danryan.us

"The expected payoff for GRIM versus GRIM is nR. The expected payoff for ALLD versus GRIM is T+(m-1)P. If nR>T+(m-1)P then ALLD cannot spread in a GRIM population when rare."

"…it is more natural to consider a repeated game with a probability w of having another round. In this case, the expected number of rounds is 1/(1-w), and GRIM is stable against invasion by ALLD provided w>(T-R)/(T-P)."

Why is EV 1/(1-w)? Draw the event tree and write out the equation for expected value. The first few terms are

Which turns out to be the Taylor series expansion for the Taylor Series geometric series 1 + w + w² + w³ + …