What Is Nash Equilibrium?
Let me explain Nash equilibrium directly to you: it's a state in game theory where changing one participant's strategy won't help if everyone else's plans stay the same.
In this setup, players know their opponents' strategies and stick to their own because it's the best response. If all are using optimal strategies, no one benefits from switching, as long as others don't change. Games can have multiple Nash equilibria or none at all.
Key Takeaways
- Nash equilibrium is a game theory theorem where sticking to your initial strategy gives the best shot at your desired outcome.
- Each player's strategy in Nash equilibrium is optimal based on others' decisions.
- The prisoner's dilemma commonly illustrates Nash equilibrium effects.
- It's often linked to dominant strategy, where your choice yields better results no matter what the opponent does.
Understanding Nash Equilibrium
Nash equilibrium gets its name from John Nash, the American mathematician who invented it. I consider it one of the core ideas in game theory, which mathematically and logically figures out what actions game participants should take for the best personal outcomes.
What makes it so crucial is how widely it applies—you can use it in economics, social sciences, and beyond. To check for a Nash equilibrium or see if one exists, just reveal each player's strategy to the others. If no one switches, you've proven it.
Nash Equilibrium vs. Dominant Strategy
You might hear Nash equilibrium compared to dominant strategy, both from game theory. In Nash equilibrium, the best move is to hold your strategy steady, knowing the opponent's, and everyone does the same.
Dominant strategy means your chosen approach gives better results than any other, no matter what the opponent picks. Remember, all game theory models assume players are rational agents—they want specific outcomes, pick the optimal ones, factor in uncertainty, and stay realistic.
These concepts are similar but not identical: Nash says no gain from changing if others don't, while dominant strategy ensures your pick is best regardless of others.
Example of Nash Equilibrium
Consider a simple game with Tom and Sam. Each can pick strategy A to gain $1 or B to lose $1. Logically, both go for A and get $1 each.
If you tell Sam’s strategy to Tom and vice versa, they'd stick with A since switching wouldn't help. The opponent's move doesn't change behavior here, so outcome A is a Nash equilibrium.
Prisoner’s Dilemma
The prisoner's dilemma is a classic game theory scenario that shows Nash equilibrium in action. Two arrested criminals are in separate cells, unable to talk. Prosecutors lack full evidence, so they offer each a deal: betray the other or stay silent.
If both betray, they each get five years. If one betrays and the other stays silent, the betrayer goes free and the silent one gets 10 years. If both stay silent, they each serve one year. The Nash equilibrium here is both betraying, even though cooperating would be better overall— but if one cooperates and the other doesn't, the cooperator loses big.
What Is a Nash Equilibrium in Simple Words?
Put simply, it's when a player sticks with their strategy because there's no benefit to changing it, considering what the opponent is doing.
How Do You Identify Nash Equilibrium?
To spot it, model all possible scenarios, see the outcomes, and pick the optimal strategies. In a two-player game, consider both sides' choices. If neither changes knowing everything, that's Nash equilibrium.
Why Is Nash Equilibrium Important?
It's key because it shows the point where players can't improve by switching strategies unilaterally.
The Bottom Line
Nash equilibrium in game theory means a player keeps their strategy, knowing the opponent's, with no reason to deviate. It's a stable state in strategic interactions.
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