Mathematics
Game Theory Equilibria through Pure and Mixed Strategies
Quick fact
John Nash proved that every finite game has at least one equilibrium when mixed strategies are allowed, even when no pure strategy equilibrium exists.
Why this is interesting
In a game of rock-paper-scissors, what is your best strategy? If you think you have a winning move, your opponent can predict it and counter it—so maybe the best choice is to be unpredictable.
Read the full explanation
Understanding Game Theory Equilibria through Pure and Mixed Strategies
Imagine two players choosing between two actions, like in a simple game of matching pennies. Each cell in a payoff matrix shows what each player gets for each combination of choices. A pure strategy is a single, deterministic choice—like always playing Heads. When each player's best response to the other is a single action, and they are mutual best responses, you have a pure Nash equilibrium. But sometimes no such pair exists, as in matching pennies. Here, whatever you do, your opponent can exploit you. The solution is to randomize. A mixed strategy assigns a probability to each pure strategy; players aim to maximize their expected payoff. In matching pennies, the equilibrium is for both players to randomize 50-50, making each indifferent between their options.
A deeper explanation
The principle behind equilibria is stability: no player wants to unilaterally change their strategy. For pure strategies, this means each player's chosen action is a best response. For mixed strategies, players randomize to keep opponents indifferent, ensuring no one can improve by switching. The key mathematical tool is expected utility: calculate the average payoff over all outcomes, weighted by probabilities. Nash's existence theorem, a fixed-point theorem (like Kakutani's), guarantees that in any finite game, a mixed-strategy equilibrium exists. Why does mixing help? It introduces uncertainty, preventing opponents from predicting and exploiting you. This mechanism is fundamental in economics (auctions), biology (evolutionary stable strategies), and computer science (algorithmic game theory).