The Competitive Facility Location Problem in a Duopoly: Connections ...

The Competitive Facility Location Problem in a Duopoly: Connections to the 1-Median Problem Daniela Saban1 and Nicolas Stier-Moses1,2 1 2

Graduate School of Business, Columbia University, USA Universidad Torcuato Di Tella, Buenos Aires, Argentina {dsaban15,stier}@gsb.columbia.edu

Abstract. We consider a competitive facility location problem on a network, in which consumers are located on the vertices and wish to connect to the nearest facility. Knowing this, competitive players locate their facilities on vertices that capture the largest possible market share. In 1991, Eiselt and Laporte established the first relation between Nash equilibria of a facility location game in a duopoly and the solutions to the 1-median problem. They showed that an equilibrium always exists in a tree because a location profile is at equilibrium if and only if both players select a 1median of that tree [4]. In this work, we further explore the relations between the solutions to the 1-median problem and the equilibrium profiles. We show that if an equilibrium in a cycle exists, both players must choose a solution to the 1-median problem. We also obtain the same property for some other classes of graphs such as quasi-median graphs, median graphs, Helly graphs, and strongly-chordal graphs. Finally, we prove the converse for the latter class, establishing that, as for trees, any median of a strongly-chordal graph is a winning strategy that leads to an equilibrium.

1

Introduction

Facility location problems deal with the optimal placement of facilities with respect to a set of customers. In the discrete version of this problem, a decisionmaker needs to select a vertex of a graph whose vertices represent the potential locations where the facility may be placed. Vertices also represent customers and have weights that encode the demand at each location. Finally, distances are captured by the topology of the graph. In the centralized problem, a decision-maker has to select a vertex that minimizes the distance that customers need to travel to visit the facility, solution normally referred to as a 1-median [7]. In the competitive version of the facility location problem, a set of players is competing to attract customers and wish to maximize market share by locating their facilities strategically in the graph. This problem was first studied by Hotelling [8] in 1929, where two players select a location on a continuous and linear market with demand uniformly distributed along it. His prediction was that at equilibrium both players locate in the 1-median of that line because otherwise they can undercut the competitor and increase the market-share. P.W. Goldberg and M. Guo (Eds.): WINE 2012, LNCS 7695, pp. 539–545, 2012. c Springer-Verlag Berlin Heidelberg 2012

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D. Saban and N. Stier-Moses

We consider a discrete version of the competitive facility location problem in a duopoly. Given a graph with weights representing demands, both players must select a vertex to locate a facility. The utility of a player is given by the total demand among vertices closest to the selected facility. To break ties, demand is split evenly for vertices that are equidistant to the two facilities. The work of Eiselt and Laporte, the first among just a few references that study the facility location game as stated here, shows that trees always admit pure-strategy Nash equilibria [4]. Indeed, a selection of facilities is a Nash equilibrium if and only if both players select a (possibly different) 1-median of the tree, which always exists. Motivated by this result, our work establishes further links between Nash equilibria of the facility location problem in a duopoly and the 1-median problem, for various classes of topologies. To the best of our knowledge, with the exception of [4], we are not aware of other results in this direction. Since it is natural for players to locate in a central location in the market, we seek to understand under what circumstances when an equilibrium exists, players have the incentive to select solutions to the 1-median problem. We provide a proof of this result for cycles, which combined with the results of [4] can be extended to cacti and other more general, but specific, topologies. This extension relies on a decomposition technique that allows one to focus in the subgraph that contains the equilibria [6]. The idea is to represent the graph as a tree of maximal bi-connected components. This representation conserves some of the relevant information about the original graph and allows one to find the components where equilibria might be located. In addition, we show that for an arbitrary graph topology, when an equilibrium exists, both players select vertices that are local optima to the 1-median problem. This result automatically translates to proving that equilibria can only be located at a 1-median for different classes of graphs where no local optima exist, such as median graphs, quasi-median graphs and Helly graphs. Those families of graphs include grids and latices, which capture the topology of many real urban networks. Finally, we generalize the result that trees always have equilibria, which are located in medians, to the class of strongly chordal graphs. That family of graphs includes trees but also other topologies such as interval graphs and block graphs. To decide if an instance of this game admits an equilibrium by exhaustive search, it is necessary to evaluate all possible deviations from each possible outcome of the game. There are O(|V |2 ) outcomes, O(|V |) deviations, and for each we must evaluate a shortest path tree to compute the market share for each player. Our results imply that it is not necessary to check every possible outcome of the game but just the combinations of winning strategies, or the 1-medians if the former are not available and a fast algorithm to compute them is available for the specific instance. To conclude, various versions of facility locations games have been studied over the last decades, differing in the number of players, the splitting techniques and the space considered to locate the facilities. For details and references, the reader is referred to [5]. Intimately related to discrete facility location games are the Voronoi games, which have been recently visited by [3,11]. In a Voronoi

The Competitive Facility Location Problem in a Duopoly

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game on a graph with several players, each player chooses a vertex and achieves a utility equal to the number of vertices that are closer to the chosen vertex than to those of the other players.

2

The Facility Location Game

Let G = (V, E) be an undirected connected weighted graph, in which each vertex represents a location and has an associated weight w(v) > 0 that quantifies de mand. We denote the demand in a set S ⊆ V of vertices by W (S) = v∈S w(v), and let W = W (V ) be the total demand. Two players compete for market share by selecting a vertex each to locate their facilities. We refer to the vertices se¯ = (x1 , x2 ), each lected by the players as x1 , x2 ∈ V , respectively. Given a profile x vertex v will split its demand evenly among the set of facilities that are closest to it; i.e., F (v, x¯) := arg mini∈{1,2} d(v, xi ), where d(·, ·) is the distance function induced by the topology of the graph where edges have unit length (this restriction is without much loss of generality since other distances can be achieved by subx) := {v ∈ V : d(xi , v) ≤ d(xj , v) ∀j = i}, dividing edges). Similarly, letting Vi (¯ a player i will receive utility ui composed by the full demand from vertices in Vi where the inequality is strict plus half of the demand from vertices in Vi where there is equality. Since u1 + u2 = W , this is a zero-sum game. We say that a profile x ¯ is a pure-strategy Nash equilibrium (PSNE) of this facility location game if ui (xi , x−i ) ≥ ui (y, x−i ) for any y ∈ V , for i = 1, 2. The main property of an equilibrium is that both players must obtain equal utility; otherwise, the player with the lowest utility would prefer to emulate the other player’s strategy and get a utility of W/2. Although there are always equilibria in mixed strategies, [6] provides examples that show that not every facility location game with two players has a PSNE. They characterized equilibria for different topologies using ad-hoc techniques. We unify some of those results, considering vertices that ensure a big-enough market share. Indeed, we say that a vertex w ∈ V is a winning strategy if the utility obtained by a player when choosing vertex w guarantees winning the game, regardless of the selection of the other player; i.e., u1 (w, v) ≥ W/2 for all v ∈ V . There is a one-to-one relationship between the location of winning strategies and that of equilibria. In fact, any equilibrium must consist of each player choosing a winning strategy. Lemma 1. For arbitrary topologies, an equilibrium of a facility location game with two players exists if and only if there exists at least a winning strategy. Proof. The result follows from the definition of a winning strategy. If x ¯ is at equilibrium, W/2 = u2 (x1 , x2 ) ≥ u2 (x1 , v) for all v ∈ V , which implies that x1 is a winning strategy because, since the game is zero-sum, W/2 ≤ u1 (x1 , v) ∀v ∈ V . To prove the converse, take a winning strategy w and consider x ¯ = (w, w). By x) ≤ u1 (w, v) for all v ∈ V . Using again that the game is definition W/2 = u1 (¯ zero-sum proves the equilibrium condition.

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1 v2

v3 10

15 v1

v4 1 1 v6

v v1 v2 v3 v4 v5 v6 D(v) 45 59 55 69 55 59

v5 10

(a) Vertices are annotated with demands. For each pair of vertices, one player can always deviate to obtain more than half of the demand.

(b) Total distances to vertices. Vertex v1 is the unique 1-median, while v3 and v5 are local medians.

Fig. 1. Instance with no equilibria and its medians

Note that winning strategies are related to dominant strategies, which refer to selections that are always optimal regardless of the opponent’s choice. Although our game does not necessarily have dominant-strategy equilibria and winning strategies are not necessarily dominant, playing a winning strategy guarantees that the player is not worse than the opponent even if the opponent deviates from the equilibrium (hence the name). To illustrate, consider the path (v1 , v2 , v3 , v4 , v5 ) of 5 vertices with unit weight. The unique winning strategy is to choose v3 (therefore, the only equilibrium is (v3 , v3 )). However, a best response to an opponent that chooses v5 is to choose v4 and hence v3 is not a dominant strategy. Winning strategies, though, are not guaranteed to exist. In the instance shown in Fig. 1, no vertex can guarantee a player a utility of W/2. Indeed, a best response to selecting a vertex with demand larger than one is to select the opposite vertex, whereas a best response to selecting a vertex with unit demand is to select the adjacent vertex. Therefore, an equilibrium for this instance does not exist. As discussed in the introduction, it is natural to select the vertex that is nearest to the demand. Hence, for a single facility located at vertex y ∈ V , we compute the total distance to it as D(y) = v∈V d(y, v)w(v). A vertex is called a 1-median of G if it minimizes D(·). In the rest of the paper, we sometimes just write median to refer to the 1-median of a graph, and we use the term median-set to refer to the set of vertices that are 1-medians.

3

Cycles and the 1-Median Problem

Although we already saw that cycles do not always admit equilibria, when demands are sufficiently large an equilibrium must exist. For instance, [6] showed that an equilibrium of a cycle can use a vertex v if and only if the demand of any subpath of cardinality |V |/2 that excludes v does not exceed W/2. That condition can be interpreted as saying that the corresponding vertices are winning strategies. To see this, note that for an arbitrary profile each player gets the demand from exactly half of the vertices (one or more vertices may be split equally depending on the parity of the cycle and on the location of both facilities). Therefore, the condition of [6] is equivalent to v being a winning strategy.


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For trees, it is known that medians and equilibria (and hence winning strategies) of the facility location game in duopolies coincide [4]. The relation of winning strategy and medians for trees follows from a result by Kariv and Hakimi that establishes that a vertex is a median if and only if removing it induces components of weight not larger than W/2 [9]. We now show a similar result for cycles: every winning strategy must be a median. Note that the converse is not true: medians always exist but winning strategies may not. Theorem 1. If a winning strategy w of a cycle exists, it must solve the 1-median problem. To prove this, we compute the difference in total distance from w to any other vertex v, representing it as a weighted sum of paths of cardinality |V |/2. Using that w is a winning strategy, we can prove that the difference is non-negative. Due to lack of space, the full proof is omitted. Combining the results for trees and for cycles, we extend the previous property to more general topologies. A cactus is a graph where every edge belongs to at most one cycle. Reducing an arbitrary cactus to a tree representing its components, as explained in [6], and then using the result for cycles, winning strategies must also solve the 1-median problem. In addition, because one can compute winning strategies for cacti in O(|V |)-time, this also provides an efficient algorithm to compute the medians of cacti that admit equilibria.

4

Local Medians

Even though it is natural to think that if an equilibrium of the facility location game exists, players will choose a 1-median solution, we do not know if this is true for arbitrary topologies. Nevertheless, we can prove that equilibria of this game translate into a local median property. Indeed, whenever a winning strategy exists, it must be a local minimum of the 1-median problem with respect to neighboring vertices. To illustrate the definition of local median, in the example of Fig. 1 there is a global median and 2 local ones (the result does not apply to the example because there are no winning strategies in it). Theorem 2. If w is a winning strategy for the the facility location game, then D(w) ≤ D(v) for all v ∈ N (w), where N (w) := {v ∈ V |vw ∈ E}. Proof. Let w be a winning strategy and let v ∈ N (w). Let dz := d(v, z) − d(w, z) for all z ∈ V . Because w and v are neighbors, dz ∈ {−1, 0, 1} ∀z. We consider x), from where x ¯ = (w, v). Since (w, w) is at equilibrium, W/2 = u2 (w, w) ≥ u2 (¯ u1 (¯ x)− u2 (¯ x) ≥ 0 because the game is zero-sum. Let Mw (resp. Mv ) be the set of vertices that are strictly closer to w than to v (resp. v to w). The result follows using that dz = 1 for z ∈ Mw and dz = −1 for z ∈ Mv , because D(v) − D(w) = w(z)(d(v, z) − d(w, z)) = W (Mv ) − W (Mw ) = u1 (¯ x) − u2 (¯ x). z∈V

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In light of this result, one would like to characterize the local minima of the 1median problem to understand the possible locations of the winning strategies. For certain families of graphs, these minima coincide with the (global) 1-medians. Indeed, [1] proved that if G is a connected graph, then the following conditions are equivalent: (a) The median-set is connected for arbitrary weights w, and, (b) The set of local medians coincide with the median-set for arbitrary rational weights w. Based on this equivalence, we obtain the following corollary. Corollary 1. Let G be a graph that belongs to a family for which, for any rational weights w, the solutions to the 1-median problem induce a connected subgraph of G. Then, every winning strategy of G solves the 1-median problem. Families of graphs satisfying this property include median graphs, quasi-median graphs, pseudo-median graphs, Helly graphs and strongly chordal graphs. A complete characterization of graphs with connected median-sets can be found in [1]. Among graphs in this family, median graphs are particularly important in location applications because they represent cities well. Median graphs satisfy that any three vertices a, b, and c have a unique median (which is a vertex that belongs to shortest paths between any two of a, b, and c). This class includes lattices, meshes, and grids, which encode the topology of many realistic networks. Notice that not all graphs have connected medians (e.g., Fig. 1a). Furthermore, median-sets may have arbitrary topologies; that is, given a graph G, there exists a graph H for which the subgraph of H induced by the median vertices is isomorphic to G [12]. This result implies that the median-set can induce a disconnected subgraph.

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Strongly Chordal Graphs

In this section we focus on strongly chordal graphs, which are relevant because they generalize many well-known classes of graphs such as trees, block graphs and interval graphs. A graph is chordal if every cycle with more than three vertices has a chord, i.e., an edge joining two non-consecutive vertices of the cycle. A p-sun is a chordal graph with a vertex set x1 , . . . , xp , y1 , . . . , yp such that y1 , . . . , yp is an independent set, (x1 , . . . , xp , x1 ) is a cycle, and each vertex yi has exactly two neighbors xi − 1 and xi , where x0 = xp . A graph is strongly chordal if it is chordal and contains no p-sun for p ≥ 3. We knew from the previous section that winning strategies in strongly chordal graphs solve the 1-median problem. We prove the converse result, establishing that graphs in that family always admit equilibria and that the equilibrium locations and 1-medians coincide. This completely extends the results for trees of [4] to this family, which is a strict superclass of trees. Theorem 3. Every connected strongly chordal graph has an equilibrium. Furthermore, there is a one-to-one correspondence between winning strategies and the solutions to the 1-median problem.


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The proof of this theorem, which is omitted due to lack of space, uses induction by identifying a vertex that cannot be a winning strategy and removing it to reduce the problem. This approach follows the methodology of Theorem 1 in [10], where it is shown that the median-set of a connected strongly chordal graph is a clique. While we use the same inductive idea, we need to rely on more complex structures. As a corollary, the set of winning strategies of a connected strongly chordal graph is, not only connected as previously discussed, but also a clique.

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Concluding Remarks

We have explored the relations between winning strategies and solutions to the 1-median problem in duopolistic facility location games. For several families of graphs, we have shown that the locations of both sets coincide or that one is inside another. We believe that both sets should coincide for some of the classes of graphs considered, and others as well. Identifying when it holds or providing counterexamples remains as open problems. In particular, it would be interesting to further extend our results to some super-class of the strongly-chordal graphs. We would have liked to prove related results for graphs of bounded treewidth, but we could not adapt the decomposition technique used to prove our result for cacti, so other ideas may be needed for that result.

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