= Trade obstruction to weighted voting
Suppose winning <coalitions> $W_1,\ldots,W_k$ and losing <coalitions> $L_1,\ldots,L_k$ contain each player the same total number of times:
$$
\sum_{j=1}^{k}\mathbf1_{\{i\in W_j\}}
=\sum_{j=1}^{k}\mathbf1_{\{i\in L_j\}}
\quad\text{for every player }i.
$$
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 $1$.
Back to article page
