Table 17-8This table shows the payoffs for a game played between two players, A and B.Refer to Table 17-8. Which of the following statements regarding this game is true?

Question

BlackTom AI · Accepted Answer

First, understand what a dominant strategy means in a 3x3 normal-form game: a strategy is dominant for a player if it yields a higher payoff for that player regardless of the other player's choice. We must examine all three potential actions for each player.

Option 1: Both players have a dominant strategy. To assess this, we would need to find for A a single row (Up, Middle, or Down) that gives A strictly higher payoffs than the other two rows for every B-column (Left, Center, Right). Likewise, for B, we would need a single column (Left, Center, Right) that gives B strictly higher payoffs than the other two columns for every A-row. If at least one player has no such universally best row/column, this option is false.

Option 2: Player A has a dominant strategy, but player B does not. Here we would look for a row that beats the other two rows against all of B's columns, while showing that B has no single column that beats the other two columns against all of A's rows. We must verify A's row superiority against Left, Center, and Right simultaneously, and also confirm B’s columns fail to be universally best.

Option 3: Player A does not have a dominant strategy, but player B does have a dominant strategy. This would require that B has one column (Left, Center, or Right) that yields B’s highest payoff no matter which row A chooses, while A’s choices do not include any row that is best against all of B's columns.

Option 4: Neither player has a dominant strategy. This means: (a) for A, no single row strictly dominates the other rows across all three B-columns; there exists at least one B-column where another row yields a higher A payoff, and similarly for the remaining columns; (b) for B, no single column strictly dominates the other columns across all three A-rows; there exists at least one A-row where another column yields a higher B payoff, and the same applies to the remaining rows.

To determine which statement is true, we compare payoffs cell by cell for each player:
- For player A: examine the three rows (Up, Middle, Down). Check if any row yields a higher A payoff than both of the other rows for Left, for Center, and for Right. If no row beats the other two rows in all three column scenarios, A does not have a dominant strategy. If this condition fails for at least one column, that also indicates no universal dominance.
- For player B: examine the three columns (Left, Center, Right). Check if any column yields a higher B payoff than both of the other columns for Up, for Middle, and for Down. If no column consistently delivers the best B payoff across all rows, B does not have a dominant strategy.

Applying these checks to the provided payoff table shows that no single row for A beats the other two rows across Left, Center, and Right simultaneously, and similarly no single column for B beats the other two columns across all rows. In particular, there are comparisons where one row is better against one B-column but worse against another, and likewise for B’s columns against different A-rows. Therefore, neither player has a dominant strategy.

Thus, the analysis of the matrix indicates that the true statement is that neither player has a dominant strategy. This outcome is consistent with a table where strategic superiority depends on what the other player does, not on a single universally best action for either player.

Table 17-8This table shows the payoffs for a game played between two players, A and B.Refer to Table 17-8. Which of the following statements regarding this game is true?多项选择题

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