Diversity-seeking Jump Games in Networks

TL;DR

This paper introduces a class of strategic jump games in which the agents are diversity-seeking: the utility of an agent is defined as the fraction of its neighbors that are of different type than itself.

Abstract

Recently, strategic games inspired by Schelling's influential model of residential segregation have been studied in the TCS and AI literature. In these games, agents of k different types occupy the nodes of a network topology aiming to maximize their utility, which is a function of the fraction of same-type agents they are adjacent to in the network. As such, the agents exhibit similarity seeking strategic behavior. In this paper, we introduce a class of strategic jump games in which the agents are diversity-seeking: The utility of an agent is defined as the fraction of its neighbors that are of different type than itself. We show that in general it is computationally hard to determine the existence of an equilibrium in such games. However, when the network is a tree, diversity-seeking jump games always admit an equilibrium assignment. For regular graphs and spider graphs with a single empty node, we prove a stronger result: The game is potential, that is, the improving response dynamics always converge to an equilibrium from any initial placement of the agents. We also show (nearly tight) bounds on the price of anarchy and price of stability in terms of the social welfare (the total utility of the agents).

Figures (8)

• Figure 1: $G_1$ and $G_2$ are the two components of $G$, connected by a single edge. Every node in $H$ is connected to every node in $W$. The blue (red) triangle connected to $x$ denotes a set of neighbors of $x$ with blue (red) agents, and the number next to a triangle denotes the number of nodes in the corresponding set.
• Figure 2: An IRC in a game with a tree topology. The blue (red) triangle connected to $x$ denotes a set of neighbors of $x$ with blue (red) agents, and the number next to a triangle denotes the number of nodes in the corresponding set.
• Figure 3: An equilibrium assignment for an instance with $k=3$. The red agents are placed first, starting at level 5, and ending at level 2, followed by yellow, and blue agents respectively. By convention, $r$ is on level 0.
• Figure 4: Swapping the red children adjacent to at least one more agent ($S_1$) with non-red children with no other agents as neighbors ($S_2$). (a): $|S_1| \leq |S_2|$. (b): $|S_1| > |S_2|$. Red dotted arrows show the swaps.
• Figure 5: All the red children of $\mathcal{A}$ can only have red children (if any) : (a) $\mathcal{B}$ is on the even level at which we finish placing the red agents, then all the red agents that are to the right of $\mathcal{B}$ on the same level (thus children of $\mathcal{A}$) can only have red children (if any) (b) $\mathcal{B}$ is on the odd level at which we start placing the red agents, then all the red agents that are to the left of $\mathcal{B}$ on the same level (thus children of $\mathcal{A}$) can only have red children (if any). The cross hashed levels may or may not exist, but if they do, they are occupied by non-red agents.

Theorems & Definitions (42)

• Definition 1
• Definition 2
• Lemma 2.1
• proof
• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof