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[18: \( 1,3,4,4,4] \) What are the weights of the voters? A has 1 vote(s). B has 3 vote(s). C has 4 vote(s). D has 4 vote(s). E has 4 vote(s). c. Are there any dictators?

Ask by Young Murray. in the United States
Feb 13,2025

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No voters are dictators because none have more than half of the total votes.

Solución

To determine if there are any dictators among the voters, we need to analyze the voting weights and the definition of a dictator in voting theory. ### Known Conditions: - Voter A has 1 vote. - Voter B has 3 votes. - Voter C has 4 votes. - Voter D has 4 votes. - Voter E has 4 votes. ### Step-by-Step Analysis: 1. **Total Votes Calculation**: We first calculate the total number of votes: \[ \text{Total Votes} = 1 + 3 + 4 + 4 + 4 = 16 \] 2. **Definition of a Dictator**: A dictator is a voter whose vote alone can determine the outcome of a decision, meaning that if the dictator votes for a particular option, that option will win regardless of how the other voters vote. 3. **Identifying a Dictator**: - For a voter to be a dictator, their votes must be greater than half of the total votes. - Half of the total votes is: \[ \text{Half of Total Votes} = \frac{16}{2} = 8 \] 4. **Comparison of Each Voter's Votes**: - Voter A: 1 vote (not a dictator) - Voter B: 3 votes (not a dictator) - Voter C: 4 votes (not a dictator) - Voter D: 4 votes (not a dictator) - Voter E: 4 votes (not a dictator) None of the voters have enough votes to exceed 8, which means none of them can unilaterally determine the outcome of a vote. ### Conclusion: There are no dictators among the voters, as no single voter has enough votes to control the outcome of the voting process.

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In voting theory, a dictator is a voter whose preference always determines the outcome regardless of the preferences of others. In the scenario provided, voter C, D, and E all have the same weight of votes (4 each), and they could form a coalition to influence decisions. However, no single voter has the power to dictate the outcome unilaterally. Thus, there are no dictators in this set of voters, as each depends on the others’ input to reach a decision. For those interested in the fascinating area of voting theory, exploring concepts like the Banzhaf Power Index and Shapley-Shubik Power Index can illuminate the dynamics of how votes are weighted and the influence individual voters wield in a group decision-making context. They highlight the complexity behind what seems like simple voting!

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