This note presents several preference-based definitions of a likely event, and shows that they induce (in the sense of Lo 2005b) the same set of possible states for biseparable preferences.

Kin Chung Lo

English

Research Papers in Economics

A robust definition of possibility for biseparable preferences 
Kin Chung Lo
Department of Economics, York University
Abstract
This note presents several preference-based definitions of a likely event, and shows that they
induce (in the sense of Lo 2005b) the same set of possible states for biseparable preferences.
I would like to thank a referee for useful comments.
Citation: Lo, Kin Chung, (2006) "A robust definition of possibility for biseparable preferences." Economics Bulletin, Vol. 4,
No. 37 pp. 1-7
Submitted: September 15, 2006.  Accepted: November 21, 2006.
URL: http://economicsbulletin.vanderbilt.edu/2006/volume4/EB-06D80017A.pdf1. Introduction
According to the expected utility model of Savage (1954), and more generally the probabilis-
tic sophistication model of Machina and Schmeidler (1992, 1995), the beliefs of a decision
maker are represented by a probability measure. For any probabilistically sophisticated de-
cision maker confronted with a ﬁnite set of states of the world, it is standard to say that a
state of the world is possible if the decision maker assigns positive probability to that state.
In contrast, for models of preference without the probabilistic sophistication property, espe-
cially Choquet expected utility (Schmeidler 1989) and maxmin expected utility (Gilboa and
Schmeidler 1989), the issue of deﬁning possibility has been controversial. Dow and Werlang
(1994) is an early paper raising the issue. Subsequently, Morris (1997), Ryan (2002) and
Lo (2005a) propose several preference-based deﬁnitions of a possible state; Lo (2005b) even
provides a recipe, which has the potential for generating diﬀerent sets of possible states from
diﬀerent deﬁnitions of a likely event.
As a concrete illustration of his recipe, Lo (2005b, Section 3) proposes a speciﬁc preference-
based deﬁnition of a likely event, and uses it to generate what he calls I-possible states. In-
terestingly, for the class of biseparable preferences (Ghirardato and Marinacci 2001), which
includes Choquet expected utility and maxmin expected utility as special cases, I-possibility
is equivalent to the notions of subjective possibility of Ryan (2002) and strict possibility of
Lo (2005a). Those equivalence results (established in Lo 2005a,b) suggest that there may
be at least a “robust” deﬁnition of possibility for biseparable preferences. The purpose of
this note is to provide further evidence. More precisely, we show that a couple of other
intuitive deﬁnitions of a likely event also induce the set of I-possible states. So, conﬁning
to biseparable preferences, the issue of deﬁning possibility may not be as controversial as it
ﬁrst appeared.
2. Likely events and possible states
This section is a brief review of Lo (2005b). Let Ω be a ﬁnite set of states of the world, and
2Ω the set of events. Objects of choice are acts, where an act is a function from Ω to R. Let
 be a preference ordering over acts. Consider the following (indirect) approach of deﬁning
a possible state. First, derive from  a subset L of 2Ω, which is interpreted as the collection
of likely events. Then possible states are induced from L as follows. (We use ⊆ for subset,
and ⊂ for strict subset. For any state ω ∈ Ω, ω also denotes the event {ω}.)
Deﬁnition 1. A state ω ∈ Ω is L-possible if there exists an event E ⊂ Ω such that E 6∈ L
and ω ∪ E ∈ L.
According to Deﬁnition 1, a state ω is L-possible if there exists an event E such that ω
has the following impact on E: E is not a likely event, but E with ω attached becomes a
likely event. As long as Ω is a likely event and ∅ is not, there exists at least one L-possible
state. If any superset of a likely event is also a likely event, then the set of all L-possible
states is equal to
∪{D|D ∈ minL}, (1)
1where
minL = {D|D ∈ L;and for all E ⊂ D,E 6∈ L}. (2)
In words, the set of L-possible states is the union of all minimal likely events. It follows that
if every element of minL is a singleton, then every L-possible state can be identiﬁed with a
likely event; on the other hand, if minL is a singleton, then L is closed under intersection,
and every element of L can be regarded as a believed event (in the sense of Morris 1997,
p. 220). In general, an event containing a L-possible state may not be a likely event, and a
likely event may not be a believed event.
Back to the derivation of L. Let  be the strict preference ordering corresponding to .
For all D ⊆ Ω and for all x,y ∈ R, xDy denotes the binary act that yields the outcome x if
the event D happens, and the outcome y otherwise. The following preference-based notion
is deﬁned and justiﬁed in Lo (1999).
Deﬁnition 2. For any D,E ⊆ Ω, D is inﬁnitely more likely than E if for all x,y,z ∈ R with
x > y and z > y, xDy  zEy.
To elaborate, an event D is inﬁnitely more likely than another event E if the decision
maker strictly prefers to bet on D rather than on E, and the strict preference persists no
matter how much bigger is the outcome for winning the E bet than that for winning the D
bet. With Deﬁnition 2, Lo (2005b) proposes
L = I ≡ {D|D is inﬁnitely more likely than Ω \ D}, (3)
where Ω \ D denotes the complement of D. Say that a state is I-possible if it satisﬁes
Deﬁnition 1, with L = I.
3. Equivalence results
Equation (3) leads us to consider the following obvious alternatives:
• I1 ≡ {D|D is inﬁnitely more likely than ∅}.
• I2 ≡ {D|Ω is inﬁnitely more likely than Ω \ D}.
• I3 ≡ {D|Ω is not inﬁnitely more likely than D}.
• I4 ≡ {D|Ω \ D is not inﬁnitely more likely than D}.
• I5 ≡ {D|Ω \ D is not inﬁnitely more likely than ∅}.
For every j = 1,...,5, the notion of Ij-possibility is parallel to that of I-possibility; to be
exact, ω is Ij-possible if it satisﬁes Deﬁnition 1, with L = Ij.
A biseparable preference ordering gets its name from the following property: For any
x,x0,y,y0 ∈ R with x ≥ y and x0 ≥ y0, and any D,E ⊆ Ω, xDy  x0
Ey0 if and only if
ρ(D)u(x) + (1 − ρ(D))u(y) ≥ ρ(E)u(x
0) + (1 − ρ(E))u(y
0), (4)
2where u:R → R is continuous and strictly increasing, and ρ : 2Ω → [0,1] is normalized and
monotone. In the literature, ρ is called a capacity (or nonadditive probability measure).
According to Equation (4),  is separable in binary acts; more precisely, a binary act is
evaluated according to its “expected utility,” where expectation is taken with respect to a
capacity in a rank dependent fashion.
We end this note with the following proposition, which establishes the equivalence of Ij-
possibility and I-possibility for biseparable preferences. Almost all the equivalence results
do not carry over to cumulative prospect theory (Tversky and Kahneman 1992) and lexico-
graphically biseparable preferences (Ryan 2002), which are two noticeable generalizations of
biseparable preferences.
Proposition. Suppose  is biseparable. Then for every j = 1,...,5, a state is Ij-possible
if and only if it is I-possible.
Proof of Proposition. As  satisﬁes Equation (4), D is inﬁnitely more likely than E if
and only if ρ(D) > 0 and ρ(E) = 0. Based on this observation, further observations can be
made. (In the sequel, minIj is as deﬁned in Equation (2), with L = Ij; similarly for minI.)
observation 1. D ∈ minI1 if and only if both Conditions A and B below are satisﬁed.
A. ρ(D) > 0.
B. For every E ⊂ D, ρ(E) = 0.
observation 2. D ∈ minI2 if and only if both Conditions A and B below are satisﬁed.
A. ρ(Ω \ D) = 0.
B. For every E ⊂ D, ρ(Ω \ E) > 0.
observation 3. D ∈ minI3 if and only if both Conditions A and B below are satisﬁed.
A. ρ(D) > 0.
B. For every E ⊂ D, ρ(E) = 0.
observation 4. D ∈ minI4 if and only if both Conditions A and B below are satisﬁed.
A. ρ(D) > 0 or ρ(Ω \ D) = 0.
B. For every E ⊂ D, ρ(E) = 0 and ρ(Ω \ E) > 0.
observation 5. D ∈ minI5 if and only if both Conditions A and B below are satisﬁed.
A. ρ(Ω \ D) = 0.
B. For every E ⊂ D, ρ(Ω \ E) > 0.
3observation 6. D ∈ minI if and only if both Conditions A and B below are satisﬁed.
A. ρ(D) > 0 and ρ(Ω \ D) = 0.
B. For every E ⊂ D, ρ(E) = 0 or ρ(Ω \ E) > 0.
The set of Ij-possible states can be derived using Equation (1), with L = Ij; likewise for
the set of I-possible states. We are now prepared to show that
• a state is I1-possible only if it is I4-possible. Fix any D ∈ minI1, and consider the
following two exhaustive cases.
case 1: Suppose ρ(ω ∪ [Ω \ D]) > 0 for every ω ∈ D. Then we can immediately
use Observations 1 and 4 to conclude that D ∈ minI4, and hence every ω ∈ D is
I4-possible. (Obviously, we rely on the fact that ρ is monotone; it is also the case in
various places below.)
case 2: Suppose there exists ω ∈ D such that
ρ(ω ∪ [Ω \ D]) = 0. (5)
Equation (5) enables us to ﬁx D∗ ⊂ D such that
ρ(Ω \ D
∗) = 0 (6)
and
ρ(Ω \ E) > 0 ∀E ⊂ D
∗. (7)
By Observation 1, D ∈ minI1 and D∗ ⊂ D imply
ρ(E) = 0 ∀E ⊆ D
∗. (8)
By Observation 4, Equations (6)–(8) imply D∗ ∈ minI4. For any ω ∈ D, if ω violates
Equation (5), then ω ∈ D∗; otherwise D∗ ⊂ D and ω 6∈ D∗ would imply ω ∪[Ω \ D] ⊆
Ω \ D∗, which can be combined with ρ(ω ∪ [Ω \ D]) > 0 to arrive at ρ(Ω \ D∗) > 0,
contradicting Equation (6). So any ω ∈ D violating Equation (5) is I4-possible. Next,
consider any ω ∈ D such that Equation (5) holds. By Observation 1, D ∈ minI1
implies
ρ(D) > 0 (9)
and
ρ(D \ ω) = 0. (10)
Equation (10) enables us to ﬁx
D
0 ⊆ ω ∪ [Ω \ D] (11)
4such that
ρ(Ω \ D
0) = 0 (12)
and
ρ(Ω \ E) > 0 ∀E ⊂ D
0. (13)
Equations (5) and (11) imply
ρ(D
0) = 0. (14)
By Observation 4, Equations (12)–(14) imply D0 ∈ minI4. Finally, Equations (9), (11)
and (12) imply ω ∈ D0. This completes the proof that any ω satisfying Equation (5)
is also I4-possible.
• a state is I4-possible only if it is I-possible. Fix any D ∈ minI4. Condition A of
Observation 4 can be broken into three exhaustive cases.
case 1: Suppose ρ(D) > 0 and ρ(Ω\D) = 0. Then we can immediately use Observa-
tions 4 and 6 to conclude that D ∈ minI, and therefore every ω ∈ D is I-possible.
case 2: Suppose ρ(D) > 0 and ρ(Ω \ D) > 0. Fix any ω ∈ D. By Observation 4,
ρ(D \ ω) = 0 and ρ(ω ∪ [Ω \ D]) > 0. (15)
By Observation 6, Equation (15) implies that there exists D∗ ∈ minI such that
D
∗ ⊆ ω ∪ [Ω \ D]. (16)
If ω 6∈ D∗, then Equation (16) would imply D ⊆ Ω \ D∗, which can be combined with
ρ(D) > 0 to arrive at ρ(Ω \ D∗) > 0, contradicting D∗ ∈ minI. So we must have
ω ∈ D∗, and therefore ω is I-possible.
case 3: Suppose ρ(D) = 0 and ρ(Ω \ D) = 0. As in Case 2 above, ﬁx any ω ∈ D
and any D∗ ∈ minI satisfying Equation (16). If ω 6∈ D∗, then (16) would imply
D∗ ⊆ Ω \ D, which can be combined with ρ(Ω \ D) = 0 to arrive at ρ(D∗) = 0,
contradicting D∗ ∈ minI. So once again we must have ω ∈ D∗, and therefore ω is
I-possible.
• a state is I-possible only if it is I2-possible. Fix any D ∈ minI. Observations 2 and
6 allow us to ﬁx D∗ ∈ minI2 such that D∗ ⊆ D. If D∗ = D, we can immediately
conclude that every ω ∈ D is I2-possible. Suppose that D∗ ⊂ D. Fix any ω such
that ω ∈ D and ω 6∈ D∗. Then ω ∪ [Ω \ D] ⊆ Ω \ D∗. Since D∗ ∈ minI2, we have
ρ(Ω \ D∗) = 0, which can be combined with ω ∪ [Ω \ D] ⊆ Ω \ D∗ to arrive at
ρ(ω ∪ [Ω \ D]) = 0. (17)
By Observation 6, D ∈ minI and Equation (17) imply
ρ(D \ ω) = 0. (18)
5By Observation 2, Equation (18) implies that there exists D0 ∈ minI2 such that
D
0 ⊆ ω ∪ [Ω \ D]. (19)
Since D ∈ minI, we have ρ(D) > 0. If ω 6∈ D0, then Equation (19) would imply
D ⊆ Ω \ D0, which can be combined with ρ(D) > 0 to arrive at ρ(Ω \ D0) > 0,
contradicting D0 ∈ minI2. So we must have ω ∈ D0 and thus ω is I2-possible.
• a state is I2-possible only if it is I1-possible. Fix any D ∈ minI2 and any ω ∈ D.
According to Observation 2,
ρ(ω ∪ [Ω \ D]) > 0. (20)
By Observation 1, Equation (20) implies that there exists D∗ ∈ minI1 such that
D
∗ ⊆ ω ∪ [Ω \ D]. (21)
Since D ∈ minI2, we have ρ(Ω \ D) = 0. If ω 6∈ D∗, then Equation (21) would imply
D∗ ⊆ Ω \ D, which can be combined with ρ(Ω \ D) = 0 to arrive at ρ(D∗) = 0,
contradicting D∗ ∈ minI1. So we must have ω ∈ D∗ and thus ω is I1-possible.1
• a state is I1-possible if and only if it is I3-possible. This follows immediately from
Observations 1 and 3.
• a state is I2-possible if and only if it is I5-possible. This follows immediately from
Observations 2 and 5.
References
Dow, J., and D. Werlang (1994) “Nash Equilibrium under Uncertainty: Breaking Down
Backward Induction” Journal of Economic Theory 64, 305-324.
Gilboa, I., and D. Schmeidler (1989) “Maxmin Expected Utility with Non-unique Prior”
Journal of Mathematical Economics 18, 141-153.
Ghirardato, P., and M. Marinacci (2001) “Risk, Ambiguity and the Separation of Utility
and Beliefs” Mathematics of Operations Research 26, 864–890.
Lo, K. C. (1999) “Nash Equilibrium without Mutual Knowledge of Rationality” Economic
Theory 14, 621–633.
—– (2005a) “More Likely than Unlikely” Mathematical Social Sciences 49, 39-53.
—– (2005b) “Likely Events and Possible States” Topics in Theoretical Economics 5,
Article 4.
1The equivalence of I-possibility, I1-possibility and I2-possibility can also be established by suitably
combining Observations 1 and 2 with Lo (2005b, p. 15, Proof of Proposition 8).
6Machina, M. J., and D. Schmeidler (1992) “A more Robust Deﬁnition of Subjective
Probability” Econometrica 60, 745–780.
—– (1995) “Bayes without Bernoulli: Simple Conditions for Probabilistically Sophisticated
Choice” Journal of Economic Theory 67, 106–128.
Morris, S. (1997) “Alternative Deﬁnitions of Knowledge” in Epistemic Logic and the Theory
of Games and Decisions by M. O. L. Bacharach et al., Eds., Dordrecht: Kluwer Academic,
217–233.
Ryan, M. (2002) “What do Uncertainty Averse Decision Makers Believe?” Economic Theory
20, 47–65.
Savage, L. (1954) The Foundations of Statistics, John Wiley: New York.
Schmeidler, D. (1989) “Subjective Probability and Expected Utility without Additivity”
Econometrica 57, 571–587.
Tversky, A., and D. Kahneman (1992) “Advances in Prospect Theory: Cumulative
Representation of Uncertainty” Journal of Risk and Uncertainty 5, 297–323.
7

A robust definition of possibility for biseparable preferences

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics