Variety-Seeking Jump Games on Graphs

TL;DR

This work investigates variety-seeking jump games on graphs, where agents seek to maximize the number of neighboring types. It shows that improving response cycles can occur, but the game is a potential game under several natural conditions, and equilibria exist on trees, cylinders, and tori via constructive methods. The paper derives tight bounds on the price of anarchy for two diversity notions—social welfare and colorful edges—demonstrating linear or near-linear degradation in general graphs and revealing tighter behavior on simple topologies. It also provides a lower bound on the price of stability, underscoring that the best possible arrangement need not be an equilibrium, and outlines open questions about broader equilibrium existence and additional diversity metrics.

Abstract

We consider a class of jump games in which agents of different types occupy the nodes of a graph aiming to maximize the variety of types in their neighborhood. In particular, each agent derives a utility equal to the number of types different from its own in its neighborhood. We show that the jump game induced by the strategic behavior of the agents (who aim to maximize their utility) may in general have improving response cycles, but is a potential game under any of the following four conditions: there are only two types of agents; or exactly one empty node; or the graph is of degree at most 2; or the graph is 3-regular and there are two empty nodes. Additionally, we show that on trees, cylinder graphs, and tori, there is always an equilibrium. Finally, we show tight bounds on the price of anarchy with respect to two different measures of diversity: the social welfare (the total utility of the agents) and the number of colorful edges (that connect agents of different types).

Figures (11)

• Figure 1: A game with an improving response cycle.
• Figure 2: (a) The assignment $\mathbf{v}$ for the tree $G'$ with a single empty node. Note that all non-red agents have utility at least 1, but some red agents have utility 0. (b) The same assignment for the tree $G$. Note that empty nodes $v_3,v_4$ and $v_5$ are adjacent to non-red agents (c) The assignment $\mathbf{v}'$: red agents at $v,v_1$ and $v_2$ with utility 0 in $\mathbf{v}$ jump to nodes $v_3,v_4$ and $v_5$ to obtain utility 1 in $\mathbf{v}'$.
• Figure 3: An example of an equilibrium assignment in a cylinder graph, when $n_k=2$.
• Figure 4: An example of an equilibrium assignment in a cylinder graph, when $n_k\ge 3$, with 5 empty nodes.
• Figure 5: An example of an equilibrium assignment for Case 2 (b). First the corners, $c_1$ and $c_2$, are covered with red agents, and the boundary is partially covered with the remaining red agents. Some blue agents are remaining after creating $L$, so they are placed adjacent to the red agents on the boundary, from nodes $v_1$ to $v_2$. Then, the agents of $L$ are then placed in a snake-like pattern starting at $v_3$ and ending at $v_4$.

Theorems & Definitions (34)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Lemma 3.5
• proof