Internalization of social cost in congestion games

Abstract

Congestion models may be studied from either the users’ point of view or the social one. The first perspective examines the incentives of individual users, who are only interested in their own, personal payoff or cost and ignore the negative externalities that their choice of resources creates for the other users. The second perspective concerns social goals such as the minimization of the mean travel time in a transportation network. This paper studies a more general setting, in which individual users attach to the social cost some weight r that is not necessarily 0 or 1. It examines the comparative statics question of whether higher r necessarily means higher social welfare at equilibrium.

Appendix: Networks

A directed or undirected two-terminal network is, respectively, a directed or undirected multigraph (where a pair of vertices may be joined by more than one edge) that is endowed with two terminal vertices, $o$ and $d$, and satisfies the following “non-redundancy” condition: each edge $e$ and each vertex $v$ is included in at least one route in the network. In the undirected case, a route is defined as a (simple) path of the form $o{e}_{1}{v}_{1}\cdots {v}_{n-1}{e}_{n}d$ ($n\ge 1$), that is, one that begins at the origin vertex $o$ and ends at the destination vertex $d$. (An alternative, simpler notation is ${e}_{1}{e}_{2}\cdots {e}_{n}$.) In the directed case, a route also has to traverse each of its edges $e$ in the direction assigned to $e$, that is, the tail and head vertices of $e$ must be its immediate predecessor and successor, respectively, in the route. This difference means that the set of routes in a directed two-terminal network is a subset of the set of routes in the network’s undirected version, which is obtained by ignoring the edge directions.

An arbitrary assignment of directions to the edges of an undirected two-terminal network $G$ does not necessarily give a directed network, as the non-redundancy condition may fail to hold. If the non-redundancy condition does hold, the resulting directed two-terminal network is said to be a directed version of $G$. Such a version always exists.

Proposition A1

Every undirected two-terminal network has at least one directed version.

Proof

The proof is by induction on the number of edges in the network. If there is only one edge, the assertion is trivial. To establish the inductive step, consider a network $G$ with more than one edge, and some edge $e$ incident with the origin $o$. If no other edge is incident with $o$, then by the induction hypothesis there exists at least one directed version of the network obtained from $G$ by contracting $e$, that is, eliminating the edge and its non-terminal vertex $v$ and replacing $v$ with $o$ as the terminal vertex of all the edges originally incident with $v$. Any such directed version gives a directed version of $G$, in which $e$ is directed from $o$ to $v$. Suppose, then, that $e$ is not the only edge incident with $o$, and consider the subnetwork ${G}_{e}$ of $G$ whose edges and vertices are all those that belong to some route in $G$ where the first edge is $e.$

Claim If a route $r$ in $G$ includes a non-terminal vertex $u$ that is in ${G}_{e}$ , then every edge (hence, also every vertex) that follows $u$ in $r$ is also in ${G}_{e}$ .

It clearly suffices to consider the edge ${e}^{{\prime}}$ that immediately follows $u$ in $r$. Suppose that ${e}^{{\prime}}$ is not in ${G}_{e}$. Let $v$ the first vertex in $r$ that follows ${e}^{{\prime}}$ and is in ${G}_{e}$. (Possibly, $v=d$.) All the edges and vertices in $r$ between $u$ and $v$ are not in ${G}_{e}$. Adding them to it creates a new subnetwork of $G$, since the addition is equivalent to adding to ${G}_{e}$ a single edge with end vertices $u$ and $v$ and then subdividing the edge, if needed, one or more times (Milchtaich 2015, Sect. 2.2). By definition, there exists in the new subnetwork a route ${r}^{{\prime}}$ that includes ${e}^{{\prime}}$. Necessarily, ${r}^{{\prime}}$ also includes $e$. The conclusion contradicts the assumption that ${e}^{{\prime}}$ is not in ${G}_{e}$, and thus proves the claim.

Consider now the collections of all edges and vertices that are obtained from those in $G$ by (i) eliminating all edges that are in ${G}_{e}$ and (ii) identifying with $d$ all non-terminal vertices that are in ${G}_{e}$. It follows from the claim that these collections constitute a network ${G}^{{\prime}}$ (with the terminal vertices $o$ and $d$). To see this, consider any edge or non-terminal vertex in them and a route $r$ in $G$ that includes it. It follows from the claim that the inclusion still holds if $r$ is replaced with its section ${r}_{ou}$, which is the path obtained from $r$ by deleting all edges and vertices that follow $u$, where $u$ is the first vertex in $r$, other than $o$, that is in ${G}_{e}$. As $u$ is one of the vertices identified with $d$, ${r}_{ou}$ is a route in ${G}^{{\prime}}$.

By the induction hypothesis, both ${G}_{e}$ and ${G}^{{\prime}}$ have directed versions, which together specify directions for all edges in $G$. It remains to show that these directions define a directed version of $G$, that is, the non-redundancy condition holds. Any edge or vertex that is or is not in ${G}_{e}$ is included in some route $r$ in the directed version of ${G}_{e}$ or some route ${r}^{{\prime}}$ in the directed version of ${G}^{{\prime}}$, respectively. By construction, $r$ is automatically a route also in $G$ and it honors the directions specified for its edges. Only the second assertion is necessarily true for ${r}^{{\prime}}$, which is a path that starts at $o$, ends at some vertex $u$ in ${G}_{e}$, and does not include any other vertex or edge that is in ${G}_{e}$. However, it is possible to extend ${r}^{{\prime}}$ to a route in $G$ that honors the edges’ directions by replacing the vertex $u$ with the section ${r}_{ud}^{{\prime\prime} }$ of some route ${r}^{{\prime\prime} }$ in the directed version of ${G}_{e}$ that includes $u$, where ${r}_{ud}^{{\prime\prime} }$ is obtained from ${r}^{{\prime\prime} }$ by deleting all the edges and vertices that precede $u$. □

Corollary A1

The collection of the undirected versions of all directed two-terminal networks coincides with the collection of all undirected two-terminal networks.

Clearly, an assignment of directions to the edges of an undirected two-terminal network $G$ defines a directed version of $G$ only if it satisfies the condition mentioned in Sect. 3.3, which is that the direction of each edge $e$ coincides with the direction in which some route in (the undirected network) $G$ traverses $e$. However, this necessary condition is not sufficient. For example, it is not difficult to see that an assignment satisfying the condition may render a non-terminal degree-two vertex the head vertex of both edges incident with it, which means that the non-redundancy condition for directed networks is violated. However, it follows from Proposition A1 that if there is only one assignment of directions that satisfies the above condition, then that assignment necessarily does give a directed version of $G$. Such uniqueness is rather special. Indeed, the following result can easily be deduced from Proposition 1 in Milchtaich (2006b).

Corollary A2

For an undirected two-terminal network $G$, the following conditions are equivalent:

• (i) For every edge $e$, all routes in $G$ that include $e$ traverse it in the same direction.

• (ii) For every assignment of allowable directions to the edges in $G$ such that the direction of each edge $e$ is that in which some route in $G$ traverses $e$, all routes in $G$ are allowable.

• (iii) $G$ is series-parallel.