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## Lexicographic probability, conditional probability, and nonstandard probability (2001)

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Venue: | In Theoretical Aspects of Rationality and Knowledge: Proc. Eighth Conference (TARK 2001 |

Citations: | 24 - 2 self |

### Citations

744 |
A Mathematical Introduction to Logic
- Enderton
- 1972
(Show Context)
Citation Context ...e integers to be a model that contains the integers and satisfies every property of the integers expressible in first-order logic. It follows easily from the compactness theorem of first-order logic [=-=Enderton 1972-=-] that, given an ordinal ff, there exists a nonstandard model of the integers that includes elements n fi , fi ! ff, such that n 0 = 0 and n fi ! n fi 0 if fi ! fi 0 . 2 2 The compactness theorem says... |

701 |
Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games
- Selten
- 1975
(Show Context)
Citation Context ...burger, and Dekel 1991b; Brandenburger, Friedenberg, and Keisler 2008; Fudenberg and Tirole 1991; Hammond 1994; Hammond 1999; Kohlberg and Reny 1997; Kreps and Wilson 1982; Myerson 1986; Selten 1965; =-=Selten 1975-=-]). It also arises in the analysis of conditional statements by philosophers (see [Adams 1966; McGee 1994]), and in dealing with nonmonotonicity in Artificial Intelligence (see, for example, [Lehmann ... |

603 | A treatise on probability - Keynes - 1921 |

410 | A definition of subjective probability.
- Anscombe, Aumann
- 1963
(Show Context)
Citation Context ...> 0. 5As pointed out by Adam Brandenburger and Eddie Dekel, this notion of equivalence is essentially the same as one implicitly used by BBD. They work with preference orders on Anscombe-Aumann acts [=-=Anscombe and Aumann 1963-=-], that is, functions from states to probability measures on prizes. Fix a utility function u on prizes. Then take ν1 ∼u ν2 if the preference order on acts generated by ν1 and u is the same as that ge... |

361 |
Reasoning About Uncertainty.
- Halpern
- 2003
(Show Context)
Citation Context ...ence of events with respect to a probability measure can be defined in any of three equivalent ways. Unfortunately, these definitions are not equivalent for other representations of uncertainty (see [=-=Halpern 2003-=-, Chapter 4] for a general discussion of this issue). The situation is perhaps simplest for nonstandard probability measures. 7 In this case, the three notions coincide, for exactly the same reasons a... |

277 | What Does a Conditional Knowledge Base Entail
- Lehmann, Magidor
- 1992
(Show Context)
Citation Context ...1a; Blume, Brandenburger, and Dekel 1991b; Hammond 1994; Myerson 1986]), the analysis of conditional statements (see [Adams 1966; McGee 1994]), and in dealing with nonmonotonicity (see, for example, [=-=Lehmann and Magidor 1992-=-]). There have been various attempts to deal with the problem of conditioning on events of measure 0. Perhaps the best known, which goes back to Popper [1968] and de Finetti [1936], is to take as prim... |

274 |
Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit, Zeitschrift für die gesamte Staatswissenschaft
- Selten
- 1965
(Show Context)
Citation Context ...lume, Brandenburger, and Dekel 1991b; Brandenburger, Friedenberg, and Keisler 2008; Fudenberg and Tirole 1991; Hammond 1994; Hammond 1999; Kohlberg and Reny 1997; Kreps and Wilson 1982; Myerson 1986; =-=Selten 1965-=-; Selten 1975]). It also arises in the analysis of conditional statements by philosophers (see [Adams 1966; McGee 1994]), and in dealing with nonmonotonicity in Artificial Intelligence (see, for examp... |

268 |
The Logic of Conditionals.
- Adams
- 1975
(Show Context)
Citation Context ...t to play a critical role in game theory (see [Blume, Brandenburger, and Dekel 1991a; Blume, Brandenburger, and Dekel 1991b; Hammond 1994; Myerson 1986]), the analysis of conditional statements (see [=-=Adams 1966-=-; McGee 1994]), and in dealing with nonmonotonicity (see, for example, [Lehmann and Magidor 1992]). There have been various attempts to deal with the problem of conditioning on events of measure 0. Pe... |

154 | Logik der Forschung, - Popper - 1934 |

104 |
Strong belief and forward induction reasoning,
- Battigalli, Siniscalchi
- 2002
(Show Context)
Citation Context ...itical role in game theory, particularly in the analysis of strategic reasoning in extensive-form games and in the analysis of weak dominance in normal-form games (see, for example, [Battigalli 1996; =-=Battigalli and Siniscalchi 2002-=-; Blume, Brandenburger, and Dekel 1991a; Blume, Brandenburger, and Dekel 1991b; Brandenburger, Friedenberg, and Keisler 2008; Fudenberg and Tirole 1991; Hammond 1994; Hammond 1999; Kohlberg and Reny 1... |

100 | Lexicographic probabilities and choice under uncertainty. - Blume, Brandenburger, et al. - 1991 |

77 |
Multistage Games with Communications.
- Myerson
- 1986
(Show Context)
Citation Context ...re 0 may seem to be of little practical interest, it turns out to play a critical role in game theory (see [Blume, Brandenburger, and Dekel 1991a; Blume, Brandenburger, and Dekel 1991b; Hammond 1994; =-=Myerson 1986-=-]), the analysis of conditional statements (see [Adams 1966; McGee 1994]), and in dealing with nonmonotonicity (see, for example, [Lehmann and Magidor 1992]). There have been various attempts to deal ... |

71 | On a new axiomatic theory of probability. - Renyi - 1955 |

68 |
Applied Nonstandard Analysis,
- Davis
- 1977
(Show Context)
Citation Context ...t field generated by adding to the reals a single infinitesimal ffl. 1 The hyperreals, nonstandard models of the reals which satisfy all the first-order properties that hold of the real numbers (see [=-=Davis 1977-=-]) are also instances of nonArchimedean fields. For most of this paper, I use only the following properties of nonArchimedean fields: 1. If IR is a non-Archimedean field, then for all b 2 IR such that... |

66 | Probabilistic Logic in a Coherent Setting, - Coletti, Scozzafava - 2002 |

47 | Belief revision in games: forward and backward induction - Stalnaker - 1998 |

46 | Lexicographic probabilities and equilibrium refinements, - Blume, Brandenburger, et al. - 1991 |

35 |
Strategic independence and perfect Bayesian equilibria
- Battigalli
(Show Context)
Citation Context ... out to play a critical role in game theory, particularly in the analysis of strategic reasoning in extensive-form games and in the analysis of weak dominance in normal-form games (see, for example, [=-=Battigalli 1996-=-; Battigalli and Siniscalchi 2002; Blume, Brandenburger, and Dekel 1991a; Blume, Brandenburger, and Dekel 1991b; Brandenburger, Friedenberg, and Keisler 2008; Fudenberg and Tirole 1991; Hammond 1994; ... |

32 | Elementary non-Archimedean representations of probability for decision theory and games.
- Hammond
- 1994
(Show Context)
Citation Context ...vents of measure 0 may seem to be of little practical interest, it turns out to play a critical role in game theory (see [Blume, Brandenburger, and Dekel 1991a; Blume, Brandenburger, and Dekel 1991b; =-=Hammond 1994-=-; Myerson 1986]), the analysis of conditional statements (see [Adams 1966; McGee 1994]), and in dealing with nonmonotonicity (see, for example, [Lehmann and Magidor 1992]). There have been various att... |

29 |
Independence on relative probability spaces and consistent assessments in game trees
- Kohlberg, Reny
(Show Context)
Citation Context ...nd Siniscalchi 2002; Blume, Brandenburger, and Dekel 1991a; Blume, Brandenburger, and Dekel 1991b; Brandenburger, Friedenberg, and Keisler 2008; Fudenberg and Tirole 1991; Hammond 1994; Hammond 1999; =-=Kohlberg and Reny 1997-=-; Kreps and Wilson 1982; Myerson 1986; Selten 1965; Selten 1975]). It also arises in the analysis of conditional statements by philosophers (see [Adams 1966; McGee 1994]), and in dealing with nonmonot... |

29 | Fine-grained opinion, probability, and the logic of full belief. - Fraassen - 1995 |

27 | Admissibility in games, - Brandenburger, Friedenberg, et al. - 2008 |

25 |
Learning the impossible. In
- McGee
- 1994
(Show Context)
Citation Context ...critical role in game theory (see [Blume, Brandenburger, and Dekel 1991a; Blume, Brandenburger, and Dekel 1991b; Hammond 1994; Myerson 1986]), the analysis of conditional statements (see [Adams 1966; =-=McGee 1994-=-]), and in dealing with nonmonotonicity (see, for example, [Lehmann and Magidor 1992]). There have been various attempts to deal with the problem of conditioning on events of measure 0. Perhaps the be... |

24 | Sequential and quasi-perfect rationalizability in extensive games, - Asheim, Perea - 2005 |

24 | Multidimensional utilities. In - Hausner - 1954 |

22 |
The Representation of Popper Measures
- Spohn
- 1986
(Show Context)
Citation Context .... As the following theorem, originally proved by Spohn [1986], shows, there are no such counterexamples if we restrict to countably additive SLPS’s and countably additive Popper spaces. Theorem 3.5: [=-=Spohn 1986-=-] For all W , the map FS→P is a bijection from SLPS c (W, F, F ′ ) to Pop c (W, F, F ′ ). Proof: Again, the difficulty comes in showing that FS→P is onto. Given a Popper space (W, F, F ′, µ), I again ... |

21 | Function theory on some nonarchimedean fields. - Robinson - 1973 |

21 | Representation of conditional probability measures on Boolean algebras. Acta Mathematica Academiae Scientiarum Hungaricae, - Krauss - 1968 |

15 | On conditional probability spaces generated by a dimensionally ordered set of measures - Rényi - 1956 |

15 | Characterizing the Common Prior Assumption.
- Halpern
- 2002
(Show Context)
Citation Context ... to the topmost information sets for player i in the game tree. There 4 This example is closely related to examples of conditional probabilities for which there is no common prior; see, for example, [=-=Halpern 2002-=-, Example 2.2]. 12may be several such topmost information sets if nature or some player j other than i makes the first move in the game. (That is why we have a forest, rather than a tree.) The immedi... |

13 | The Logic of Scientific Discovery. Revised edition - Popper - 1968 |

11 |
The consistent preferences approach to deductive reasoning in games, Theory and Decision Library,
- Asheim
- 2006
(Show Context)
Citation Context ...ss preferences that cps’s cannot. However, as Asheim and Perea [2005] point out, in finite spaces, cps’s can also represent conditional preferences that cannot be represented by LPS’s and NPS’s. See [=-=Asheim 2006-=-] for a detailed discussion of the expressive power of these representations with respect to conditional preferences. Although NPS’s are the most expressive of the three approaches I have considered, ... |

10 |
A note on stochastic independence without Savage-null events.
- Battigalli, Veronesi
- 1996
(Show Context)
Citation Context ...ically independent iff it can be represented by some (real-valued) utility function u and a nonstandard probability measure ν such that X1, . . . , Xn are approximately independent with respect to ν [=-=Battigalli and Veronesi 1996-=-]. A second notion they consider is a weak notion of product measure that requires only that there exist measures ν1, . . . , νn such that st (ν(w1, . . . , wn)) = st (ν1(w1) × · · · ν(wn)). As we hav... |

9 | Epistemic conditions for iterated admissibility. Unpublished manuscript - Brandenburger, Keisler - 2000 |

9 |
Non-Archimedean Probabilities, and Lexicographic Expected Utility
- Hammond
- 1999
(Show Context)
Citation Context ...; Battigalli and Siniscalchi 2002; Blume, Brandenburger, and Dekel 1991a; Blume, Brandenburger, and Dekel 1991b; Brandenburger, Friedenberg, and Keisler 2008; Fudenberg and Tirole 1991; Hammond 1994; =-=Hammond 1999-=-; Kohlberg and Reny 1997; Kreps and Wilson 1982; Myerson 1986; Selten 1965; Selten 1975]). It also arises in the analysis of conditional statements by philosophers (see [Adams 1966; McGee 1994]), and ... |

8 | Sur la structure des espaces de probabilite conditionnelle. - Csaszar - 1955 |

8 | Les probabilités nulles. Bulletins des Science Mathématiques (première partie - Finetti - 1936 |

7 | Representation of conditional probabilities.” - Fraassen - 1976 |

6 | Sur les espaces simples de probabilités conditionelles - Rényi - 1964 |

6 | The logic of scientific discovery (2nd ed - Popper - 1992 |

5 |
Models and Ultraproducts: An Introduction (third revised printing
- Bell, Slomson
- 1974
(Show Context)
Citation Context ...ultrafilter is a maximal filter, that is, a filter that is not a strict subset of any other filter. It is not hard to show that if U is an ultrafilter on S, then for all U ` S, either U 2 U or U 2 U [=-=Bell and Slomson 1974-=-]. Suppose F is either IR or a non-Archimedean field, J is an arbitrary set, and U is an ultrafilter on J . Define an equivalence relationsU on F J by taking (a j : j 2 J)sU (b j : j 2 J) if fj : a j ... |

4 | Les probabilités nulles,’ Bulletin des SciencesMathématiques - Finetti - 1936 |

4 | Interactive beliefs, epistemic independence, and strong rationalizability
- Battigalli, Siniscalchi
- 1999
(Show Context)
Citation Context ...me extent on the notion of extended probability used. Similarly, a number of characterizations of solution concepts depend on independence (see, for example, [Battigalli 1996; Kohlberg and Reny 1997; =-=Battigalli and Siniscalchi 1999-=-]). Again, the results of this paper show that these notions can be somewhat sensitive to exactly how uncertainty is represented, even with a finite state space. While I do not present any new game-th... |

3 |
Subjective expected lexicographic utility with infinite state sets
- Fishburn, Lavalle
- 1998
(Show Context)
Citation Context ...e I have restricted to real-valued random variables). There is a tradition in decision theory going back to Hausner [1954] and continued recently in a sequence of papers by Fishburn and Lavalle (see [=-=Fishburn and Lavalle 1998-=-] and the references therein) and Hammond [1999] of considering nonstandard or lexicographically-ordered utilities. I have not considered the relationship between these ideas and the ones considered h... |

2 | Preference-based belief operators - Asheim, Søvik - 2005 |

2 | The Logic of Scientific Discovery (2nd ed.). London: Hutchison. The first version of this book appeared as Logik der Forschung - Popper - 1968 |

1 |
Perfect Bayesian reasoning
- Fudenberg, Tirole
- 1991
(Show Context)
Citation Context ...games (see, for example, [Battigalli 1996; Battigalli and Siniscalchi 2002; Blume, Brandenburger, and Dekel 1991a; Blume, Brandenburger, and Dekel 1991b; Brandenburger, Friedenberg, and Keisler 2008; =-=Fudenberg and Tirole 1991-=-; Hammond 1994; Hammond 1999; Kohlberg and Reny 1997; Kreps and Wilson 1982; Myerson 1986; Selten 1965; Selten 1975]). It also arises in the analysis of conditional statements by philosophers (see [Ad... |

1 | Learning the impossible - unknown authors - 1994 |

1 | Sur la structure des espace de probabilité conditionelle - unknown authors - 1955 |

1 | Fine-grained opinion, probability, and the logic of full belief - Fraassen - 1995 |