Nash Equilibrium and Pareto Efficient
Nash Equilibrium: A Nash Equilibrium occurs in game theory when players in a strategic game choose strategies that are best for them, given the strategies chosen by the other players. In this state, no player can benefit by changing their strategy while the others keep theirs unchanged.
Visual Aid: Imagine two companies, A and B, deciding whether to set a high price or a low price for their products. The outcome of their choices can be represented in a payoff matrix:
Company B: High Price | Company B: Low Price | |
A: High Price | (3, 3) | (1, 4) |
A: Low Price | (4, 1) | (2, 2) |
In this example:
(3, 3) means both companies set a high price, receiving moderate profits.
(1, 4) means Company A sets a high price while Company B sets a low price, where Company A profits less.
(4, 1) means Company A sets a low price while Company B sets a high price, where Company B profits less.
(2, 2) means both set a low price, resulting in lower profits for both.
The Nash Equilibrium occurs at (2, 2), where both choose a low price. If either company tries to change to a high price while the other stays low, it will decrease their own profits. Thus, no player has an incentive to deviate from their strategy, leading to stability.
Nash Equilibrium
A Nash Equilibrium occurs in game theory when players in a strategic game choose strategies that are best for them, given the strategies chosen by the other players. In this state, no player can benefit by changing their strategy while the others keep theirs unchanged.
Visual Aid
Imagine two companies, A and B, deciding whether to set a high price or a low price for their products. The outcome of their choices can be represented in a payoff matrix:
Company B: High Price Company B: Low Price | ||
Company A: High Price | (3, 3) | (1, 4) |
Company A: Low Price | (4, 1) | (2, 2) |
Interpreting the Payoff Matrix
(3, 3): Both companies set a high price, receiving moderate profits.
(1, 4): Company A sets a high price, while Company B sets a low price; Company A profits less.
(4, 1): Company A sets a low price, while Company B sets a high price; Company B profits less.
(2, 2): Both set a low price, resulting in lower profits for both.
Finding the Nash Equilibrium
To visually identify the Nash Equilibrium:
Analyze each cell of the matrix to see if either player can improve their payoff by changing their strategy while the other keeps theirs unchanged.
The Nash Equilibrium occurs at (2, 2), where both choose a low price. In this equilibrium:
If Company A changes to a high price while Company B stays low, Company A's profit decreases to (1).
Conversely, if Company B changes to a high price while Company A stays low, Company B's profit decreases to (1).
Thus, neither company has an incentive to deviate from the strategy of setting a low price, leading to stability.
Practice Problem for Finding Nash Equilibrium
Consider two airlines, Airline X and Airline Y, deciding whether to set a high ticket price or a low ticket price for their flights. The following information applies:
If both airlines set a high ticket price, they each earn 4 units of profit.
If Airline X sets a high price while Airline Y sets a low price, Airline X earns 2 units of profit and Airline Y earns 5 units of profit.
If Airline X sets a low price while Airline Y sets a high price, Airline X earns 5 units of profit and Airline Y earns
Pareto Efficient: