Consider the game described by the following payoff matrix. Which of the following statements is TRUE?

Player 2

Left

Middle

Right

Player 1

Top

25, 30

16, 32

30, 23

Center

34, 43

18, 23

35, 12

Bottom

43, 15

21, 33

36, 38

Question

Consider the game described by the following payoff matrix. Which of the following statements is TRUE?

Player 2

Left

Middle

Right

Player 1

Top

25, 30

16, 32

30, 23

Center

34, 43

18, 23

35, 12

Bottom

43, 15

21, 33

36, 38

BlackTom AI · Accepted Answer

Question restatement: We are given a 3x3 payoff matrix with Player 1 choosing Top, Center, or Bottom and Player 2 choosing Left, Middle, or Right. The options ask which statement is TRUE about Nash equilibria, maximin, and efficiency in this game.

Option 1: The maximum amount Player 2 would pay Player 1 to induce it to choose Center is $5.
- To assess how much transfer would be needed to induce Player 1 to choose Center, we compare P1’s payoffs for Center with his best responses under each possible P2 column.
  • If P2 plays Left, P1’s payoffs are Top 25, Center 34, Bottom 43. P1’s best response is Bottom with 43, so to switch P1 to Center from Bottom would require at least 9 in compensation (43 − 34 = 9).
  • If P2 plays Middle, P1’s payoffs are Top 16, Center 18, Bottom 21. P1’s best response is Bottom with 21, so to move to Center would require at least 3 (21 − 18 = 3).
  • If P2 plays Right, P1’s payoffs are Top 30, Center 35, Bottom 36. P1’s best response is Bottom with 36, so to move to Center would require at least 1 (36 − 35 = 1).
- The minimum transfer needed in each case to make Center a best response is 9, 3, and 1 respectively. The question asks for the maximum amount Player 2 would need to pay to induce Center; taking the worst case among these (the largest of 9, 3, 1) yields 9. Therefore, $5 is not correct. The claim underestimates the required transfer.

Option 2: This game does not have any Nash equilibrium.
- A Nash equilibrium is a pair of strategies where each player’s action is a best response to the other. Checking potential best responses:
  • If Player 2 plays Left, Player 1’s best response is Bottom (43). Given Bottom, Player 2’s best response is Right (38). Thus (Bottom, Right) is a candidate NE and needs checking: given Right, P1’s best response is Bottom (36 over 30 and 35). Since Bottom is best for P1 when Right, (Bottom, Right) satisfies mutual best responses and is a Nash equilibrium.
- Therefore, the statement claiming there is no Nash equilibrium is false.

Option 3: The maximin outcome and the efficient outcome coincide in this game.
- Maximin for Player 1 is found by taking the maximum of the row-minima: Row minima are Top: 16, Center: 18, Bottom: 21; the maximin value for P1 is 21 (associated with the Bottom row). For Player 2, the column minima are Left: 15, Middle: 23, Right: 12; the maximin value for P2 is 23 (associated with the Middle column). The corresponding purported maximin outcome would be (Bottom, Middle) with payoffs (21, 33).
- The efficient (Pareto-dominant) outcome is the profile with the highest sum of payoffs that is not Pareto-dominated by any other outcome. The (Bottom, Right) outcome yields (36, 38) with a total of 74, which dominates other outcomes in sum without leaving both players worse off, and is in fact a safe NE. The maximin outcome (Bottom, Middle) has a lower sum (21 + 33 = 54) and is not the same as the efficient outcome. Hence this statement is false.

Option 4: The Nash equilibrium outcome is also the efficient outcome in this game.
- We identified a Nash equilibrium at (Bottom, Right) with payoffs (36, 38). To determine efficiency, compare with other outcomes: the largest total payoff among all profiles is indeed 74 at (Bottom, Right) (36 + 38). No other outcome Pareto-dominates this one, since any other outcome either lowers Player 1’s payoff, Player 2’s payoff, or both. Therefore, (Bottom, Right) is Pareto efficient as well as a Nash equilibrium, making this statement true.

Summary of reasoning for each option:
- Option 1 is incorrect because the required transfer to center ranges up to 9 in the Left case (and lower in others), not $5.
- Option 2 is incorrect because a Nash equilibrium (Bottom, Right) indeed exists.
- Option 3 is incorrect because the maximin outcome does not coincide with the efficient outcome; they occur at different strategy profiles with different payoffs.
- Option 4 is correct because the Nash equilibrium outcome (Bottom, Right) is also Pareto efficient, given that no other outcome makes at least one player better off without making the other worse off.

COMM_V 295 101 102 2025W1 2025W1 COMM 295 Final (December 13, 2025)- Requires Respondus LockDown Browser

Consider the game described by the following payoff matrix. Which of the following statements is TRUE? Player 2 Left Middle Right Player 1 Top 25, 30 16, 32 30, 23 Center 34, 43 18, 23 35, 12 Bottom 43, 15 21, 33 36, 38

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