Graphs and Path Equilibria

This chapter introduces the terminology of dalographs which refers to finite, arclabelled, directed graphs with non-zero outdegree, i.e. each of whose node has an outgoing arc. An embedding of arc-labelled digraphs into dalographs shows that the non-zero-outdegree constraint may not yield a serious loss of generality. The paths that are considered in this chapter are infinite. Indeed, finite paths and infinite paths are of slightly different “types”. Considering both may hinder an algebraic approach of the system. However, another embedding allows representing finite paths in a dalograph as infinite paths in another dalograph. This shows that the infiniteness constraint may not yield a serious loss of generality either. Note that the non-zero-outdegree constraint ensures existence of infinite paths, starting from any node. This uniformity facilitates an algebraic approach of the system. The paths considered in this chapter are non-self-crossing, which somehow suggests consistency. This sounds desirable in many areas, but it may be an actual restriction in some others.

In this formalism, a path induces an ultimately periodic sequence of labels (of arcs that are involved in the path). An arbitrary binary relation over ultimately periodic sequences of labels is assumed and named preference. This induces a binary relation over paths, which is also named preference. It is defined as follows. Given two paths starting from the same node, one is preferred over the other if the sequence of labels that is induced by the former is preferred over the sequence of labels that is induced by the latter. Maximality of a given path in a graph means that no path is preferred over the given path. A strategy is an object built over a dalograph. It amounts to every node choosing an outgoing arc. This way, a strategy induces paths starting from any given node. An equilibrium is a strategy inducing optimal paths for any node.

The proof of equilibrium existence is structured as follows. First, a seekingforward function is defined so that given a node it returns a path. Given a node, the function chooses a path that is maximal (according to the definition of the previous paragraph), and the function “follows” the path until the remaining path is not maximal (among the paths starting from the current node). In this case, the function chooses a maximal path starting from the current node and proceeds as before. All of this is done under the constraint that a path is non-selfcrossing. Under some conditions, this procedure yields a path that is maximal not only at its starting node, but also at all nodes along the path. Such a path is called a hereditary maximal path. Equipped with this lemma, the existence of an equilibrium for every dalograph is proved as follows by induction on the number of arcs in the dalograph.

-Compute a hereditary maximal path in the dalograph.

-Remove the arcs of the dalograph that the path ignored while visiting adjacent nodes and get a smaller dalograph. -Compute an equilibrium on this smaller dalograph and add the ignored arcs back. This yields an equilibrium for the original dalograph.

The sufficient condition for equilibrium existence involves a notion lying between strict partial order and strict total order, namely strict weak order, which is discussed in [1], for instance. This chapter requires a few preliminary results about strict weak orders. Moreover, the definition of the seeking-forward function requires the design of a recursion principle that is also used as a proof principle in this chapter. To show the usefulness of this sufficient condition, this chapter provides a few examples of non-trivial relations that meet the requirements of this condition: lexicographic extension of a strict weak order, Pareto-like order, and two limit-set-oriented orders. Then, as an application to network routing, one derives a sufficient condition for a routing policy to guarantee existence of stable routing solutions.

The proof of the necessary condition for equilibrium existence involves various closures of binary relations. Most of the closures defined here are related to properties that are part of the sufficient condition. For instance, the sufficient condition involves transitivity of some preference (binary relation), and the necessary condition involves the transitive closure of the preference.

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