Weighted voting game ($v(S)=1_{\{{\sum }_{i\in S}{w}_i>t\}}$)

A weighted voting game assigns nonnegative weights $w_i$ to players and a threshold $t\ge 0$, 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.

Suppose winning coalitions $W_1,\dots ,W_k$ and losing coalitions $L_1,\dots ,L_k$ contain each player the same total number of times:No real-weight, weak-quota weighted voting game represents those outcomes. Summing weights over the winning family gives at least $kq$, while summing over the losing family gives strictly less than $kq$. Equal player multiplicities make the two sums identical, a contradiction. This certificate works without assuming nonnegative weights.

For example, the winners $\{1,2,3\},\{1,4,5\}$ and losers $\{1,2,4\},\{1,3,5\}$ form a two-against-two obstruction. A monotone game whose minimal winning coalitions are the first two sets realizes this certificate, even though every winner contains player $1$.

A zero-one coalitional game has this property if exchanging one player each way between two disjoint winning coalitions leaves at least one winning. Any weighted voting game satisfies it: the exchange preserves their aggregate weight, which is at least twice the quota, whereas two losing coalitions would have aggregate weight strictly below twice the quota.

The disjointness restriction is substantive. If every winning coalition contains a common player, the condition is vacuous. It is consequently weaker than restrictions allowing exchanges between overlapping winning coalitions, and does not characterize weighted voting games.