 Weighted voting game

A weighted voting game assigns nonnegative weights $w_i$ to players and a threshold $t\ge 0$, and declares a coalition winning whenNonnegative weights make the rule monotone. The empty coalition loses, and the grand coalition wins when $t<{\sum }_i{w}_i$; under that nontrivial condition this is a simple cooperative game.

If all weights are integers, the strict threshold is equivalent to the weak quota ${\sum }_{i\in S}{w}_i\ge \lfloor t\rfloor +1$. It becomes quota $t+1$ when $t$ is itself an integer. This distinction matters for noninteger thresholds.

For example, three unit weights and $t=1$ give two-of-three majority. Nonnegative weights do not imply a convex cooperative game: $v(\{1,2\})+v(\{2,3\})=2$, while $v(\{1,2,3\})+v(\{2\})=1$, violating the supermodular inequality.