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Extensive-Form Games

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Extensive-Form Games

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Now · 🎯 Learning Objectives

Extensive-Form Games

🎯 Learning Objectives

Draw and interpret game trees for sequential games
Distinguish perfect from imperfect information
Define information sets and their role in game analysis
Convert extensive-form games to normal form
Identify strategies in extensive-form games

4.1 Intuition: Sequential Decisions

Many strategic situations unfold over time — one player moves, then another responds. Chess, bargaining, entry deterrence: all involve sequential moves where later players observe earlier moves.

🔑 Key Insight: In sequential games, commitment, threats, and promises matter because players can react to what others have done.

4.2 Formal Definition

An extensive-form game is a tuple

\langle N, H, P, \iota, \{u_i\} \rangle

Component	Symbol	Meaning
Players	$N$	Set of players (plus "Nature" for chance moves)
History	$H$	Sequence of actions from start to current point
Player function	$P$	Which player moves at each history
Information sets	$\iota$	What each player knows when moving
Payoffs	$u_i$	Payoffs at terminal histories

Game Tree

An extensive-form game is represented as a tree: (Diagram) Components:

Nodes: Decision points for players
Branches: Available actions
Terminal nodes: Outcomes with payoffs
Information sets: Dashed lines connecting nodes the player cannot distinguish

4.3 Perfect vs. Imperfect Information

Perfect Information

Every player knows all previous moves when making a decision. Every information set contains exactly one node. Example: Chess, Go, tic-tac-toe.

Imperfect Information

Some players do not know all previous moves. Information sets contain multiple nodes the player cannot distinguish. Example: Card games (you don't know your opponent's hand), simultaneous moves (modelled as imperfect information — Player 2 doesn't know Player 1's choice).

Converting Simultaneous to Extensive Form

The simultaneous Prisoner's Dilemma can be represented as: (Diagram) The dashed line connecting the two P2 nodes indicates P2 doesn't know whether P1 played C or S — this is imperfect information.

4.4 Strategies in Extensive-Form Games

A strategy for a player specifies an action at every information set belonging to that player — even those that are never reached. Example: In the game tree above, Player 2's strategy is a pair

(a_C, a_S)

specifying what to do after each possible P1 move, even though only one will occur.

Normal-Form Representation

Every extensive-form game can be converted to normal form. For the Prisoner's Dilemma in extensive form:

P1 \ P2	(C, C)	(C, S)	(S, C)	(S, S)
C	(-5, -5)	(-5, -5)	(0, -10)	(0, -10)
S	(-10, 0)	(-1, -1)	(-10, 0)	(-1, -1)

P2's strategies: first action at the top (after C), second at the bottom (after S).

4.5 Information Sets

An information set for player

i

is a collection of decision nodes such that:

Player $i$ moves at every node in the set
The set of available actions is identical at each node
Player $i$ cannot distinguish between nodes in the set (Diagram)

⚠️ A player's strategy must specify the same action at every node in an information set — because the player can't tell which node they're at.

📊 Formula Summary

Concept	Definition
Perfect information	Every info set is a singleton
Imperfect information	Some info sets contain >1 node
Strategy	A plan mapping each info set to an action
Normal-form representation	Converting extensive form to payoff matrix

✅ Practice Questions

Q1: Draw the game tree for the following sequential game: Player 1 chooses In or Out. If Out, game ends with (1, 1). If In, Player 2 chooses Fight or Accommodate. If Fight, (0, 0); if Accommodate, (3, 2).

Solution

Diagram

(Diagram) Q2: How many strategies does each player have in the game above? Solution

Player 1: 2 strategies (In, Out). Player 2: 2 strategies (Fight after In, Accommodate after In) — but if Out is played, P2's choice doesn't matter. Q3: Convert the game in Q1 to normal form. Solution

> \begin{array}{c|cc} & \text{Fight} & \text{Accommodate} \\ \hline \text{In} & (0, 0) & (3, 2) \\ \text{Out} & (1, 1) & (1, 1) \end{array} >

Note: When P1 plays Out, the payoff is (1,1) regardless of P2's choice. Join Discord PreviousMixed Strategy EquilibriumNextInformation Sets