I believe they are saying that there are multiple rounds, each with different games - some with honest optimal strategies and some with dishonest optimal strategies. A dishonest person can always choose the optimal strategy for each game, but the honest person can only choose the best honest strategy. So in aggregate the dishonest person comes out ahead.
Ok but people are both honest and dishonest so how do you decide what type a person is?
Life has a lot of iterated games. A reputation for dishonesty can sometimes end up following you around.