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

We start by laying out the key elements of the game from the given payoff matrix. The outcomes are the nine cell payoffs, each a pair (P1, P2):

Top-Left (25,30), Top-Middle (16,32), Top-Right (30,23);
Center-Left (34,43), Center-Middle (18,23), Center-Right (35,12);
Bottom-Left (43,15), Bottom-Middle (21,33), Bottom-Right (36,38).

Option A: The maximum amount Player 2 would pay Player 1 to induce Center is $5.
- This kind of statement attempts to quantify a transfer to change P1’s action to Center. To assess it, we’d need to know the baseline payoff for P1 under the current or likely equilibrium path and compare Center’s payoff to alternatives in a way that isolates the effect of a transfer. However, simply looking at the Center row, P1’s payoffs are 34 (Left), 18 (Middle), and 35 (Right). The minimum in Center is 18, and there is no single transfer amount that universally “induces” Center across all P2 strategies without changing incentives elsewhere. Moreover, identifying a precise transfer of $5 would require specifying the reference outcome and how transfers are implemented in the game’s equilibrium concept. In short, this statement mixes a monetary transfer with strategic choice in a way that is not supported by the payoff structure alone, and the $5 figure is not a well-founded, intrinsic property of the game.

Option B: The maximin outcome and the efficient outcome coincide in this game.
- Compute maximin for Player 1: for each row, take the worst payoff across columns. Top: min(25,16,30) = 16; Center: min(34,18,35) = 18; Bottom: min(43,21,36) = 21. The maximin value is 21, achieved by the Bottom row (specifically Bottom against the column that minimizes P1’s payoff, which is Bottom-Middle giving P1 = 21). Now check Pareto (efficient) outcomes: an outcome is Pareto efficient if no other outcome makes at least one player better off without making the other worse off. The outcome Center-Left (34,43) is strictly better for both players than Bottom-Middle (21,33) on both coordinates; thus Bottom-Middle is not Pareto efficient. Since the maximin outcome (Bottom-Middle, 21 for P1) is not Pareto efficient, the two concepts do not coincide. Therefore this statement is false.

Option C: The Nash equilibrium outcome is also the efficient outcome in this game.
- Determine the Nash equilibrium(s) by checking best responses. For Player 1, given any column chosen by Player 2, the best responses are:
  - If P2 chooses Left, P1’s best is Bottom (43 vs 25, 34).
  - If P2 chooses Middle, P1’s best is Bottom (21 vs 16, 18).
  - If P2 chooses Right, P1’s best is Bottom (36 vs 30, 35).
  So P1’s dominant strategy is Bottom.
- For Player 2, given P1’s row:
  - If P1 is Top, P2’s best is Middle (32 > 30, 23).
  - If P1 is Center, P2’s best is Left (43 > 23, 12).
  - If P1 is Bottom, P2’s best is Right (38 > 15, 33).
- The intersection of P1’s Bottom with P2’s best response to Bottom is (Bottom, Right). Thus (Bottom, Right) is a Nash equilibrium with payoffs (36,38).
- Check Pareto efficiency of this NE outcome: Compare (36,38) with other outcomes. There is no outcome that makes both players at least as well off with one strictly better; for instance Center-Left (34,43) is better for P2 but worse for P1 (34 < 36). No cell dominates (36,38) for both players. Hence the Nash equilibrium is Pareto efficient. Therefore this statement is true.

Option D: This game does not have any Nash equilibrium.
- As shown above, (Bottom, Right) is a clear Nash equilibrium. Therefore, this statement is false.

In summary, option A presents an unsupported transfer claim, option B incorrectly ties maximin to Pareto efficiency, option C correctly identifies a Nash equilibrium that is Pareto efficient, and option D incorrectly denies existence of a Nash equilibrium. The true statement, based on the analysis of best responses and Pareto efficiency, is that the Nash equilibrium outcome is also the efficient outcome, which aligns with option C, while the other statements do not hold under standard game-theoretic definitions.

COMM_V 295 105 106 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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