pure versus mixed strategies
Imagine a penalty kick in football. If the kicker always shoots left, the goalkeeper learns it and dives left every time, and the kicker is doomed. In situations like this, sticking to one fixed move is a losing plan — being predictable is the problem. The cure is to keep the opponent guessing by deliberately randomizing. Game theory has a precise vocabulary for this: a pure strategy is committing to one specific move; a mixed strategy is choosing among your moves according to set probabilities — say, shoot left 60 percent of the time and right 40 percent.
More carefully, a pure strategy picks a single action with certainty (always 'High price'). A mixed strategy is a probability recipe over your pure strategies; you then let chance pick the actual move so even you don't decide in advance. The crucial insight is that some games have no Nash equilibrium in pure strategies at all — rock-paper-scissors is the classic case, since whatever fixed throw you favor can be beaten. John Nash's great theorem says that if you allow mixed strategies, every finite game has at least one equilibrium. In the rock-paper-scissors equilibrium each player throws each option one-third of the time, making themselves perfectly unpredictable.
Mixed strategies show up wherever being readable would be costly: sports, military feints, tax audits (the agency audits a random fraction so no one can safely cheat), and security patrols. A common misunderstanding is that mixing means 'acting irrationally' or 'just guessing'. It is the opposite — randomizing with exactly the right probabilities is the rational response to a clever opponent who would exploit any pattern. The right mix is the one that leaves your opponent indifferent, so they cannot profit by reading you.
A tax authority cannot audit every return, so it audits a random share each year. Because each taxpayer faces a real but uncertain chance of being checked, cheating stops paying off on average — the agency's mixed strategy keeps everyone honest more cheaply than auditing all.
Random tax audits: a mixed strategy that deters cheating without checking every return.
A mixed strategy is not 'random for its own sake'. The specific probabilities are chosen to make the opponent indifferent between their replies — a precise, optimal recipe, not mere whim.