Network Preference Dynamics using Lattice Theory

TL;DR

The paper develops a lattice-theoretic framework for dynamic, decentralized updating of agent preferences on networks, using an information order and meet/join operations to model revision and aggregation. It introduces a general message-passing scheme where agents exchange preference-relations, aggregate neighbor information via lattice-polynomial operators, and update their own preferences with structure-preserving maps. Existence of equilibrium (stable) preference profiles is established via the Tarski fixed-point theorem, with the equilibria forming a complete lattice and supporting greatest/least fixed points; convergence results are provided for inflationary/deflationary update schemes on finite option sets. Numerical experiments on random graphs demonstrate convergence to stable profiles, reveal monotone non-decreasing disagreement (Dirichlet energy), and show clustering of disagreements according to agents’ stubbornness, highlighting the model's potential for guiding design in distributed decision-making systems.

Abstract

Preferences, fundamental in all forms of strategic behavior and collective decision-making, in their raw form, are an abstract ordering on a set of alternatives. Agents, we assume, revise their preferences as they gain more information about other agents. Exploiting the ordered algebraic structure of preferences, we introduce a message-passing algorithm for heterogeneous agents distributed over a network to update their preferences based on aggregations of the preferences of their neighbors in a graph. We demonstrate the existence of equilibrium points of the resulting global dynamical system of local preference updates and provide a sufficient condition for trajectories to converge to equilibria: stable preferences. Finally, we present numerical simulations demonstrating our preliminary results.

Figures (1)

• Figure 1: On a $k$-regular graph ($k=4$) with $N =20$ nodes with a randomly-selected initial preference profile, we plot (a) the Kendall tau distance $d(\pi_i(t),\pi_j(t))$ for every $(i,j) \in \mathcal{E}$ for both $t=0$ (initial disagreement) and $t=15$ (final disagreement) with a heat-map on the edges of the graph. We also plot (b) the Dirichlet energy of the trajectory (black) as well as the energies of other trajectories from different initial conditions (blues).

Theorems & Definitions (27)

• Definition 1: Lattices
• Lemma 1
• proof
• Definition 2: Transitive Closure
• Theorem 1
• proof
• Proposition 1
• proof
• Definition 3
• Proposition 2