2001 LexicographicProbabilityConditi

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The relationship between Popper spaces (conditional probability spaces that satisfy some regularity conditions), lexicographic probability systems (LPS's) [Blume, Brandenburger, and Dekel 1991a; Blume, Brandenburger, and Dekel 1991b], and nonstandard probability spaces (NPS's) is considered. If countable additivity is assumed, Popper spaces and a subclass of LPS's are equivalent; without the assumption of countable additivity, the equivalence no longer holds. If the state space is finite, LPS's are equivalent to NPS's. However, if the state space is infinite, NPS's are shown to be more general than LPS's.

1 Introduction

Probability is certainly the most commonly-used approach for representing uncertainty and conditioning the standard way of updating probabilities in the light of new information. Unfortunately, there is a well-known problem with conditioning: Conditioning on events of measure 0 is not defined. That makes it unclear how to proceed if an agent learns something to which she initially assigned probability 0. Although consideration of events of measure 0 may seem to be of little practical interest, it turns 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; 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 and Magidor 1992]).

There have been various attempts to deal with the problem of conditioning on events of measure 0. Perhaps the best known involves conditional probability spaces (CPS's). The idea, which goes back to Popper [1934, 1968] and de Finetti [1936], is to take as primitive not probability, but conditional probability. If � is a conditional probability measure on a space W, then � (V j U) may still be undefined for some pairs V and U, but it is also possible that � (V j U) is defined even if � (U jW) = 0. A second approach, which goes back to at least Robinson [1973] and has been explored in the economics literature [Hammond 1994; Hammond 1999], the AI literature [Lehmann and Magidor 1992; Wilson 1995], and the philosophy literature (see [McGee 1994] and the references therein) is to consider nonstandard probability spaces (NPS's), where there are infinitesimals that can be used to model events that, intuitively, have infinitesimally small probability yet may still be learned or observed.

There is a third approach to this problem, which uses sequences of probability measures to represent uncertainty. The most recent exemplar of this approach, which I focus on here, are the lexicographic probability systems (LPS's) of Blume, Brandenburger, and Dekel [1991a, 1991b] (BBD from now on). However, the idea of using a system of measures to represent uncertainty actually was explored as far back as the 1950s by Rffenyi [1956] (see Section 3.4). A lexicographic probability system is a sequence h�0; �1;::: i of probability measures. Intuitively, the first measure in the sequence, �0, is the most important one, followed by �1, �2, and so on. One way to understand LPS's is in terms of NPS's. Roughly speaking, the probability assigned to an event U by a sequence such as h�0; �1i can be taken to be �0 (U) + ��1 (U), where � is an infinitesimal. Thus, even if the probability of U according to �0 is 0, U still has a positive (although infinitesimal) probability if �1 (U) > 0.

What is the precise relationship between these approaches? The relationship between LPS's and CPS's has been considered before. For example, Hammond (1994) shows that conditional probability spaces are equivalent to a subclass of LPS's called lexicographic conditional probability spaces (LCPS's) if the state space is finite and it is possible to condition on any nonempty set.1 As shown by Spohn [1986], Hammond's result can be extended to arbitrary countably additive Popper spaces, where a Popper space is a conditional probability space where the events on which conditioning is allowed satisfy certain regularity conditions. As I show, this result depends critically on a number of assumptions. In particular, it does not work without the assumption of countable additivity, it requires that we extend LCPS's appropriately to the infinite case, and it is sensitive to the choice of conditioning events. For example, if we consider CPS's where the conditioning events can be viewed as information sets, and so are are not closed under supersets (this is essentially the case considered by Battigalli and Sinischalchi (2002)), then the result no longer holds.

2 Conditional, lexicographic, and nonstandard probability spaces

In this section I briefly review the three approaches to representing likelihood discussed in the introduction.

2.1 Popper spaces

A conditional probability measure takes pairs [math]\displaystyle{ U,V }[/math] of subsets as arguments; [math]\displaystyle{ \mu(V,U) }[/math] is generally written [math]\displaystyle{ \mu(V \mid U) }[/math] to stress the conditioning aspects. The first argument comes from some algebra [math]\displaystyle{ \mathcal{F} }[/math] of subsets of a space [math]\displaystyle{ W }[/math]; if [math]\displaystyle{ W }[/math] is infinite, [math]\displaystyle{ \mathcal{F} }[/math] is often taken to be a [math]\displaystyle{ \sigma-algebra }[/math]. (Recall that an algebra of subsets of W is a set of subsets containing W and closed under union and complementation. A [math]\displaystyle{ \sigma-algebra }[/math] is an algebra that is closed under union countable.) The second argument comes from a set F0 of conditioning events, that is, that is, events on which conditioning is allowed. One natural choice is to take F0 to be F ??


But it may be reasonable to consider other restrictions on F0. For example, Battigalli and Sinischalchi (2002) take F0 to consist of the information sets in a game, since they are interested only in agents who update their beliefs conditional on getting some information. The question is what constraints, if any, should be placed on F0. For most of this paper, I focus on Popper spaces (named after Karl Popper), defined next, where the set F0 satisfies four arguably reasonable requirements, but I occasionally consider other requirements (see Section 3.3).

Definition 2.1: A conditional probability space (cps) over (W; F) is a tuple (W; F; F0; �) such that F is an algebra over W, F0 is a set of subsets of W (not necessarily an algebra over W) that does not contain;, and �: [[F�F0 !]] [ 0; 1] satisfies the following conditions:

CP1. [math]\displaystyle{ \mu(U \mid U) = \ 1 \ \text{if} \ U \in F' }[/math].

CP2. � (V1 [V2 j U) = � (V1 j U) + � (V2 j U) if V1 \ V2 =;, U 2 F0, and V1; V2 2 F.

CP3. � (V j U) = � (V jX) � � (X j U) if V � X � U, U; X 2 F0, V 2 F.

Note that it follows from CP1 and CP2 that � (� j U) is a probability measure on (W; F) (and, in particular, that � (j U) = 0) for each U 2 F0. A Popper space over (W; F) is a conditional probability space (W; F; F0; �) that satisfies three additional conditions: (a) F0 � F, (b) F0 is closed under supersets in F, in that if V 2 F0, V � V 0, and V 0 2 F, then V 0 2 F0, and (c) if U 2 F0 and � (V j U) 6= 0 then V \ U 2 F0. If F is a �-algebra and � is countably additive (that is, if � ([ Vi j U) = P1 i=1 � (Vi j U) if the Vi's are pairwise disjoint elements of F and U 2 F0), then the Popper space is said to be countably additive. Let Pop (W; F) denote the set of Popper spaces over (W; F). If F is a �-algebra, I use a superscript c to denote the restriction to countably additive Popper spaces, so Popc (W; F) denotes the set of countably additive Popper spaces over (W; F). The probability measure � in a Popper space is called a Popper measure. The last regularity condition on F0 required in a Popper space corresponds to the observation that for an unconditional probability measure �, if � (V j U) 6= 0 then � (V \ U) 6= 0, so conditioning on V \ U should be defined. Note that, since this regularity condition depends on the Popper measure, it may well be the case that (W; F; F0; �) and (W; F; F0; �) are both cps's over (W; F), but only the former is a Popper space over (W; F). Popper [1934, 1968] and de Finetti [1936] were the first to formally consider conditional probability as the basic notion, although as Rffenyi [1964] points out, the idea of taking conditional probability as primitive seems to go back as far as Keynes [1921]. [[CP1{3]] are essentially due to Rffenyi [1955]. [[Van Fraassen [1976]] defined what I have called Popper measures; he called them Popper functions, reserving the name Popper measure for what I am calling a countably additive Popper measure. Starting from the work of de Finetti, there has been a general study of coherent conditional probabilities. A coherent conditional probability is essentially a cps that is not necessarily a Popper space, since it is defined on a set F �F0 where F0 does not have to be a subset of F); see, for example, [Coletti and Scozzafava 2002] and the references therein. Hammond (1994) discusses the use of conditional probability spaces in philosophy and game theory, and provides an extensive list of references.

2.2 Lexicographic probability spaces Definition 2.2: A lexicographic probability space (LPS) (of length �) over (W; F) is a tuple (W; F; ~�) where, as before, W is a set of possible worlds and F is an algebra over W, and ~� is a sequence of finitely additive probability measures on (W; F) indexed by ordinals < �. (Technically, ~� is a function from the ordinals� to probability measures on (W; F).) I typically write ~� as (�0; �1;:::) or as (�: < �). If F is a �-algebra and each of the probability measures in ~� is countably additive, then ~� is a countably additive LPS. Let LPS (W; F) denote the set of LPS's over (W; F). Again, if F is a �-algebra, a superscript c is used to denote countable additivity, so LPSc (W; F) denote the set of countably additive LPS's over (W; F). When (W; F) are understood, I often refer



 AuthorvolumeDate ValuetitletypejournaltitleUrldoinoteyear
2001 LexicographicProbabilityConditiJoseph Y. HalpernLexicographic Probability, Conditional Probability, and Nonstandard Probability