Select one option. Exactly one option is true.

In a simple game, if coalition A is winning and i is a player not in A, then coalition
𝐴
∪
{
𝑖
}
is:

Question

Select one option. Exactly one option is true.

In a simple game, if coalition A is winning and i is a player not in A, then coalition 
𝐴
∪
{
𝑖
}
is:

BlackTom AI · Accepted Answer

We are told: In a simple game, coalition A is winning and i is a player not in A. We are to analyze what A ∪ {i} is.

Option 1: minimal winning
- A is winning, but minimal winning means that removing any member from A would make it losing. The fact that A is winning does not imply that A ∪ {i} has any relation to being minimal. In fact, adding a player to a winning coalition does not create a minimality condition; minimality would typically apply to the original coalition, not the enlarged one. Therefore A ∪ {i} being minimal winning is not supported by the given information and is unlikely in general.

Option 2: blocking
- A blocking coalition is a set of players whose presence guarantees the blocking of a winning coalition or prevents passage, depending on the formal definition used. The fact that A is winning already ensures a majority, and adding i cannot convert A ∪ {i} into a blocking coalition under the standard definitions in simple voting games. Thus this option contradicts monotonicity of winning coalitions.

Option 3: losing
- In a simple voting game with monotonicity, if A is winning, then any superset of A (i.e., A plus additional players) is also winning. Since i is outside A, the coalition A ∪ {i} has at least as many votes as A, so it cannot be losing. Therefore this option is inconsistent with the basic property of simple games.

Option 4: winning
- Given that A is winning and i is not in A, the coalition A ∪ {i} includes all members of A plus an additional player. In monotonic (simple) voting games, adding players cannot turn a winning coalition into a losing one; it typically preserves or reinforces status as winning. Therefore A ∪ {i} remains winning under the standard assumptions.

Summary of reasoning across options:
- For minimal winning: the claim does not follow from the information and is not guaranteed by monotonicity.
- For blocking: inconsistent with the idea that supersets of a winning coalition remain at least as powerful as the original.
- For losing: directly contradicts monotonicity of winning coalitions in simple games.
- For winning: aligns with monotonicity—supersets of a winning coalition are winning.

Overall, the only option that aligns with the standard properties of a simple game given A is winning and i is outside A is that A ∪ {i} is winning.

VPM Spring 2025 VPM Practice Final

Select one option. Exactly one option is true. In a simple game, if coalition A is winning and i is a player not in A, then coalition 𝐴 ∪ { 𝑖 } is:

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