Nash-equilibrium Seeking Algorithm for Power-Allocation Games on Networks of International Relations

TL;DR

The paper addresses how countries strategically allocate limited power in networked international relations by extending a signed-graph power-allocation framework. It introduces a revised static game with flexible preferences and a constructive algorithm that guarantees a pure-strategy Nash equilibrium, then generalizes to a dynamic co-evolution model with almost-sure convergence. Empirical validation using WWII-era data shows the model can predict survivability with over 70% accuracy, while dynamic simulations align power changes with states of safety and network structure. An inverse-inference framework for recovering edge weights from equilibrium trajectories further enhances interpretability and potential policy relevance.

Abstract

In the field of international security, understanding the strategic interactions between countries within a networked context is crucial. Our previous research has introduced a ``games-on-signed graphs'' framework~\cite{LiMorse2022} to analyze these interactions. While the framework is intended to be basic and general, there is much left to be explored, particularly in capturing the complexity of strategic scenarios in international relations. Our paper aims to fill this gap in two key ways. First, we modify the existing preference axioms to allow for a more nuanced understanding of how countries pursue self-survival, defense of allies, and offense toward adversaries. Second, we introduce a novel algorithm that proves the existence of a pure-strategy Nash equilibrium for these revised games. To validate our model, we employ historical data from the year 1940 as the game input and predict countries' survivability. Our contributions thus extend the real-world applicability of the original framework, offering a more comprehensive view of strategic interactions in a networked security environment.

Figures (2)

• Figure 1: A) is the graph of international relationships in 1940, where the green edges represent that the two countries are friendly countries and the red edges represent that they are antagonistic. B)- G) represent respectively the distribution histogram of survival likelihood from 1939 to 1944.
• Figure 2: A) is the heat map of $p_{avg}$ and $(q_e,q_n)$, where the color of each point represents the number of average power. B)- E) represent scatter plots of the average power and the frustration of network given $(q_e,q_n)$, where $(q_e,q_n)=(0.2,0.2)$ in B), $(q_e,q_n)=(0.2,0.8)$ in C), $(q_e,q_n)=(0.8,0.2)$ in D), and $(q_e,q_n)=(0.8,0.8)$ in E). F) is the heat map of Gini coefficient and $(q_e,q_n)$, where the color of each point represents the number of Gini coefficient. G)- J) represent scatter plots of the average power and the frustration of network given $(q_e,q_n)$, where $(q_e,q_n)=(0.2,0.2)$ in G), $(q_e,q_n)=(0.2,0.8)$ in H), $(q_e,q_n)=(0.8,0.2)$ in I), and $(q_e,q_n)=(0.8,0.8)$ in J)

Theorems & Definitions (15)

• Definition 1
• Definition 2
• Remark 1
• Theorem 1: Existence of Nash Equilibrium
• Definition 3: NSND strategy matrix
• Definition 4: Preferable adjustment
• Lemma 1: Properties of preferable adjustment
• Theorem 2: Number of safe countries in a fully antagonistic world
• proof
• Lemma 2