The table below is not a game in strategic form because the payoffs should be different for the different actions U and D. [table]
U | (1,1) | (0,0) D | (0,0) | (0,0)
[/table]

Question

BlackTom AI · Accepted Answer

First, consider what it means for a table to be in strategic (normal) form. A strategic-form representation lists each player's strategies (rows and columns) and the corresponding payoffs for every combination of those strategies. It does not require that the payoffs differ across actions within a single player's strategy; it only requires that a payoff for each strategy profile is specified.

Option 1: True. The statement says the table is not a game in strategic form because the payoffs should be different for the different actions U and D. This claim is incorrect. A strategic-form game does not mandate that payoffs differ across the actions within a single row (or column). It only requires that payoffs are defined for each combination of actions. Here, the table does provide payoffs for all profiles: (U,U)=(1,1), (U,D)=(0,0), (D,U)=(0,0), (D,D)=(0,0). Even though some payoffs are identical across different profiles, the representation is still a valid normal-form (strategic-form) game.

Option 2: False. This is the correct choice because the reason given in the statement (that payoffs must be different for U and D) is not a requirement for strategic-form games. A game can have identical payoffs in multiple cells and still be represented in strategic form. The only requirement is that a payoff pair is specified for every combination of strategies, which this table satisfies. The left-hand payoffs depend on the joint strategies, and the table indeed provides those payoffs for all possible actions, even if some are the same across some cells.

Why the other potential concerns are incorrect:
- The possibility that there are no distinct best responses due to identical payoffs does not disqualify a representation from being strategic form; it may simply indicate a degenerate or coordination-type game.
- The structure shown still aligns with a bi-matrix game where players’ payoffs are determined by the chosen actions U or D, and the payoffs in each cell are specified, satisfying the formal definition of strategic form.

In summary, the assertion in the question is false because strategic-form games do not require payoffs to differ across the actions U and D; the provided table is indeed a valid strategic-form representation.

The table below is not a game in strategic form because the payoffs should be different for the different actions U and D. [table] U | (1,1) | (0,0) D | (0,0) | (0,0) [/table]判断题

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