Repeated games may violate the expected utility theorem

In a game, we are not always able to get any pure strategy equilibrium, and instead often we get a mixed strategy equilibrium. Rock-paper-scissors is a classic game with a mixed strategy equilibrium. However, when we are actually playing the game, we sometimes do not randomly play our strategy as what the mixed strategy equilibrium says, we often try to work out the other player's next strategy based on his previous play and play our counter strategy. It is a mixed strategy equilibrium, as the other player will do exactly the same. However, we could imagine that the probabilities of the three strategies are no longer the same. Moreover, the game repeats over and over again, the probability of each strategy will vary from round to round. Such action violates the expected utility theorem, as the utility of each strategy should be the same based on the expected utility theorem, but sometimes, we do find that some strategy dominates another strategy when we actually make our decision. Therefore, in real life, especially when we are playing a repeated zero sum game, backward induction is not effective to help us to find the best strategy and the equilibrium, as people are taking action based on historical records. In order to find the best strategy or the equilibrium, we have to notice that players do not make decisions independently to maximize the so-called expected utility, they are more likely to take actions based on their opponents' historical strategies.