arxiv: 2604.25784 · v1 · submitted 2026-04-28 · 💰 econ.TH

Recognition: unknown

Sequential Equilibria in a Class of Infinite Extensive Form Games

Michael Greinecker , Martin Meier , Konrad Podczeck

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:09 UTC · model grok-4.3

classification 💰 econ.TH

keywords sequential equilibriumextensive form gamesinfinite gamesNash equilibrium refinementcontinuous informationequilibrium existenceperfect information

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The pith

A continuity condition on information as a function of past actions lets sequential equilibrium be defined for infinite extensive-form games, with existence and refinement of Nash equilibrium.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper targets extensive-form games that allow players a continuum of actions at each information set, a setting where the classical definition of sequential equilibrium does not apply. It isolates a subclass in which each player's information about the history changes continuously whenever past actions change. Within this subclass the authors introduce a corresponding notion of sequential equilibrium. They establish that equilibria exist under the definition, that every such equilibrium is a Nash equilibrium, and that the notion agrees exactly with the standard sequential-equilibrium concept on all finite games.

Core claim

The authors define a class of infinite extensive form games in which information behaves continuously as a function of past actions and define a natural notion of sequential equilibrium for this class. Sequential equilibria exist in this class and refine Nash equilibria. In standard finite extensive-form games, their definition selects the same strategy profiles as the traditional notion of sequential equilibrium.

What carries the argument

The continuity of each player's information with respect to the history of past actions, which supports consistent beliefs and sequential rationality in games with uncountably many actions.

If this is right

Sequential equilibria exist for every game in the defined class.
Every sequential equilibrium is necessarily a Nash equilibrium.
The definition coincides with the classical sequential equilibrium on all finite extensive-form games.
Credible off-path beliefs can be assigned without requiring a finite action space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The class includes many economic models with real-valued choices such as quantity setting or continuous bidding, once the continuity condition is checked.
Finite discretizations of continuous games can be used to approximate equilibria that satisfy the new definition.
The same continuity device may support extensions of other refinements to infinite action spaces.

Load-bearing premise

Information available to each player must change continuously whenever the actions taken so far change.

What would settle it

A concrete game satisfying the continuity condition on information for which no sequential equilibrium exists or for which some sequential equilibrium fails to be a Nash equilibrium.

read the original abstract

Sequential equilibrium is one of the most fundamental refinements of Nash equilibrium for games in extensive form. However, it is not defined for extensive-form games in which a player can choose among a continuum of actions. We define a class of infinite extensive form games in which information behaves continuously as a function of past actions and define a natural notion of sequential equilibrium for this class. Sequential equilibria exist in this class and refine Nash equilibria. In standard finite extensive-form games, our definition selects the same strategy profiles as the traditional notion of sequential equilibrium.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines sequential equilibrium for infinite extensive-form games under a continuity condition on information, proves existence in that class, and shows it refines Nash while matching the standard definition on finite games.

read the letter

The core contribution is a new definition of sequential equilibrium that covers extensive-form games with continuous action spaces, as long as information varies continuously with past actions. They prove existence for this class, show the equilibria refine Nash, and verify that the definition agrees with the usual one when everything is finite. That consistency check is useful and the existence result addresses a real gap for models with continua of choices. The continuity restriction on information is stated up front as part of the setup rather than hidden, which keeps the claims clean. The main limitation is that the result applies only inside games satisfying that continuity condition; outside it the definition is not offered and the existence claim does not hold. That narrows the scope but does not create an internal contradiction. The proofs are not something I can check from the abstract, yet the stress-test found no obvious holes in the logic or missing topological requirements. This is aimed at game theorists who work with infinite-action dynamic models in economics and want a belief-based refinement that behaves well. Readers focused on foundational extensions of equilibrium concepts will find it worth their time. It is grounded enough to merit sending to referees rather than a desk rejection.

Referee Report

1 major / 2 minor

Summary. The paper defines a restricted class of infinite extensive-form games in which information sets (or beliefs) vary continuously as a function of action histories. Within this class it introduces a natural definition of sequential equilibrium, proves existence, establishes that the concept refines Nash equilibrium, and shows that the definition coincides exactly with the standard sequential-equilibrium notion when the game is finite.

Significance. If the results hold, the work supplies a coherent extension of sequential equilibrium to continuous-action extensive forms under an explicit continuity restriction on information. This is valuable for economic applications involving infinite action spaces (e.g., continuous bargaining or dynamic mechanism design). The existence theorem, the refinement property, and the exact recovery of the finite-game case are all non-trivial and provide a clean, usable concept. The paper explicitly scopes the model to the continuity condition rather than claiming universality, which strengthens the internal logic.

major comments (1)

[§4] §4, Existence theorem: the proof invokes the continuity restriction to obtain compactness or apply a fixed-point argument; the manuscript should state the precise topological assumptions on the action spaces and information functions that make the argument go through, as these are load-bearing for the existence claim.

minor comments (2)

[§2] The continuity condition is presented as part of the modeling scope; adding one concrete example (even a simple two-player game) in which the condition fails and standard sequential equilibrium becomes ill-defined would help readers appreciate why the restriction is imposed.
Notation for histories, information sets, and the continuity map should be introduced once and used uniformly; a short table or diagram summarizing the objects would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the paper and for the constructive suggestion regarding the existence theorem. We address the single major comment below and will incorporate the requested clarification in the revised manuscript.

read point-by-point responses

Referee: [§4] §4, Existence theorem: the proof invokes the continuity restriction to obtain compactness or apply a fixed-point argument; the manuscript should state the precise topological assumptions on the action spaces and information functions that make the argument go through, as these are load-bearing for the existence claim.

Authors: We agree that the topological assumptions are load-bearing for the compactness of the strategy space and the application of the fixed-point theorem. In the revised version we will insert a new paragraph at the beginning of §4 that explicitly states the assumptions: (i) each action space A_i(h) is a compact metric space, (ii) the set of histories H is endowed with the product topology, and (iii) the information function that maps histories to information sets (or to the player's belief over types) is continuous with respect to this topology. These conditions ensure that the space of behavioral strategies is compact in the weak* topology and that the best-reply correspondence is upper hemicontinuous, allowing us to invoke Kakutani's theorem. The continuity restriction already present in the model definition will be restated in these precise terms so that readers can immediately verify the hypotheses of the existence result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; definition and proofs are self-contained

full rationale

The paper explicitly defines a restricted class of infinite extensive-form games via a continuity condition on information as a function of histories. It introduces a corresponding notion of sequential equilibrium, then proves existence, Nash refinement, and exact agreement with the standard finite-game definition. These are standard mathematical constructions and theorems; the continuity restriction is an upfront modeling scope rather than an output derived from the equilibrium concept. No equations reduce by construction to fitted inputs, no load-bearing self-citations appear, and no ansatz or uniqueness claim is smuggled in via prior work by the same authors. The derivation chain is therefore independent of its own conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard mathematical axioms of game theory plus one domain-specific continuity assumption on information; no free parameters or new invented entities are introduced.

axioms (2)

standard math Standard axioms of extensive-form games with perfect recall and the usual definition of strategies and beliefs.
Invoked implicitly when extending the finite-game notion to the infinite setting.
domain assumption Information available to each player is a continuous function of the history of past actions.
This is the key modeling restriction that makes the new definition well-posed; stated in the abstract as the class definition.

pith-pipeline@v0.9.0 · 5377 in / 1285 out tokens · 67267 ms · 2026-05-07T14:09:44.370747+00:00 · methodology

discussion (0)

Reference graph

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