Chapter 3. Competitive or Voting Locationwith Proportional Choice
Chapter 3 Competitive or Voting Location with Proportional Choice This chapter is concerned with a competitive or voting location problem on networks under a proportional choice rule that has previously been intro- duced by Bauer et al. (1993) (cf. already Sections 2.1 and 2.2). Its contribu- tion is threefold: • First, as in Kress and Pesch (2012b), we reﬁne a discretization result of Bauer et al. (1993) by proving convexity and concavity properties of related expected payoﬀ functions and we answer the long time open question whether 1-suboptimal points are always vertices by providing a counterexample on a tree network. These results are presented in Section 3.2.1 and are complemented by examples and counterexamples in Section 3.2.2. • Second, in Section 3.2.3, we are concerned with implementational issues of an algorithm for determining 1-suboptimal points of a network. • Third, we will show that the choice rule of Bauer et al. (1993) can be extended to include edge demand in Section 3.3. In order to be able to deﬁne the proportional choice rule and its properties in detail, however, we will ﬁrst have to introduce the concept of bottleneck points in Section 3.1 (cf. also Hakimi, 1964; Hooker et al., 1991). The chapter closes with a conclusion and remarks on future research in Section 3.4. 52 Chapter 3. Competitive or Voting Location with Proportional Choice 3.1 Bottleneck Points Let N = (V,E, λ) be a ﬁnite network. Furthermore, let [u, v] ∈ E be any edge and i ∈ N , i /∈ (u, v)...
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