Measuring Concentration of Power in Approval Voting Games

Abstract: The ratio of voting power between a permanent member and a non-permanent member of the United Nations Security Council varies substantially across indices: approximately 100 to 1 according to the Shapley-Shubik index, 10 to 1 according to the Banzhaf index, and 2.5 to 1 according to the Deegan-Packel index. Such comparisons depend on the choice of power index and are meaningful only in settings where players are divided into two types. To address these limitations, this paper proposes and characterizes a function that measures the level of power concentration in monotonic approval voting games. The proposed measure assigns a single value to each voting game, reflecting the extent to which voting power is unevenly distributed among players. The proposed measure is proportional to the sum of squared Deegan-Packel power indices and can also be interpreted as the degree of overlap among minimal winning coalitions. An application to the United Nations Security Council is also provided.

• The paper introduces an axiomatic framework for power concentration in approval voting games, establishing four foundational axioms.
• It characterizes a unique measure as a quadratic aggregation of Deegan-Packel power indices, drawing parallels with the Herfindahl-Hirschman index.
• Applying the measure to the UNSC, the study quantitatively compares how changes in voting thresholds impact power diffusion.

Power Concentration Measurement in Approval Voting Games

Introduction and Motivation

The measurement of power concentration in approval voting games is a critical topic in cooperative game theory and political science, particularly relevant to collective bodies like the United Nations Security Council (UNSC). Traditional power indices—such as the Shapley-Shubik, Banzhaf, and Deegan-Packel indices—produce widely diverging evaluations of voting power inequality. These discrepancies are particularly evident in real-world cases such as the UNSC, where the ratio of permanent to non-permanent member voting power ranges from 100:1 (Shapley-Shubik) to 2.5:1 (Deegan-Packel), illustrating antagonistic perspectives depending on the index used. This underscores the need for a theoretically justified, unified measure of power concentration that is meaningful across a broad class of approval voting games, including those with more than two player types and nontrivial coalition structures.

Theoretical Framework and Axiomatic Basis

The paper introduces an axiomatic framework for power concentration in monotonic approval voting games. It grounds the analysis in minimal winning coalitions, which uniquely specify any such game. The measure of power concentration is defined as a real-valued function on the set of all finite monotonic voting games, reflecting how much voting power is aggregated in a few hands versus being diffused.

Four foundational axioms are proposed:

1. Weak Symmetric Game Property (WSYM): All-player unanimity and all-inclusive winning rule games should have identical concentration.
2. Duplication (DUP): Replicating a player (in the extreme case, splitting a singleton into two) proportionally dilutes concentration.
3. Minimal Winning Coalition Property (MWC): Games with identical sets of minimal winning coalitions must have identical concentrations, ensuring null players do not affect concentration calculations.
4. Average (AVE): Concentration for a given game must aggregate, using a specific overlap-weighted average, the concentration of its constituent unanimity games (associated to each minimal winning coalition).

Replication, as formalized, requires that splitting each voter into $t$ identical agents (preserving coalition structures) scales concentration by $1/t$, distinguishing this approach from those sensitive to the count or identity of null players.

Characterization: The Deegan-Packel–Herfindahl Concentration Index

Under these axioms, the paper proves a unique characterization: concentration must be proportional to the sum of squared Deegan-Packel (DP) power indices across all players. Explicitly,

$\mu(N,W) = a \cdot \sum_{i \in N} DP_i(N,W)^2$

where $DP_i(N,W)$ is the canonical DP index for player $i$ in game $(N,W)$.

This construction is parallel to the Herfindahl-Hirschman (HH) index used in industrial economics, repurposing it as a measure of concentration over the DP allocation of winning power. The DP index's focus on minimal winning coalitions naturally aligns with MWC and AVE, and the quadratic HH-aggregation is uniquely compelled by the axioms—higher powers violate the averaging property.

Notably, the measure can be re-expressed purely in terms of the overlaps of minimal winning coalitions, providing a coalition-structural perspective: $$\mu(N,W) = \frac{1}{|M(W)|^2} \sum_{(S,T) \in M(W)^2} \frac{|S \cap T|}{|S|\cdot|T|}$$ This captures the intuition that high overlap equates to high concentration, as the same players recurrently appear in decisive configurations.

Alternative and Related Measures

The paper analyzes several natural alternatives:

• Herfindahl over other power indices (e.g., Shapley-Shubik): Applying the HH construction to indices like Shapley-Shubik leads to measures satisfying several axioms but not the AVE property, and with more intricate expressions involving inclusion-exclusion over subsets of minimal winning coalitions.
• Higher powers of DP (beyond square): For exponents $k>2$, these measures always violate the AVE axiom, emphasizing the singularity of quadratic aggregation for reasonable concentration assessment.
• Direct application of set overlap indices (Dice, Jaccard): These alternatives are structurally simpler but fail key axioms (DUP and AVE), and nullify concentration for any game with disjoint minimal winning coalitions, a highly undesirable property for practical analysis.

Application to the United Nations Security Council

The significance of the measure is exemplified through an application to the UNSC. The Deegan-Packel–Herfindahl measure operationalizes as

$$\mu(n_1, k_1; n_2, k_2) = \frac{1}{(k_1 + k_2)^2} \left(\frac{k_1^2}{n_1} + \frac{k_2^2}{n_2}\right)$$

where $n_1$, $1/t$0 are the number and requirement for permanent members, and $1/t$1, $1/t$2 for non-permanent.

For the current UNSC ($1/t$3, $1/t$4, $1/t$5, $1/t$6), the calculated concentration is approximately $1/t$7. Systematic analysis across all threshold values reveals non-monotonicity: for instance, reducing $1/t$8 from $1/t$9 to $\mu(N,W) = a \cdot \sum_{i \in N} DP_i(N,W)^2$0 decreases concentration less than increasing $\mu(N,W) = a \cdot \sum_{i \in N} DP_i(N,W)^2$1 from $\mu(N,W) = a \cdot \sum_{i \in N} DP_i(N,W)^2$2 to $\mu(N,W) = a \cdot \sum_{i \in N} DP_i(N,W)^2$3. The measure's minimum, for fixed $\mu(N,W) = a \cdot \sum_{i \in N} DP_i(N,W)^2$4, $\mu(N,W) = a \cdot \sum_{i \in N} DP_i(N,W)^2$5, is achieved when relative approval requirements are balanced across groups ($\mu(N,W) = a \cdot \sum_{i \in N} DP_i(N,W)^2$6), corresponding to maximal dispersion of power.

Considering UNSC reform proposals, expanding both permanent and non-permanent seats while keeping voting thresholds constant leads to a halving of the concentration measure, reflecting significant power diffusion. This quantitative evaluation provides an objective basis for comparing voting system design options with respect to concentration.

Conclusion

The paper establishes that only the quadratic aggregation of Deegan-Packel power satisfies a natural set of axioms for measuring power concentration in approval voting games. This index robustly distinguishes games with structurally concentrated coalition requirements from those with widespread, diffused power. Practically, it enables the principled comparison of institutional reforms, with clear implications for political design and the assessment of fairness. Theoretically, it motivates further research into complementary notions such as "decisiveness" (closely related to the number and cardinality of winning coalitions), for which the paper suggests the development of a formal measure as a prospect for future work.