Embezzlement-based nonlocal game that cannot be played optimally with finite amount of entanglement

We introduce a three-player nonlocal game, such that the optimal winning probability of 1 can only be achieved in the limit of strategies using arbitrarily high dimensional entangled states.  This game is explicit, and has very few classical questions and answers.  Our game is based on the coherent state exchange game introduced in arXiv:0804.4118, which in turns is based on embezzlement of entanglement due to van Dam and Hayden.  We discuss the main ideas behind each of these ingredients, and how they can be put together to obtain a quantitative tradeoff in the winning probability vs the dimension of the entangled state shared by the players.