A formal game-theory preprint proposes agnostic sequential equilibrium, or ASE, a refinement for judging strategies in imperfect-information games. The proposal is designed to avoid imposing one arbitrary off-equilibrium belief system: it asks whether a prescribed action can be profitably replaced under any minimally sufficient explanation for reaching an information set.
This is a methods result, not an experiment. The unit of analysis is formal strategy behavior in extensive-form games. The setting includes finite game trees, possible chance nodes, perfect recall, and either pure or behavior strategies. No empirical participants, observations, or dataset are involved.
A different way to handle beliefs
Off-equilibrium beliefs are the paper’s central concern. In a dynamic game, they are the beliefs assigned after a path that the strategy profile does not prescribe. Sequential equilibrium, perfect Bayesian equilibrium, and related refinements serve as theoretical comparators. ASE is proposed to avoid making one arbitrary off-equilibrium belief system the foundation of the test.
To build the test, the paper defines strong consistency. Strongly consistent beliefs are generated from action sets that are minimally sufficient to make an information set reachable. In ordinary terms, the construction keeps the action sets needed to account for reaching that point rather than fixing one broader belief system in advance.
The resulting belief sets have a precise relationship. At every information set, strongly consistent beliefs are contained in fully consistent beliefs, and fully consistent beliefs are contained in the convex hull of the strongly consistent beliefs. The convex hull is the range formed by combining those strongly consistent beliefs.
What the proposed test requires
The paper argues that strongly consistent beliefs are structurally consistent and do not require future deviations in the continuation game. The ASE condition requires that the prescribed action cannot be profitably replaced at any information set under any minimally sufficient action set that makes that information set reachable.
A theorem gives the condition a local form, sometimes described as a one-deviation property. Checking each information set suffices to rule out profitable changes to a player’s strategy at that information set and at all later information sets.
Another result connects ASE to sequential equilibrium. A strategy profile is an ASE exactly when it is a sequential equilibrium for every belief system fully consistent with that profile. In practical terms, the profile must satisfy the test across all such fully consistent systems rather than under one selected system.
The framework’s boundaries
The definition applies without modification to games with continuum action spaces, where the available actions form a continuum. The paper also says that this scope does not overcome all conceptual difficulties in those games.
Strategic reasoning creates another qualification. Restrictions based on strategic interests may invalidate strongly consistent beliefs or prune the belief set. The analysis notes that pruning can enlarge the ASE set, so removing beliefs can enlarge, rather than shrink, the set of qualifying strategy profiles.
When no single profile survives
The paper’s set-valued extension is called agnostic sequential polyequilibrium, or ASPE. It works with a nonempty set of strategy profiles rather than requiring one profile. A strategy excluded from that set must have a retained substitute that performs at least as well under every relevant profile and minimally sufficient action set. When the ASPE is a singleton, it is exactly an ASE.
ASPE always exists because the set containing all strategy profiles is a trivial ASPE, even when a single ASE does not exist. The existence result is therefore set-valued rather than a guarantee that one ASE can be selected.
An illustrative game shows the distinction. When α > 2, no ASE exists, but a nontrivial ASPE fixes player 1 at Out and leaves player 2’s strategy unspecified. The example shows how a set-valued answer can retain a meaningful strategy outcome without specifying every player’s strategy.
A formal proposal, not a behavioral test
The result should be read as a formal refinement, not as evidence about behavior. Its unit of analysis is strategy behavior in formal games, and the supplied analysis includes no empirical participants, observations, or dataset. The conclusions are conditional on the formal assumptions, including finite game trees and perfect recall where specified.
The paper presents strategic-interest restrictions and continuum-action games as areas where conceptual refinement is still needed. It also shows that a single ASE may not exist in some games, which is the problem ASPE addresses with a set-valued solution.
The supplied document reports no funding or funder statement. Its metadata classifies the item as an arXiv preprint, while the front matter presents it as an authored manuscript.
Paper data and sources
Original title: Agnostic Sequential Rationality
Authors: Igal Milchtaich
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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