There may be two disjoint (i.e., having no players in common) losing coalitions in a simple game.

Question

BlackTom AI · Accepted Answer

Consider the concept being tested: whether two losing coalitions in a simple game can be disjoint.
Option 1 (True): This statement is plausible because a simple game is monotone, meaning once a coalition is winning, any superset remains winning, and losing coalitions may be quite small. In many standard examples, there exist multiple losing coalitions that do not share any players with each other. For instance, in a 3-player majority game (quota of 2), coalitions of size 0 or 1 are losing. The singletons {1} and {2} are both losing coalitions and they are disjoint. This demonstrates that two disjoint losing coalitions can indeed exist.
Option 2 (False): To assess this, one would need a property that forbids the existence of two disjoint losing coalitions. However, such a restriction is not a general feature of simple games. The monotonicity condition does not imply that losing coalitions cannot be disjoint; it only constrains how winning coalitions behave under taking supersets, not how losing coalitions relate to each other. Therefore, this option runs counter to common constructions of simple games and is not universally true.
In summary, because a simple game can have multiple non-overlapping losing coalitions (as shown by the 3-player majority example), the claim that there may be two disjoint losing coalitions is correct. The incorrect reasoning would require an impossibility that is not implied by the definition of a simple game, which is not the case here.

VPM Spring 2025 Final B QB

There may be two disjoint (i.e., having no players in common) losing coalitions in a simple game.

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