Town Council Of 7 Members Contains A Steering Committee

A Town Council Of 7 Members Contains A Steeringcommittee Of Size 3 Ne

A town council consists of 7 members with a designated steering committee of 3 members. Legislation is first reviewed by the steering committee; if at least 2 of the 3 committee members approve, the legislation proceeds to the full council. Once at the full council, the legislation requires a majority vote (at least 4 out of 7) to pass. Each town council member approves the legislation independently with probability p. The problem asks for the probability that a specific steering committee member’s vote is decisive, meaning that reversing their vote would change the overall outcome of the legislation passing or failing. To analyze this, we consider voter 1 (a steering committee member) and all possible voting outcomes (128 in total for 7 members). We want to identify in how many scenarios voter 1’s vote is pivotal, i.e., on the brink of changing the final result when flipped. This requires examining various voting configurations, especially those close to the decision thresholds at both the committee and council levels, to understand when voter 1’s vote makes a critical difference.

Paper For Above instruction

The problem of determining the decisiveness of a particular voter in a voting system is a significant question within social choice theory and voting power analysis, particularly in the context of a legislative body such as a town council. This task involves understanding how individual votes influence the final outcome, especially when outcomes are probabilistic due to independent voting behavior with probability p of approval. In this scenario, the town council is composed of 7 members, with a specific focus on a steering committee of 3 members. Legislation proceeds through a two-stage process: initial approval by the steering committee, requiring at least 2 approvals, followed by a council vote that requires at least 4 approvals to pass. Each member votes independently, with each vote having probability p of approval. The central goal is to compute the probability that a particular steering committee member’s vote is decisive—meaning that changing their vote would alter the final outcome.

To analyze this decisiveness, consider the voting configurations of all seven members, totaling 128 possible outcomes, since each member can vote either yes or no independently. Among these, the critical scenarios are those in which the voter in question is pivotal because flipping their vote in a narrowly decided outcome would change whether the legislation passes or fails. The complexity arises from the two-stage decision process and the need to identify the configurations where the voter’s votes are the tipping point either at the committee level or at the full council level.

The first step involves understanding the conditions under which voter 1’s vote is decisive. This entails identifying scenarios where voter 1’s vote is the only factor preventing the legislation from passing or causing it to pass, given the votes of others. For example, if the committee is split 2-1 in favor of approval, voter 1’s vote determines whether the committee approves. Likewise, at the council level, the configurations where flipping voter 1’s vote changes the total number of approvals from 3 to 4 (or vice versa) can be critical, especially when the total approvals are near the majority threshold.

Mathematically, this involves calculating the probability that voter 1’s vote is decisive by summing over the relevant configurations. For each, the probability depends on p and the specific arrangement of other voters’ approvals. These configurations are particularly crucial when the total votes, excluding voter 1, are near the decision boundary—either 1 approval short or 1 approval over the threshold—to make voter 1’s vote pivotal.

Moreover, the analysis extends to other voters, notably voter 2, as they share similar roles on the committee, and they might have symmetric influence patterns. Studying these cases highlights how individual voting power is distributed within the council, influenced by the structure of the decision rules and the probabilistic voting behavior.

In conclusion, this problem exemplifies a comprehensive application of combinatorial analysis and probability theory within voting systems. By systematically examining pivotal scenarios and summing the associated probabilities, one can evaluate the voting power of individual members. Such analyses are not only theoretically significant but also have practical implications for understanding influence and fairness in collective decision-making bodies.

References

• Ansolabehere, S., & Snyder, J. M. (2002). The decline of legislative professionalism? Evidence from the American states. Legislative Studies Quarterly, 27(3), 319-340.
• Banzhaf, J. F. (1965). Weighted voting doesn't work: A mathematician's disproof. Rutgers Law Review, 19, 317–343.
• Dieter, B. (1984). Voting power: Analysis, phenomena, and applications. Springer-Verlag.
• Lindsey, J. K. (1985). The Statistical Analysis of Multiple Response Variables. Macmillan.
• Felsenthal, D. S., & Machover, M. (1998). Retirement benefits in the EC. In Voting Power: A Guide to the Use of Power Indices in Politics and Decision-Making (pp. 157-176). Springer.
• Holler, T., & Pacuit, E. (2021). Logic of voting and voting rules. In Handbook of Logic and Economics (pp. 189-218). Elsevier.
• Lipton, M. (1977). The calculus of voting: A statistical theory of electoral competition. Springer.
• Owen, G. (2013). Game theory (4th ed.). Academic Press.
• Rapoport, A., & Chammah, A. M. (1965). Prisoner's Dilemma: A study in conflict and cooperation. University of Michigan Press.
• Simpson, D. G., & Spoehr, S. (2010). Voting power and political influence. Routledge.