A Nash Equilibrium is a state in a game where no player has an incentive to unilaterally change their strategy, assuming the other players’ strategies remain unchanged.
It is a “stable” outcome because each player’s choice is the best possible response to the choices of the other players.
Key Concepts
- Rational Players: The concept assumes that all players are rational and will choose the strategy that maximizes their own payoff.
- No Incentive to Deviate: The defining characteristic of a Nash Equilibrium is that if a player were to change their strategy on their own, they would end up with a worse outcome.
- Simultaneous Decisions: The game is often modeled with players making their decisions at the same time, without knowing the other players’ choices beforehand.
The Prisoner’s Dilemma Example
The Prisoner’s Dilemma is the classic illustration of a Nash Equilibrium.
Two suspects, A and B, are arrested and interrogated in separate rooms. They can either confess to the crime or remain silent. The outcomes are:
- If both confess: Each gets 5 years in prison.
- If both remain silent: Each gets 1 year in prison. (This is the best outcome for both combined).
- If A confesses and B remains silent: A goes free, and B gets 10 years.
- If B confesses and A remains silent: B goes free, and A gets 10 years.
Let’s find the Nash Equilibrium:
- From A’s perspective:
- If B confesses, A is better off confessing (5 years) than remaining silent (10 years).
- If B remains silent, A is better off confessing (0 years) than remaining silent (1 year).
- In both scenarios, A’s best choice is to confess.
- From B’s perspective:
- The same logic applies. Regardless of what A does, B’s best choice is to confess.
The Nash Equilibrium is the outcome where both players confess. This results in a suboptimal outcome (5 years each), but it’s the stable one because neither player can improve their situation by changing their strategy alone. This illustrates how individual rationality can lead to a collectively poor result.
Nash Equilibrium vs. Pareto Efficiency
While a Nash Equilibrium represents a stable outcome, it is not always the best possible outcome for all players. This is a crucial distinction.
A Nash Equilibrium is a stable state where no individual player can improve their outcome by changing their strategy.
A Pareto Efficient outcome is a state where it's impossible to make any player better off without making at least one other player worse off.
In the Prisoner’s Dilemma, the Nash Equilibrium (both confess) is not Pareto efficient because there is a better outcome for both players: if they both remain silent, they both get a better outcome (1 year vs. 5 years). The silent/silent outcome is Pareto efficient, but it’s not a Nash Equilibrium because each prisoner has an incentive to confess and betray the other for a better individual outcome.