Is extensive form the same as normal form?

No. Normal form presents strategies and their payoffs; extensive form makes decision histories and information structure explicit.

Can it represent chance?

Yes. Chance nodes use specified probabilities for outcomes, distinct from players choosing actions.

Why are information sets necessary?

They prevent a strategy from conditioning on distinctions the acting player cannot observe. Without them, the represented game can give a player unavailable information.

AI Glossary / Graphs and symbolic AI / Intermediate

Extensive form

Definition

Extensive form is a way of representing a game that explicitly describes the sequence of decisions, who makes them, what each player knows when choosing, and the payoffs of possible outcomes. It commonly uses a rooted game tree, with branches for actions and terminal nodes for completed outcomes.

What it means

Many strategic situations depend on timing and observation. Choosing a price before seeing a competitor's offer is different from choosing after observing it. An extensive-form representation makes such distinctions part of the model instead of leaving them implicit.

The tree describes possible histories, not just a single predicted sequence. A player may have several decision points, and chance can determine events with specified probabilities. Information sets group decision points that a player cannot distinguish at the moment of choice. These sets are essential when the diagram's author can see more than the player actually observes.

How it works

Begin at the root, representing the initial situation. Assign each decision node to the player acting there, or to chance. Label outgoing edges with the available actions or chance outcomes. Give each terminal node a payoff for every player, using consistent preferences and units.

Then define information sets. At nodes in the same information set, a player faces the same available actions and cannot base its choice on knowing which of those nodes has occurred. A pure strategy specifies an action at every information set belonging to that player, including ones that may never be reached during play.

Solving the game requires additional concepts. Backward induction works for appropriate finite perfect-information games, while imperfect-information games require analysis that respects shared information sets. Drawing a tree does not itself select an equilibrium or validate assumptions about other players' preferences.

Three practical examples

1. A negotiation model records an initial offer, the recipient's response, and any later counteroffer. The order matters because later decisions can depend on earlier observed actions.

2. A bidding simulation represents private information and public announcements separately. Information sets prevent an AI bidder from using another participant's unobserved choice as if it were visible.

3. A game-playing agent considers a random deal followed by player decisions. Chance probabilities describe the deal, while player strategies govern deliberate actions; these two kinds of uncertainty should not be merged casually.

Think of it this way

Think of a branching script for a play, with a note beside each scene saying which actor must choose the next line and which earlier events that actor knows. The audience might see every scene, but an actor's character may not. The analogy explains information restrictions; formal games also need precise action sets, payoffs and strategy definitions that an ordinary script lacks.

Worked example

Consider a two-player game with payoffs written as (Player 1, Player 2). Player 1 chooses Stay or Enter. Stay ends the game with (2, 2). If Player 1 enters, Player 2 observes entry and chooses Resist or Welcome. Resist produces (0, 0); Welcome produces (3, 1).

At Player 2's decision node, Welcome gives that player 1 rather than 0, so it is the preferred action under the stated payoffs. Anticipating this choice, Player 1 compares Enter's payoff 3 with Stay's payoff 2 and chooses Enter. Backward induction therefore gives Enter followed by Welcome in this finite perfect-information example.

Player 2's strategy still specifies Welcome even when describing a hypothetical path where Player 1 stays and that decision node is never reached. Replacing Player 2's payoffs or hiding earlier information could change the analysis. The numerical result follows from this invented model; it is not a forecast of how real negotiators behave.

Illustrative case study

Fictional case study: Aspen Training builds a negotiation exercise for an AI assistant. The first diagram gives the assistant a different response depending on a private choice made by the other participant. A reviewer points out that the assistant receives no observation revealing that choice.

The designers connect the indistinguishable decision points into one information set and revise the strategy representation accordingly. This removes an unrealistic information advantage from the exercise. The case illustrates model correction, without claiming that the revised assistant is strategically optimal or effective in real negotiations.

Common misconceptions and mistakes

A path is not a complete strategy: it describes one realized history, while a strategy also specifies actions in other contingencies. A tree can represent simultaneous or hidden choices by using information sets, so a drawn sequence does not automatically imply that later players observe earlier actions.

Perfect information and perfect recall are different properties. Perfect recall concerns remembering previously known information and one's own actions; it does not reveal opponents' secrets. Also, a player's decision should not be replaced by a random probability distribution unless that distribution is justified as part of the chosen model or strategy.

Business use

Extensive form helps teams examine negotiation, competition and strategic interaction when timing affects incentives. Define what each participant observes, when commitments become visible and which choices remain available. A mistaken information assumption can matter more than adding another branch to the tree.

Keep payoff estimates and behavioral assumptions explicit. Use sensitivity analysis when preferences are uncertain, and distinguish a solution under rational-choice assumptions from a prediction about actual participants. For large games, the full tree may be impractical to enumerate, making abstraction and specialized representations important parts of the modeling work.

Check your understanding

Why does Player 2 choose Welcome in the worked example, even though Player 1 receives more from that outcome?

Answer: Player 2 compares its own payoffs: 1 from Welcome versus 0 from Resist. The model does not say Player 2 minimizes Player 1's payoff; this is not a zero-sum game.

Related terms

Game tree
Information sets
Strategy
Normal form
Perfect information
information set
payoff
backward induction
perfect recall
equilibrium

Learn next

Draw a small observed-action game and then hide one action by grouping the appropriate nodes into an information set. Compare the feasible strategies before studying equilibrium methods for imperfect-information games.

Imperfect information

Sources

Primary source: Ioannis Anagnostides, Carnegie Mellon Computational Game Solving, “Extensive-Form Games and Counterfactual Regret Minimization”: https://www.cs.cmu.edu/~sandholm/cs15-888F25/Lecture5-Slides.pdf . The source supports the tree, information-set and strategy definitions. The worked game and fictional case are original teaching examples.

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