Repeated games and nonlinear utility
Let's assume a trivial two-player game where each player has two options A and B; and the payout is +1/-1 if players pick the same and -1/+1 if players pick differently.
Let's assume that the game is repeated 100 times with the strategies chosen and committed to beforehand. This means that if your opponent picks a fixed strategy, and you choose a fixed strategy, then the expected result is 50% chance of +100 and 50% chance of -100, depending on what strategy they've chosen. However, if you choose a 50/50 mixed strategy (random), then the expected result is close to 0 - between -10 and +10 with a ~95% probability or so, and you choose a 50/50 mixed strategy that's strictly alternating, then you get an expected result of exactly 0 with absolute certainty.
Thing is, many theories of utility (and behavioral economics experiments) expect that it can be nonlinear, and that "50% chance of +100 and 50% chance of -100" and "certain 0" are not equivalent, there can be a strong preference for one or the other depending on circumstances. So, if the utility of one player "prefers" reducing variance and avoiding the risk of a large loss, then a mixed strategy is strongly preferrable.