# How to write down a normal form of the game with information set?

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

(The picture is modified from a picture from Wikipedia.)

• 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. Commented Nov 28, 2019 at 3:31
• Thank you for the information. Would the current game work? Commented Nov 28, 2019 at 11:12
• @HerrK. So the strategies to be included in the normal form is: Player 1: Up, Uq, Dp, Dq; Player 2: T, B? Commented Nov 28, 2019 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.

• The note lists the same steps that I do... Commented Nov 28, 2019 at 18:08
• @Giskard: Yes, but with a little bit more detail :) Commented Nov 28, 2019 at 19:16

Step 1.

Determine strategies.

Step 2.

Calculate payoffs for strategy profiles.

Step 3.

Write normal form.

• 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"). Commented Nov 27, 2019 at 22:16
• No. If you do not understand what strategies are, please consult your textbook. Commented Nov 27, 2019 at 22:17
• 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? Commented Nov 27, 2019 at 22:21
• Yes: your mistake is that these are not the strategies with imperfect information. Please consult your textbook. Commented Nov 27, 2019 at 22:23
• Does it mean that I need to make assumption about belief before deriving the appropriate strategy? Commented Nov 27, 2019 at 22:32