Suppose we have some mixed strategy for this hat puzzle. For each of the $8$ possible hat assignments and each of the $3$ people involved, we can ask about the probability that the person guesses correctly conditional on that hat assignment, and the probability that they guess incorrectly. After specifying this, we'll have assigned a probability to every possible (world state, person 1 accuracy, person 2 accuracy, person 3 accuracy) tuple, where the world state can be in one of 8 arrangements, and each person's accuracy can be graded as "correct", "silent", or "incorrect".

For a single person, their expected probability of being correct (conditional on venturing a guess) is always 0.5 - that is, the total probability of all outcomes in which they guess correctly must equal the total probability of all outcomes in which they guess incorrectly.

In order for an outcome to lead to a success, we need at least one correct guess and no incorrect guesses. So the probability of success in the puzzle is at most the sum of P(correct) across all three people.

Meanwhile, an unsuccessful outcome can have at most three incorrect guesses, because there are only three people. So the probability of failure is at least one-third as large as the sum of P(incorrect) across all three people.

But for each person, P(correct) = P(incorrect)! So we know that the success probability is at most three times the failure probability, which bounds our success rate by 0.75.