How can we write down a normal form for this following game with information set?

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    $\begingroup$ A player cannot have different actions at different nodes of an information set. In particular, Player 2 cannot have {T,B} at the top node of his information set and {L,R} at the bottom node. $\endgroup$ – Herr K. Nov 28 '19 at 3:31
  • $\begingroup$ Thank you for the information. Would the current game work? $\endgroup$ – Aqqqq Nov 28 '19 at 11:12
  • $\begingroup$ @HerrK. So the strategies to be included in the normal form is: Player 1: Up, Uq, Dp, Dq; Player 2: T, B? $\endgroup$ – Aqqqq Nov 28 '19 at 11:16

The dimension of the normal form game derived from an extensive form is given by the number of pure strategies each player has.

Generally speaking, the number of pure strategies a player has in an extensive form game equals the product of the number of actions at each information set where she moves. Suppose a player moves at $N$ information sets on a game tree, and at each information set $n$ there are $m_n$ actions, then she has $m_1\times m_2\times\cdots\times m_N$ number of pure strategies.

In your example, $N=2$, $m_1=m_2=2$ for player 1; and $N=1$, $m_1=2$ for player 2. So the normal form game corresponding to your game tree should be a $4\times2$ matrix.

This note describes a step-by-step procedure that's relatively easy to follow.

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  • $\begingroup$ The note lists the same steps that I do... $\endgroup$ – Giskard Nov 28 '19 at 18:08
  • $\begingroup$ @Giskard: Yes, but with a little bit more detail :) $\endgroup$ – Herr K. Nov 28 '19 at 19:16

Step 1.

Determine strategies.

Step 2.

Calculate payoffs for strategy profiles.

Step 3.

Write normal form.

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  • $\begingroup$ Would it be the same as if the information set was not there? (I mean "as if the two nodes in the information set are in separate information set"). $\endgroup$ – Aqqqq Nov 27 '19 at 22:16
  • $\begingroup$ No. If you do not understand what strategies are, please consult your textbook. $\endgroup$ – Giskard Nov 27 '19 at 22:17
  • $\begingroup$ I only know that their strategies are respectively: Player 1: Up, Uq, Dp, Dq; Player 2: TL, TR, BL, BR. Can you tell me my mistake? $\endgroup$ – Aqqqq Nov 27 '19 at 22:21
  • $\begingroup$ Yes: your mistake is that these are not the strategies with imperfect information. Please consult your textbook. $\endgroup$ – Giskard Nov 27 '19 at 22:23
  • $\begingroup$ Does it mean that I need to make assumption about belief before deriving the appropriate strategy? $\endgroup$ – Aqqqq Nov 27 '19 at 22:32

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