First They Came for the Others: A Theory of Divide-and-Conquer

Technical Summary: "First They Came for the Others: A Theory of Divide-and-Conquer"

Problem Statement
The paper addresses a central puzzle in conflict theory and collective action: why potential victims of aggression often fail to mount a united front, allowing an aggressor to conquer them sequentially rather than simultaneously. While standard models attribute such failures to mechanical coordination problems, conflicting interests, or free-rider issues, this paper posits that a distinct constraint drives the phenomenon: epistemic friction regarding the aggressor's underlying intent.

The core problem is an inference problem faced by a "bystander" victim ($V_2$) when an aggressor ($A$) attacks a first victim ($V_1$). $V_2$ must determine whether the attack is a localized grievance (targeting only $V_1$) or the opening move of a systemic campaign (targeting both). If $V_2$ interprets the attack as localized, she abstains from joining the fight, anticipating that $V_1$ will surrender alone. This allows the aggressor to proceed with a divide-and-conquer (D&C) strategy. The paper seeks to formalize the conditions under which this inferential friction leads to sequential conquest, even when joint resistance is materially superior for the victims.

Methodology
The authors develop a Perfect Bayesian Equilibrium (PBE) model of sequential conflict involving one attacker and two potential victims.

• Information Structure: The attacker privately observes her benefits ($b_1, b_2$) from conquering $V_1$ and $V_2$. The victims know the distribution of these benefits but not the realization. The support of the distribution assumes $V_1$ is the "natural" first target ($b_1 \ge b_2$).
• Game Timeline:
  1. $A$ observes types and decides whether to attack $V_1$.
  2. If attacked, $V_1$ chooses to surrender or fight.
  3. If $V_1$ fights, $V_2$ observes the attack and $V_1$'s decision, then chooses to join or stay out.
  4. If $V_2$ stays out, $A$ fights $V_1$ alone. If $V_1$ surrenders, $A$ may subsequently attack $V_2$.
• Key Assumptions:
  • Assumption 1: Joint resistance is effective ($p_2 \ell + d < \ell$), but solitary resistance is worse than surrender ($p_1 \ell + d > \ell$).
  • Regularity: The model employs a "no signaling what you don't know" restriction. $V_2$ updates beliefs based on $A$'s attack and equilibrium strategy but does not update further based on off-path deviations by $V_1$.
• Equilibrium Selection: When multiple equilibria exist, the paper selects the attacker-preferred equilibrium (the D&C outcome), as it yields higher payoffs for the aggressor than collective resistance.

Key Results and Comparative Statics
The analysis yields two canonical equilibria: a Divide-and-Conquer (D&C) equilibrium where $V_2$ stays out and $V_1$ surrenders, and a Collective-Resistance equilibrium where $V_2$ joins and $V_1$ fights. The selection between them depends on whether the posterior probability of $V_2$ being a future target exceeds a "danger threshold" ($\kappa$).

1. The Cost of Attack Paradox:
   Contrary to intuition, a higher cost of attack ($c$) can facilitate divide-and-conquer.

   • Mechanism: A high-cost attack is "unthinkable" under normal circumstances. When observed, $V_2$ rationally infers that the attacker must have a highly particularized, exceptional reason to incur such a cost. This lowers the posterior probability that the attack is systemic ($\phi(c)$ decreases in $c$).
   • Result: As $c$ rises, the D&C equilibrium becomes more likely to be selected, potentially increasing the probability of an initial attack and the eventual conquest of both victims.

2. Correlation of Fates:
   The correlation ($\rho$) between the benefits of conquering $V_1$ and $V_2$ is critical.

   • Mechanism: High correlation implies that if $V_1$ is a target, $V_2$ is likely one too. Low correlation allows $V_2$ to rationalize the attack as specific to $V_1$.
   • Result: Lower correlation makes D&C easier to sustain. This formalizes the "Niemöller inference": collective resistance is harder to sustain when victims perceive their fates as unlinked.

3. Asymmetry and Specialness:
   If $V_1$ is perceived as an exceptional target (e.g., due to unique historical or ethnic claims), the posterior risk for $V_2$ decreases.

   • Result: Greater perceived "specialness" of the first victim expands the parameter space where D&C is sustainable.

Extensions and Richer Environments
The paper extends the baseline model to four additional dimensions:

• Behavioral Responses:
  • Wishful Thinking: If $V_2$ derives utility from believing she is safe, she may sustain a D&C equilibrium even when the objective Bayesian posterior suggests otherwise.
  • Magical Thinking: If $V_2$ believes her neutrality will protect her (a causal misperception), the range of costs supporting D&C expands.
• Rhetoric and Propaganda:
  • Particularizing Signals: Verifiable evidence framing the attack as a localized grievance lowers $V_2$'s posterior risk, inducing D&C.
  • Systemic Signals: Counter-propaganda linking the attack to a broader campaign raises the posterior risk, inducing collective resistance.
• Treaty Commitments:
  • Neutrality/Non-Aggression Pacts: Paradoxically, a treaty between the aggressor and $V_2$ can increase the likelihood of $V_1$ being attacked. By raising the cost of attacking $V_2$ (or the cost of $V_2$ joining), the treaty reassures $V_2$ that she is not next, leading her to stay out.
  • Collective-Defense Treaties: These impose a cost on $V_2$ for failing to join, effectively removing the option to wait for information. This can eliminate the D&C equilibrium and ensure collective resistance.
• Multiple Victims:
  In a three-victim setting, the existence of a credible downstream alliance (e.g., $V_3$ protecting $V_2$) can weaken immediate resistance by $V_2$. If $V_2$ expects protection later, she may delay joining $V_1$, facilitating the aggressor's sequential conquest.

Significance and Claims
The paper claims to contribute to the literature on conflict, collective action, and signaling by identifying inferential friction as a distinct mechanism for the failure of collective resistance.

• Theoretical Contribution: It shifts the focus from mechanical coordination failures to the informational problem of interpreting an aggressor's intent. It demonstrates that the failure to unite does not require victims to be indifferent or irrational; they may simply be rationally uncertain about whether the first attack reveals a threat to themselves.
• Policy and Historical Implications: The model provides a formal framework for understanding historical cases of appeasement and sequential conquest (e.g., the Munich Agreement, the annexation of Crimea, "salami tactics" in authoritarian regimes). It suggests that effective collective defense requires not only the capacity to fight but also a shared interpretation of the first attack as systemic.
• Modest Scope: The authors do not claim to explain all instances of collective action failure, nor do they argue that behavioral biases are the primary driver. Instead, they show how standard Bayesian updating, combined with specific structural parameters (costs, correlation, and information structures), is sufficient to generate divide-and-conquer outcomes. The paper emphasizes that institutions (treaties) and rhetoric can either widen or close the "epistemic wedge" that allows aggressors to succeed.