The price of imperfect competition for a spanning network

Highlights

• We study the price paid by a buyer who procures a network to span a set of nodes.

• We consider both a first-price auction and the pivotal mechanism.

• Irrespective of bidding licenses and cost structure, and efficient tree is built.

• For metric costs the surcharge essentially cannot exceed 100% of the true efficient cost.

• The surcharge increases swiftly when sellers can only provide a subset of all edges.

Abstract

A buyer procures a network to span a given set of nodes; each seller bids to supply certain edges, then the buyer purchases a minimal cost spanning tree. An efficient tree is constructed in any equilibrium of the Bertrand game.

We evaluate the price of imperfect competition (PIC), namely the ratio of the total price that could be charged to the buyer in some equilibrium, to the true minimal cost. If each seller can only bid for a single edge and costs satisfy the triangle inequality, we show that the PIC is at most 2 for an odd number of nodes, and at most $2\frac{n-1}{n-2}$ for an even number n of nodes. Surprisingly, this worst case ratio does not improve when the cost pattern is ultrametric (a much more demanding substitutability requirement), although the overhead is much lower on average under typical probabilistic assumptions. But the PIC increases swiftly when sellers can only provide a subset of all edges.

Introduction

Bertrand competition for differentiated commodities typically implies some welfare losses, as in the Hotelling model. We consider a special case where it does not, and where the surplus that the competing firms are able to extract admits a simple upper bound.

A buyer procures a network spanning a given set of nodes, while the sellers bid for different edges of the network. Efficiency requires to build a minimal cost spanning tree. This familiar optimization problem has a variety of applications (including rail infrastructure, the Internetʼs backbone, water distribution, etc.; see Sharkey, 1995, for a survey).

If one seller is the sole bidder for a certain edge, or a group of edges, he can typically bid higher than their true cost. But this overhead is bounded above because the edges are partial substitutes: for any edge e, there are several alternative paths ensuring the connection of eʼs two end nodes. We wish to evaluate the welfare consequences of this imperfect competition between the sellers.

We observe first that irrespective of the ownership structure and of the cost pattern, in equilibrium an efficient (minimal cost) spanning tree is constructed (Proposition 1). This is very close to being a special case of the efficiency result in the matroid markets of Chen and Karlin (2007) – on which more below.

Our main concern is to evaluate the surcharge to the buyer procuring the network. We provide tight bounds when the cost of the edges satisfy the triangle inequality, i.e., for any three edges $e,f,g$ forming a triangle the cost of one edge is not larger than the sum of the costs of the other two. We speak then of metric costs. This familiar assumption is realistic when costs are a subadditive function of the Euclidean length of an edge, or some other measure of spatial distance between its end nodes (for instance the shortest length of a path connecting the nodes in an arbitrary network).¹ It is also automatic when all sellers have access to the same technology to build edges, but each one is only licensed to bid for certain edges: it is then feasible to connect the end nodes of e by building f and g.

Our first main result (Theorem 1) is that if each seller bids for a single edge, there is at least one bidder for each edge, and true costs are metric, the buyerʼs surcharge essentially cannot exceed 100% of the true minimal cost. However our second main result (Theorem 2) says that the overhead grows rapidly if some edges are not available (no one is bidding for those edges): if at least half of the edges are missing, the worst case ratio of the overhead to the true cost is $n-1$, where n is the number of nodes to connect.

It is instructive to compare our results to those of the literature evaluating the frugality of some related auction mechanisms (Archer and Tardos, 2001, Archer and Tardos, 2002, Karlin et al., 2005, Talwar, 2003). The procurement of a spanning tree among sellers who each own an edge of the network is a special case of the procurement of a matroid basis; other examples include the procurement of a path between two given nodes of the network (Archer and Tardos, 2002, Elkind et al., 2004, Immorlica et al., 2005), and more generally the procurement of a team to perform a complex task (Karlin et al., 2005). Like us, these papers are poised to evaluate the worst possible surcharge to the buyer. However, the payments to the various sellers are more complicated than in the simple first-price auction of Bertrand competition: they are given by the canonical VCG mechanism (known as the pivotal mechanism since Green and Laffont, 1979), in which each winning edge is paid its true cost, plus the extra cost incurred if we cannot use that very edge in our spanning tree. Under this payment scheme bidding oneʼs true cost is a dominant strategy for each seller. Thus, implementation of the mechanism is prior-free, in sharp contrast with our equilibrium analysis of the Bertrand game requiring complete information among sellers.²

When we assume, as we do in our main results, that a seller bids for a single edge, it turns out that the two games, Bertrand bidding and pivotal, are essentially equivalent: the buyerʼs surcharge in the pivotal game is precisely the same as in the most expensive equilibrium (equivalently, in its unique equilibrium in undominated strategies) of the Bertrand game (see Section 6.1). Thus, our results can be used indifferently in both contexts.

There have been at least three attempts in the literature to measure the frugality of a mechanism to procure a spanning tree. They differ in the choice for the benchmark cost to which the total payment of the seller is compared. In Archer and Tardos, 2001, Archer and Tardos, 2002, like here, the benchmark is the true minimal (efficient) cost. They show that the worst case ratio (total charge to efficient cost) is again $n-1$, where n is the number of nodes to connect.³ Our Theorem 1 qualifies this negative result when costs are metric.

Subsequently, Talwar (2003) uses for benchmark the cheapest cost of a spanning tree with no edge in common with the efficient tree. This curious proposal may lead in our model to a frugality ratio smaller than one⁴! Finally in Karlin et al. (2005) the benchmark is the solution of a linear program that, in the spanning tree problem, coincides with the most expensive equilibrium (for the buyer), so that the frugality ratio is one, tautologically.

We submit that from the buyerʼs perspective, the only meaningful benchmark is the true efficient cost. To avoid confusion with the multiple frugality indices, and to convey the basic economic intuition, we propose to call price of imperfect competition (PIC) the ratio of the worst buyerʼs charge in some equilibrium (or the actual charge in the pivotal mechanism), to the efficient cost.⁵

Summary of results. Section 2 introduces the minimum cost spanning tree problem and the Bertrand game of procurement. Proposition 1 states that an efficient tree is built in all relevant equilibria. Starting with Section 3 we assume that each seller bids for only one edge. We provide a general formula for the PIC that uses only the network structure and arbitrary costs (Proposition 2, Corollary 1).

We assume in Section 4 that costs are metric, and show (Theorem 1) that if all edges have at least one bidder the worst case PIC is 2 for an odd number of nodes, and $2\frac{n-1}{n-2}$ for an even number n. However a single missing edge raises the worst case PIC to 3 (Proposition 5), and more generally the PIC may reach $n-1$ as soon as half of the edges are missing (Theorem 2).

In Section 5 we assume that costs are ultrametric, i.e., if edges e, f, g form a triangle, $c_e⩽\max \{c_f,c_g\}$. This is stronger, and implies much closer substitutability than the triangle inequality. Ultrametric costs are a natural restriction in some environments. Consider a set of nodes for which the connection cost is determined by the compatibility of certain attributes that are ordered in decreasing levels of complexity. For instance the nodes are scattered in North America (consisting of 3 countries), and an edge has the largest cost ${\theta }_1$ if the two end nodes are in two different countries, the second largest cost ${\theta }_2$ if they are in the same country but different states or provinces, the third largest cost if in different counties of a given state or province, and so on.⁶

Ultrametric costs also play an important role in the design of rules sharing the cost of an optimal spanning tree between the different nodes (see Norde et al., 2004, Moretti et al., 2004, Bergantinos and Vidal-Puga, 2007, Bogomolnaia and Moulin, 2010).

Surprisingly, the worst case PIC does not improve (decrease) when the cost pattern is ultrametric, although it is clearly better on average. We propose some numerical computations to justify the latter.

In Section 6 we show first that if the familiar pivotal mechanism replaces our Bertrand game, it delivers the equilibrium of the latter most expensive for the buyer, therefore all our results apply. Next we explain how much of our results are preserved in the more general context of an “auction for a matroid basis” discussed above. Finally we probe the robustness of our results when a single seller can be the exclusive bidder on several edges. The news is good only if those edges are mutually disconnected.

Proofs are gathered in Appendix A.