[Paper Review] Power Distribution in Randomized Weighted Voting: the Effects of the Quota

Joel Oren, Yuval Filmus+2 more|arXiv (Cornell University)|Aug 3, 2014

Game Theory and Voting Systems2 references3 citations

TL;DR

This paper analyzes how changes in the quota—threshold for passing legislation—affect Shapley values in randomized weighted voting games with stochastically generated weights. Using the Balls and Bins model with uniform and exponentially decaying probabilities, it shows that quota choices near multiples of m/n can cause Shapley values to differ by up to 2x, even when weights are nearly identical, and fully characterizes Shapley values for super-increasing weight sequences.

ABSTRACT

We study the Shapley value in weighted voting games. The Shapley value has been used as an index for measuring the power of individual agents in decision-making bodies and political organizations, where decisions are made by a majority vote process. We characterize the impact of changing the quota (i.e., the minimum number of seats in the parliament that are required to form a coalition) on the Shapley values of the agents. Contrary to previous studies, which assumed that the agent weights (corresponding to the size of a caucus or a political party) are fixed, we analyze new domains in which the weights are stochastically generated, modelling, for example, elections processes. We examine a natural weight generation process: the Balls and Bins model, with uniform as well as exponentially decaying probabilities. We also analyze weights that admit a super-increasing sequence, answering several open questions pertaining to the Shapley values in such games.

Motivation & Objective

• To study the impact of quota changes on Shapley values in weighted voting games where agent weights are randomly generated.
• To model real-world electoral processes by assuming weights are stochastically assigned via the Balls and Bins process with uniform or exponential decay probabilities.
• To characterize Shapley values as a function of quota in super-increasing weight sequences, resolving open questions in prior work.
• To identify conditions under which consecutive agents have equal Shapley values in such games.
• To demonstrate that small quota changes near critical thresholds can cause large disparities in power, even with nearly equal weights.

Proposed method

• Models weight generation using the Balls and Bins process, where each ball (voter) is independently assigned to a bin (party) according to a fixed probability distribution.
• Analyzes the Shapley value as a function of the quota in both uniform and exponentially decaying probability settings.
• Identifies a periodic fluctuation pattern in Shapley values with cycles of length m/n, where m is total weight and n is number of agents.
• Uses the concept of 'critical thresholds' near small multiples of m/n to analyze power imbalances due to quota proximity.
• Applies asymptotic analysis and limit theorems to derive closed-form expressions for Shapley values in the infinite case.
• Proves continuity of the Shapley value function from both the left and right at all quota values, ensuring stability analysis is valid.

Experimental results

Research questions

• RQ1 How does changing the quota affect Shapley values when agent weights are stochastically generated rather than fixed?
• RQ2 What is the impact of quota proximity to multiples of m/n on power distribution in the Balls and Bins model with uniform weights?
• RQ3 Can the Shapley value be fully characterized as a function of quota in super-increasing weight sequences?
• RQ4 Under what conditions do consecutive agents have equal Shapley values in super-increasing games?
• RQ5 How do noise effects and weight fluctuations influence power disparities despite similar weights?

Key findings

• When the quota is bounded away from small multiples of m/n, Shapley values of all agents are likely to be very close to each other.
• If the quota is near a small multiple of m/n, the highest Shapley value can be roughly double the lowest, even with nearly identical weights.
• For exponentially decaying probabilities, the analysis reduces to characterizing Shapley values in super-increasing weight sequences.
• The paper provides a closed-form formula for Shapley values in super-increasing sequences as a function of the quota.
• The Shapley value function is continuous from both the left and right at all quota values, ensuring stable behavior.
• The limit of the Shapley value as the quota approaches 0 or the total weight tends to 0, confirming power vanishes at extremes.