1 General concept

An information set is a collection of possible states, histories, or observations that an agent cannot distinguish from one another at a given moment. It captures the information available to that agent and provides a formal way to represent limited perception, incomplete data, or hidden aspects of a situation.

The concept is used across mathematics and economics to describe decision-making when the full underlying state is not directly observable. Rather than assuming complete knowledge, an information set identifies the alternatives that remain plausible from the agent’s point of view.

1.1 Basic definition

In its simplest form, an information set is a subset of all possible states or histories that are indistinguishable to the decision-maker. If two situations generate the same evidence for the agent, they belong to the same information set.

This idea is often formalized as a partition of the state space or history space. Each element of the partition represents one informationally accessible class of possibilities.

1.2 Intuitive interpretation

Intuitively, an information set describes “what the agent knows at the moment.” It does not necessarily specify which state is true; instead, it records which states the agent cannot tell apart.

For example, a player in a card game may know the cards already revealed but not the opponent’s hidden hand. All hidden hands consistent with the visible cards form the player’s current information set.

1.3 Relation to knowledge and uncertainty

Information sets are closely connected to both knowledge and uncertainty. Knowledge narrows the range of possibilities, while uncertainty remains over the states inside the set.

The larger the information set, the less precise the agent’s information. A smaller set indicates more knowledge, and in the extreme case a singleton information set corresponds to full certainty about the relevant state.

2 Information sets in game theory

In game theory, information sets are used to describe what players can observe when making choices in strategic settings. They are especially important in extensive-form games, where play unfolds over time and later actions may depend on earlier hidden moves.

These sets allow analysts to distinguish between situations in which players have complete visibility and those in which some actions or outcomes are concealed.

2.1 Extensive-form games

Extensive-form games represent a game as a tree of possible histories. Each node corresponds to a point in the play of the game, and information sets group nodes that a player cannot distinguish when it is their turn to act.

This structure makes it possible to model sequential decision-making under imperfect information.

2.1.1 Nodes within an information set

A node within an information set is one of several possible positions in the game tree that look identical to the player whose turn it is. The player knows they are at one of these nodes, but not which one specifically.

Because of this uncertainty, a single action choice must be made across all nodes in the same set. The player’s strategy therefore cannot depend on the exact node if the node is not observable.

2.1.2 Player awareness and indistinguishability

An information set reflects the player’s awareness at a particular point in play. If two nodes lie in the same information set, the player has no way to distinguish them using the available information.

This indistinguishability shapes strategic reasoning. A player must choose actions based on what can be inferred from the visible history rather than on hidden details of the game state.

2.2 Perfect and imperfect information

A game has perfect information when every decision node is fully observed by the player who acts there. In that case, each information set contains only one node.

By contrast, imperfect information arises when some nodes are grouped together because the player lacks complete observation. Many games of card play, bargaining, and signaling involve imperfect information.

2.3 Strategy selection under information sets

A strategy in an extensive-form game specifies what a player will do at every information set they may face. Since the player cannot tell nodes in the same set apart, the same planned action must apply throughout that set.

This means strategies are defined over available information rather than over hidden states. The structure ensures that a strategy is feasible given the agent’s knowledge at the time of choice.

2.4 Beliefs and mixed strategies

When a player faces an information set containing multiple possible nodes, they may attach beliefs to those alternatives. These beliefs indicate how likely each node is, conditional on the information available.

Mixed strategies also interact with information sets by introducing randomness into action choice. Together, beliefs and mixed strategies help represent behavior when outcomes depend both on hidden information and on probabilistic decision rules.

3 Information sets in probability and decision theory

In probability theory and decision theory, information sets describe what events or variables an agent can observe. They provide the basis for conditional reasoning, where conclusions are drawn relative to partial knowledge.

This framework is useful whenever agents update beliefs after receiving information or make choices without observing the full underlying process.

3.1 Sigma-algebras and observable events

In measure-theoretic probability, an information set is often represented by a sigma-algebra of observable events. The sigma-algebra specifies which events can be distinguished using the available information.

Events outside that collection are not directly observable, so probabilities are evaluated relative to what the agent can see. This creates a formal link between information and measurable uncertainty.

3.2 Conditional probability

Conditional probability expresses the chance of an event given the information the agent has. The relevant conditioning event or sigma-algebra corresponds to the agent’s information set.

As the information set becomes more informative, conditional probabilities may change because fewer states remain possible. This makes conditioning a central tool for analyzing rational inference under uncertainty.

3.3 Bayesian updating

Bayesian updating describes how an agent revises beliefs when new information arrives. The original information set is replaced or refined by a new one, and prior probabilities are adjusted accordingly.

In this framework, information sets mark the boundaries of what is known at each stage. The updated belief distribution reflects both the prior and the newly observed evidence.

3.4 Partial observability

Partial observability occurs when an agent sees only some signals or outcomes rather than the full state. Information sets then collect all states consistent with those observations.

This approach is common in models where decisions must be taken with incomplete data. It allows analysts to study behavior when the true state is hidden but the observable consequences are not.

4 Information sets in dynamic optimization

In dynamic optimization, information sets determine what a decision-maker knows at each point in time when choosing a control or action. The agent’s current information influences feasible plans, expected payoffs, and future expectations.

These models are common in economics, operations research, and control theory, where choices unfold over time under uncertainty.

4.1 State space and histories

A dynamic model may describe the world using a state space or a sequence of histories. An information set identifies which states or histories remain possible given the observations made so far.

This is especially important when the current state is not directly observed. The agent must base decisions on the set of histories consistent with past signals and outcomes.

4.2 Feasible actions at a given time

At any time period, the available actions may depend on the information available to the agent. An action is feasible if it can be selected using only the knowledge contained in the current information set.

This requirement ensures that the decision rule does not rely on hidden variables. It also aligns the model with realistic constraints on perception and inference.

4.3 Information constraints in control problems

In control problems, the controller often does not observe the full system state. Instead, decisions are made using partial observations or summaries of past data.

Information constraints limit how precisely the controller can respond to the environment. The quality of the information set therefore affects both the control policy and the resulting performance.

4.4 Stochastic dynamic programming

Stochastic dynamic programming handles optimization problems in which future outcomes are random and information changes over time. The information set at each stage determines what the decision-maker knows when evaluating current and future actions.

The recursive structure of dynamic programming can be adapted to incomplete information by conditioning on the current information set. This allows the agent to plan optimally despite uncertainty about the true state.

5 Formal properties

Information sets have several formal properties that make them useful in mathematical analysis. These properties clarify how information is organized, refined, and related to rational choice.

They also help distinguish between different levels of knowledge and different informational environments.

5.1 Partition structure

In many settings, information sets form a partition of the state space or history space. Each state belongs to exactly one information set, and the sets do not overlap.

This partition structure provides a clean representation of indistinguishability. It ensures that the agent’s informational perspective is well defined at each point in time.

5.2 Refinement and coarsening

A refinement of an information set divides it into smaller, more precise subsets. Coarsening combines several sets into a larger, less informative one.

Refinement corresponds to receiving more detailed evidence, while coarsening corresponds to losing precision or using a more aggregated description. These operations are central to comparing levels of information.

5.3 Common knowledge and private information

Information sets help distinguish private information from shared knowledge. Private information is available only to one agent, whereas common knowledge is understood by all agents and is mutually recognized.

The structure of information sets can therefore influence coordination, signaling, and belief formation. Common knowledge often simplifies analysis, while private information introduces strategic uncertainty.

5.4 Consistency with rational behavior

A rational agent’s choices should be consistent with the information available to them. This means the chosen action should depend only on the current information set and not on hidden distinctions the agent cannot observe.

Such consistency is important in economic models because it prevents unrealistic decision rules. It also ensures that predictions reflect the informational limits built into the model.

6 Examples

Examples help illustrate how information sets work in practice. They show how the same formal idea appears in games, uncertainty, and sequential decisions.

6.1 Simple card game example

Suppose a player sees their own card but not the opponent’s. From the player’s viewpoint, all possible opponent cards compatible with the visible information belong to the same information set.

The player must choose an action without knowing which hidden card is actually held. Their strategy therefore depends on beliefs about the likelihood of each possibility.

6.2 Decision-making with hidden variables

Consider a manager choosing inventory levels without directly observing future demand. The manager may know past sales, seasonal indicators, and current stock, but not the exact demand shock.

All demand states consistent with the observed signals form the manager’s information set. Decisions are then based on expected outcomes conditional on that set.

6.3 Extensive-form game tree example

In a game tree, two different nodes may appear identical to a player if the relevant earlier move was concealed. Those nodes are placed in the same information set.

When the player reaches that stage, the same action must be chosen at each node in the set. This representation captures uncertainty about the path taken through the game.

Several nearby concepts are often used alongside information sets. They overlap in meaning but arise in different formal frameworks.

7.1 Information partition

An information partition is the collection of subsets that divides the state space according to what an agent can distinguish. It is one of the most common formal representations of information sets.

7.2 Knowledge set

A knowledge set is the set of states an agent considers possible given what they know. It is closely related to an information set and is often used in epistemic and decision-theoretic contexts.

7.3 Belief state

A belief state summarizes an agent’s probabilistic assessment of the possible underlying states. It is often constructed from an information set together with a probability distribution.

7.4 Observation function

An observation function maps hidden states to observed signals or messages. It helps generate information sets by identifying which states produce the same observation.