Condorcet Winners for Public Goods - Semantic Scholar

Annals of Operations Research 137, 229–242, 2005 c 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.

Condorcet Winners for Public Goods LIHUA CHEN Guanghua School of Management, Peking University, Beijing 100080, P.R. China XIAOTIE DENG Department of Computer Science, City University of Hong Kong, Hong Kong, P.R. China QIZHI FANG Department of Mathematics, Ocean University of China, Qingdao 266071, P.R. China FENG TIAN Institute of Systems Science, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, P.R. China

Abstract. In this work, we consider a public facility allocation problem decided through a voting process under the majority rule. A location of the public facility is a majority rule winner if there is no other location in the network where more than half of the voters would have been closer to than the majority rule winner. We develop fast algorithms for interesting cases with nice combinatorial structures. We show that the computing problem and the decision problem in the general case, where the number of public facilities is more than one and is considered part of the input size, are all NP-hard. Finally, we discuss majority rule decision making for related models. Keywords: public goods, Condorcet winner, majority equilibrium, complexity, algorithm

1.

Introduction

Majority rule is arguably the best decision mechanism for public decision makings. Informally, a majority winner has the property that no other solutions would please more than half of the voters in comparison to it. On the other hand, it is well known that a majority winner may not always exist as shown in the famous Condorcet paradox where three agents have three different orders of preferences, A > B > C, B > C > A, C > A > B among three alternatives A, B and C. In this work we are interested in computational aspects for majority winners in public decision making. We focus on a public facility location problem. Demange (1983) reviewed continuous and discrete spatial models of collective choice, aiming at a characterization of the location problem of public services as a result of public voting process. To facilitate a rigorous study of the related problem, Demange proposed four types of Condorcet winners and discussed corresponding results (Romero, 1978; Hansen and Thisse, 1981) concerning conditions for their existences. We consider a weighted version of the discrete

230

CHEN ET AL.

model of Demange, represented by a network G = ((V, w), (E, l)) linking communities together. Each vertex i ∈ V represents a community, and w(i) represents the number of voters that reside there. For each e ∈ E, l(e) > 0 represents the distance between two ends of the road e = (i, j) connecting the two communities i and j. The location of a public facility such as library, community center, etc., is to be determined by the public via a voting process under the majority rule. We consider a special type of utility function: each member of the community is interested in minimizing the distance of its location to that of the public facility. While each desires to have the public facility to be close to itself, the decision has to be agreed upon by a majority of the votes. Following Demange (1983), a location x ∈ V is a strong (resp. weak) Condorcet winner if, for any other y ∈ V , the total weight of vertices that is closer to x than to y is more (resp. no less) than the total weight of vertices that is closer to y than to x. Similarly, it is a quasi-Condorcet winner if we change “closer to x than to y” to “closer to x than to y or of the same distance to x as to y”. Of the four types of majority winner, strong Condorcet winner is the most restrictive of all, and weak quasi-Condorcet winner is the least restrictive one and the other two are between them. For discrete models considered by Romero (1978), Hansen and Thisse (1981), it was known that, the order induced by strict majority relation (the weak Condorcet order) in a tree is transitive. Therefore, a weak Condorcet winner in any tree always exists. In addition, Demange extended the existence condition of a weak Condorcet winner to all single peaked orders on trees (Demange, 1982). Note that an order on a tree is single peaked if, along each single path on the tree, the preference strictly increases up to a peak and then strictly decreases. Our study will focus on the algorithmic and complexity aspects of Condorcet winners, and discuss trees and cycles as well as the cactus graph, that is, a connected graph in which each block is an edge or a cycle. In general, a location choice rule is a systematic way to map preferences into locations. Among many properties proposed for a choice rule, the incentives property of strategy-proofness (i.e., an agent should never be able to manipulate the choice rule by misreporting his preferences to it) has been extensively studied (Moulin, 1980; Border and Jordan, 1983; Barbera, Masso, and Serizawa, 1998). Schummer and Vohra has recently studied the public location choice rule on a continuous (non-discrete) network model (Schummer and Vohra, 2002). They characterized a class of onto choice rules that satisfy the strategy-proofness condition when each agent’s preference over points on a continuous network is determined by the distance between his location and the points. In our model, the domain and the range of the majority rule are restricted to the set of vertices of a network. Therefore, our setting can be viewed as a discrete version of Schummer and Vohra’s model. Our study distinguishes from previous work in our focus on algorithmic issues. Recently, there has been a growing research effort in re-examination of concepts in human and social sciences via computational complexity approach, e.g., cooperative games (Megiddo, 1978; Deng and Papadimitriou, 1994; Fang and Zhu, 2002), organizational structure (Deng and Papadimitriou, 1999), arbitrage (Deng, Li, and Wang, 2000), as well as general equilibrium (Deng, Papadimitriou, and Safra, 2003).

CONDORCET WINNER FOR PUBLIC GOODS

231

In Section 2, we introduce the formal formulation of the public facility location problem with a single facility in a network. Obviously, enumerating through all n locations allows us to have a polynomial time algorithm to find a majority winner location. The issue here is how to improve the time complexity. In particular, we are interested in classifying the types of networks for which a Condorcet winner can be found in linear time. As a warm-up example, we present the solution for trees in Section 3. We present a linear algorithm for finding weak quasi-Condorcet winners of a tree with vertex-weight and edge-length functions; and prove that in this case, weak quasi-Condorcet winners are the points which minimize the total weight-distance to individuals’ locations. In Section 4, we give a sufficient and necessary condition for a point to be a weak quasiCondorcet winner for cycles in the case the edge-length function is a constant, and present a much more interesting linear time algorithm. This is further improved to obtain a linear time algorithm for cactus graphs (Section 5). We also discuss the property of (group) strategy-proofness for the majority rule of choosing weak (quasi) Condorcet winners on a tree and a cycle respectively. In Section 6, we present NP-hard proofs for the problem of finding a majority winner and the corresponding decision problem when the number of public facilities is taken as the input size, not a constant. Finally, in Section 7, our approach is extended to a more general public decision making process, the public road repair problem, pioneered by Tullock (1959) to study redistribution of tax revenue under a majority rule. We conclude with remarks and discussions on related issues. 2.

Definition

Demange (1983) surveyed and discussed some spatial models of collective choice, and gave some results concerning the transitivity of the majority rule and the existence of a majority winner. In this paper, we consider a public facility location problem with a single facility in a graph. Let G = (V, E) be an undirected graph of order n with a weight function ω that assigns to each vertex v of G a non-negative weight ω(v), and a length function l that assigns to each edge e of G a positive length l(e). If P is a path of G, then we denote by l(P) the sum of lengths of all edges of P. We denote by dG (u, v) the length of a shortest path joining two vertices u and v in G, and call it distance between u and v in G. For any R ⊆ V , we set ω(R) = v∈R ω(v). In particular, if R = V , we write ω(G) instead of ω(V ). A vertex v of G is said to be pendant if v has exactly one neighbor in G. Given a graph G = (V, E) with V = {v1 , v2 , . . . , vn }, each vi ∈ V has a preference order ≥i on V induced by the distance on G. That is, we have x ≥i y if and only if dG (vi , x) ≤ dG (vi , y) for two vertices x, y ∈ V . The following definition is an extension of that given in Demange (1983). Definition 1. Given a graph G = (V, E) and a profile (≥i )vi ∈V on V , a vertex v0 ∈ V is called:

232

CHEN ET AL.

(1) Weak quasi-Condorcet winner, if for every u ∈ V distinct of v0 , ω({vi ∈ V : v0 ≥i u}) ≥

ω(G) ω(G) , i.e. ω({vi ∈ V : u >i v0 }) ≤ . 2 2

(2) Strong quasi-Condorcet winner, if for every u ∈ V distinct of v0 , ω({vi ∈ V : v0 ≥i u}) >

ω(G) ω(G) , i.e. ω({vi ∈ V : u >i v0 }) < . 2 2

(3) Weak Condorcet winner, if for every u ∈ V distinct of v0 , ω({vi ∈ V : v0 >i u}) ≥ ω({vi ∈ V : u >i v0 }). (4) Strong Condorcet winner, if for every u ∈ V distinct of v0 , ω({vi ∈ V : v0 >i u}) > ω({vi ∈ V : u >i v0 }). Example. Denote by K 2 and K 3 the complete graphs of orders 2 and 3, respectively. Suppose that the length function on the edge set and the weight function on the vertex set in K 2 and K 3 are constant. Then K 2 has weak Condorcet winners, and hence has also weak quasi-Condorcet winners, but has no strong Condorcet winners and strong quasiCondorcet winners; K 3 has strong quasi-Condorcet winners, weak Condorcet winners and weak quasi-Condorcet winners, but has no strong Condorcet winners. In this paper, we will mainly consider algorithms for finding weak quasi-Condorcet winners of a tree, a cycle and, more generally, a cactus graph. The properties and algorithms for the other three types of Condorcet winners can be discussed in a similar way. Remark 1. Barycenter of a graph. For each vertex u of a graph G = (V, E), denote sG (u) = ω(v)dG (u, v). v∈V

Intuitively, sG (u) is the weighted total distance from members of other communities to u. A vertex u 0 is called a barycenter if sG (u 0 ) = minv∈V sG (v). Though “weak Condorcet winner” and “barycenter” are two different concepts for general graphs, we will show that they are the same for trees. Remark 2. Strategy-proofness and group-strategy-proofness. Let G = (V, E) be an undirected graph and N = {1, 2, . . . , n} be a set of n voters. In general, when the preference on V of each voter is single peaked and uniquely determined by its peak on G, a public facility choice rule is a function π : V n → V mapping voters’


233

peaks into vertices on the graph. In our model, there is a weight function ω that assigns to each voter i ∈ N a positive weight ω(i) representing the voter i’s decision power. Given a profile of the peaks (residencies) of n voters p = ( p1 , p2 , . . . , pn ) ∈ V n , π( p) is defined to be the corresponding majority rule winners. In the rest of this paper, we denote by p; pk , k the profile obtained by substituting voter k’s peak pk with pk in p, and denote by p; p S , S the profile obtained by substituting the peaks p S = { pi : i ∈ S ⊆ N } with peaks p S = { pi : i ∈ S ⊆ N } in p. Among many properties of choice rules, we are interested in the well known incentives property of strategy-proofness: any agent should never be able to manipulate the choice rule by misreporting his preferences to it. Formally, a choice rule π : V n → V is strategy-proof if for every voter i ∈ N , and for every profile of peaks p = ( p1 , p2 , . . . , pn ) ∈ V n , ∀ pi ∈ V, dG ( pi , π( p)) ≤ dG ( pi , π( p; pi , i)). A choice rule π is group-strategy-proof if for every coalition S ⊆ N , and for every profile of peaks p = ( p1 , . . . , pn ) ∈ V n ,

∃ p S ∈ V |S| , such that for any i ∈ S, dG ( pi , π( p; p S , S)) ≥ dG ( pi , π ( p)), and at least one of the above inequalities is strict. The concepts of strategy-proofness and group-strategy-proofness can be viewed as corresponding to the stability of Nash equilibrium and strong equilibrium. We will discuss these properties for majority rules in our models. 3.

Weak quasi-Condorcet winner of a tree

Romero (1978) and Hansen and Thisse (1981) pointed out that the family of orders induced by a distance on a tree guarantees the existence of weak Condorcet winners. Furthermore, weak Condorcet winners are the points that minimize the total distance to the individuals’ locations (Demange, 1983). In this section we propose a linear algorithm for finding weak quasi-Condorcet winners on a tree with vertex-weight and edge-length functions; and prove that in this model, weak quasi-Condorcet winners are the same as points which minimize the total weight-distance to the individuals’ locations. Given two vertices v, x ∈ V , the set of quasi-friend vertices of v relative to x is defined as FG (v, x) = {u : dG (u, v) ≤ dG (u, x)}; and the set of hostile vertices of v relative to x is defined as HG (v, x) = {u : dG (u, v) > dG (u, x)}.

234

CHEN ET AL.

By the definition of weak quasi-Condorcet winner, a vertex v0 ∈ V is a weak quasiCondorcet winner of G, if for any vertex x = v0 , ω(FG (v0 , x)) ≥

1 ω(G), or equivalently, ω(FG (v0 , x)) ≥ ω(HG (v0 , x)). 2

Theorem 3.1. Every tree has one weak quasi-Condorcet winner, or two adjacent weak quasi-Condorcet winners. We can find it or them in linear time. Proof. Let T = (V, E) be a tree of order n, and ω(v) and l(e) be the weight function and length function on V and E, respectively. Without loss of generality, we assume that ω(v) > 0 for each vertex v ∈ V . We prove the theorem by induction on n. When n = 1, the conclusion is trivial. When n = 2, we set T = ({u, v}, uv). If ω(u) = ω(v), assuming without loss of generality, that ω(u) < ω(v), then it is easy to see that v is the unique weak quasi-Condorcet winner of T . If ω(u) = ω(v), then both of u and v are weak quasi-Condorcet winners of T . Now suppose that n ≥ 3. Let u 1 , u 2 , . . . , u k (k ≥ 2) be all the pendant vertices of T . Assume, without loss of generality, that u 1 is one vertex with minimum weight among these pendant vertices. Hence w(u 1 ) < 12 ω(T ). Let u 1 v1 be the corresponding pendant edge. Denote T ∗ = T − u 1 , and define the weight function ω∗ on T ∗ as follows. ω(u 1 ) + ω(v1 ), if v = v1 ; ∗ ω (v) = ω(v), if v = v1 . Note that FT (u 1 , v1 ) = {u 1 } and HT (u 1 , v1 ) = V (G) − {u 1 }. Because ω(u 1 ) < 12 ω(T ), we have ω(FT (u 1 , v1 )) < ω(HT (u 1 , v1 )), and hence u 1 is not a weak quasi-Condorcet winner of T . We show the following Claim. If v0 is a weak quasi-Condorcet winner of T ∗ , then v0 is also a weak quasiCondorcet winner of T . Proof of Claim. Notice that for any vertex u = u 1 , dT (u, v1 ) = dT ∗ (u, v1 ) and dT (u, u 1 ) = dT (u, v1 ) + l(u 1 v1 ). Thus, (i) v1 ∈ FT (v0 , u) if and only if v1 ∈ FT ∗ (v0 , u); Similarly, v1 ∈ HT (v0 , u) if and only if v1 ∈ HT ∗ (v0 , u). (ii) If v1 ∈ FT ∗ (v0 , u), then u 1 ∈ FT (v0 , u); Similarly, if v1 ∈ HT ∗ (v0 , u), then u 1 ∈ HT (v0 , u). (iii) ω∗ (FT ∗ (v0 , u)) = ω(FT (v0 , u)); ω∗ (HT ∗ (v0 , u)) = ω(HT (v0 , u)). By the definition of weak quasi-Condorcet winner, if v0 is a weak quasi-Condorcet winner of T ∗ , then ω∗ (FT ∗ (v0 , u)) ≥ ω∗ (HT ∗ (v0 , u)). Thus ω(FT (v0 , u)) ≥ ω(HT (v0 , u)). That is, v0 is also a weak quasi-Condorcet winner of T . Theorem 3 now follows from the Claim.


235

Thus we can propose the following linear algorithm for finding weak quasi-Condorcet winners of a tree T of order n: Step 1. Take the pendant vertex v of T such that w(v) < 12 w(T ); Step 2. T − v ⇒ T , w(v) + w(u) ⇒ w(u), where u is the (unique) neighbor of v; Step 3. n − 1 ⇒ n; if n = 1, or, n = 2 and the two vertices have the same weights, then stop; otherwise go to Step 1. Theorem 3.2. Let T be a tree. Then v0 is a weak quasi-Condorcet winner of T if and only if v0 is a barycenter of T . Proof. Let T1 , T2 , . . . , Tk be the subtrees of T −v0 and let u i be the (unique) vertex of Ti adjacent to v0 (i = 1, 2, . . . , k). Obviously, FT (v0 , u i ) = V (T ) − V (Ti ), HT (v0 , u i ) = V (Ti ) for all i = 1, 2, . . . , k. (i) By the definition of weak quasi-Condorcet winner, we get that v0 is a weak quasiCondorcet winner of T if and only if for each i = 1, 2, . . . , k, ω(V (T ) − V (Ti )) ≥ ω(Ti ), i.e., ω(Ti ) ≤ 12 ω(T ). Denote T¯i = T − V (Ti ), i = 1, 2, . . . , k. It is easy to see that sT (v0 ) = sT¯i (v0 ) + sTi (u i ) + ω(Ti )dT (v0 , u i ) and sT (u i ) = sT¯i (v0 ) + sTi (u i ) + ω(T¯i )dT (v0 , u i ). (ii) By the definition of barycenter, we get that v0 is a barycenter of T if and only if ω(Ti ) ≤ ω(T¯i ), i.e., ω(Ti ) ≤ 12 ω(T ) for each i = 1, 2, . . . , k. From (i) and (ii), we get the conclusion of Theorem 3.2.

Given a set of voters N and a profile of peaks p of N on a tree T , denote the set of weak quasi-Condorcet winners by C p , which contains one vertex or two adjacent vertices according to Theorem 1. Let π : V n → V denote the majority rule of choosing weak quasi-Condorcet winners. Then for any profile of peaks p of N , π( p) may be any member of C p . In the following theorem we define dT (v, π ( p)) as the distance between vertex v and the set C p , that is, dT (v, π ( p)) = mint∈C p dT (v, t) for any vertex v of T . Theorem 3.3. Let T = (V, E) be a tree, N = {1, 2, . . . , n} be a set of voters with positive weight ω : N → R + . The majority rule π : V n → V of choosing weak quasi-Condorcet winners of T satisfies the property of (group) strategy-proofness.

236

CHEN ET AL.

Proof. Suppose the contrary, given a profile p = ( p1 , p2 , . . . , pn ) ∈ V n , there is a voter k ∈ N having an incentive to announce pk instead of pk , that is, dT ( pk , π( p)) > dT ( pk , π( p; pk , k)).

(3.1)

Denote by c1 ∈ C( p) and c2 ∈ C( p; pk ,k) the vertices nearest to peak pk , respectively. (3.1) implies that c2 ∈ C p . Let P(c1 , c2 ) be the (unique) path connecting c1 and c2 in T , a1 be the vertex adjacent to c1 on P(c1 , c2 ), T1 and T2 be the two connected components of T \ e(c1 , a1 ) concluding c1 and a1 , respectively. Then c2 ∈ T2 , and pk ∈ T2 by our assumption. Following the definition of c1 , it is easy to see that a1 ∈ C p . Hence for the profile p, pi ∈T2 ω(i) < ω(N )/2, which implies that ω(i) ≥ ω(i) > ω(i) + ω(k) ≥ ω(i) + ω(k). (3.2) c1 >i c2

c1 >i a1

a1 >i c1 , i =k

c2 >i c1 , i =k

On the other hand, by our assumption c2 ∈ C p; pk ,k and pk ∈ T2 , we have pk ∈ T2 and ω(i) ≤ ω(i) + ω(k), c1 >i c2

c2 >i c1 , i =k

which is contrary to (3.2). Therefore, the majority rule π satisfies the property of strategyproofness. The proof of group-strategy-proof is similar, which is omitted. 4.

Weak quasi-Condorcet winner of a cycle

In this section, we characterize weak quasi-Condorcet winners and discuss the related algorithm for cycles. Throughout this section, we assume that the edge length function of a cycle is constant (without loss of generality, equal to 1). For convenience, we denote a cycle of order n by Cn = v1 v2 . . . vn v1 , whose consecutive vertices are adjacent; and denote a path vi vi+1 . . . v j of Cn and its length by Cn [vi , v j ] and l(Cn [vi , v j ]), respectively. A path of length k is called a k-interval of Cn . For a real number α, denote by α the greatest integer no more than α. Theorem 4.1. Let Cn be a cycle of order n. Then v ∈ V (Cn ) is a weak quasi-Condorcet winner of Cn if and only if the weight of each n+1 -interval containing v is at least 2 1 w(Cn ). 2 Proof. By the definition of weak quasi-Condorcet winner, v is a weak quasi-Condorcet winner of Cn if and only if for all vertices u ∈ V (Cn ) distinct of v, 1 w(FCn (v, u)) ≥ w(Cn ). 2 It is easy to verify that FCn (v, u) is an interval containing v of length n+1 . As vertex u 2 runs over all vertices except v, these intervals are exactly all n+1 -intervals containing 2 1 v, and satisfy the condition that the weight of each interval is at least 2 w(Cn ).


237

Corollary 4.2. Let Cn [ p, q] be a n+1 -interval of Cn . If u∈Cn [ p,q] w(u) < 12 w(Cn ), 2 then the path Cn [ p, q] contains no weak quasi-Condorcet winner of Cn . Corollary 4.3. Suppose that vi , v j ∈ V (Cn ) are both weak quasi-Condorcet winners of Cn . If l(Cn [vi , v j ]) ≤ l(Cn [v j , vi ]), then all vertices of path Cn [vi , v j ] are weak quasi-Condorcet winners of Cn . are both weak quasi-Condorcet winners of Cn , then all Corollary 4.4. If v1 and v n+2 2 vertices of Cn are weak quasi-Condorcet winners. Corollary 4.5. Let S be the set of all weak quasi-Condorcet winners of Cn . If S = ∅, then the subgraph induced by S is connected. Based on Theorem 4.1, we propose the following linear algorithm for determining whether a vertex is a weak quasi-Condorcet winner of a cycle. Denote Cn = v1 v2 . . . vn v1 and v = v n+1 . 2 Step 1. Calculate W1 = w(v1 ) + w(v2 ) + · · · + w(v n+1 ) and W (Cn ) = w(v1 ) + w(v2 ) 2 + · · · + w(vn ); Step 2. For i = 1, 2, . . . , n+1 − 1, calculate Wi+1 = Wi − w(vi ) + w(v n+1 ); 2 2 +i 1 n+1 n+1 Step 3. If Wi ≥ 2 w(Cn ) for each i = 1, 2, . . . , 2 , then v = v 2 is a weak quasi-Condorcet winner of Cn ; otherwise, v = v n+1 is not a weak quasi-Condorcet 2 winner. Furthermore, we have Theorem 4.6. The problem of finding a weak quasi-Condorcet winner of a cycle with vertex-weight function is solvable in linear time. Let N be a set of voters with weight function ω : N → R + , and C be a cycle. If ω(i) < ω(N )/2 for each i ∈ N , then there always exists a profile of peaks p of N on C such that the corresponding weak quasi-Condorcet winners do not exist. If there is a voter i ∗ ∈ N with a power weight ω(i ∗ ) ≥ ω(N )/2, then the peak of voter i ∗ must be a weak quasi-Condorcet winner for any profile p on C. That is, the voter i ∗ has a decisive power on the majority rule, which is called the “cycle dictator”. In this case, we have the following result. Theorem 4.7. Let C = (V, E) be a cycle, and N = {1, 2, . . . , n} be a set of voters with weight function ω : N → R + . If there is a voter i ∗ ∈ N such that ω(i ∗ ) ≥ ω(N )/2, then for any profile of peaks p = ( p1 , . . . , pn ) on C, i ∗ is always a weak quasi-Condorcet winner, and the majority rule π : V n → V of choosing weak quasi-Condorcet winners of C satisfies the property of (group) strategy-proofness.

238

5.

CHEN ET AL.

Weak quasi-Condorcet winner of a cactus graph

In this section we discuss the problem of finding weak quasi-Condorcet winners of a cactus graph. We also assume that the edge length function of a graph is constant throughout this section. Given a graph G = (V, E), a vertex v of G is a cut vertex if E(G) can be partitioned into two nonempty subsets E 1 and E 2 such that the induced graphs G[E 1 ] and G[E 2 ] have just the vertex v in common. A block of G is a connected subgraph of G that has no cut vertices and is maximal with respect to this property. Every graph is the union of its blocks. A block of G is said to be pendant if it contains exactly one cut vertex. Definition 2. A graph G is called a cactus graph, if G is a connected graph in which each block is an edge or a cycle. Note that if a cactus graph G of order n ≥ 3 is not a cycle, then G contains at least two pendant blocks (edges and/or cycles). Based on the algorithms given for trees and of cycles, we discuss the algorithm of finding weak quasi-Condorcet winners of a cactus graph G as follows. Case 1. G is a block. Then G is an edge or a cycle. We can find the weak quasi-Condorcet winners of G by the algorithms given in Section 3 and Section 4. Case 2. G is not a block. Let B1 , B2 , . . . , Bk (k ≥ 2) be the pendant blocks of G, and bi be the cut vertex contained in Bi (i = 1, 2, . . . , k). Assume without loss of generality, that w(B1 − b1 ) ≤ w(B2 − b2 ) ≤ · · · ≤ w(Bk − bk ). Since 2w(B1 − b1 ) + w(b1 ) ≤ w(B1 − b1 ) + w(b1 ) + w(B2 − b2 ) ≤ w(G), w(B1 − b1 ) < 12 w(G). Let G ∗ = G − (V (B1 ) − b1 ) and define the weight function w ∗ on G ∗ as follows: w(B1 ), if v = b1 ; ∗ w (v) = w(v), if v = b1 . Thus, from a similar argument as in the proof of Theorem 1, the set of weak quasiCondorcet winners of G ∗ is the same as that of G. Note that |V (G ∗ )| < |V (G)|, we will get Case 1 after repeating applications of the algorithm in Case 2 in no more than |V (G)| times. Thus we have Theorem 5.1. The problem of finding a weak quasi-Condorcet winner of a cactus graph with vertex-weight function is solvable in linear time. Moreover, we can see that if weak quasi-Condorcet winners exist for a cactus graph G, then all weak quasi-Condorcet winners lie in a block (an edge or a cycle) of G, and as a generalization of Corollary 4.5, the subgraph induced by weak quasi-Condorcet winners of G is connected.


6.

239

Complexity issues for public location in general networks

In general, the problem can have various extensions. There may be a number of public facilities to be allocated during one voting process. The community network may be of general graphs. The public facilities may be of the same type, or they may be of different types. The utility functions of the voters may be of different forms. Our discussion in this section will take such variations into consideration. However, in our discrete model, we keep the restriction that the public facilities will be located at vertices of the graph. We have the following general results for any of the four types of Condorcet winners. Theorem 6.1. If there are a bounded constant number of public facilities to be located at one voting process under the majority rule, then a Condorcet winner (of any of the four types) can be computed in polynomial time. Proof. We present the proof for the strong Condorcet solution. (Other cases can be proven similarly.) Let k be the bounded constant for the number of public facilities to be located in the network G = (V, E, w), |V | = n. The total number of possible choices for the locations of k publics facilities is n k . We construct an auxiliary directed graph D = (U, A) as follows: The vertex set U consists of n k points each representing a set of locations for the k public facilities. An arc (u, v) ∈ A if and only if the total weight of communities that prefers the set of locations u to the set of locations v is more than one half of the total weight. Therefore, a vertex in D with in-degree zero, that is, without any incoming arc, is a Condorcet solution. In the construction of the auxiliary graph D, every vertex in D has to be compared with every other vertex, and each comparison can be done by an algorithm of time complexity O(n) by passing through all n vertices in the original network G. So the total time complexity of computing a strong Condorcet winner is O(n 2k+1 ). Theorem 6.2. If the number of public facilities to be located is not a constant but considered as the input size, the problem of computing a Condorcet winner is NP-hard; and the corresponding decision problem: deciding whether a candidate set of public facilities is a Condorcet winner, is co-NP-complete. Proof. We present the proof also for the strong Condorcet solution. We obtain an NPhard proof by reduction from EXACT COVER BY 3-SETS. We are given a family F = {S1 , . . . , Sm } of subsets of Q = {1, 2, . . . , 3n}, where every set in F has three elements, and we are asking whether a subfamily F ⊆ F with n sets exists that covers Q without overlap. We construct a network G = (V, E) with edge length function l and vertex weight function ω as follows. The vertex set V of G consists of 3 types: (1) X of m elements (one for each 3-set in F), and ∀v ∈ X , ω(v) = 0; (2) n sets of vertices Yi , i = 1, 2, . . . , n, each Yi consists of 3n vertices (one for each element in

240

CHEN ET AL.

Q), and ∀v ∈ Yi (i = 1, 2, . . . , n), ω(v) = 1; (3) Z of n vertices z 1 , z 2 , . . . , z n , and ∀v ∈ Z , ω(v) = 0. The edge set E of G consists of 2 parts: (1) There is an edge between a vertex u in X and a vertex v in Yi (i = 1, 2, . . . , n), if the element corresponding to v is in the 3-set corresponding to u. Denote the set of these edges by E 0 , and all edges in E 0 have the same length 1. (2) For each i = 1, 2, . . . , n, there is an edge between vertex z i and every vertex in Yi , and denote the set of these edges by E i . For each set E i , only one edge in E i has the length 2 and the other edges have the the same length 1. The voters’ utility function is the minimum distance to one of n public facilities to be located in the network, and the strong Condorcet winner is required to be a subset of n vertices of V under the majority rule. It is not hard to see that all n vertices of the strong Condorcet winner will belong to X if there is an exact cover F ⊆ F for the original EXACT COVER BY 3-SETS problem, and vice versa. For the decision problem, we set the vertex set Z to be the candidate set for the strong Condorcet winner. It can be shown that Z is not a strong Condorcet winner if and only if there is an exact cover F ⊆ F of Q. 7.

Remarks and further discussion

The study on public decision making has been an important area in political economics and majority rule has had fundamental influence in the study of this area (Black, 1958; Buchanan and Tullock, 1962; Plott, 1967; Hansen and Thisse, 1981; Romero, 1978; Demange, 1983; Schummer and Vohra, 2002). In this work, we apply a computational complexity approach to the study of public facility location problem decided via a voting process under the majority rule. Our study follows the network model that has been applied to the study of similar problems in economics (Romero, 1978; Hansen and Thisse, 1981; Schummer and Vohra, 2002). We prove that the general problem is NP-hard and establish efficient algorithms to interesting networks as used in the study of strategyproof model for public facility location problem (Schummer and Vohra, 2002). Our mathematical results depend on understanding of combinatorial structures of underlying networks. Our approach can be applied to more general public decision making processes. As a demonstration, in the following, we discuss majority winners for a public road repair problem, pioneered by Tullock (1959) to study redistribution of tax revenue under a majority rule system. In the public road repair problem, an edge weighted graph G = (V, E, w) is used to represent a network of local roads. There is a distinguished vertex S ∈ V representing the entry point to the highway system, and the weight of each edge representing the cost of repairing. The majority decision problem involves a set of agents A ⊆ V situated at vertices of the network who would choose a subset F of edges (F


241

is called a solution of this problem). The cost of repairing F, which is the sum of the weights of edges in F, will be shared by all n agents, each an n-th of the total. An agent benefits more under solution H than under solution F if the shortest path (if exists) from its location to the entry point in H is no longer than in F and it pays the cost of repairing in solution H no more than in solution F, and at least one of the inequalities is strict. Solution H dominates solution F, if there is a majority of agents who would benefit more under H than under F. A solution not dominated by any other solution is called a majority stable solution under the majority rule (or a majority winner). More formally, a majority stable solution under the majority rule is a collection F of edges that connects to S a subset A1 ⊂ A of agents with |A1 | > |A|/2 such that no other solution H connecting to S a subset of agents A2 ⊂ A with |A2 | > |A|/2 satisfies the conditions that w(H ) ≤ w(F), and that for each agent in A2 , its shortest path to S in solution H is no longer than that in solution F, and at least one of the inequalities is strict. We are able to show that the public road repair problem is in general NP-hard. On the other hand, we obtain a polynomial time algorithm when the network is a tree. Theorem 7.1. It is strongly NP-hard to compute a majority stable solution under the majority rule of the public road repair problem. Theorem 7.2. When the network is a tree, there is a polynomial time algorithm that finds a majority stable solution under the majority rule of the public road repair problem. Many problems open up from our study. The complexity study for other rules for public facility location is very interesting and deserves further study. And it would be interesting to extend our study to other areas and problems of public decision making process. Acknowledgment The results reported in this work are supported by a RGC CERG grant (CityU 1081/02E) and a SRG grant (7001514) of City University of Hong Kong. The authors would like to thank the anonymous referees for their careful reviews of the manuscript and constructive suggestions. References Black, D. (1958). The Theory of Committees and Elections. Cambridge University Press. Barbera, S., J. Masso, and S. Serizawa. (1998). “Strategy-Proof Voting on Compact Ranges.” Games Econ. Behavior 25, 272–291. Border, K. and J. Jordan. (1983). “Straightforward Elections, Unanimity and Phantom Voters.” Rev. Econ. Stud. 50, 153–170. James M. Buchanan and G. Tullock. (1962). The Calculus of Consent, Logical Foundations of Constitutional Democracy. The University of Michgan Press.

242

CHEN ET AL.

Demange, G. (1983). Spatial Models of Collective Choice, in Locational Analysis of Public Facilities, J.F. Thisse and H.G. Zoller (eds.), North-Holland Publishing Company. Demange, G. (1982). “Single Peaked Orders on a Tree.” Math. Soc. Sci. 3, 389–396. Deng, X., Z. Li, and S. Wang. (2000). “On Computation of Arbitrage for Markets with Friction.” Lecture Notes in Computer Science 1858, 310–319. Deng, X. and C. Papadimitriou. (1994). “On the Complexity of Cooperative Game Solution Concepts.” Mathematics of Operations Research 19, 257–266. Deng, X. and C. Papadimitriou. (1999). “Decision Making by Hierarchies of Discordant Agents.” Mathematical Programming 86, 417–431. Deng, X., C. Papadimitriou, and S. Safra. (2003). “On the Complexity of Price Equilibrium.” Journal of Computer and System Sciences 67, 311–324. Fang, Q. and S. Zhu. (2002). “Linear and Integer Programming Techniques for Cooperative Games.” International Journal of Foundations of Computer Science 13, 653–666. Hansen, P. and J.-F. Thisse. (1981). “Outcomes of Voting and Planning: Condorcet, Weber and Rawls Locations.” Journal of Public Economics 16, 1–15. Megiddo, N. (1978). “Computational Complexity and the Game Theory Approach to Cost Allocation for a Tree.” Mathematics of Operations Research 3, 189–196. Moulin, H. (1980). “On Strategy Proofness and Single Peakedness.” Pub. Choice 35, 437–455. Plott, C.R. (1967). “A Notion of Equilibrium and its Possibility Under Majority Rule.” American Economic Review 57, 787–806. Romero, D. (1978). Variations sur l’effet Condorcet, Thèse de 3ème cycle, Université de Grenoble. Schummer, J. and R.V. Vohra. (2002). “Strategy-Proof Location on a Network.” Journal of Economic Theory 104, 405–428. Tullock, G. (1959). “Some Problems of Majority Voting.” J. Polit. Econo. 67, 571–579.