Nonatomic Non-Cooperative Neighbourhood Balancing Games

Abstract

We introduce a game where players selfishly choose a resource and endure a cost depending on the number of players choosing nearby resources. We model the influences among resources by a weighted graph, directed or not. These games are generalizations of well-known games like Wardrop and congestion games. We study the conditions of equilibria existence and their efficiency if they exist. We conclude with studies of games whose influences among resources can be modelled by simple graphs.

Figures (5)

• Figure 1: Costs functions for the example in Subsection \ref{['sub:firstex']}. The cost $C_1$ (resp. $C_2$) is represented by the dotted curve (resp. solid). The $x$-axis is the mass of vertex $1$.
• Figure 2: We consider a game with $2$ vertices. The curves represent the costs of the two vertices as a function of the mass on vertex $1$. There are three equilibria in this case: $x_1 = 0$, $x_1 = \gamma$ (second intersection of the costs curves) and $x_1=r$. Equilibria $x_1 = 0$ and $x_1 = r$ are strong; however $x_1 = \gamma$ is not strong since a small quantity $\epsilon$ of mass can always move from $2$ to $1$ and improve its cost.
• Figure 4: With $n=2$ and $r=1$ (so that $x_2=1-x_1$), consider $\alpha_{1,2}=\alpha_{2,1}=1$, $f_1(x_1)=1$ and $f_2(x_2)=1+x_2$. This gives $C_1({\mathbf x})=2-x_1$, $C_2({\mathbf x})=2$, and $\Phi({\mathbf x})=x_1+(x_2+\frac{1}{2}x_2^2)+x_1x_2=(3-x_1^2)/2$ (this potential function is given by Equation \ref{['eq:formulePhi']} in the upcoming Proposition \ref{['prop:potentialFormula']}; we can check that this is indeed a potential. Otherwise, other potentials are equal to this one up to an additive constant; also note that a potential can be expressed with $x_1$ alone, with $x_2$ alone, or with both $x_1$ and $x_2$). There are two equilibria, one in $x_1=1$ which is a minimum of the potential function, and one in $x_1=0$, which is a maximum.
• Figure 5: The costs and equilibria for three different values of $b_2$.
• Figure 6: A family of graphical games reaching the maximum price of stability of $2$ for affine vertex-cost functions. With $n=2$ and $r=1$, consider $\alpha_{1,2}=\alpha_{2,1}=1$, $f_1(x_1)=1+2\lambda$ and $f_2(x_2)=(2+\lambda) x_2 + \lambda$. This gives $C_1({\mathbf x})=2+2\lambda -x_1$ and $C_2({\mathbf x})=2+2\lambda-(1+\lambda)x_1$. The only equilibrium of the game has cost $2+2\lambda$, while the best utilitarian social cost is obtained for $x_1=1$ and is equal to $1+2\lambda$ (this social cost is decreasing on $[0,1]$ if $0 < \lambda < 1$). Thus, we can reach the bound of $2$ for $PoS(G)$ by letting $\lambda$ go to $0$.

Theorems & Definitions (16)

• Proposition 2.1
• Theorem 3.1
• Proposition 3.2
• Corollary 3.3
• Corollary 3.4
• Proposition 3.5
• Proposition 3.6
• Proposition 3.7
• Proposition 3.8
• Lemma 3.9