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)

Question

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)

BlackTom AI · Accepted Answer

To approach this payoff matrix question, I’ll first restate the setup and then analyze best responses for each player across all strategy profiles.

Question restatement: We have a 3-by-3 payoff matrix with Player 1 choosing rows A, B, C and Player 2 choosing columns A, B, C. Each cell gives a pair (P1 payoff, P2 payoff):
- A,A: (3,2)  A,B: (3,1)  A,C: (2,2)
- B,A: (1,3)  B,B: (2,2)  B,C: (0,2)
- C,A: (5,1)  C,B: (8,1)  C,C: (1,7)

Step-by-step analysis of options (best-responses):
1) For each fixed column (i.e., given Player 2’s choice), determine Player 1’s best response (the row that gives the highest P1 payoff):
- If Player 2 plays A: P1 payoffs are A→3, B→1, C→5. The best response for P1 is C (payoff 5).
- If Player 2 plays B: P1 payoffs are A→3, B→2, C→8. The best response for P1 is C (payoff 8).
- If Player 2 plays C: P1 payoffs are A→2, B→0, C→1. The best response for P1 is A (payoff 2).
Thus, P1’s best-response mapping is: BR1(A) = C, BR1(B) = C, BR1(C) = A.

2) For each fixed row (given Player 1’s choice), determine Player 2’s best response (the column that gives the highest P2 payoff):
- If Player 1 plays A: P2 payoffs are A→2, B→1, C→2. The best responses for P2 are A or C (tie at 2).
- If Player 1 plays B: P2 payoffs are A→3, B→2, C→2. The best response for P2 is A (payoff 3).
- If Player 1 plays C: P2 payoffs are A→1, B→1, C→7. The best response for P2 is C (payoff 7).
Thus, P2’s best-response mapping is: BR2(A) ∈ {A, C}, BR2(B) = A, BR2(C) = C.

3) Nash equilibria are the profiles where each player’s move is a best response to the other’s move. Check each profile (row, column) against the BR mappings:
- (A,A): P1’s BR to A is C, so A is not a best response. Not a NE.
- (A,B): P1’s BR to B is C, so B is not a best response. Not a NE.
- (A,C): P1’s BR to C is A, so A is a best response. P2’s BR to A includes C (since A yields 2 and C yields 2 for P2). Column C is a best response for P2 when P1 plays A. This profile satisfies both players’ best-response conditions, so it is a Nash equilibrium.
- (B,A): P1’s BR to A is C, so B is not a best response. Not a NE.
- (B,B): P1’s BR to B is C, so B is not a best response. Not a NE.
- (B,C): P1’s BR to C is A, so B is not a best response. Not a NE.
- (C,A): P1’s BR to A is C, so C is a best response. P2’s BR to C is C, but the chosen column is A, which is not a BR when P1 plays C. Not a NE.
- (C,B): P1’s BR to B is C, so C is a best response. P2’s BR to C is C, but the chosen column is B, which is not a BR when P1 plays C. Not a NE.
- (C,C): P1’s BR to C is A, so C is not a best response. Not a NE.

From this systematic check, only the profile (A,C) satisfies both players’ best-response conditions, indicating a unique Nash equilibrium at that profile.

Note on the provided content: The question presents an answer field claiming “The nash equilibrium is (A,C).” while the options list is empty. The reasoning above derives the NE directly from the payoff structure, independent of listed options, and shows why (A,C) is the equilibrium profile.

5284AGBM6Y Agent-based Modelling Practice Quiz v2

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)

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