Nash equilibrium, game theory, pure strategy, mixed strategy: In game theory, a Nash equilibrium occurs when each player in a game chooses their best strategy, taking into account the strategies of the other players. A pure strategy is a fixed plan of action that a player chooses to follow regardless of the actions of the other players. A mixed strategy, on the other hand, is a strategy that involves randomizing one’s actions according to a probability distribution.
Pure Nash Strategy vs. Mixed Strategy
In game theory, a Nash equilibrium is a set of strategies for all players in a game such that no player can improve their outcome by unilaterally changing their strategy. A pure Nash strategy is a strategy where each player chooses a single action. A mixed strategy is a strategy where each player chooses a probability distribution over the possible actions.
Pure Nash Strategy
- Each player chooses a single action.
- No player can improve their outcome by unilaterally changing their strategy.
Mixed Strategy
- Each player chooses a probability distribution over the possible actions.
- No player can improve their outcome by unilaterally changing their probability distribution.
Comparison of Pure and Mixed Strategies
| Feature | Pure Strategy | Mixed Strategy |
|---|---|---|
| Number of actions | 1 | Any number |
| Predictability | High | Low |
| Complexity | Simple | Complex |
| Effectiveness | Can be effective in simple games | Can be effective in complex games |
When to Use a Pure Nash Strategy
- The game is simple and has few actions.
- The players are rational and have perfect information.
When to Use a Mixed Strategy
- The game is complex and has many actions.
- The players are not rational or do not have perfect information.
Example
Consider the following game:
| Player 1 | Player 2 | Payoff to Player 1 |
|---|---|---|
| A | A | 1 |
| A | B | 0 |
| B | A | 0 |
| B | B | 1 |
The pure Nash equilibrium in this game is for both players to choose A. However, if Player 2 is not rational or does not have perfect information, Player 1 may be better off choosing a mixed strategy. For example, Player 1 could choose to play A with probability 3/4 and B with probability 1/4. This would prevent Player 2 from exploiting Player 1’s strategy.
Question 1:
- What is the fundamental difference between a pure Nash strategy and a mixed strategy?
Answer:
- A pure Nash strategy is a deterministic strategy in which a player selects a specific action with certainty.
- A mixed strategy, on the other hand, is a randomized strategy in which a player selects actions with varying probabilities.
Question 2:
- Explain the concept of a trembling hand perfect equilibrium (THPE) and its relationship to mixed strategies.
Answer:
- A trembling hand perfect equilibrium (THPE) is a subgame perfect equilibrium in which players use mixed strategies.
- THPE occurs when no player can improve their payoff by deviating from their randomized strategy, even if they believe their opponents may make small mistakes.
Question 3:
- How can you determine if a game has a pure strategy Nash equilibrium?
Answer:
- To determine if a game has a pure strategy Nash equilibrium, you can use the Nash equilibrium theorem.
- This theorem states that every finite game with a finite number of players and actions has at least one pure strategy Nash equilibrium.
Well there you have it, folks! Now you can sound like a pro when you’re talking about game theory. Just remember, if you’re ever stuck in a tough spot, a mixed strategy might just be the way to go. And hey, thanks for stopping by! Be sure to come back soon for more awesome game theory content.