Escalation of a local conflict into a Cold War

Since the Russia–Ukraine conflict of 2022, the world has been increasingly divided. Such division can be enormously costly to multiple layers of the world such as loss of international trade opportunities, increased vulnerability of emerging economies with accumulated debt, and risks of reduced cooperation for pressing global issues including climate change (Georgieva 2023). The increasing division is not necessarily preordained by the existing scholarship of “nationalist” models based upon the costs and benefits of conflicts and trade. For example, Jackson and Nei (2015) theoretically and empirically find that increasing trade between two countries makes it more likely that the two countries will avoid conflicts. The natural question is then, why is the world increasingly divided after one local conflict despite the increasing costs of division? This note provides a driving force for such division by studying a simple signed network game first proposed by Hiller (2017). In this game, a country chooses to be an ally of every other country. I show that the local conflict together with a particular strategy by one of the countries involved in the conflict is sufficient to lead to a dichotomized world.

In particular, I demonstrate that if (i) this specific country (call it A) chooses to be an enemy of another specific country, and if (ii) country A employs a policy that a friend of my enemy is my enemy, then the world will always be divided into two groups, within which countries are allies and across which countries are enemies. I demonstrate that any constrained efficient way of partitioning countries into two that maximizes total welfare is stable in the sense that no one set of countries together wants to deviate from the state. Stability may support the imposition of countervailing measures and working against a potential conflict.

With the main result, I will briefly develop my model with consideration of (1) the choice of neutral states and (2) different preference structures. That is, I will analyze strategies as we can infer from voting strategies. For instance, at the United Nations General Assembly (UNGA), India has abstained from voting for Russia to be suspended from the Human Rights Council and from voting against Russia’s annexation of a portion of Ukraine. Many countries adopt this neutral position, and President Macron has argued that the United Nations cannot dismiss Russia in order to stay neutral about defending Russia. I will discuss how the results of my model would change in the absence of such a strategy.

Furthermore, the value of a link and its surplus-sharing structure between two countries may depend on particular characteristics of countries such as their sizes. I will explain under conditions that are sufficient for the result of my model to carry through with the twists in the preferences. Moreover, Russia has so far labeled many of the countries that impose economic sanctions on Russia and support Ukraine as “unfriendly countries,” while it has not regarded those countries that support the allies of Ukraine as “unfriendly countries” yet. This results of my model imply that once it fully employs the strategy, this may lead to a dichotomized world for an extended period. Given that Russia appears to be moving toward fully adopting the strategy, which will be discussed later, I will provide possible policy scenarios in my model.

Before delving into the details of my model, I acknowledge its limitations here. As empirically found by Maoz et al. (2007), while the friend-of-my-enemy-is-my-enemy type triad relations are prevalent in international relations, there are many triad relations that do not follow such rules, both historically (such as the time span of 1816–2001 studied in Maoz et al. 2017) and contemporarily. In the context of the current Ukraine–Russia conflict, India is an example and is not regarded as an enemy by Russia in spite of being a Quad member. While this could be explained by the aforementioned India’s neutral stance and/or the aforementioned Russia’s partial reinforcement of the strategy, there can be many other factors that result in the example. Turkey is another example of the limitation.

To reiterate, my main model is a simplification of a complex reality. For example, the celebrated “domino theory” turned out to be overly celebrated and only accounted for 11% of the adoption of democracy by countries between 1850 and 2000 (Leeson and Dean 2009) since there is a hypothetical world that ignores a number of countervailing influences. Similarly, my note considers a hypothetical situation that considers an extreme case of one country fully committing to the above strategy and abstracts away from a number of other important factors such as regime types and economic sanctions. While I propose towards the end a simple extension of my model to incorporate such complexity for future theoretical and empirical studies, the purpose of this paper is neither to explain all historical international relations nor to construct a complex model for precise predictions on future dyadic or triadic international relations. Rather, the objective of this paper is to provide a simple model and succinctly portray how a local conflict can have a butterfly effect and affect the global structure of international relations.

The paper situates itself within signed network theory and international relations. Classic works by Cartwright and Harary (1956) and Lee et al. (1994) use Heider-based rules (e.g., the enemy of my enemy is my friend) to analyze how network preference structures can yield dichotomies, but they do not incorporate strategic incentive structures over countries’ behavior. Koiemzidi (2023) introduces incentives in an unweighted network framework but assumes all countries follow Heider-based rules. Hiller (2017) develops a seminal signed network game with homogeneous preferences and provides conditions for dichotomy under Nash and strong Nash equilibria. Jackson and Nei (2015) study how trade and alliance formation affect the likelihood of war, showing trade can stabilize networks and prevent conflict. In contrast, this note focuses on how, given antagonism, a single country’s adoption of a friend-of-my-enemy-is-my-enemy policy can suffice to produce a global dichotomy, and it analyzes stability of coalitions (a concept between Nash equilibrium and strong Nash equilibrium) while allowing for heterogeneous preference structures. The contribution is thus in applying signed network game insights to current events and policy, rather than advancing the technical theory of such games.

The study presents a simple signed network game modeling international relations.

• Players: A finite set of countries N = {1, …, n}, n ≥ 3.
• Strategies: Each country r chooses a vector g_r = (g_{r1}, …, g_{rn}) with g_{rj} ∈ {−1, 1}, where 1 denotes attempting a friendly (ally) link to j and −1 denotes attempting a hostile (enemy) link. The realized undirected link between i and j is positive only if both directed choices are positive; if either directs a negative link, the realized link is negative. Thus, every pair is either positively or negatively linked.
• Weights and value: A signed weighted graph H = (N, L, w) assigns weights w_{ij} to each undirected link {i, j}, representing the bilateral surplus (positive for positive links, negative for hostile links). The value function v(G) is additive: v(G) = Σ_{i<j} w_{ij}. This rules out complementarities across links.
• Allocation rule: Each country i’s payoff is Y_i(g) = Σ_j g_{ij} w_{ij}. This aligns individual incentives with the additive total surplus generated by the realized signed network.
• Stability (coalitional): A signed network g is stable if there exists no coalition S ⊆ N and alternative network g′ such that at least one member of S is strictly better off (Y_i(g′) > Y_i(g)) and all other members of S are weakly better off (Y_j(g′) ≥ Y_j(g)). This concept lies between standard Nash equilibrium and strong Nash equilibrium, allowing coordinated deviations by groups.

Key assumptions modeling a local conflict and a unilateral strategy:

• Assumption 1 (Exogenous dyadic hostility): A specific country 2 extends a negative link to country 1 regardless of 1’s action, i.e., g_{21} = −1, capturing a localized conflict between 1 and 2.
• Assumption 2 (Friend-of-my-enemy-is-my-enemy policy): Country 2 labels as enemies any countries that are friends with its enemy (country 1). Operationally, if a country forms or maintains friendship with 1, country 2 extends a negative link to that country.

Analysis proceeds by characterizing constrained efficient signed networks (partitions that maximize the additive total surplus subject to the above realization rules), examining cycles (triads), and proving stability under the allocation rule. The paper derives lemmas and a proposition (proofs in the appendix) establishing that under Assumptions 1 and 2, constrained efficient networks eliminate negative 3-cycles, induce a clean partition of countries into two opposing camps, and are stable against coalitional deviations.

Extensions considered methodologically include: allowing a neutral link type (δ_{ij} = 0) that avoids negative surplus across camps (except with country 2), and relaxing the equal surplus-sharing assumption to pre-specified bilateral sharing rules that can depend on dyadic characteristics (e.g., sizes) but not on network position or third parties (no externalities/complementarities).

• A single local conflict combined with one country’s adoption of a friend-of-my-enemy-is-my-enemy policy is sufficient to divide the entire network into two antagonistic groups (dichotomy): allies within groups and enemies across groups.
• Lemma 1: Under Assumptions 1 and 2, any triadic cycle in a constrained efficient signed network is positive. This implies that even if bilateral weights are intrinsically positive, the unilateral policy by country 2 forces negative links to appear beyond the initial dyad, generating a domino effect.
• Lemma 2 (from Davis, 1967): Given the assumptions, cross-set ties follow the induced partition structure, supporting the dichotomy characterization.
• Proposition 1: Any constrained efficient signed network is stable under the coalitional stability concept with the given allocation rule. Hence, no coalition has profitable deviations.
• Corollary: All stable signed networks under Assumptions 1 and 2 feature a dichotomy (two-group partition of countries).
• Extensions: Allowing neutral links leads to equilibria where members of the anti-2 camp maintain negative links only with country 2 and neutral links otherwise, mitigating cross-camp hostility. Results are robust to heterogeneous, pre-specified bilateral surplus-sharing rules that do not depend on network position or third parties.
• Policy implications: Preventing full adoption of the friend-of-my-enemy policy by the instigating country or ending the initial dyadic conflict can forestall global dichotomization. Otherwise, the model predicts persistent two-bloc polarization akin to a Cold War.

The research question asks why a single local conflict (e.g., Russia–Ukraine) could precipitate a broad, costly geopolitical bifurcation despite trade-based incentives for peace. The signed network game shows that if one principal actor in the conflict adopts a friend-of-my-enemy-is-my-enemy policy, the realized network rules (where any negative proposal makes the tie negative) force third countries to choose sides, eliminating the feasibility of maintaining friendships with both adversaries. This mechanism endogenously produces a dichotomy: within-group positive ties and across-group negative ties, even when many bilateral relationships are intrinsically valuable.

Significance: By focusing on coalitional stability rather than individual deviations (Nash) or overly demanding strong Nash, the paper captures the real-world possibility of group coordination and communication among states. The stability of constrained efficient partitions implies the emergent blocs are resilient to coalitional reshuffling, offering an explanation for persistent polarization reminiscent of the Cold War.

Robustness and extensions enrich the interpretation: allowing neutral ties can soften cross-bloc hostility (except toward the instigating country), and heterogeneous bilateral surplus-sharing does not overturn the dichotomy as long as sharing rules are dyadic, pre-specified, and independent of network position.

Comparative perspective: Standard Nash equilibrium is ill-suited here because unilateral deviations can unrealistically violate the enforced policy constraint, and it underestimates inter-country coordination. Strong Nash can fail to select constrained efficient outcomes due to requiring strict improvement for all deviators. The chosen stability concept balances realism and tractability.

Policy relevance: To avoid bifurcation, either resolve the initiating dyadic conflict or deter the instigating country from fully enforcing the friend-of-my-enemy policy (e.g., through credible signals, sanctions relief conditional on de-escalation, or diplomatic arrangements). Otherwise, the model anticipates entrenched, costly polarization across the international system.

The paper introduces a simple, policy-relevant signed network game explaining how a local conflict, combined with one actor’s friend-of-my-enemy-is-my-enemy policy, can escalate into a global dichotomy of two antagonistic blocs. It proves that under these conditions, constrained efficient partitions are coalitionally stable and that all stable networks display dichotomy. Extensions show that permitting neutral ties or adopting heterogeneous but dyad-specific surplus-sharing rules does not overturn the main result.

Future research directions include: incorporating richer externalities and complementarities across links; modeling dynamics and incomplete enforcement of the policy; allowing regime type, sanctions, and institutional effects; and empirically testing the model’s predictions using temporal network data on alliances, conflicts, trade, and UN voting behavior.

• The model abstracts from many real-world complexities: regime types, multilateral institutions, sanctions, and economic complementarities are omitted by the additive value function and dyadic allocation rule.
• It assumes binary relations (friend/enemy), with neutrality only in an extension; degrees of alignment are not captured.
• The key mechanism relies on one country fully enforcing a friend-of-my-enemy-is-my-enemy policy; in practice, enforcement may be partial or inconsistent.
• Empirical counterexamples exist where triads do not follow Heider-type balance (historically and contemporarily, e.g., India and Turkey in the current context), limiting generalizability.
• The analysis is static (steady-state) and does not model transition dynamics or shocks.
• The allocation rule and efficiency concept preclude complementarities and externalities across ties, potentially understating coalition formation complexity.
• Additional antagonisms within the same camp are largely abstracted away and only discussed conditionally (e.g., enemy-of-my-enemy accommodations).