by optimizing suitable payoff functions. The resulting game has nice properties, so that a computational procedure is performed by using genetic algorithm approach. We consider the case where the design variables are the coordinates of n points in a region of the plane and we look for the optimal configuration of the points under some constraints. The problem arises from a concrete situation: find the optimal location of n receivers able to pick up particles of cosmic ray in a given platform (some experiment to measure the quantities of gamma rays are ongoing in Nepal thanks to the altitude of the region). Theoretical and computational results are presented for this location problem.