The problem with being predictable
If an opponent can guess your next move, they can counter it. This is true in poker, in a fighting game, in a penalty shootout-style duel and in many board games. The remedy from game theory is to mix your plays: choose among several options according to set proportions, so that no single response by the opponent does better than any other.
The general vocabulary of payoffs and equilibrium is covered in the introduction to game theory. This article applies it to specific gaming situations, focusing on one question: how should you vary your play, and in what proportions?
Start with rock-paper-scissors
Rock-paper-scissors is the cleanest example. Each option beats one and loses to another. Suppose you always play rock. After a few rounds, a thoughtful opponent plays paper every time, and you lose every round. Any fixed habit can be exploited in the same way.
Now suppose you pick each option one third of the time, unpredictably. Whatever the opponent does, you win a third of rounds, lose a third and tie a third, so your long-run average is level. You cannot be beaten on average, but you also do not win on average against this simple model. That is what an equilibrium mix provides: safety, not profit.
If instead your opponent is the one with a habit, you can take advantage. Imagine they play rock half the time and paper and scissors a quarter each, as a made-up example. If you always play paper, you win against rock (half the rounds), tie against paper (a quarter) and lose against scissors (a quarter). You come out ahead by half minus a quarter, which is a quarter of the rounds on average. But that only works while they keep the habit. If they notice and shift, your fixed choice becomes the weakness. This is the trade-off between exploiting a pattern and staying unexploitable, and spotting the pattern in the first place is the subject of opponent patterns.
A worked example: attack and defence
Here is a small made-up game with illustrative numbers. An attacker chooses to strike Left or Right. A defender simultaneously chooses to guard Left or Right. The table gives the attacker’s chance of success.
| Defender guards Left | Defender guards Right | |
|---|---|---|
| Attack Left | 20% | 80% |
| Attack Right | 90% | 40% |
If the attacker always goes Left, the defender guards Left and the attacker succeeds 20% of the time. If the attacker always goes Right, the defender guards Right and the attacker succeeds 40%. Either habit lets the defender choose a good reply.
What if the attacker mixes? Let q be the share of the time the attacker goes Left. The attacker wants a mix where the defender gains nothing by choosing one guard over the other.
- If the defender guards Left, the attacker’s success is 0.2q + 0.9(1 - q) = 0.9 - 0.7q.
- If the defender guards Right, the attacker’s success is 0.8q + 0.4(1 - q) = 0.4 + 0.4q.
Set these equal: 0.9 - 0.7q = 0.4 + 0.4q, so 0.5 = 1.1q, and q = 5/11, about 45%.
Check: with q = 5/11, guarding Left gives 0.9 - 0.7 x 5/11 = 0.9 - 0.318 = about 0.582. Guarding Right gives 0.4 + 0.4 x 5/11 = 0.4 + 0.182 = about 0.582. The two agree, so the defender has no preferred guard. The attacker succeeds about 58% of the time whichever way the defender guards, which is better than either fixed habit (20% or 40%) achieves against a thoughtful defender.
One feature of this result is worth noticing: the mix is set by what makes the defender indifferent. Your proportions depend on the payoffs the opponent faces, not only on your own.
These numbers are an illustration. Real games rarely come with exact percentages.
Applying mixing in poker
Poker is the best-known case. If you only bet when you have a strong hand, opponents fold to your bets and you win little. If you bet strong hands and weak hands in a sensible proportion, opponents cannot tell which they face. The question of how often a defender must continue to make bluffing unprofitable is the topic of minimum defense, and bluffing itself is covered in the article on that subject.
A key idea is that mixing is about proportions across many hands, not about acting randomly on each hand. Players sometimes use a hidden source of randomness, such as the second hand of a watch, to choose between two close options. Doing so is a way to avoid unintentional patterns, which people are poor at hiding.
Mixed play also helps with your table image, because opponents cannot easily build a sharp picture of someone who varies.
Fighting games, shooters and board games
The same logic appears elsewhere:
- Fighting games: mixing between a strike and a throw, or between attacking and waiting, so the opponent cannot always block the right one.
- Team games: varying which approach you use to a target, so defenders cannot always prepare.
- Hidden-move board games: choosing between a safe route and a risky route in proportions that keep the opponent guessing.
In each case the mix is set by the strategic situation, and a very large imbalance, such as using one option nearly always, usually signals a weakness.
When mixing is not the answer
Mixing is a defence against an opponent who is trying to exploit you. It is not always the best choice:
- Against a weak opponent who makes obvious mistakes, a direct exploitative approach can earn more.
- If you cannot estimate the proportions, a rough mix is better than none, but do not claim precision you do not have.
- In games of pure chance, such as slot machines or roulette, there are no opponents to mislead, so mixing has no meaning. The house edge applies whatever you do, and the mathematics of negative expectation means that regular play tends to lose money.
Testing how a mix performs can be done by simulation, as in strategy testing and Monte Carlo simulation, which let you see the long-run results of a plan without risking money.
Takeaway
Mixing plays protects you from being exploited by making your choices unpredictable in the proportions that remove an opponent’s best reply. The result is safety rather than a guaranteed win, and its exact proportions depend on the payoffs in the specific game. Chance still plays its part, and anyone playing for money should remember that losses are possible.