Relationship Design for Socially-Aware Behavior in Static Games

TL;DR

This work addresses designing socially-aware behavior in static games by incorporating bounded rationality through quantal response equilibria (QRE) and inter-agent relationships captured as a graph of weighted adjacencies. It introduces a relationship game where costs are modified by a weighted combination of relationship graphs, and optimizes the weights $oldsymbol{w}$ to minimize the social cost $J(oldsymbol{x}^*,V)$ with the equilibrium $oldsymbol{x}^*$ derived from the QRE of the induced game $G_{ ilde{c}}$, handling multiple equilibria via Min-Max and Min-Min formulations. A differentiable, bi-level optimization pipeline is developed: compute the QRE via a nonlinear least-squares reformulation and obtain $ abla_{oldsymbol{w}}J$ through implicit differentiation of the KKT system, then apply two projected gradient-descent algorithms with $L_2$ normalization and a convergence criterion. The framework is validated on two congestion scenarios (two-lane and ambulance-priority), showing convergence to equilibria with the intended social costs and revealing emergent altruistic behavior as weights adapt to favor prioritized agents. This has practical significance for coordinating autonomous agents with humans in shared environments by shaping interaction topologies to reduce social costs and improve system efficiency.

Abstract

Autonomous agents can adopt socially-aware behaviors to reduce social costs, mimicking the way animals interact in nature and humans in society. We present a new approach to model socially-aware decision-making that includes two key elements: bounded rationality and inter-agent relationships. We capture the interagent relationships by introducing a novel model called a relationship game and encode agents' bounded rationality using quantal response equilibria. For each relationship game, we define a social cost function and formulate a mechanism design problem to optimize weights for relationships that minimize social cost at the equilibrium. We address the multiplicity of equilibria by presenting the problem in two forms: Min-Max and Min-Min, aimed respectively at minimization of the highest and lowest social costs in the equilibria. We compute the quantal response equilibrium by solving a least-squares problem defined with its Karush-Kuhn-Tucker conditions, and propose two projected gradient descent algorithms to solve the mechanism design problems. Numerical results, including two-lane congestion and congestion with an ambulance, confirm that these algorithms consistently reach the equilibrium with the intended social costs.

Figures (4)

• Figure 1: Descriptive illustrations for the two congestion game scenarios.
• Figure 2: Multiplicity of equilibria in congestion with ambulance with $\lambda=1.0$ and $\mathbf{w}=[0.5, 0.5, w_3, 0.5]$, where $w_3$ is sampled in the range $0 \leq w_3 \leq 3$ with increments of 0.01. At each $w_3$ value, we test $300$ different seeds with a tolerance threshold of $0.0001$ to identify unique values.
• Figure 3: The convergence of the iterates in \ref{['algo:minmax']} (Min-Max) and \ref{['algo:downstairs']} (Min-Min) on both congestion game examples. Termination points are highlighted with circles. The plots show the social cost at the current equilibrium $J(\mathbf{w}, \sigma) = J(\mathbf{x}, V)$ with respect to the iteration number.
• Figure 4: Superposed relationship graphs for selected iterations of the Min-Min optimization for the congestion with ambulance example, where $\lambda = 0.5$. Node colors represent the identity matrix for selfish costs, while undirected lines indicate bidirectional relationships between regular cars (Rs).

Theorems & Definitions (1)

• Definition 1: Relationship Game