Spatial Interaction Models

Facility location and allocation problems have been a major area of research for decades, which has led to a vast and still growing literature. Although there are many variants of these problems, there exist two common features: finding the best locations for one or more facilities and allocating demand points to these facilities. A considerable number of studies assume a monopolistic viewpoint and formulate a mathematical model to optimize an objective function of a single decision maker. In contrast, competitive facility location (CFL) problem is based on the premise that there exist competition in the market among different firms. When one of the competing firms acts as the leader and the other firm, called the follower, reacts to the decision of the leader, a sequential-entry CFL problem is obtained, which gives rise to a Stackelberg type of game between two players. A successful and widely applied framework to formulate this type of CFL problems is bilevel programming (BP). In this chapter, the literature on BP models for CFL problems is reviewed, existing works are categorized with respect to defined criteria, and information is provided for each work.

We consider how three firms compete in a Salop location model and how cooperation in location choice by two of these firms affects the outcomes. We consider the classical case of linear transportation costs as a two-stage game in which the firms select first a location on a unit circle along which consumers are dispersed evenly, followed by the competitive selection of a price. Standard analysis restricts itself to purely competitive selection of location; instead, we focus on the situation in which two firms collectively decide about location, but price their products competitively after the location choice has been effectuated. We show that such partial coordination of location is beneficial to all firms, since it reduces the number of equilibria significantly and, thereby, the resulting coordination problem. Subsequently, we show that the case of quadratic transportation costs changes the main conclusions only marginally.

Cooperative and non-cooperative games arising from location problems on an undirected graph with facilities of different types are considered. The nodes of the graph correspond to players. Distances between the players are measured along the edges of the graph.The graph is assumed to be connected and the distance between two nodes is the length of a shortest path from one node to the other. Each player has a need for a facility of each type. The intensity of the need of each player for a certain facility depends on the type of the facility. A cooperative game in which the number of facilities that a coalition is allowed to build depends on the size of the coalition is introduced. A coalition can only build a facility in a node corresponding to a member of the coalition. A cooperative game arising from such a facility location problem need not be balanced. Conditions that guarantee the balancedness of the game are discussed. A non-cooperative game in which the pure strategies of the players corresponds to choosing a facility to be built in their geographical location is introduced. Conditions that guarantee that the Nash equilibria of this game correspond to the complete set of facilities being chosen are studied.

In this chapter we present a survey of location methods based on Nash equilibria for the design of experiments. The solution of the location problem is given in the bi-dimensional case by means of a potential formulation and a Nash game. The most important definitions and proofs are reported. Two main application fields are employed to stress the capability of an ad hoc numerical methodology involved in the solution of the location problem. The first one refers to optimal (constrained) location of sensors collecting cosmic rays for astrophysics experiments. The second one concerns the design of experiment in aerospace engineering related to set of flight tests within the flight envelope of an airplane. An outlook on location-allocation problem in economics is considered for a linear city with congestion in the conclusions.

Facility location models deal, for the most part, with the location of plants, warehouses, distribution centers, retail facilities among others. In this chapter we review the game theoretical concept of the leader-follower in facility location models which addresses specific circumstances: (i) competitive location of two facilities anywhere on the plane; (ii) covering a large area by chain facilities so that a future competitor will not be able to attract much demand; (iii) competitive location of two facilities applying the gravity (Huff) rule; (iv) competitive location of multiple facilities using the cover-based rule; and (v)locating facilities on the nodes of a network to cover as much demand as possible following a removal of a link by a follower.

This paper first presents a standard competitive duopoly location model on a linear market and derives an equilibrium solution as well as a solution for the sequential von Stackelberg game. The heart of the contribution then investigates scenarios, in which the duopolists face or follow asymmetric situations or strategies. In particular, we examine situations, in which the duopolists have different objectives, models, in which firms apply different pricing policies, and instances, in which the competitors have different capabilities.

The so-called leader-follower (or Stackelberg) problem is researched. A chain, the leader, wants to locate a single new facility in a region of the plane. After that, as a reaction, the competitor chain, the follower, will locate a single new facility too, knowing the decision taken by the leader. Several variants of the problem are analyzed. In the simplest one, the objective of both the leader and the follower is to maximize the market share, the qualities of the facilities to be located are given beforehand, and the demand is fixed (no costs are considered). In the second one, the qualities of the facilities to be located are considered variables of the problem, and costs related both to location and quality are taken into account; the demand is fixed as in the first model. Finally, the last model extends the previous one considering that the demand varies depending on the location and the quality of the facilities. Exact (for the first problem) and heuristic (for the second and third problems) approaches proposed for the aforementioned location models are described and analyzed. High performance computing approaches for the heuristic methods are also reviewed. A new exact branch-and-bound method for the last two problems is also suggested.

In this paper we consider the problem of locating emergency units in a given area, defining a suitable class of TU-games. The Shapley value results to be a very good solution for this class of games. A simple formula for computing the Shapley value is derived. A real-world example and some comments conclude.

We develop a theory of an equilibrium-econometric analysis of rental housing markets with indivisibilities. It provides a bridge between a (competitive) market equilibrium theory and a statistical/econometric analysis. The listing service of apartments provides the information to both economic agents and an econometric analyzer: each economic agent uses a small part of the data from the service for his economic behavior, and the analyzer uses them to estimate the market structure. It is argued that the latter may be done by assuming that the economic agents take the standard price-taking behavior. We apply our theory to the data in the rental housing markets in the Tokyo area, and examine the law of diminishing marginal utility for household. It holds strictly with respect to the consumption, less with commuting time-distance, and much less with the sizes of apartments.

We consider spatial competition when consumers are arbitrarily distributed on a compact metric space. Retailers can choose one of finitely many locations in this space. We focus on symmetric mixed equilibria which exist for any number of retailers. We prove that the distribution of retailers tends to agree with the distribution of the consumers when the number of competitors is large enough. The results are shown to be robust to the introduction of (1) randomness in the number of retailers and (2) different ability of the retailers to attract consumers.

In this paper, we study some of classical results in facility location games. Shapley value and equal surplus sharing rules are considered. It is seen that these rules do not have a population monotonic allocation schemes (PMAS). Further, we introduce facility location interval games and their properties. Finally, we conclude this paper.

In this paper we consider a Hotelling model on the linear city, where the location is not a free good. We assume that firms play a location-cum-price game, and that the game is played into two steps. After the first step, in which the classical duopoly game is played, we suppose that in a second step a third firm enters the market and that the incumbents are allowed to react to this entry. In both steps firms have to face a cost for location, for which we consider two different cases.

This paper addresses a location-price problem on a transportation network. We suppose that the competing firms select their facility locations, and then they compete on delivered prices with the aim of profit maximization. The firms sell an homogeneous product and the customers buy from the one that offers the lowest price. Under some general conditions, for any locations of the facilities, the existence of a unique price equilibrium is shown. Then the location price problem is reduced to a location game if the competing firms set the equilibrium prices. The aim of this paper is to study this location game for any number of competing firms which locate multiple facilities. For essential products, it is proved that the global minimizers of the social cost are location Nash equilibria. In particular, there exists at least one global minimizer of social cost at the nodes of the network if marginal delivered costs are concave. In this case, an Integer Linear Programming (ILP) formulation is proposed to minimize the social cost. For non essential products, the minimizers of social cost may not be location Nash equilibria. Then a best response algorithm is proposed to find a location Nash equilibrium. If marginal delivered costs are concave, an ILP formulation is given for profit maximization of one firm, assuming that the locations of its competitors are fixed. Finally, the selection of a location Nash equilibrium when there are multiple location Nash equilibria is discussed.

A location problem occurs whenever a set of users have to agree on the position of one or several facilities in order to provide some service for them. The goal is to minimize the overall service cost and depending on the framework space, nature of the service and the globalizing cost function many different models appear: median, center, ordered median, coverage, hub-location, etcetera. Any of these problems has produced a large body of literature in order to find optimal or approximate solutions to their corresponding optimization problems. However, even knowing the exact solution of those problems there is another interesting problem that deserves the attention of researchers: How to share the cost of implementing such an optimal solution among the users of that system? This chapter addresses this question for several well-known location problems that appears in location problems in the continuous setting.