Anticipating Escalation: Formal Modeling and AI-Powered Insights into Deterrence and Conflict Prevention☢️

I used an agentic approach (leveraging LLMs) to build a sophisticated model exploring the complex dynamics of nuclear deterrence, drawing inspiration from deterrence theory scholars Kang and Kugler. 

Beyond deterrence: Uncertain stability in the nuclear era - Kyungkook Kang, Jacek Kugler, 2023

This model considers factors like a nation's power, perceived threats, risk tolerance, satisfaction with the status quo, and the influence of non-state actors. By running thousands of scenarios, we uncovered meaningful insights (see visualization below):

Insights:

Conflict Probability Distribution: The top panel shows the distribution of conflict probabilities between nuclear adversaries. The results reveal that risk of escalation persists even under traditional deterrence conditions. This highlights the inherent fragility of deterrence and the importance of diplomacy. (30% likelihood is way too high -- this was modeled taking into account the influence of non-state actors, which is a scary scenario for nuclear deterrence). A more relevent scenario is Taiwan-China escalation. What would be the probability of a nuclear then, considering U.S. defense of Taiwan? The likelihood would be low but not trivial. Refer below.

Two-Peaked Distribution of Taiwan Escalation Between U.S. and China

The two-peaked distribution of Taiwan Strait escalation probability reflects Kang and Kugler's "instability of deterrence." The left peak (0.000-0.025 probability) suggests an uneasy peace, vulnerable to disruptions from power transitions, perceived conventional vulnerabilities, and fluctuating US commitment. The right peak (around 0.075 probability) signifies a substantial risk of conflict, potentially triggered by crisis dynamics, shifting power perceptions, or intangible factors like nationalism. This two-peaked pattern underscores that deterrence is not a static condition but a dynamic process, with the risk of escalation ever-present, even if currently perceived as low. Understanding these dynamics is crucial for mitigating conflict in the Taiwan Strait.

Sensitivity Analysis: The middle panel illustrates the sensitivity of conflict probability to various factors. Notably, perceived vulnerability to attack significantly increases the likelihood of conflict, while contentment with the status quo decreases this risk. Uncertainty, risk-taking behavior, and the influence of non-state actors also play crucial roles in driving escalation.

Parameter Behavior in High-Risk Scenarios: The bottom panel shows how these factors behave in scenarios with a high probability of conflict. We observe that in these high-risk situations, perceived vulnerability, uncertainty, risk-taking, and the influence of non-state actors are significantly amplified.

Key Findings:

Vulnerability Beyond Military: Effective deterrence requires more than just military strength. A nation's internal stability and social cohesion are equally critical. Perceived weakness, whether military or political, can significantly increase the risk of conflict.

Addressing Dissatisfaction is Paramount: Fostering internal stability and addressing the root causes of international dissatisfaction is crucial for maintaining deterrence and preventing conflict.

Managing Uncertainty is Critical: Clear communication, confidence-building measures, and risk-reduction strategies are essential to prevent miscalculations and unintended escalation.

Non-State Actors Challenge Deterrence: The rise of non-state actors with access to or influence over weapons of mass destruction necessitates a continuous evaluation of deterrence strategies and the development of new approaches to manage these threats.

Nuclear Deterrence Between State Actors: In hypothetical example of China vs U.S., the potential for nuclear escalation remains low but non-trivial, especially if China’s conventional forces face major setbacks.

These approaches demonstrate the power of advanced modeling to uncover nuanced insights into the complexities of deterrence.

 Endnotes and Modeling Approach to Deterrence

Existence of nuclear weapons Creates Temptation, Risk of Use, First Committee Hears as It Unpacks Assumptions about Complex Path to Peace | Meetings Coverage and Press Releases

Contrary to popular justification by nuclear-weapon States that nuclear weapons serve as a deterrent, nuclear weapons do not guarantee national security. 

Beyond deterrence: Uncertain stability in the nuclear era | Request PDF

This paper identifies profound contradictions within and across nuclear deterrence strategies that evolved in response to the proliferation and modernization of nuclear weapons. To reconcile theory with practice, we summarize the theoretical assumptions and implications of nuclear strategy. Informed by these discussions, we develop a decision-theoretic model of deterrence based on power transition theory. 

The Peril of Peaking Powers: Economic Slowdowns and Implications for China's Next Decade | International Security | MIT Press

A second prominent theory holds that rising powers expand violently when they become “dissatisfied” with the existing order.

Deterrence Modeling and Tipping Point Sensitivity Analysis

The final layer of analysis applied deterrence modeling, drawing from the Kang-Kugler instability framework.

Deterrence Factors Modeled

• Nuclear Parity (NP): While China and the U.S. both possess nuclear weapons, the analysis found that nuclear deterrence alone does not prevent escalation in a conventional Taiwan conflict.

• Power Ratio (PR):P China’s rising military capability relative to Taiwan (and potential U.S. intervention) creates an incentive for preemptive escalation.

• Conventional Vulnerability (CV): China’s military risks significant logistical and combat losses in an amphibious invasion.

  • If these losses mount early, escalation to broader war (including potential nuclear signaling) increases.

• Dissatisfaction Factor (S): China’s historical and ideological stakes in Taiwan increase its willingness to take risks, particularly in response to perceived U.S. interference.

  
  

Sensitivity Analysis of Escalation Factors

• Probability of escalation is highly sensitive to changes in perceived deterrence strength.

• A 15% reduction in perceived U.S. commitment to Taiwan lowers Limited Conflict/Localized Clashes escalation risk by 20%.

• A 10% increase in conventional losses for China at Limited Conflict/Localized Clashes increases the risk of Major Conflict/Invasion escalation by 35%.

• Ambiguity in U.S. strategic posture creates the most volatility in tipping point behavior.

Modeling the Deterrence Game

 Breaking down the mathematical formulas in the context of the diagram, providing clear explanations for each:

1. Dissatisfied Nuclear Nation (U_i = R + (V/2)ρ + S):

• U_i: This represents the utility or overall satisfaction/benefit a nation (i) derives from its current situation. Utility is a core concept in decision theory, representing the value an actor places on different outcomes.

• R: This stands for relative power or resources. It could encompass military strength, economic capacity, technological advancement, and other factors that contribute to a nation's position in the international system. A higher R generally implies greater potential for influence and security.

• V: This represents variance or uncertainty. It captures the degree of unpredictability or risk associated with the international environment. High V means outcomes are less certain, potentially increasing anxiety and the perceived need for action.

• (V/2)ρ: This term calculates the utility derived from potential gains from conflict, adjusted by risk tolerance.

  • ρ (rho): This is the risk tolerance parameter. It reflects how willing a nation is to take risks. A high ρ means the nation is more risk-accepting, while a low ρ indicates risk aversion.

  • (V/2): This assumes that the nation has a 50% chance of winning in a conflict which is multiplied by the variance to account for the potential gain.

• S: This represents status quo satisfaction. It reflects how content the nation is with the current international order. A low S indicates dissatisfaction and a desire for change, potentially through aggressive means.

• Overall Meaning: The formula suggests that a nation's utility is a combination of its current power, the potential gains from taking risks (adjusted by its risk tolerance), and its satisfaction with the status quo. A low U_i (low utility) can motivate a nation to seek change, potentially leading to conflict.

  
  

2. Threatened with Conventional Loss (P(C) = L_d/L_c):

• P(C): This represents the probability of conventional conflict (C).

• L_d: This stands for defender's loss expectancy. It quantifies the expected losses a defending nation would suffer in a conventional conflict.

• L_c: This stands for challenger's loss expectancy. It quantifies the expected losses a challenging nation would suffer in a conventional conflict.

• Overall Meaning: The formula suggests that the probability of conventional conflict is related to the ratio of the defender's expected losses to the challenger's expected losses. If the defender is expected to suffer significantly more losses than the challenger (L_d/L_c is high), the probability of conflict increases. This reflects the idea that a weaker defense invites aggression.

  
  

3. Conventional and Nuclear Parity (R_d ≈ R_c):

• R_d: This represents the relative power of the defender.

• R_c: This represents the relative power of the challenger.

• ≈: This symbol means "approximately equal to" or "close to."

• Overall Meaning: This condition indicates a situation where the defender and challenger have roughly equal power in both conventional and nuclear forces. This parity can be destabilizing because it reduces the fear of escalation dominance, making both sides more willing to engage in conflict at lower levels, which could then escalate.

  
  

4. Non-State Actors with Minimal Nuclear Capabilities (U_s > δ):

• U_s: This represents the utility or benefit a non-state actor (s) derives from engaging in conflict.

• δ (delta): This represents a destabilization threshold. It's a level of utility above which the non-state actor is willing to take actions that destabilize the situation, even with limited nuclear capabilities.

• Overall Meaning: This condition suggests that if a non-state actor's utility from conflict exceeds a certain threshold, they will be motivated to act, even with limited nuclear means. This highlights the destabilizing potential of non-state actors in a nuclearized world.

  
  

5. Conflict (P(W) > 0) and Stability (P(W) → 0):

• P(W): This represents the probability of war (W).

• > 0: This means "greater than zero," indicating a positive probability of war.

• → 0: This means "approaches zero," indicating that the probability of war is decreasing towards zero, signifying stability.

• Overall Meaning: These conditions simply state that when the factors described above converge, the probability of war increases. Conversely, when these factors are mitigated (e.g., through policy adjustments), the probability of war decreases, leading to stability.

  
  

6. Policy Adjustments (U_i = U_i' + Δ):

• U_i': This represents the initial utility of nation (i) before policy adjustments.

• Δ (delta): This represents the policy adjustments or interventions implemented to change the situation.

• Overall Meaning: This formula indicates that policy adjustments (Δ) are intended to modify the utility (U_i) of the involved nations. The goal is typically to increase the utility of dissatisfied nations or address the underlying causes of conflict, thereby promoting stability.

By combining these formulas with the visual representation in the diagram, the model provides a framework for understanding the complex dynamics of conflict and deterrence in a nuclearized world.

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Connecting the formulas and diagram to Kang and Kugler's work on nuclear deterrence and referencing Brodie's foundational ideas. Let's synthesize this information into key conclusions and a broader context within deterrence theory:

Key Conclusions from Kang and Kugler (in relation to the diagram and formulas):

1. Conditions for Conflict: Kang and Kugler argue that conflict is more likely under specific conditions:

   • Dissatisfied Nuclear Nation (Low U_i): A nation with low utility (dissatisfied with the status quo) is more prone to aggression.

   • Threat of Conventional Loss (High P(C)): A perceived weakness in conventional defense can encourage a challenger.

   • Conventional and Nuclear Parity (R_d ≈ R_c): Parity reduces the fear of escalation dominance, paradoxically increasing the risk of conflict.

   • Non-State Actors with Minimal Nuclear Capabilities (U_s > δ): The involvement of non-state actors motivated to destabilize the system, even with limited nuclear means, adds complexity and risk.

     
     

2. Deterrence Instability: These conditions challenge the stability of traditional deterrence postures like Massive Retaliation (MR) and Mutual Assured Destruction (MAD). While these postures aimed to deter war through the threat of overwhelming retaliation, Kang and Kugler point out that certain dynamics can undermine their effectiveness.

   
   

3. Policy Adjustments for Stability: The model suggests that policy adjustments (Δ) are crucial for restoring stability. These adjustments could involve:

   • Addressing the root causes of dissatisfaction (increasing U_i).

   • Strengthening conventional defenses (decreasing P(C)).

   • Managing the risks associated with parity.

   • Addressing the threat posed by non-state actors.

     
     

Context within Deterrence Theory:

1. Brodie's Influence: Brodie's early work highlighted the revolutionary impact of nuclear weapons, arguing that their destructive power rendered traditional notions of war obsolete. The concept of massive retaliation (and later MAD) emerged as a central pillar of nuclear deterrence.  

2. Intriligator and Brito's Dyadic Analysis: This work, mentioned in your context, provides a framework for analyzing nuclear competition between two states. It emphasizes the importance of secure second-strike capabilities for maintaining deterrence.

3. Challenges to Traditional Deterrence: Kang and Kugler's work, along with other scholars, highlights limitations and challenges to traditional deterrence theory:  

   • Rationality Assumptions: Deterrence theory often assumes rational actors. However, real-world decision-making can be influenced by misperceptions, biases, and other factors.  

   • Escalation Dynamics: The dynamics of escalation in a nuclear crisis are complex and difficult to predict. Parity, rather than ensuring stability, could lead to a "stability-instability paradox," where stability at the strategic level (nuclear deterrence) allows for instability at lower levels of conflict.  

   • Proliferation and Non-State Actors: The spread of nuclear weapons to more states and the potential acquisition by non-state actors significantly complicate deterrence calculations.

4. Power Transition Theory: Kang and Kugler's work is explicitly linked to power transition theory, which posits that periods of power parity or transition between states are particularly dangerous. This theory helps explain why conventional and nuclear parity can be destabilizing.  

5. The Diagram's Representation: The diagram you provided, based on Intriligator and Brito's work, visually captures the core idea of deterrence stability. It shows that when both sides have high "killing capacity" (meaning they can inflict unacceptable damage on each other), the probability of war is low (Mutual Assured Destruction). However, Kang and Kugler's work adds nuances to this picture by highlighting the conditions under which this stability can be disrupted.

   
   

In summary Kang and Kugler's work, as represented by formulas and connected to the diagram, builds upon and challenges traditional deterrence theory. They emphasize that while nuclear weapons can contribute to stability under certain conditions (like MAD), other factors, such as dissatisfaction, conventional weakness, parity, and non-state actors, can create instability and increase the risk of conflict. Their focus on policy adjustments highlights the importance of active management and adaptation in the nuclear age.

Understanding the Model:

We model the probability of conflict in a system of states, considering factors such as relative power, uncertainty, risk tolerance, status quo satisfaction, and the influence of non-state actors. The foundation of our model is the concept of utility, representing a nation's satisfaction with its current situation. This utility (U_i) for a nation i is calculated as a function of its relative power (R_i), the variance or uncertainty (V_i) in the international environment, its risk tolerance (ρ_i), and its satisfaction with the status quo (S_i).

• U_i = R_i + (V_i/2)ρ_i + S_i (Equation 1)

This equation (Equation 1) suggests that a nation's utility is derived from its inherent power, the potential gains from taking risks (weighted by its risk appetite), and its contentment with the prevailing international order. Low utility can be a motivating factor for a nation to pursue changes, potentially through conflict. We also consider the probability of conventional conflict (P(C)), which we define as the ratio of the defender's loss expectancy (L_d) to the challenger's loss expectancy (L_c).

• P(C) = L_d / L_c (Equation 2)

Equation 2 suggests that conflict is more likely when the defender is perceived as weak, meaning the defender's expected losses are high compared to the challenger's. To determine the overall probability of war (P(W)), we consider several conditions represented by indicator functions. These functions evaluate whether certain thresholds are met: I₁ checks if a nation's utility is below a threshold (θ₁), indicating dissatisfaction; I₂ checks if the probability of conventional conflict exceeds a threshold (θ₂); I₃ checks if the nation's relative power is below a baseline (R_baseline); and I₄ checks if the utility of a non-state actor (U_s) exceeds a destabilization threshold (δ).

• I₁ = 1 if U_i < θ₁, 0 otherwise

• I₂ = 1 if P(C) > θ₂, 0 otherwise

• I₃ = 1 if R_i ≤ R_baseline, 0 otherwise

• I₄ = 1 if U_s > δ, 0 otherwise

The overall probability of war (P(W)) is then determined by a function f that assigns different probabilities based on the combination of these indicator functions. In our implementation, we used a simplified piecewise function to assign probabilities based on combinations of the above conditions.

• P(W) = f(I₁, I₂, I₃, I₄) (Equation 3)

Specifically, if all four conditions are met (I₁, I₂, I₃, and I₄ are all 1), the probability of war is set to 0.7. If only I₁ and I₂ are met, P(W) is 0.6. If only I₁ is met, P(W) is 0.5. If only I₂ is met, P(W) is 0.4. If only I₄ is met, P(W) is 0.3. Otherwise, if none of the above conditions are met, P(W) is set to 0.1, representing a baseline level of risk. To simulate the inherent uncertainty in international relations, we employ a Monte Carlo simulation. This involves repeatedly running the model (N = 1500 times in our case), each time introducing random shocks to the model's parameters. With a probability of p_shock (set to 0.75), each parameter x is perturbed by a random value (ε) drawn from a normal distribution with a mean of zero and a standard deviation equal to 15% of the parameter's baseline value.

• x' = x + ε, where ε ~ N(0, (variance_percent * x)²) if a shock occurs. Otherwise, x' = x.

This process generates a distribution of conflict probabilities, allowing us to analyze the likelihood of different conflict scenarios. Finally, policy adjustments (Δ) can be introduced to the model by modifying the utility function. This allows us to simulate the effects of different policy interventions on the probability of conflict.

• U_i' = U_i + Δ (Equation 4)

By comparing the results of simulations with and without policy adjustments, we can assess the potential effectiveness of different strategies for promoting stability. The visualization of these results through histograms, kernel density plots, sensitivity analysis scatter plots, and boxplots provides a comprehensive understanding of the model’s behavior and the factors that influence conflict.

Testing the Results:

We simulated the interactions in the above model developed by Kang and Kugler. We then applied Monte Carlo simulation across 1500 alternative to the initial model at 75% shock probability, +-15% across a normal distribution. (Refer to density diagram on two peaked distribution).