= Weighted voting game
{title2=$v(S)=1_{\{\sum_{i\in S}w_i>t\}}$}
A <weighted voting game> assigns nonnegative weights $w_i$ to players and a threshold $t\ge0$, and declares a <coalition> winning when
$$
v(S)=\mathbf1_{\{\sum_{i\in S}w_i>t\}}.
$$
Nonnegative weights make the rule monotone. The empty <coalition> loses, and the grand <coalition> wins when $t<\sum_iw_i$; 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.
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