The problem with a best move
Suppose one option in a game is strictly better than the others no matter what the opponent does. Then you simply pick it every time, and there is nothing to discuss. The interesting cases are the ones where the right choice depends on what the opponent picks, and the opponent can learn from your past behaviour.
In that setting, any fixed pattern is information. If you always take the same route, an opponent who notices can prepare for it and your most reliable play becomes your most predictable one. Mixed strategies are the formal answer: instead of choosing one option, you choose a probability for each option, and you pick randomly according to those probabilities.
A worked example
Take a simplified, hypothetical attack-and-defend situation. The attacker picks High or Low. The defender guards either High or Low. The attacker’s reward, in arbitrary points, is:
| Defender guards High | Defender guards Low | |
|---|---|---|
| Attacker goes High | 0 | 3 |
| Attacker goes Low | 1 | 0 |
A guarded attack scores nothing. An unguarded High attack is worth 3 and an unguarded Low attack is worth 1.
Suppose the attacker chooses High with probability p. If the defender guards High, the attacker’s average reward is p x 0 + (1 - p) x 1 = 1 - p. If the defender guards Low, it is p x 3 + (1 - p) x 0 = 3p.
The attacker wants a mix that does not give the defender anything to exploit, which means making both averages equal: 1 - p = 3p, so p = 1/4. At that mix, the attacker earns 0.75 points on average whatever the defender guards (3 x 0.25 = 0.75 and 1 - 0.25 = 0.75).
The surprising part is that High has the bigger payoff, yet the attacker uses it only a quarter of the time. That is because the defender is more inclined to guard it, so overusing it hands the defender an easy counter.
The defender can work the same way. If the defender guards High with probability q, the attacker’s average from going High is (1 - q) x 3 and from going Low is q x 1. Setting 3 - 3q = q gives q = 3/4. So the defender guards High three times in four, and the attacker’s average is again 0.75. Neither side can do better by changing their mix alone, which is what makes this pair of mixes stable.
What the example shows
Two lessons carry over to real play.
First, the mix is set by the payoffs, not by your preferences. The goal is to make the opponent indifferent between their options, so that no reading of your habits is profitable. Your own indifference comes from their payoffs, and theirs from yours.
Second, being unpredictable is not the same as being random in a loose sense. It means your frequencies are right and your individual choices cannot be forecast. Spreading choices evenly would be wrong in the example above, even though it feels fair.
Why people struggle to do it
Humans are poor randomisers. We avoid repeating ourselves, alternate more than chance would, and drift towards what worked a moment ago. An attentive opponent can notice this, which is why the study of opponent habits matters on both sides of the table.
Practical ways players approximate a mix include tying choices to something unpredictable, such as a detail of the clock or a prior game’s count, and using a rough target such as “about one time in four”. The aim is not mathematical precision, which is rarely available in real games, but breaking the pattern an opponent could exploit.
Where the idea applies
Mixing shows up wherever guessing is built in. Fighting games with simultaneous attack and defence options, hidden-card games, and decisions about where to attack or defend all have this structure. Resource-hiding in games with hidden information is a related case, since the other side has to guess what you hold. Competitive formats with a pick and ban phase involve it too, since teams avoid becoming predictable in what they take or remove.
Limits of the idea
The calculation assumes a sensible opponent and a stable game. Real situations differ in several ways:
- Payoffs are usually uncertain, so the exact mix is an estimate.
- If an opponent has a clear habit, deviating from the mix to exploit it can be better than staying balanced.
- Some options carry costs not captured in a single number, such as exposing your plan.
- In a game played once, or with no way for the opponent to adapt, there is little value in randomising.
Treat a mixed strategy as a defensive baseline: it protects you against being read, and you can move away from it when evidence about the opponent justifies doing so. It does not promise wins, only that you are not handing opponents an easy edge.