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arxiv: 1907.09106 · v1 · pith:P2YHT2NDnew · submitted 2019-07-22 · 💻 cs.GT · cs.AI· cs.LO

A Conceptually Well-Founded Characterization of Iterated Admissibility Using an "All I Know" Operator

Joseph Y. Halpern (Cornell University) , Rafael Pass (Cornell University) This is my paper

Pith reviewed 2026-05-24 18:07 UTC · model grok-4.3

classification 💻 cs.GT cs.AIcs.LO
keywords iterated admissibilityall I know operatorepistemic game theorylexicographic probability sequencesapproximate beliefSamuelson's concernprobability structuresweak dominance
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The pith

An 'all I know' operator characterizes iterated admissibility while addressing Samuelson's concern about higher-order beliefs.

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

The paper establishes a characterization of iterated admissibility that uses the 'all I know' operator to capture the idea that each agent knows only the relevant rationality assumptions at every level. This version resolves Samuelson's worry that higher-order reasoning would exclude the very strategies used to justify lower-level choices. The characterization is first stated with lexicographic probability sequences. It is then restated using approximate belief and approximately all I know so that the same outcome holds inside ordinary probability structures.

Core claim

Iterated admissibility is the set of strategy profiles that survive iterated deletion of weakly dominated strategies; it equals the profiles that arise when each player knows that all players are rational and that all they know is precisely this rationality fact, with the construction extended to all finite orders. The construction works directly in complete LPS structures and, after replacing exact belief with approximate belief, also works in standard probability structures.

What carries the argument

The 'all I know' operator, which asserts that an agent's information is exhausted by a given set of propositions about rationality and beliefs.

Load-bearing premise

The notions of approximate belief and approximately all I know can be defined inside ordinary probability structures so that the resulting type spaces still produce exactly the iterated-admissibility outcome without reintroducing Samuelson's problem.

What would settle it

A concrete finite game together with a probability structure in which the approximate all-I-know condition holds yet some player assigns positive probability to a strategy that was eliminated at an earlier round of weak dominance deletion.

read the original abstract

Brandenburger, Friedenberg, and Keisler provide an epistemic characterization of iterated admissibility (IA), also known as iterated deletion of weakly dominated strategies, where uncertainty is represented using LPSs (lexicographic probability sequences). Their characterization holds in a rich structure called a complete structure, where all types are possible. In earlier work, we gave a characterization of iterated admissibility using an "all I know" operator, that captures the intuition that "all the agent knows" is that agents satisfy the appropriate rationality assumptions. That characterization did not need complete structures and used probability structures, not LPSs. However, that characterization did not deal with Samuelson's conceptual concern regarding IA, namely, that at higher levels, players do not consider possible strategies that were used to justify their choice of strategy at lower levels. In this paper, we give a characterization of IA using the all I know operator that does deal with Samuelson's concern. However, it uses LPSs. We then show how to modify the characterization using notions of "approximate belief" and "approximately all I know" so as to deal with Samuelson's concern while still working with probability structures.

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.

Referee Report

1 major / 0 minor

Summary. The paper extends prior work on an 'all I know' epistemic characterization of iterated admissibility (IA). It first gives an LPS-based version that addresses Samuelson's concern (higher-order beliefs failing to consider strategies used at lower levels of the IA process). It then introduces notions of 'approximate belief' and 'approximately all I know' inside standard probability structures so that the resulting type spaces still deliver the IA outcome without reintroducing Samuelson's concern or generating higher-order inconsistencies.

Significance. If the constructions are sound, the work supplies a probability-based characterization of IA that is conceptually cleaner than the LPS version while still handling a noted conceptual objection. This would strengthen the epistemic foundations of iterated admissibility in the style of Brandenburger-Friedenberg-Keisler without requiring complete type structures.

major comments (1)
  1. [Abstract] Abstract (final paragraph): the central claim that the approximate-belief and approximately-all-I-know operators can be defined inside ordinary probability structures while still validating IA and avoiding both Samuelson's concern and higher-order inconsistencies is load-bearing, yet no model definitions, type-space construction, or verification that the fixed-point remains exactly the IA strategies are supplied; without these the soundness of the modification cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and comments on the manuscript. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final paragraph): the central claim that the approximate-belief and approximately-all-I-know operators can be defined inside ordinary probability structures while still validating IA and avoiding both Samuelson's concern and higher-order inconsistencies is load-bearing, yet no model definitions, type-space construction, or verification that the fixed-point remains exactly the IA strategies are supplied; without these the soundness of the modification cannot be assessed.

    Authors: The abstract is a concise summary of the paper's results. The full model definitions of approximate belief and approximately-all-I-know, the constructions of the relevant type spaces within standard probability structures, and the verification that the fixed point of the resulting epistemic conditions coincides exactly with the IA strategies (while avoiding Samuelson's concern and higher-order inconsistencies) are all supplied in the body of the manuscript, following the LPS-based characterization. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper builds a characterization of iterated admissibility first via the all-I-know operator in LPS structures (addressing Samuelson's concern) and then via newly introduced approximate-belief and approximately-all-I-know notions inside ordinary probability structures. These approximate notions are defined directly in the paper rather than being fitted to data or reduced to prior self-citations by construction. The central equivalence claims are therefore independent conceptual adjustments, not tautological renamings or self-referential fits. Self-citation to the authors' earlier all-I-know work supplies background but is not load-bearing for the new approximate operators or the final IA validation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The characterizations rest on standard epistemic-logic axioms for knowledge and belief operators plus the prior definition of the all I know operator; no free parameters, new physical entities, or ad-hoc postulates are described in the abstract.

axioms (1)
  • standard math Standard axioms of epistemic logic (knowledge and belief operators satisfy the usual modal properties)
    The all I know operator and its approximate variant are defined on top of these background properties.

pith-pipeline@v0.9.0 · 5751 in / 1383 out tokens · 35725 ms · 2026-05-24T18:07:39.336062+00:00 · methodology

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Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    C. E. Alchourr ´on, P. G¨ardenfors & D. Makinson (1985):On the logic of theory change: partial meet functions for contraction and revision. Journal of Symbolic Logic 50, pp. 510–530, doi:10.2307/2274239

    work page doi:10.2307/2274239 1985

  2. [2]

    L. E. Blume, A. Brandenburger & E. Dekel (1991): Lexicographic probabilities and choice under uncer- tainty. Econometrica 59(1), pp. 61–79, doi:10.2307/2938240

    work page doi:10.2307/2938240 1991

  3. [3]

    L. E. Blume, A. Brandenburger & E. Dekel (1991): Lexicographic probabilities and equilibrium refinements. Econometrica 59(1), pp. 81–98, doi:10.2307/2938241

    work page doi:10.2307/2938241 1991

  4. [4]

    Keisler (200 8)

    A. Brandenburger, A. Friedenberg & J. Keisler (2008): Admissibility in games . Econometrica 76(2), pp. 307–352, doi:10.1111/j.1468-0262.2008.00835.x

    work page doi:10.1111/j.1468-0262.2008.00835.x 2008

  5. [5]

    C. F. Camerer, T.-H. Ho & J.-K. Chong (2004): A cognitive hierarchy model of games. Quarterly Journal of Economics 119, pp. 861–897, doi:10.1162/0033553041502225. 232 A Conceptually Well-Founded Characterization of Iterated Admissibility

    work page doi:10.1162/0033553041502225 2004

  6. [6]

    Catonini & N

    E. Catonini & N. de Vito (2018): Cautious belief and iterated admissibility. Unpublished manuscript

    work page 2018

  7. [7]

    Darwiche & J

    A. Darwiche & J. Pearl (1997): On the logic of iterated belief revision . Artificial Intelligence 89(1–2), pp. 1–29, doi:10.1016/S0004-3702(96)00038-0

    work page doi:10.1016/s0004-3702(96)00038-0 1997

  8. [8]

    J. Y . Halpern (2003): Reasoning About Uncertainty . MIT Press, Cambridge, MA. A second edition was published in 2017

    work page 2003

  9. [9]

    J. Y . Halpern (2010): Lexicographic probability, conditional probability, and nonstandard probability . Games and Economic Behavior 68(1), pp. 155–179, doi:10.1016/j.geb.2009.03.013

    work page doi:10.1016/j.geb.2009.03.013 2010

  10. [10]

    J. Y . Halpern & G. Lakemeyer (2001): Multi-agent only knowing. Journal of Logic and Computation 11(1), pp. 41–70, doi:10.1093/logcom/11.1.41

    work page doi:10.1093/logcom/11.1.41 2001

  11. [11]

    J. Y . Halpern & R. Pass (2009): A logical characterization of iterated admissibility . In: Theoret- ical Aspects of Rationality and Knowledge: Proc. Twelfth Conference (TARK 2009) , pp. 146–155, doi:10.1145/1562814.1562836

    work page doi:10.1145/1562814.1562836 2009

  12. [12]

    J. Y . Halpern & R. Pass (2009): A logical characterization of iterated admissibility and extensive-form ra- tionalizability. Unpubished manuscript. A preliminary version, with the title ”A logical characterization of iterated admissibility”, appears in Proc. Twelfth Conference on Theoretical Aspects of Rationality and Knowledge (TARK), 2009, pp. 146–155

    work page 2009

  13. [13]

    Katsuno & A

    H. Katsuno & A. Mendelzon (1991): On the difference between updating a knowledge base and revising it. In: Principles of Knowledge Representation and Reasoning: Proc. Second International Conference (KR ’91), pp. 387–394

    work page 1991

  14. [14]

    H. J. Keisler & B.S. Lee (2015): Common assumption of rationality. Unpublished manuscript

    work page 2015

  15. [15]

    B. S. Lee (2016): Admissibility and assumption . Journal of Economic Theory 163, pp. 42–72, doi:10.1016/j.jet.2016.01.006

    work page doi:10.1016/j.jet.2016.01.006 2016

  16. [16]

    H. J. Levesque (1990): All I know: a study in autoepistemic logic. Artificial Intelligence 42(3), pp. 263–309, doi:10.1016/0004-3702(90)90056-6

    work page doi:10.1016/0004-3702(90)90056-6 1990

  17. [17]

    Lo (1999): Nash equilibrium without mutual knowledge of rationality

    K.C. Lo (1999): Nash equilibrium without mutual knowledge of rationality . Economic Theory 14(3), pp. 621–633, doi:10.1007/s001990050344

    work page doi:10.1007/s001990050344 1999

  18. [18]

    D. G. Pearce (1984): Rationalizable strategic behavior and the problem of perfection. Econometrica 52(4), pp. 1029–1050, doi:10.2307/1911197

    work page doi:10.2307/1911197 1984

  19. [19]

    Perea (2012): Epistemic Game Theory

    A. Perea (2012): Epistemic Game Theory . Cambridge University Press, Cambridge, U.K., doi:10.1017/CBO9780511844072

    work page doi:10.1017/cbo9780511844072 2012

  20. [20]

    Samuelson (1992): Dominated strategies and common knowledge

    L. Samuelson (1992): Dominated strategies and common knowledge. Games and Economic Behavior 4, pp. 284–313, doi:10.1016/0899-8256(92)90020-S

    work page doi:10.1016/0899-8256(92)90020-s 1992

  21. [21]

    Stahl (1995): Lexicgraphic rationalizability and iterated admissibility

    D. Stahl (1995): Lexicgraphic rationalizability and iterated admissibility. Economic Letters 47, pp. 155–159, doi:10.1016/0165-1765(94)00530-F

    work page doi:10.1016/0165-1765(94)00530-f 1995

  22. [22]

    Yang (2016): Weak assumption and iterative admissibility

    C. Yang (2016): Weak assumption and iterative admissibility. Journal of Economic Theory 158, pp. 87–101, doi:10.1016/j.jet.2015.03.009

    work page doi:10.1016/j.jet.2015.03.009 2016