Optimal Teaming for Coordination with Bounded Rationality via Convex Optimization

TL;DR

This paper studies optimal teaming for coordination on a network of $N$ agents with binary actions under bounded rationality modeled by log-linear learning. The designer selects the weight matrix $W$ to maximize the stationary probability of convergence to a Nash equilibrium while penalizing connectivity by $(\rho/2) 1^T W 1$; the resulting optimization is convex. A key finding is that, with no sparsity constraints, the optimal $W$ takes the uniform off-diagonal form $W^* = w^*(11^T - I)$ with $w^* > 0$, and $w^*$ decreases as the rationality parameter $\beta$ increases. The paper extends the analysis to arbitrary sparsity patterns, showing convexity remains but optimal weights can be asymmetric depending on the pattern; numerical examples compare topologies (line, star, hybrid) and illustrate topology-dependent performance. Overall, the work provides a principled method to balance learning probability and communication cost in networks of boundedly rational agents, with implications for distributed task allocation and network design.

Abstract

Teaming is the process of establishing connections among agents within a system to enable collaboration toward achieving a collective goal. This paper examines teaming in the context of a network of agents learning to coordinate with bounded rationality. In our framework, the team structure is represented via a weighted graph, and the agents use log-linear learning. We formulate the design of the graph's weight matrix as a convex optimization problem whose objective is to maximize the probability of learning a Nash equilibrium while minimizing a connectivity cost. Despite its convexity, solving this optimization problem is computationally challenging, as the objective function involves the summation over the action profile space, which grows exponentially with the number of agents. Leveraging the underlying symmetry and convexity properties of the problem, when there are no sparsity constraints, we prove that there exists an optimal solution corresponding to a uniformly weighted graph, simplifying to a one-dimensional convex optimization problem. Additionally, we show that the optimal weight decreases monotonically with the agent's rationality, implying that when the agents become more rational the optimal team requires less connectivity.

Figures (5)

• Figure 1: Framework for learning to coordinate over a network with bounded rationality. The teaming problem is to choose the weights that maximize the probability of learning to coordinate at a Nash equilibrium.
• Figure 2: A coordination game with parameter $\theta$ between two players.
• Figure 3: Numerical results for the optimal graph weight when $N=20$, $\theta=N/2+1$, and $\rho=1,5,10$ as a function of the rationality $\beta$: (left) shows the total optimal cost, (middle) shows the optimal edge weight, and (right) shows the resulting probability of playing the Nash equilibrium $a^\star=\mathbb{0}$.
• Figure 4: Examples of Networks with Incomplete Sparsity Patterns.
• Figure 5: Optimized objective function value as a function of rationality parameter $\beta$ for three graph topologies.

Theorems & Definitions (12)

• Remark 1: Payoff Interpretation
• Definition 1: Exact potential games
• Proposition 1
• Proposition 2
• Proposition 3
• proof
• Remark 2
• Theorem 1
• proof
• Theorem 2