If you knew the mixed solution for a game, it sounds like just as bad of a situation as if you knew a pure solution. You still wouldn't be making any decisions, just randomizing across a set of choices. But this is NOT correct; there's still a lot for you to do in the case of a mixed solution. To understand why we'll have to look more closely on what playing "optimally" really means.

Playing Optimally

We said that if you had several possible mixed strategies, the one that lets you play optimally we'll call a mixed solution (this is also a Nash Equilibrium). There's a lot of potential confusion there because the word optimal has two meanings: an ordinary English meaning and a specific mathematical definition. This article is always referring to the mathematics meaning, NOT the everyday usage of the word that means "the best way to play." The math meaning is that playing optimally is playing least exploitably.

Let's see what playing exploitably looks like. If you were playing rock, paper, scissors and you decided to play rock 100% of the time, that is extremely exploitable. Your opponent could pick up on that and shift to playing paper 100% of the time. Your opponent can exploit your strategy so fully that your win rate goes down to 0%. If instead you play rock only 80% of the time (and paper 10%; scissors 10%), that's still a bad idea but it's a bit less exploitable. Your opponent could still play paper 100% of the time, but at least you'll win 10% of the time, rather than 0%.

If you want to be the least exploitable possible, you'll have to play each option 33% of the time. If you do that, there's no strategy your opponent can use to do better than you. That's the optimal mixed strategy to simple RPS.

Optimal Is Not "Best"

Playing optimally sounds like the best you can do, but if your goal is to win a tournament, then playing optimally is very likely not to be the best idea. Imagine you entered a rock, paper, scissors tournament and face a player who is known to play rock 100% of the time and they do exactly that against you. If you play optimally, you'll play each option 33%, so each hand of RPS there's a 33% chance you'll lose. Meanwhile, another player in the tournament could choose to play 100% paper when facing the 100% rock player. Your so-called optimal strategy has a much higher chance of losing and getting you eliminated from the tournament than if you had played 100% paper, too.

By choosing to play optimally, you gave up a massive advantage that was right there for you to take. Your opponent was ridiculously exploitable, but you chose not to capitalize on it. That's poor play if your goal is to win the tournament. This is an extreme example but the concept is still true even if the opponent was playing 40% rock, or even 35%.

What if you do play 100% paper against the 100% rock player, but after several rounds of play they change their strategy? It's possible that they could exploit you because now you strayed from optimal play. Yes, that's correct, but it's still worth it to try. If you're worried about your opponent changing their strategy to exploit you, then you don't have to go all the way from 33% paper to 100% paper. If you went up to, say, 40% then you're more likely to win this match than someone who stuck to 33%, but you're still not all that exploitable. Also, how good is your opponent at a) recognizing that you strayed from optimal and b) correctly implementing a strategy against that? It's entirely possible that you are better at those things, in which case you should definitely exploit their strategy. As they slowly adjust to that, you adjust faster.