Final answer:
In the weighted voting system [54: 59, 52:1], Statement 1 is false because the 52-weight voter is not a dummy; they hold significant voting power. Statement 2 is true, as the 1-weight voter is indeed a dummy, having no power to influence the outcome independently or within a coalition.
Step-by-step explanation:
When analyzing the weighted voting system [54: 59, 52:1], we look at the power each voter has in influencing the outcome. The quota for this system is 54, meaning that at least 54 votes are needed for a decision or a motion to pass. In this context, a 'dummy' refers to a voter who has no real power in affecting the outcome of the vote. Looking at the weights of the voters, we have one voter with 59 votes and another with 52 votes. Together they make up 111 votes, which is more than double the quota. Therefore, either of these voters has considerable power and can form a winning coalition with or without the help of the 1-weight voter. This means that the 52-weight voter is not a dummy, as they have significant power in the voting system.Statement 1, claiming that the 52-weight voter is a dummy, is false. With their 52 votes, this voter can control the outcome because their vote is crucial for achieving or exceeding the quota of 54. The coalition of the 59-weight voter and the 52-weight voter leaves only 1 weight outside, which is not enough to affect the decision independently, illustrating the power of the 52-weight voter.
On the other hand, Statement 2 suggests that the 1-weight voter is a dummy. This statement is true. The 1-weight voter, by themselves, cannot contribute to reaching the quota, as their weight is insufficient, nor can they change the outcome of a vote when added to any coalition that already meets the quota. Consequently, the 1-weight voter lacks both the capacity to influence the outcome independently and to be a critical part of any coalition. They are essentially powerless, which is the defining characteristic of a dummy in a weighted voting system.