Here's the strategy with a one-in-a-million chance of failure. Choose a number between one and one million. On your chosen round, send the strategy instructions

Here's the strategy with a one-in-a-million chance of failure. Choose a number between one and one million. On your chosen round, send the strategy instructions, "Pick red next round and end the game." If they send strategy instructions before your chosen round, comply. If they send a message of deference before your chosen round, follow up immediately with your strategy instructions. With infinite rounds, arbitrary random numbers are a way to break symmetry.