The model of congestion games is widely used to analyze games related to traffic and communication. A central property of these games is that they are potential games and hence posses a pure Nash equilibrium. In reality it is often the case that some players cooperatively decide on their joint action in order to maximize the coalition's total utility. This is by modeled by Coalitional Congestion Games. Typical settings include truck drivers who work for the same shipping company, or routers that belong to the same ISP. The formation of coalitions will typically imply that the resulting coalitional congestion game will no longer posses a pure Nash equilibrium. In this paper we provide conditions under which such games are potential games and posses a pure Nash equilibrium.
Congestion games, introduced by Rosenthal [5], form a very natural model for studying many real-life strategic settings: Traffic problems, load balancing, routing, network planning, facility locations and more. In a congestion game players must choose some subset of resources from a given set of resources (e.g., a subset of edges leading from the Source to the Target on a graph). The congestion of a resource is a function of the number of players choosing it and each player seeks to minimize his total congestion accross all chosen resources. In many modeling instances players and the decision making entity have been thought of as one and the same. However, in a variety of settings this may not be the case.
Consider, for example, a traffic routing game where each driver chooses his route in order to minimize his travel time, while accounting for congestion along the route caused by other drivers. Now, in many cases drivers are actually employees in shipping firms, and in fact it is in the interest of the shipping firm to minimize the total travel time of its fleet. Similarly, routers in a communication network participate in a congestion game. However, as various routers may belong to the same ISP we are again in a setting where coalitions naturally form. This motivated Fotakis et al. [2] and Hayrapetyan et al. [3] to introduce the notion of Coalitional Congestion Games (CCG). In a CCG we think about the coalitions as players and each coalition maximizes its total utility. The coalitional congestion game inherits its structure from the original game, once the coalitions of players from the original congestion games (now, becoming the players of the coalitional congestion game) have been identified.
The most notable property of congestion games is that they have posses pure Nash equilibrium. This has been shown by Rosenthal in [5]. Later, Monderer and Shapley [4] formally introduce potential games and show the equivalence of these two classes. The fact that potential games posses a pure Nash equilibrium is straightforward. Unfortunately, the statement that a CCG is a potential game or that it possesses a pure Nash equilibrium is generally false. In this paper we investigate conditions under which this statement is true. We focus on a subset of congestion games called simple congestion games, where each player is restricted to choose a single resource.
Our main contributions are:
Whenever each coalition contains at most two players the CCG induced from a simple congestion game possesses a pure-strategy Nash equilibrium (Theorem 1).
If some coalition contains three players, then there may not exist a pure-strategy Nash equilibrium (Example 1).
If a the congestion game is not simple then there may not exist a pure-strategy Nash equilibrium (Example 2); and 4. Suppose there exists at least one singleton coalition and at least one coalition composed of two players, then a coalitional congestion game induced from a simple congestion game is a potential game if and only if cost functions are linear (Theorem 2).
Our results extend and complement the results in Fotakis et al. [2] and Hayrapetyan et al. [3]. For example, Fotakis et al. [2] show that if the resource cost functions are linear then the coalitional congestion game is a potential game. We show that, with some additional mild conditions on the partition structure, this is also a necessary condition. Hayrapetyan [3] shows that if the underlying congestion game is simple and costs are weakly convex then the game possesses a pure Nash equilibrium. We demonstrate additional settings where this holds.
Section 2 provides a model of a coalitional congestions games, section 3 discusses the conditions for the existence of a pure Nash equilibrium in such games and section 4 discusses the conditions for the existence of a potential function. All proofs are relegated to an appendix.
Let G = {N, S, U} be a non-cooperative game in strategic form. Let C = {C 1 , . . . , C n c } be a Partition of N into n c nonempty sets. Hence:
The game G and the partition C form a Coalitional Non-Cooperative (CNC) Game G C = {N C , S c , U c } defined as follows:
• N C is the set of agents which are the elements of C.
• The strategy space is
, since we only changed the order of the coordinates. Thus, we can look on s c as a vector in S.
• The utility function is defined as follows:
In the context of G C , G is the Underlying Game and a player in G is referred to as a sub agent. As always, a Pure Nash Equilibrium of the game
) the (possibly empty) set of Pure Nash equilibrium strategy profiles in G C .
A congestion game is a game G = {N, R, Σ, P } where N is the finite set agents, R is the finite set resources, P = {P r } r∈R are the resource costs functions, where P r : [1, . . . , n] → R and Σ = × i∈N Σ i , where Σ i ⊆ 2 R , is i’s strategy space. Agent i selects s i ∈ Σ i and pays
, where c(s) r = j∈N I {r∈s j } is the number of agents who select r in s. In ut
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