Hamilton's Rule for Enabling Altruism in Multi-Agent Systems

TL;DR

This work addresses enabling altruistic behavior in autonomous multi-agent systems by transplanting Hamilton's rule from biology into control theory. It redefines agent fitness as productivity $P_i(x_i,u_i)$ and uses a graph-based networked-dynamics model together with Collaborative Control Lyapunov Functions (CCLFs) to quantify how an agent's action affects both its own and neighbors' progress toward goals, formalizing an altruism condition with $r_{ij} B_j(u_i) \ge C_i(u_i)$. The authors show that, under undirected graphs and positive task importances, the weighted total goal-reaching is nonincreasing when agents follow the altruism rule, ensuring convergence, and demonstrate this on a multi-agent way-point navigation scenario. The framework provides a principled mechanism for sacrificing individual costs to improve team performance in robot networks and opens paths toward predictive planning in dynamic environments. Key contributions include the HO-CLF construction, the decomposition of neighbor influence into $a_{ij}$ and $b_i$, and the demonstration that the total weighted objective converges under the proposed altruism conditions.

Abstract

This paper explores the application of Hamilton's rule to altruistic decision-making in multi-agent systems. Inspired by biological altruism, we introduce a framework that evaluates when individual agents should incur costs to benefit their neighbors. By adapting Hamilton's rule, we define agent ``fitness" in terms of task productivity rather than genetic survival. We formalize altruistic decision-making through a graph-based model of multi-agent interactions and propose a solution using collaborative control Lyapunov functions. The approach ensures that altruistic behaviors contribute to the collective goal-reaching efficiency of the system. We illustrate this framework on a multi-agent way-point navigation problem, where we show through simulation how agent importance levels influence altruistic decision-making, leading to improved coordination in navigation tasks.

Figures (3)

• Figure 1: An example of the altruism condition from \ref{['eq:alt_cond_example']} applied to two agents with goals to swap positions. We plot the trajectories of both agents, where in Figure \ref{['fig:red_eq_blue']} the importance values are equal and in Figure \ref{['fig:red_g_b']} the importance of the blue agent is set to be less than the red agent, with Figure \ref{['fig:red_g_b_offset']} showing an example with agents swap positions slightly offset.
• Figure 2: Two examples of trajectories for an 8-agent system where a star of the corresponding color shows each agent's goal, with agent dynamics defined by \ref{['eq:uncomfortable_dyn']} subject to the altruism condition defined by \ref{['eq:hamiltons_rule_CCLF']}. In Figure \ref{['fig:all_eq']}, all agents are assigned equal importance values, whereas in Figure \ref{['fig:blue_g_orange_g_all']} we assign the greatest importance to the blue and orange agents, where $w_{\text{blue}}=10^6$ and $w_{\text{orange}}=10^3$, with all other agent importance being set at $1$.
• Figure 3: Plots of $\phi^2_i(\mathbf{x}_i^+, u_i, u_{\mathcal{N}_i})$ for all $i \in[n]$ (top) and $\sum_{i \in [n]} w_i \phi^2_i(\mathbf{x}_i^+, u_i, u_{\mathcal{N}_i})$ (bottom) over time for the 8-agent simulation shown in Figure \ref{['fig:blue_g_orange_g_all']}. Note that, despite some agents moving away from their goal, Theorem \ref{['thm:sum_phi_0']} still holds for all time.

Theorems & Definitions (6)

• Definition 1: Control Lyapunov Function ogren2001control
• Definition 2: Collaborative Control Lyapunov Function
• Proposition 1
• proof
• Theorem 1
• proof