Consociational Structuring

• Consociational structuring is a formal mechanism for power-sharing among distinct groups in multi-group societies, defined through non-cooperative coalition-formation games.
• It employs explicit coalition-formation rules and mixed-strategy Nash equilibria to capture intra- and inter-coalition externalities within bounded deviation settings.
• The framework underlines institutional design by quantifying power-sharing robustness via stability criteria such as mutual veto and proportionality in coalition configurations.

Consociational structuring refers to formalized mechanisms for power sharing among distinct groups, especially in the context of multi-group societies or polities. The concept is rigorously modeled within a non-cooperative game-theoretic framework that generalizes how coalitions form among self-interested agents in the presence of intra- and inter-coalition externalities. Recent developments formalize consociational structuring as a coalition-formation process embedded in a family of finite non-cooperative games, parameterized by the maximum coalition size, and equipped with explicit coalition-formation rules and stability criteria applicable to real-world power-sharing institutions (Levando, 2017).

1. Non-Cooperative Coalition-Formation Games

The rigorous foundation for consociational structuring is a family of finite games $G(K)$, structured as follows. The player set $N = \{1,2,\dots,n\}$ represents the agents (e.g., groups or political parties). For each fixed integer $K$, $1 \leq K \leq n$, $K$ designates both:

• The maximum permissible size of any coalition (block) that can be formed.
• The size of the largest subset of agents able to coordinate a joint deviation.

Partitions of $N$ into coalition structures, $\mathcal{P}(K)$, are defined as the family of all partitions with blocks of size at most $K$: $$\mathcal{P}(K)=\Big\{P=\{g_1,\ldots,g_m\} : \bigsqcup_{j=1}^m g_j=N,\, |g_j|\leq K\,\forall j\Big\}.$$ As $K$ increases, the allowable coalition structures become more inclusive: $N = \{1,2,\dots,n\}$0. For each $N = \{1,2,\dots,n\}$1, player $N = \{1,2,\dots,n\}$2 chooses an action $N = \{1,2,\dots,n\}$3 for that coalition structure. The overall strategy set for $N = \{1,2,\dots,n\}$4 is

$N = \{1,2,\dots,n\}$5

This construction encodes both a proposed partition and intra-partition action.

Each agent's payoff, $N = \{1,2,\dots,n\}$6, is a function of the realized partition $N = \{1,2,\dots,n\}$7 and the joint action profile $N = \{1,2,\dots,n\}$8, capturing both intra- and inter-coalition externalities.

2. Coalition-Formation Rules and Equilibrium

Divergent agent proposals require a deterministic coalition-formation rule (social mechanism) $N = \{1,2,\dots,n\}$9. For every agent profile $K$0, $K$1 produces a unique partition $K$2, ensuring well-posedness.

The game thus defined, $K$3, generalizes traditional cooperative games by embedding strategic actions and explicit formation rules. Unlike the characteristic function form, externalities across and within coalitions are intrinsic to the payoff structure.

A mixed-strategy Nash equilibrium for $K$4 is a profile $K$5 such that for every $K$6 and every admissible deviation (depending on $K$7), no player can improve their expected payoff by unilaterally or jointly deviating. The existence of a mixed-strategy Nash equilibrium in each $K$8 follows from the finiteness of the action sets and standard fixed-point arguments.

3. Stability Criteria in Coalition Structures

The classical Nash equilibrium only addresses unilateral deviations. Consociational structuring, however, requires stability against joint deviations up to some maximal size—directly relevant for institutional resilience and focal-point selection in social and political systems.

A baseline equilibrium $K$9 in $1 \leq K \leq n$0 is said to be stable up to $1 \leq K \leq n$1 if:

• $1 \leq K \leq n$2 persists as an equilibrium in every $1 \leq K \leq n$3 for $1 \leq K \leq n$4.
• The support (i.e., the set of pure strategies played with positive probability) remains the same across these values.

Formally, for all $1 \leq K \leq n$5, $1 \leq K \leq n$6, and $1 \leq K \leq n$7,

$1 \leq K \leq n$8

with $1 \leq K \leq n$9.

This stability concept differs from the core, strong Nash equilibrium, coalition-proof Nash equilibrium, and the Shapley value, in that it explicitly models externalities and restricts deviations to those of bounded size. In particular, there are no recursive or higher-order stability refinements—joint deviations up to $K$0 suffice (Levando, 2017).

4. Nested Games, Equilibrium, and Cooperative Substructures

The nested family $K$1 induces a hierarchy of equilibrium structures, where the support of equilibrium distributions can also be chosen nested, reflecting how feasible coalition structures and stability can evolve as $K$2 grows.

A proposition formalizes: for each $K$3, there exists a mixed-strategy equilibrium $K$4 in $K$5, and choices can be made so supports are nested in $K$6.

"Complete cooperation" in a coalition $K$7, $K$8, is defined by: (a) all $K$9 only assign positive probability to strategies that propose partitions containing $N$0 (ex ante), and (b) with probability one, the realized partition contains $N$1 (ex post). This models endogenous, self-enforcing coalition formation within the non-cooperative paradigm (Levando, 2017).

5. Consociational Structuring for Power-Sharing Institutions

The framework directly maps to consociational power-sharing, particularly in settings with multiple salient groups (e.g., ethnolinguistic, religious, or partisan cleavages):

• Grand coalition & mutual veto: Setting $N$2 enables the formation of all-inclusive (grand) coalitions. The coalition-formation rule $N$3 can encode mutual vetoes, requiring unanimity among designated groups for critical decisions.
• Proportionality rules: Within coalition blocks, strategies can represent bids for offices (e.g., ministerial portfolios); $N$4 plus $N$5 can implement proportionality directly through the mechanism of partition formation and payoff allocation.
• Stability analysis: The stability criterion provides a quantitative measure: if an equilibrium in $N$6 is stable up to $N$7, then at least $N$8 groups are required to coordinate a successful challenge to the arrangement. This quantifies the robustness of institutional power-sharing.
• Two-community model: With $N$9 (e.g., two communities $\mathcal{P}(K)$0 with three parties each), $\mathcal{P}(K)$1 restricts coalitions to within-group; raising $\mathcal{P}(K)$2 to $\mathcal{P}(K)$3 allows cross