Histories and Information in Extensive-Form Games

In many real-life situations, students, decisions happen one after another, not all at once. 🎯 You may speak, wait, observe what others do, and then choose your next move. This is exactly the kind of situation studied in extensive-form games. In this lesson, you will learn how history records what has happened so far, how information tells players what they know when they move, and how both shape the strategies players can use.

What you will learn

By the end of this lesson, students, you should be able to:

• Define a history in an extensive-form game.
• Identify what players know at decision points.
• Describe how information affects strategy choice.

Why histories matter in sequential games

Think about a game like chess, bargaining, or even choosing whether to study after seeing your quiz grade. Each new choice depends on what happened earlier. In game theory, the sequence of past actions is called a history.

A history is the record of moves that have been made up to a certain point in the game. It includes everything that has already happened, but not future actions. For example, in a simple sequential game:

• Player 1 chooses Left or Right.
• Then Player 2 chooses after seeing Player 1’s move.

If Player 1 chose Left, then the history at Player 2’s decision point is the sequence of actions that led there, such as $\text{Left}$.

Histories are important because later choices are based on earlier ones. If a company sees a competitor cut prices, it may respond differently than if prices stayed high. The past shapes the present. 📈

A terminal history is a history where the game ends. A nonterminal history is a history where the game continues and another player may still move. These ideas help us describe the game tree accurately.

Decision points and what players know

In an extensive-form game, a player does not choose randomly from nowhere. Each choice happens at a decision point, sometimes called a node in the game tree. At that point, the player may know some past actions, but not necessarily all of them.

The key idea is this: information determines what the player can condition their decision on.

Suppose a student group project has two steps:

1. One student chooses whether to start early or wait.
2. Another student decides whether to help, but may or may not know what the first student did.

If the second student can observe the first student’s action, then the history is known exactly. If not, the second student has less information and must choose without knowing the full past.

In game theory, the set of histories a player cannot distinguish between is called an information set. At any one decision point, if a player cannot tell exactly which history occurred, those possible histories are grouped together in the same information set. This means the player knows that one of those histories has happened, but not which one.

For example, if a driver chooses between two routes and then another driver responds without knowing which route was taken, the responding driver’s information set contains both possible histories. The response must be based on that limited knowledge.

History, information sets, and the game tree

Extensive-form games are usually drawn as game trees 🌳. The tree starts with the initial state and branches out as players make choices. Each path from the root to a point in the tree represents a history.

Here is the basic idea:

• The root represents the empty history, before any action has occurred.
• Each branch adds a new action to the history.
• A node represents a decision point reached after some history.
• A player at a node chooses an action based on what they know.

Let’s use a simple example.

Example: Two-step pricing decision

A store owner, Player 1, chooses between:

• $H$ = high price
• $L$ = low price

Then a competitor, Player 2, chooses between:

• $C$ = compete aggressively
• $N$ = do nothing

If Player 2 sees Player 1’s price, then Player 2 knows whether the history is $H$ or $L$. If Player 2 does not observe it, then both histories are in the same information set.

This matters because Player 2’s action may depend on the observed price. If the price is high, competing aggressively might be profitable. If the price is low, the best response may be different.

So the game tree is not just about what can happen. It is also about what each player knows when making choices.

What a strategy must include

A strategy in an extensive-form game is a complete plan that tells a player what to do at every information set where they might move. This is a big difference from just choosing one action.

Why? Because a player may face different situations depending on the history. A strategy must say what to do in each possible situation.

For example, imagine Player 2 may move after either $A$ or $B$ has happened, but Player 2 can tell the difference. Then Player 2’s strategy must specify:

• what to do after history $A$
• what to do after history $B$

If Player 2 cannot tell the difference, then the strategy must choose one action for the whole information set.

This shows how information shapes strategy choice. More information usually allows more tailored behavior. Less information means the player must make a plan that works across multiple possible histories.

A practical analogy is a soccer goalkeeper. The keeper reacts differently if they can see the shooter’s body angle and foot position. If they cannot see clearly, they must choose a move based on incomplete information. The available strategy depends on what the keeper knows at that moment. ⚽

Information and rational choice

Information affects not only what actions are available, but also which action is reasonable.

Suppose a player knows that the other player has already chosen a strong move. Then the best response may be different than if the other player chose a weak move. In this way, past actions change the meaning of future choices.

A player with more information can often make choices that are more precise. But more information does not automatically mean a better outcome in every case. It just means the player can condition their choice on a more detailed history.

Here is a simple example:

• If Player 1 invests in a project, Player 2 may want to cooperate.
• If Player 1 does not invest, Player 2 may walk away.

If Player 2 sees whether the investment occurred, then Player 2 can choose differently in each case. If Player 2 cannot see it, then Player 2 must pick one action without knowing the true history.

This is why economists care about information: it affects incentives, predictions, and equilibrium outcomes.

Perfect and imperfect information

Two important ideas in extensive-form games are perfect information and imperfect information.

• A game has perfect information if, at every decision point, the player who moves knows the full history of past actions.
• A game has imperfect information if at least one player must choose without knowing the full history.

Chess is a classic example of a game with perfect information because both players see all moves. Poker is an example of imperfect information because players do not see the other players’ private cards.

In imperfect-information games, information sets become very important. A player may know the structure of the game and the possible moves, but not the exact history. That is why many strategic choices in real life involve guessing, inference, and beliefs.

Real-world example: negotiation with hidden information

Imagine two classmates negotiating who will present first in a project. One student can either volunteer or wait. If one volunteers, the other sees it and responds. But if both send messages privately first, one student may not know what the other has promised.

The second student’s choice depends on the history they can observe. If they know the first student volunteered, they may choose differently than if they do not know. The decision is shaped by both the actual past and the information available at the moment of choice.

This kind of situation appears in business deals, job searches, political campaigns, and online auctions. The past matters, but only the part that is observed can directly affect the player’s action.

Conclusion

Histories and information are central to understanding extensive-form games. A history is the sequence of actions that has happened so far. A player’s information tells them which histories they can distinguish when it is their turn to move. If a player can tell the exact past, they have perfect information at that point. If they cannot, the possible histories are grouped into an information set.

Because strategies must specify what to do at every information set, information directly affects strategy choice. In sequential games, the best action is often not just about the current move. It depends on what happened before and what the player knows now. That is the power of history in game theory. ✅

Study Notes

• A history is the sequence of actions that have occurred up to a point in an extensive-form game.
• The empty history is the starting point before any action is taken.
• A terminal history ends the game; a nonterminal history allows play to continue.
• Players make choices at decision points or nodes in a game tree.
• What a player knows when choosing is called their information.
• An information set groups together histories that a player cannot distinguish.
• A strategy must specify an action at every information set where the player may move.
• In perfect information games, players know the full history at each move.
• In imperfect information games, at least one player does not know the full history.
• More information can change which strategy is best, because future choices depend on the observed past.
• Real-world examples include pricing, bargaining, sports, poker, and project planning.