Strategic thinking and the prisoners' dilemma
Game theory studies decisions where your best choice depends on what others do. It began with mathematicians like John von Neumann and John Nash.
What is a game?
In game theory, a game has players, the strategies available to each, and payoffs for every combination of choices. Players are assumed to want the best payoff for themselves. Games model oligopoly pricing, arms races, negotiations, elections, and even animal behaviour.
Payoff matrices
A simple two-player game is shown in a payoff matrix: rows are one player's strategies, columns the other's, and each cell lists both players' payoffs. Reading the matrix lets you work out each player's best response to each of the other's choices.
Dominant strategies
A dominant strategy is best for a player no matter what the other does. If both players have dominant strategies, the outcome is easy to predict: each plays its dominant strategy. Not every game has dominant strategies.
The prisoners' dilemma
Two suspects are questioned separately. If both stay silent, each gets 1 year. If one confesses and the other stays silent, the confessor goes free and the other gets 10 years. If both confess, each gets 5 years. Whatever the other does, confessing is better for each, so both confess and get 5 years, though both staying silent would have given each just 1.
The lesson of the dilemma
The prisoners' dilemma shows that individually rational choices can produce collectively worse outcomes. It explains why cartels break down, why countries overfish shared seas, why arms races happen, and why cooperation needs trust, enforcement, or repeated interaction.
Nash equilibrium
A Nash equilibrium is a set of strategies where no player can improve their payoff by changing strategy alone, given the others' choices. In the prisoners' dilemma, both confessing is the Nash equilibrium. John Nash proved that every game with a finite number of players and strategies has at least one equilibrium if players can randomise.
Worked example: advertising
Two firms each choose to advertise or not. If neither advertises, each earns 10 million. If both advertise, each earns 6 million. If only one advertises, it earns 12 million and the other 4 million. Advertising is dominant for each, so both advertise and earn 6 million, worse than the 10 million they would earn by both not advertising.
Games without dominant strategies
In some games, the best choice depends on the other player. In a coordination game, such as choosing which side of the road to drive on, there are two equilibria (both left or both right), and the challenge is agreeing which. Conventions and laws often solve coordination problems.
Key takeaways
Games have players, strategies, and payoffs, often shown in a matrix. A dominant strategy is best regardless of rivals' choices. In the prisoners' dilemma, individual rationality leads to a worse joint outcome. A Nash equilibrium is where no player gains by changing alone. Coordination games have several equilibria; conventions help choose.
Flashcard
Study of strategic decisions
Check your understanding
In the prisoners' dilemma, why do both prisoners confess even though staying silent is better for both?