Is sequence form the same as a sequence model in machine learning?

No. Here the term refers to a game representation, not a neural architecture for text or time series.

Can it handle hidden information?

Yes, in games satisfying the relevant assumptions, through correctly defined information sets.

Is it itself an equilibrium algorithm?

No. It supplies a representation used by algorithms; the suitable computation depends on the game and desired solution concept.

AI Glossary / Graphs and symbolic AI / Advanced

Sequence form

Definition

Sequence form is a compact representation of an extensive-form game that describes each player's strategy through probabilities associated with sequences of that player's own moves. In the standard formulation, players have perfect recall, and these probabilities satisfy linear consistency constraints.

What it means

An extensive-form game represents decisions over time, including what each player knows when choosing. A conventional normal-form representation lists complete contingent plans: what a player would do at every relevant information set, even ones never reached.

That list can grow enormously. Sequence form avoids explicitly enumerating every complete plan. It records the player's contribution to reaching particular sequences of their own actions, together with constraints that make those numbers a valid strategy representation.

A sequence is not simply a chronological transcript containing every player's moves. Chance and opponents also affect whether an outcome occurs, but their probabilities are handled separately. This distinction makes the representation useful for calculation and prevents a common misreading of its entries as complete outcome probabilities.

How it works

The empty sequence has realization weight one. Each action extends the player's own preceding sequence. At an information set, the weights of its possible extensions sum to the weight of that preceding sequence; every weight must be nonnegative.

For a behavioral strategy, an own-sequence weight is the product of that player's action probabilities along the sequence. Conditional action probabilities can be recovered by dividing an extension's weight by its positive parent weight. When the parent weight is zero, that division is unavailable; choices there require a valid convention without changing the realization plan's reachable behavior.

Perfect recall means the player does not forget their own past actions and information. It does not mean the player sees everything opponents do. Under the standard assumptions, sequence form preserves strategically relevant possibilities while keeping representation size tied to the explicit game tree. For two-player zero-sum games with perfect recall, equilibrium computation can be formulated through linear programming. These qualifications should accompany the computational claim.

Three practical examples

A game-theory researcher represents a small imperfect-information card game without listing every complete contingent plan.
A simulation team models a sequential competitive allocation exercise, making each participant's observations explicit before calculating strategies.
An AI course compares normal form with sequence form to show how representation affects the size of an equilibrium calculation.

Think of it this way

Think of recording a traveler's branch choices rather than writing a separate full instruction booklet for every possible combination of choices. The record still needs rules ensuring that later choices fit earlier ones.

The analogy only concerns representation. In a game, several decision makers and chance determine which branches are actually reached, and information sets can group situations that a player cannot distinguish.

Worked example

Own-action weights are not all outcome probabilities

In an invented game, player One first chooses A with probability 0.60 or B with probability 0.40. After A, suppose One may later choose C with conditional probability 0.25 or D with probability 0.75 at a particular information set.

The own-sequence weights are x(empty) = 1, x(A) = 0.60, x(B) = 0.40, x(AC) = 0.15 and x(AD) = 0.45. The constraints hold: 0.60 + 0.40 = 1 and 0.15 + 0.45 = 0.60. Recovering the conditional C probability gives 0.15/0.60 = 0.25.

If an opponent independently chooses the branch that reaches this later information set with probability 0.50, the corresponding full path probability for AC is 0.60 × 0.50 × 0.25 = 0.075, assuming no additional chance factors. That differs from x(AC). Also, the five sequence weights sum to 2.60, not one: they include nested prefixes rather than mutually exclusive outcomes.

Illustrative case study

Fictional case study: Alder Strategy Lab builds a teaching simulation in which two teams make decisions with partially hidden information. Its initial implementation enumerates complete plans, including choices at many situations that will never occur together.

The developers switch to sequence-form constraints and compare the resulting values with direct calculations on a tiny version of the game. They discover that one reported “outcome probability” included only one team's actions. Incorporating the other team's contribution fixes that interpretation.

The lab also checks information sets: a team must not receive different decision options based on hidden facts it cannot observe. The fictional lesson is that compact representation helps computation only when the underlying game and the meaning of its probabilities are modeled correctly.

Common misconceptions and mistakes

Every sequence weight is the probability of reaching that node. Other players and chance may also contribute to node reachability.
All sequence weights should sum to one. Prefixes and their extensions overlap; the relevant constraints are local flow equations.
Perfect recall means perfect information. A player can remember everything they observed while still lacking information about opponents.
Compact representation makes every large game easy. The explicit game tree may itself be huge, and solution properties depend on the game class.

Business use

Sequence form is mainly useful for specialist modeling and algorithm design, rather than everyday spreadsheet planning. It can help teams understand the computational implications of sequential strategic interaction and hidden information.

Before commissioning a model, define players, available actions, observations, chance events and payoffs. Ask which solution concept is being calculated and whether the zero-sum and perfect-recall assumptions are appropriate. An equilibrium of an artificial model does not automatically predict how real participants will behave or establish that the model captures the business problem.

Check your understanding

A player's sequence weight for AC is 0.15, but reaching the relevant branch also requires an opponent action with probability 0.50. With no other chance factors, is the full path probability 0.15?

Answer: No. Under the stated behavioral probabilities it is 0.15 × 0.50 = 0.075.

Related terms

Information sets
Linear programming
Extensive-form game
normal-form game
information set
perfect recall
behavioural strategy
realization plan
Nash equilibrium

Learn next

Read information set and perfect recall. Draw a two-stage game, mark exactly what each player observes, and write the nonnegative sequence weights and local sum constraints before attempting an equilibrium calculation.

Sources

Bernhard von Stengel, Equilibrium Computation for Two-Player Games, section 4: https://ranger.uta.edu/~weems/NOTES6319/PAPERSONE/vonStengel.pdf. The primary author's sequence-form definitions, realization-plan constraints, perfect-recall scope and two-person zero-sum linear-programming result were reviewed. The A/B/C/D calculation and fictional laboratory scenario are original teaching examples.

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