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

The question describes a simple cooperative game with the monotonicity property: if a coalition A is winning, then any coalition that strictly contains A is also winning. Here, i is a player not in A, so A∪{i} is a strict superset of A. Given monotonicity, A∪{i} must be at least as good as A and cannot be losing if A is winning.

Option analysis given the provided context:
- If the only fact we have is that A is winning and i is not in A, then by monotonicity A∪{i} cannot be losing; it should be winning as well. Therefore, the statement that A∪{i} is winning is consistent with the standard definition of a simple game.

Why this reasoning matters: in simple (monotonic) voting games, adding players to a winning coalition cannot turn it into a losing one. The coalition only gains power or remains at least as effective, so its status as winning is preserved or enhanced.

Note on missing options: the prompt indicates exactly one option is true, and the provided answer field says 'winning'. Since no alternative options are given to compare, we focus on why 'winning' is the correct characterization under monotonicity. If other options existed (e.g., 'losing', 'inactive', 'critical'), they would be evaluated against the monotonicity axiom and the definition of a winning coalition, but such options are not available here.

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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