Suppose two players are playing the repeated Prisoner's Dilemma with an unknown number of stages; after each stage, a lottery is conducted, such that with probability 1 - β the game ends with no further stages conducted, and with probability β the game continues to another stage, where β ∈ [0, 1) is a given real number.

Each player's goal is to maximize the sum total of payoffs received over all the stages of the game.

Prove that if β is sufficiently close to 1, the strategy vector in which at the first stage every player plays C, and in each subsequent stage each player plays C if the other player played C in the previous stage, and he plays D otherwise, is an equilibrium.

This strategy is called the Tit-for-Tat strategy.