1. I agree with what you say. There exists a low-order strategy that has the same payoff against the low-order strategy your opponent uses, and you can use that

1. I agree with what you say. There exists a low-order strategy that has the same payoff against the low-order strategy your opponent uses, and you can use that if you know your opponent's strategy. However, in many games there aren't low-order strategies that would work well against any low-order strategy so you need to know the opponent's strategy to choose a proper low-order strategy. Alternatively you could use a higher order strategy that learns the opponent's strategy and adapts to it. 2. Well, I was thinking that a strategy would be a mapping from game history to a probability distribution over possible moves. Should this be considered in some other way? Sure, you can just play according to the Nash equilibrium and you can't be exploited, but probably a more interesting case is when both players try to outsmart the other player and not just aim for a draw.