Mixed Strategies and Being Unpredictable

In Rock-Paper-Scissors, is there a best move? No single one — whatever you pick, a mind-reading opponent beats it every time. But there's still a correct way to play: pick each option with probability one-third, every round, independent of anything. Not because a specific mix "feels" balanced, but because it's the unique way to play that leaves your opponent with literally nothing to exploit, no matter what they know about your general strategy — as long as they can't see the specific outcome of your randomization in advance.

Why a pure choice fails

If you always throw Rock, an opponent who figures that out (even just by watching a pattern) beats you every single time with Paper. Any predictable rule — even something like "Rock, then Paper, then Scissors, repeating" — is still crackable once spotted. A mixed strategy replaces "which move do I make" with "what probability do I assign to each move," and the goal is to choose those probabilities so that the opponent's expected payoff is identical no matter which of their own moves they pick in response — at that point, they have no move that exploits you better than any other, because your unpredictability has erased the target.

Worked example

Simplify to a game with two choices, Bluff and Fold, where you're the bettor and the opponent decides whether to Call. Your payoffs: Bluff-then-Called costs you 4, Bluff-then-Not-Called wins you 6, Fold always costs you 1 regardless of the opponent's choice (folding forfeits a fixed small amount). You must choose the probability $p$ of bluffing (and $1-p$ of folding) so the opponent can't exploit your consistent tendency. The opponent's payoff from calling depends on how often you bluff: if you bluff too often, calling becomes profitable for them every time; if you never bluff, calling is a waste for them but folding never wins you anything back. The equilibrium mix is the $p$ where the opponent is exactly indifferent between calling and not calling, since if they strictly preferred one, you'd be handing them a free exploit. Setting the opponent's expected payoff from Call equal to their payoff from Not Call and solving typically yields something like $p = 0.4$: you should bluff 40% of the time, fold 60%, using something like a hidden die roll to decide each instance — a fixed 40/60 rule that's still unpredictable to the opponent because they never see the underlying randomizer, only the pattern of your final actions.

What this means in practice

Traders use mixed strategies whenever being predictable is itself costly — varying order sizes, timing, and venues so competitors and predatory algorithms can't front-run a recognizable pattern. The equilibrium mix isn't chosen for variety's sake; it's the specific proportion that makes every one of the opponent's responses equally unrewarding, which is the only state where a rational opponent truly cannot do better by studying you.