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

A weighted voting game assigns player weights and a threshold, and declares coalitions winning when their total weight passes that threshold. The strict $>t$ convention is equivalent to quota $t+1$ for integer weights; another common convention writes a weak inequality directly. Nontrivial examples are simple cooperative games. Having nonnegative weights does not guarantee a convex cooperative game: two-of-three majority supplies a counterexample.

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.