# 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. – Herr K. Nov 28 '19 at 3:31
• Thank you for the information. Would the current game work? – Aqqqq Nov 28 '19 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? – 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.

• The note lists the same steps that I do... – Giskard Nov 28 '19 at 18:08
• @Giskard: Yes, but with a little bit more detail :) – Herr K. Nov 28 '19 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"). – Aqqqq Nov 27 '19 at 22:16
• No. If you do not understand what strategies are, please consult your textbook. – Giskard Nov 27 '19 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? – Aqqqq Nov 27 '19 at 22:21
• Yes: your mistake is that these are not the strategies with imperfect information. Please consult your textbook. – Giskard Nov 27 '19 at 22:23
• Does it mean that I need to make assumption about belief before deriving the appropriate strategy? – Aqqqq Nov 27 '19 at 22:32