Question

Microeconomics

Posted 5 months ago

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- 5. The following table depicts two firms in a single-stage duopoly game. Each firm makes its decision without knowledge of the other firm's decision. The payoffs for each firm represent economic profits and each firm strictly prefers more economic profit than less.
- If $X$ is greater than $\$ 4,500$, how many Nash equilibriums exist in this game? Would this game be considered a prisoner's dilemma?
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Answer from Sia

Posted 5 months ago

Solution

a

Identify Nash Equilibriums: A Nash equilibrium occurs when each player's strategy is optimal given the other player's strategy. We need to analyze each cell in the payoff matrix to determine if any player can improve their payoff by unilaterally changing their strategy

b

Analyze Payoffs:
- If Tasha produces 300 flowers and Joshua produces 200 flowers, Tasha gets $3,500 and Joshua gets$4,500.
- If Tasha produces 300 flowers and Joshua produces 300 flowers, Tasha gets $2,000 and Joshua gets$2,000.
- If Tasha produces 200 flowers and Joshua produces 200 flowers, Tasha gets $2,000 and Joshua gets$4,500.
- If Tasha produces 200 flowers and Joshua produces 300 flowers, Tasha gets $X and Joshua gets$X

c

Determine Nash Equilibriums:
- If $X > 4,500$, then Tasha producing 200 flowers and Joshua producing 300 flowers is not a Nash equilibrium because Joshua can increase his payoff by switching to producing 200 flowers.
- The other cells need to be checked similarly to see if any player can improve their payoff by changing their strategy

d

Conclusion on Nash Equilibriums:
- If $X > 4,500$, the Nash equilibriums are:
- Tasha produces 300 flowers, Joshua produces 200 flowers.
- Tasha produces 300 flowers, Joshua produces 300 flowers.
- Tasha produces 200 flowers, Joshua produces 200 flowers

e

Prisoner's Dilemma Analysis:
- A game is considered a prisoner's dilemma if both players have a dominant strategy that leads to a suboptimal outcome for both.
- In this case, if $X > 4,500$, the dominant strategy for both players is not clear, and the game does not fit the classic prisoner's dilemma structure

Answer

There are three Nash equilibriums in this game if $X > 4,500$. This game is not considered a prisoner's dilemma.

Key Concept

Nash Equilibrium

Explanation

A Nash equilibrium occurs when each player's strategy is optimal given the other player's strategy. In this game, if $X > 4,500$, there are three Nash equilibriums, and the game does not fit the classic prisoner's dilemma structure.

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