Extensive-Form Games
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# 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 Int...
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 :
| Component | Symbol | Meaning |
|---|---|---|
| Players | Set of players (plus "Nature" for chance moves) | |
| History | Sequence of actions from start to current point | |
| Player function | Which player moves at each history | |
| Information sets | What each player knows when moving | |
| Payoffs | 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 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 is a collection of decision nodes such that:
- Player moves at every node in the set
- The set of available actions is identical at each node
- Player 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).
SolutionDiagram(Diagram) Q2: How many strategies does each player have in the game above? SolutionPlayer 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
Note: When P1 plays Out, the payoff is (1,1) regardless of P2's choice. Join Discord PreviousMixed Strategy EquilibriumNextInformation Sets