The Science of Strategy
Game theory is the study of mathematical models of strategic interaction among rational decision-makers. It has applications in economics, political science, psychology, logic, and biology.
Components of a Game
1.Players: The decision-makers in the game.
2.Strategies: A complete plan of action a player will take, given the set of circumstances that might arise within the game.
3.Payoffs: The outcome or consequence (utility, prize, profit) that a player receives from a particular combination of strategies. Payoffs are often represented in a payoff matrix.
The Prisoner's Dilemma
This is the most famous example in game theory.
Scenario: Two members of a criminal gang are arrested and imprisoned in separate rooms. The prosecutors lack sufficient evidence to convict them on the principal charge, but they have enough to convict both on a lesser charge.
Strategies for each prisoner:
1.Cooperate (with the other prisoner, by staying silent).
2.Defect (betray the other prisoner, by confessing).
Payoff Matrix (years in prison):
| | Prisoner B Stays Silent | Prisoner B Betrays |
| :--- | :--- | :--- |
| Prisoner A Stays Silent | A: 1 year, B: 1 year | A: 3 years, B: 0 years |
| Prisoner A Betrays | A: 0 years, B: 3 years | A: 2 years, B: 2 years |
Analysis: From each prisoner's individual perspective, betraying is always the better strategy, regardless of what the other does. If B stays silent, A gets 0 years by betraying instead of 1. If B betrays, A gets 2 years by betraying instead of 3.
Outcome: Both prisoners, acting in their own rational self-interest, will betray each other and both serve 2 years, even though they could have both served only 1 year if they had cooperated.
Nash Equilibrium
A Nash Equilibrium is a set of strategies, one for each player, such that no player has an incentive to unilaterally change their strategy. Each player is playing their best possible strategy, given the strategies of all other players.
In the Prisoner's Dilemma, the (Betray, Betray) outcome is a Nash Equilibrium. Given that Prisoner B is betraying, Prisoner A's best move is to also betray. Given that Prisoner A is betraying, Prisoner B's best move is to also betray. Neither has an incentive to change their strategy alone.