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Concept of Nash Equilibrium
1. Introduction
Nash Equilibrium (NE) is a fundamental concept in game theory, introduced by John Nash in 1950. It describes a situation where no player has an incentive to deviate from their chosen strategy, given the strategies of all other players.
🔹 Key Idea: Each player is making the best decision they can, assuming others stick to their strategies.
2. Formal Definition
A Nash Equilibrium is a strategy profile (S1∗,S2∗,…,Sn∗)(S_1^*, S_2^*, …, S_n^*) such that for each player ii: Ui(Si∗,S−i∗)≥Ui(Si,S−i∗)∀SiU_i(S_i^*, S_{-i}^*) \geq U_i(S_i, S_{-i}^*) \quad \forall S_i
where:
- Si∗S_i^* = Player ii’s equilibrium strategy
- S−i∗S_{-i}^* = Strategies of all other players
- Ui(Si,S−i)U_i(S_i, S_{-i}) = Player ii’s payoff
At NE, no player benefits by changing their strategy alone.
3. Examples of Nash Equilibrium
Example 1: Prisoner’s Dilemma
Two criminals are arrested and separately decide to confess (Defect, D) or remain silent (Cooperate, C).
| Player 1 / Player 2 | Cooperate (C) | Defect (D) |
|---|---|---|
| Cooperate (C) | (3,3) | (0,5) |
| Defect (D) | (5,0) | (1,1) |
🔹 Finding Nash Equilibrium:
- If Player 1 cooperates, Player 2 prefers defect (5 > 3).
- If Player 1 defects, Player 2 also defects (1 > 0).
- (D, D) is the Nash Equilibrium because no player benefits by switching alone.
Example 2: Battle of the Sexes
A couple wants to spend time together but prefers different activities.
| Husband / Wife | Opera | Football |
|---|---|---|
| Opera | (2,1) | (0,0) |
| Football | (0,0) | (1,2) |
🔹 Finding Nash Equilibria:
- If Husband chooses Opera, Wife’s best response is Opera (1 > 0).
- If Husband chooses Football, Wife’s best response is Football (2 > 0).
- Two Nash Equilibria: (Opera, Opera) and (Football, Football).
This shows coordination problems, where multiple equilibria exist.
Example 3: Coordination Game
Two companies must decide whether to adopt a new technology (A) or stick with the old one (B).
| Firm 1 / Firm 2 | Adopt (A) | Old Tech (B) |
|---|---|---|
| Adopt (A) | (3,3) | (0,2) |
| Old Tech (B) | (2,0) | (1,1) |
🔹 Finding Nash Equilibrium:
- If Firm 1 adopts (A), Firm 2’s best response is A (3 > 0).
- If Firm 1 sticks to B, Firm 2’s best response is B (1 > 0).
- Two Nash Equilibria: (A, A) and (B, B).
This represents network effects, where firms benefit from coordinating choices.
4. Types of Nash Equilibrium
- Pure Strategy Nash Equilibrium
- Players choose one action with certainty.
- Example: (D, D) in Prisoner’s Dilemma.
- Mixed Strategy Nash Equilibrium
- Players randomize over strategies with certain probabilities.
- Used when no pure strategy Nash equilibrium exists.
🔹 Example: Rock-Paper-Scissors
No pure strategy NE, but in mixed strategy NE, players choose Rock, Paper, or Scissors with equal probability (1/3, 1/3, 1/3).
5. How to Find Nash Equilibrium?
Method 1: Best Response Analysis
- Identify each player’s best response to the opponent’s strategy.
- Find where both players’ best responses coincide → This is Nash Equilibrium.
Method 2: Eliminating Dominated Strategies
- Eliminate strategies that are always worse (dominated).
- Check the remaining strategies for equilibrium.
Method 3: Solving Equations (For mixed strategies)
- Set up probability equations where each player’s expected payoff is the same.
6. Importance of Nash Equilibrium
- Used in Economics & Business → Pricing strategies, auctions, firm competition.
- Political Science → Voting behavior and negotiations.
- Sports & Military → Game plans and tactical decisions.
7. Limitations of Nash Equilibrium
- Multiple equilibria → Hard to predict which one will be chosen.
- Not always socially optimal → In Prisoner’s Dilemma, NE (D, D) is worse than (C, C).
- Players must be rational → Assumes perfect reasoning, which is unrealistic in real-world behavior.
8. Conclusion
- Nash Equilibrium occurs when no player has an incentive to deviate alone.
- Can exist in pure or mixed strategies.
- Important in real-world decision-making (markets, politics, games).
- Not always the best outcome for society (e.g., Prisoner’s Dilemma).