  We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game
has two players, Proposer and Decider. Starting from an empty graph on $n$
vertices, on each turn Proposer proposes a potential edge and Decider
simultaneously decides (without knowing Proposer's choice) whether to add it to
the graph. Proposer cannot propose an edge which would create a triangle in the
graph. The game ends when Proposer has no legal moves remaining, and Proposer
wins if the final graph has independence number at least $s$. We prove a
threshold phenomenon exists for this game by exhibiting randomized strategies
for both players that are optimal up to constants. Namely, there exist
constants $0<A<B$ such that (under optimal play) Proposer wins with high
probability if $s<A\sqrt{n}\log{n}$, while Decider wins with high probability
if $s>B\sqrt{n}\log{n}$. This is a factor of $\Theta(\sqrt{\log{n}})$ larger
than the lower bound coming from the off-diagonal Ramsey number $r(3,s)$.
