"A lie’s true power cannot be accurately measured by the number of people who believe its deception when it is told, it must be measured by the number of people who will go out after hearing it trying to convince others of its truth.” -- Dennis Sharpe
The Soviets probably had nothing to do with the hoax either.
If the Bay of Pigs invasion was a hoax, as I suspect it was, the Soviets would have known, Castro would have known, and the Pentagon would have known. This leads me to suspect that, given that everyone knew everything, that the entire Cuban Missile Crisis was a hoax. By “hoax” I mean no showdown, no Soviet vessels turning around, no blockade, no missiles. It was all a TV show.