Let p be the probability that player 1 chooses U and let q be the probability that player 2 chooses L. Select the correct statement. (Hint: before applying the short cut method, observe the payoff matrix carefully.)

L R
U 1,4 8,1
D 0,4 7,1

Question

Let p be the probability that player 1 chooses U and let q be the probability that player 2 chooses L. Select the correct statement. (Hint: before applying the short cut method, observe the payoff matrix carefully.)

L	R
U	1,4	8,1
D	0,4	7,1

BlackTom AI · Accepted Answer

The problem asks us to select the correct statement about Nash equilibria given the payoff matrix.
First, I restate the payoff structure to keep the analysis clear. The matrix shows payoffs for (Player 1, Player 2):
- If Player 1 plays U and Player 2 plays L: (1, 4)
- If Player 1 plays U and Player 2 plays R: (8, 1)
- If Player 1 plays D and Player 2 plays L: (0, 4)
- If Player 1 plays D and Player 2 plays R: (7, 1)

To find Nash equilibria, we examine best responses for each player.
- Player 1’s best response:
  - If Player 2 plays L: payoffs for Player 1 are U→1 and D→0, so U is strictly best.
  - If Player 2 plays R: payoffs for Player 1 are U→8 and D→7, so U is still strictly best.
  Therefore, no matter what Player 2 does, Player 1’s best response is U. Consequently, in any NE we must have p = 1 (Player 1 always plays U).

- Player 2’s best response given p = 1 (Player 1 plays U with certainty):
  - If Player 2 plays L: payoff to Player 2 is 4.
  - If Player 2 plays R: payoff to Player 2 is 1.
  Since 4 > 1, Player 2’s best response is L, so q = 1 (Player 2 always plays L).

From these best-response analyses, there is a unique intersection of best responses at (p, q) = (1, 1), i.e., (U, L) is the Nash equilibrium.
Now consider the given statement: “There is only one Nash equilibrium in which p=q=1.”
- The analysis shows that indeed p must be 1 and q must be 1 in any Nash equilibrium, and the pair (p, q) = (1, 1) is the only such intersection of best responses.
- Therefore, the statement aligns with the derived equilibrium structure: there is exactly one Nash equilibrium and it has p = q = 1.

In a broader sense, this means there are no alternative mixed equilibria: any deviation from p = 1 for Player 1 would violate his best-response condition, and any deviation from q = 1 for Player 2 would violate her best-response condition given p = 1.

23506 Strategic Decision Making and Game Theory - Autumn 2025 Online quiz 3

Let p be the probability that player 1 chooses U and let q be the probability that player 2 chooses L. Select the correct statement. (Hint: before applying the short cut method, observe the payoff matrix carefully.) L R U 1,4 8,1 D 0,4 7,1

View Explanation

Log in for full answers

Similar Questions

The advertising agency raises its prices, causing a firm that chooses to run ads to earn $200 less in profits. Once the firms adjust to the higher advertising prices, what is the total combined profit of the two firms in the new Nash equilibrium?

Consider a matrix form game between two players, A and B. Player A can choose between Up and Down, and Player B can choose between Left and Right. If the payoffs are represented using this matrix, what is the pure Nash equilibrium?

In the following duopoly game, the two firms can either set the price of their product high or low. For each firm, the final profit depends on the price they set as well as the price the other firm sets. The game is represented in the table below. The Nash equilibrium for this game is for:

Consider a sequential-move game with two players. The game tree with order of payoffs (Player 1, Player 2) is provided above. Choose the answer that is consistent with it.

Consider a simultaneous-move game with two players. The payoff table with order of payoffs (Player 1, Player 2) is provided above. Choose the statement that is consistent with it.

Consider the payoff matrix below. Which is correct for the following payoff matrix? Player 2 A B C Player 1 A (3,2) (3,1) (2,2) B (1,3) (2,2) (0,2) C (5,1) (8,1) (1,7)

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler