91Ó°ÊÓ

The weighted voting system \([27: 10,8,6,4,2]\) represents a partnership among five people \(\left(P_{1}, P_{2}, P_{3}, P_{4},\right.\) and \(\left.P_{5}\right) .\) You are \(P_{5},\) the one with two votes. You want to increase your power in the partnership and are prepared to buy one share (one share equals one vote) from any of the other partners. Partners \(P_{1}, P_{2},\) and \(P_{3}\) are each willing to sell cheap \((\$ 1000\) for one share), but \(P_{4}\) is not being quite as cooperative-she wants \(\$ 5000\) for one of her shares. Given that you still want to buy one share, from whom should you buy it? Use the Banzhaf power index for your calculations. Explain your answer.

Step by step solution

01

Calculate initial Banzhaf Index

Find out how many times each player is a critical player, i.e., his/her vote is decisive to reach or exceed the quota (here 27). Calculate the Banzhaf Index as follows: \(Banzhaf Index (P_i) = \frac{Number Of Times P_i Is A Critical Player}{Total Number Of Times All Players Are Critical}\)

02

Calculate new Banzhaf Index after buying a share from P1

Assuming P5 buys a share from P1, re-calculate the Banzhaf Index for each player. Note that P5 now has 3 votes, and P1 has 9 votes.

03

Calculate new Banzhaf Index after buying a share from P2

Assuming P5 buys a share from P2, re-calculate the Banzhaf Index for each player. Note that P5 now has 3 votes, and P2 has 7 votes.

04

Calculate new Banzhaf Index after buying a share from P3

Assuming P5 buys a share from P3, re-calculate the Banzhaf Index for each player. Note that P5 now has 3 votes, and P3 has 5 votes.

05

Compare the changes in Banzhaf Index for P5

The player P5 should buy the share from the partner that yields the largest increase in their Banzhaf Index.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Weighted Voting System

A weighted voting system is a decision-making process where different voters have varying levels of influence based on their 'weight,' or number of votes. In the given example, the voting system is represented as \([27: 10,8,6,4,2]\). Here, the number before the colon (27) is the quota required to make a decision. The numbers after the colon represent the votes each player holds, with \(P_1\) having 10 votes, \(P_2\) 8 votes, and so on.

• This system allows for a more nuanced and balanced approach to decision-making.
• Each participant's power is proportional to their weight.
• It is crucial that the total vote exceeds the quota to ensure a decision is reached.

Understanding this context helps make informed strategic decisions, especially when participants might want to gain more influence by acquiring additional votes.

Critical Player

A critical player in a voting session is one whose vote is essential to garner the necessary support (quota) for a proposal to pass. In our example, the quota is 27, meaning the votes must total at least 27 for approval.

• If removing a player's vote results in dropping below the quota, they are termed as critical.
• The role of a critical player is dynamic and can change with shifting vote allocations.

In the problem at hand, understanding who the critical players are helps each participant strategize their offers or requests for additional votes.

Voting Power

Voting power measures how much influence a player truly has in the decision-making process. The Banzhaf Power Index is a common way to quantify this.

• To compute the Banzhaf Power Index, count the number of times a player's vote is critical.
• The index is the fraction of times a player is critical out of all critical instances across players.
• Higher indices indicate greater influence.

In our scenario, players should aim to maximize their Banzhaf Power Index to have more sway in outcomes. This enables better negotiation and decision leverage.

Quota

The quota is the minimum number of votes needed for a decision to be approved in a weighted voting system. In this case, the quota is 27.

• It ensures that a significant portion of voters agree before an action is taken.
• High quotas lead to more collaborative decision-making, requiring broader agreement.
• Adjusting the quota can change the dynamics of who is a critical player.

Understanding the quota is essential for strategy, helping players identify how they might affect changes or coalitions to meet or block decisions.