When any fixed choice can be punished
Some games have a clear best move. Many do not. Consider a game where you must pick between two options at the same moment as a rival, and where the rival wins if they match you and you win if you do not. Whatever you pick consistently, the rival learns and counters. If you always choose the left option, the rival chooses left every time. Any fixed pattern becomes a weakness once someone notices.
A mixed strategy answers this. Instead of committing to one option, you commit to probabilities: for example, choose left 60% of the time and right 40% of the time. A choice of one option with certainty is called a pure strategy. A mixed strategy is a probability distribution over pure ones.
Examples appear in sports (where to aim a penalty kick or serve), in card play (how often to bluff), and in many simultaneous-choice mechanics in tabletop games. In each case the underlying reason is the same: if the other side can predict you, they can respond in a way that costs you.
The classic example: matching
In the simplest version, two players each show heads or tails. If the faces match, player one wins 1 from player two. If they differ, player two wins 1 from player one. No pure choice is stable. If player one always shows heads, player two shows heads. If player two does that, player one switches to tails, and the chase continues forever.
The stable answer is for each to play 50-50. Against a 50-50 rival, every choice you make wins half the time and loses half the time, so your average result is zero whatever you do. You cannot be exploited, and neither can they.
The indifference principle
The surprising feature of equilibrium mixing is how you find the probabilities. You choose your mix so that your rival is indifferent among their own options. When the rival is indifferent, none of their pure choices beats any other against your mix, and so you cannot be exploited. Your mix is set by the rival’s payoffs, not your own.
Take a lopsided version. Player Row and player Column each choose Heads (H) or Tails (T) at the same time. The payoffs to Row, with Column receiving the negative, are:
| Column H | Column T | |
|---|---|---|
| Row H | +2 | -1 |
| Row T | -1 | +1 |
Row would love to hit the +2, so you might expect Row to play H most of the time. Watch what the algebra says.
Let q be the chance Column plays H. Row’s average result is:
- from playing H: 2q + (-1)(1 - q) = 3q - 1
- from playing T: (-1)q + 1(1 - q) = 1 - 2q
Column wants to make Row indifferent, so sets these equal: 3q - 1 = 1 - 2q, which gives 5q = 2 and q = 2/5 = 0.4. Then Row’s average is 3(0.4) - 1 = 0.2 for either choice.
Now let p be the chance Row plays H. Column’s average result, which is the negative of Row’s, is:
- from playing H: -(2p + (-1)(1 - p)) = -(3p - 1) = 1 - 3p
- from playing T: -((-1)p + 1(1 - p)) = -(1 - 2p) = 2p - 1
Setting equal: 1 - 3p = 2p - 1, so 5p = 2 and p = 0.4. Row plays H only 40% of the time, even though H holds the big prize.
Check the value: Row’s average is p(3q - 1) + (1 - p)(1 - 2q) = 0.4(0.2) + 0.6(0.2) = 0.2. Row has a small edge of 0.2 per round in this game, and neither side can change that by altering their mix alone.
Why does Row play H less than half the time? Because the large payoff is visible to Column too. If Row played H often, Column would respond with T and punish them. Column plays H only 40% of the time for a similar reason. Large rewards attract predictable behaviour, and predictable behaviour gets countered.
Why humans are bad at randomizing
Telling yourself to be unpredictable rarely works. People tend to avoid repeating themselves, switch too often, and fall into subtle habits. Because many people expect streaks to be “unlikely”, they produce sequences with too few runs, which is a close cousin of the gamblers fallacy. A rival who studies you may spot these regularities, which is where pattern recognition cuts against you.
Better approaches, where the rules allow them, are to use an external source of randomness. A die can give you probabilities such as one-third or one-sixth. A glance at a clock’s second hand can approximate a coin flip. If you want a 40% chance of heads, roll a ten-sided die and play heads on a 1, 2, 3 or 4. A tabletop group with dice at hand has the tools to do this. For formal software, the role of random number generators is to produce the needed unpredictability.
Mixing is not the same as being random about everything
A mixed strategy is a deliberate probability, not a lack of thought. You still decide which options belong in the mix and in what proportions. Options that are dominated stay out entirely. You may also change your mix if the rival is not playing the equilibrium. If you see a rival leaning toward one option, the best response is to lean toward its counter, trading some safety for an advantage. The risk is that you become exploitable yourself, and a thoughtful rival may be setting a trap.
This trade-off is the heart of chance as strategy: randomness is a defensive tool, and also a way to avoid giving away information.
Games with more than two options
With three or more options, the same principle holds: mix so that the rival is indifferent among the options they use. The classic three-option game is the subject of the rock paper scissors article, where the mix is one-third each by symmetry. When payoffs are not symmetric, the mix can look odd, as in the example above.
Practical points
- Use mixing where predictability is punished: simultaneous choices, repeated encounters, and bluffing decisions.
- Do not mix where one option clearly dominates. Randomizing between a good and a bad option only gives away value.
- Calculate from the rival’s payoffs, not only your own, when setting the proportions.
- Keep a record if you can. Notes on a rival’s tendencies tell you when to leave the equilibrium.
- Remember sample size. A rival who looks like they favour one option after five rounds may simply have had a run of chance results. Small samples mislead.
The aim is not to be cute with randomness. It is to make sure that no pattern in your play can be turned into a reliable advantage for someone else.