= Convex simple games are unanimity games
Every nontrivial <simple cooperative game> that is a <convex cooperative game> is a <unanimity game>. If two <coalitions> win, the <supermodular> inequality forces their intersection to win as well. Intersecting the finite nonempty family of <winning coalitions> gives a winning <set> $K\ne\varnothing$. Every winning <coalition> contains $K$; conversely monotonicity makes every superset of $K$ win. Thus the game is $u_K$. Conversely, every <unanimity game> is a <supermodular set function>, as a direct check of the winning and losing cases shows.
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