On a branch-and-bound approach for a Huff-like Stackelberg location problem

Modelling the location decision of two competing firms that intend to build a new facility in a planar market can be done by a Huff-like Stackelberg location problem. In a Huff-like model, the market share captured by a firm is given by a gravity model determined by distance calculations to facilities. In a Stackelberg model, the leader is the firm that locates first and takes into account the actions of the competing chain (follower) locating a new facility after the leader. The follower problem is known to be a hard global optimisation problem. The leader problem is even harder, since the leader has to decide on location given the optimal action of the follower. So far, in literature only heuristic approaches have been tested to solve the leader problem. Our research question is to solve the leader problem rigorously in the sense of having a guarantee on the reached accuracy. To answer this question, we develop a branch-and-bound approach. Essentially, the bounding is based on the zero sum concept: what is gain for one chain is loss for the other. We also discuss several ways of creating bounds for the underlying (follower) sub-problems, and show their performance for numerical cases

Determination of stable coalitions in a CO2 emission game

This paper presents an implementable description of a game on Kyoto protocol coalition formation. Concepts from literature about the so-called CO2 emission game where stability of coalition structures in an Open Membership Game and Exclusive Membership Game are applied, are translated into a new notation for specifying the stability and facilitating the implementation into computer coding. We consider multiple coalitions that groups of world regions can join. Implementation aspects are outlined and results are show

Game theory at work: OR models and algorythms to solve multi-actor heterogeneous decision problems

Key words: Game theory, operations research, optimisation methods, algorithms. The objective of this thesis is to explore the potential of combining Game Theory (GT) models with Operations Research (OR) modelling. This includes development of algorithms to solve these complex OR models for different empirical situations. The challenge is to get GT “at work” by applying such models and techniques to practical cases. Four different cases with a challenge on the development of algorithms are studied. A first case illustrates a multiple coalition formation game in which membership rules and different transfer schemes are described. Given the GT model and the OR model, the goal is to develop methods for checking stability of coalition structures. A new mathematical programming formulation, crucial for the development of the algorithms, is elaborated. Available data is used to determine which stable coalitions appear and which procedures (transfer schemes) can be used to make coalitions stable. Also the influence of membership rules (whether actors are free to become a member) is investigated. Main conclusion is that transfer schemes are useful to be implemented to obtain stable coalitions. Moreover, different membership rules, e.g. veto or majority voting of current members, generate different results with and without transfer schemes. A second case studies a model of coalition formation in politics with n parties trying to form a government. Given the number of parties n and policy dimension m (number of items), computational algorithms are developed to compute all possible majority coalitions and preferences of parties over those coalitions. Application to Dutch data and theoretical examples leads to testing of hypotheses with surprising results with respect to coalition formation such as: being a first mover is not necessarily advantageous, being less flexible is not necessarily advantageous, forming a minimal winning coalition is not necessarily advantageous. A third case describes a two-stage location-quantity game where n > 2 firms are competing on m > 2 markets. The space where the firms can locate are nodes on a network. Analytical solutions for the supplying decisions and properties for determining the number of suppliers to each market are derived. In finding the equilibria, a complete enumeration algorithm and a local search algorithm are used. Two cases are elaborated to illustrate the procedures and the analytical results. The last case deals with a competitive facility location problem in which the concept of Stackelberg leader-follower problem is applied. The follower problem and leader problem are global optimisation problems. Branch-and-Bound (B&B) algorithms that guarantee to find the optimum of both problems are designed

Methods for computing Nash equilibria of a location-quantity game

A two-stage model is described where firms take decisions on where to locate their facility and on how much to supply to which market. In such models in literature, typically the market price reacts linearly on supply. Often two competing suppliers are assumed or several that are homogeneous, i.e., their cost structure is assumed to be identical. The focus of this paper is on developing methods to compute equilibria of the model where more than two suppliers are competing that each have their own cost structure, i.e., they are heterogeneous. Analytical results are presented with respect to optimality conditions for the Nash equilibria in the two stages. Based on these analytical results, an enumeration algorithm and a local search algorithm are developed to find equilibria. Numerical cases are used to illustrate the results and the viability of the algorithms. The methods find an improvement of a result reported in literature

Methods for computing Nash equilibria of a location quantity game, Submitted to Operations Research

A two stage model is described where firms take decisions on where to locate their facility and on how much to supply to which market. In such models in literature, typically the market price reacts linearly on supply. Often two competing suppliers are assumed or several that are homogeneous, i.e. their cost structure is assumed to be identical. The focus of this paper is on developing methods to compute equilibria of the model where more than two suppliers are competing that each have their own cost structure, i.e. they are heterogeneous. Analytical results are presented with respect to optimality conditions for the Nash equilibria in the two stages. Based on these analytical results, algorithms are developed to find equilibria. Numerical cases are used to illustrate the results and the viability of the algorithms. The method finds an improvement of a result reported in literature