Microeconomic Theory: Simultaneous-Move Games, Dominant Strategies, Nash Equilibrium, and Maximin Strategies

Q1. Matching Pennies Payoff Matrix: What is the value of d?

Background

Topic: Game Theory – Payoff Matrices

This question tests your understanding of how to fill in payoffs in a simultaneous-move game matrix, specifically the matching pennies game. You need to determine the payoff for Even when Odd plays Head and Even plays Tail.

Key Terms and Formulas:

• Payoff Matrix: A table that shows the payoffs for each player for every possible combination of strategies.

• Matching Pennies: A zero-sum game where one player wins if the pennies match, and the other wins if they do not.

Step-by-Step Guidance

1. Recall the rules: If the pennies do not match (one Head, one Tail), Odd wins (+1), Even loses (−1).

2. Identify the cell: Odd plays Head, Even plays Tail. This is the (Head, Tail) cell in the matrix.

3. Assign the payoffs: Odd gets +1, Even gets −1. The value of d is Even's payoff in this cell.

Try solving on your own before revealing the answer!

Final Answer: d = -1

Even loses when the pennies do not match, so the payoff is -1 for Even in this scenario.

Q2. How many strategies does Odd have in Matching Pennies?

Background

Topic: Game Theory – Strategies

This question checks your understanding of what constitutes a strategy in a simultaneous-move game. Each player chooses between two options: Head or Tail.

Key Terms:

• Strategy: A complete plan of action for a player, specifying what they will do in every possible situation.

Step-by-Step Guidance

1. List Odd's possible actions: Head or Tail.

2. Count the number of distinct strategies available to Odd.

Try solving on your own before revealing the answer!

Final Answer: 2 strategies

Odd can choose either Head or Tail, so there are two strategies.

Q3. Rock-Paper-Scissors Payoff Matrix: What is the value of N?

Background

Topic: Game Theory – Payoff Matrices

This question asks you to determine the payoff for Player 2 when Player 1 chooses Scissors and Player 2 chooses Rock in a standard rock-paper-scissors game.

Key Terms and Formulas:

• Payoff Matrix: Shows the outcomes for each player for every combination of strategies.

• Rock-Paper-Scissors Rules: Rock beats Scissors, Scissors beats Paper, Paper beats Rock. Winner gets +1, loser gets -1, tie is 0.

Step-by-Step Guidance

1. Identify the cell: Player 1 chooses Scissors, Player 2 chooses Rock.

2. Apply the rules: Rock beats Scissors, so Player 2 wins (+1), Player 1 loses (−1).

3. Find the value of N, which is Player 2's payoff in this cell.

Try solving on your own before revealing the answer!

Final Answer: N = 1

Player 2 wins when playing Rock against Scissors, so the payoff is 1.

Q4. What does a dominant strategy produce?

Background

Topic: Dominant Strategies in Game Theory

This question tests your understanding of the definition of a dominant strategy and its implications for payoffs in a game.

Key Terms:

• Dominant Strategy: A strategy that yields the highest payoff for a player, no matter what the other players do.

Step-by-Step Guidance

1. Recall the definition: A dominant strategy is always the best response, regardless of opponents' actions.

2. Consider what this means for payoffs: It must produce the highest possible payoff for every possible strategy of the other players.

Try solving on your own before revealing the answer!

Final Answer: The highest payoff for every possible strategy of the other players.

A dominant strategy always yields the best outcome for the player, no matter what the opponents do.

Q5. Which firm has a dominant strategy in the rebate game?

Background

Topic: Dominant Strategies in Oligopoly Pricing Games

This question involves analyzing a payoff matrix for two firms (ABC and XYZ) deciding whether to offer a rebate. You are asked to determine which firm, if any, has a dominant strategy.

Key Terms:

• Dominant Strategy: The best action for a player, regardless of what the other player does.

Step-by-Step Guidance

1. For each firm, compare the payoffs for offering a rebate versus not offering a rebate, given each possible action of the other firm.

2. Check if one action always yields a higher payoff for a firm, regardless of the other firm's choice.

3. Repeat for both ABC and XYZ.

Try solving on your own before revealing the answer!

Final Answer: Both ABC and XYZ have a dominant strategy, which is Offer Rebate.

Offering a rebate always gives a higher payoff for both firms, regardless of the other firm's action.

Q6. In the following simultaneous-move game, which statement is correct?

Background

Topic: Identifying Dominant Strategies in Payoff Matrices

This question asks you to analyze a payoff matrix and determine which player, if any, has a dominant strategy.

Key Terms:

• Dominant Strategy: The best action for a player, regardless of the other player's choice.

Step-by-Step Guidance

1. For Player 2, compare the payoffs for choosing U versus D, given each possible action of Player 1.

2. If one action always yields a higher payoff, that is the dominant strategy.

3. Repeat for Player 1.

Try solving on your own before revealing the answer!

Final Answer: "D" is a dominant strategy for Player 2.

Player 2 always does better by choosing D, regardless of Player 1's action.

Q7. Which statement is correct about dominant strategies in the following game?

Background

Topic: Dominant Strategies in Simultaneous-Move Games

This question asks you to determine if either player has a dominant strategy in the given payoff matrix.

Key Terms:

• Dominant Strategy: The best action for a player, regardless of the other player's choice.

Step-by-Step Guidance

1. For each player, compare the payoffs for each strategy, given the possible actions of the other player.

2. If neither player has a strategy that is always best, then neither has a dominant strategy.

Try solving on your own before revealing the answer!

Final Answer: Neither player has a dominant strategy.

Each player's best response depends on the other player's action.