Abstract for
The Valuation of Choices in The Logic of Decision
(154 words)
Here is an example of an easy choice. When you visit a restaurant, would you
prefer to have an entree selected from the menu by a lottery, or would you prefer to
choose your dinner for yourself? While most individuals would have no hesitation
identifying the opportunity to choose for oneself as preferable, most theories of rational
decision are incapable of even posing the question.
In this paper I will explain how within one influential system of analyzing
decisions under risk, we may represent decision problems themselves as events which
stand in relations of relative preference to other events.
First I will show how to
construct an algebra of choices from the algebra of events. Then I will show that the
algebra of choices can be interpreted as including the original algebra of events. To
conclude, I will compare the choice algebras constructed here to the representation of
decision problems proposed by other authors.
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The Valuation of Choices in The Logic of Decision
Ken Presting
presting@mindspring.com
Here is an example of an easy choice. When you visit a restaurant, would you
prefer to have an entree selected from the menu by a lottery, or would you prefer to
choose your dinner for yourself? While most individuals would have no hesitation
identifying the opportunity to choose for oneself as preferable, most theories of rational
decision are incapable of even posing the question. This is perhaps understandable,
since the problem traditionally1 called "decision making under risk" has not included an
analysis of the value of choosing for oneself. The fact that in some situations one may
choose for oneself has been represented only in the context of the discussion, not in the
formal structure of the theory.
By contrast, the theory of two-person zero-sum games has given explicit
attention to the value of being in a game. Once a game has been "solved," that is, once
an optimal strategy has been identified, then the value of the game is the expected
utility of playing the optimal strategy.
In this paper I will explain how within one influential system of analyzing
decisions under risk, we may represent decision problems themselves as events which
stand in relations of relative preference to other events.
I will use Richard Jeffrey's "The
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Logic of Decision" as my starting point, primarily because the abstract mathematical
foundation of that system lends itself to extensions.
Two basic concepts of Jeffrey's system are a Boolean algebra of events, and a
preference relation defined over those events. First I will show how to construct an
algebra of choices from the algebra of events. The algebra of choices represents all the
possible decision problems which can be constructed from the given algebra of events.
Then I will show that the algebra of choices can be interpreted as including the original
algebra of events. Next, I will show that the preference relation may be naturally
extended to compare the values of choice events to the simple events. To conclude, I
will compare the choice algebras constructed here to the representation of decision
problems proposed by other authors.
Another useful feature is that the Desirability function of Jeffrey’s system
effectively ranks the values of propositions in a one-dimensional continuous interval,
like an interval in the real line. The values of propositions form equivalence classes, in
which each point in a real interval is associated to a mutually indifferent set of
propositions. Decision problems will also be mapped to equivalence classes, some of
which correspond to real numbers. However, it will appear that certain cases of
problems with infinitely many options correspond not to a specific real number, but to
an open interval of reals.
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Choice Events
The event of having a choice is represented by a set. This "option set" set may
be infinite, but may not be empty. The set represents a “menu”, and when a decisionmaking subject is presented with a menu, a subset of the menu items are selected. This
"chosen subset" may be a singleton, and is also permitted to be empty 2.
Formally, we begin with a complete and atom-free Boolean algebra of events,
denoted by ℰ. Our option sets will be all the subsets of ℰ which exclude the null event
“F”; in other words, the power set P(ℰ-{F}). Call the collection of option sets ℭ. We
assume that the impossible event “F” can never be chosen. Note that in The Logic of
Decision, “F” is excluded from the domain of the preference relation.
The choice algebra ℭ is again a complete Boolean algebra, under the set
operations of union, intersection, and complementation. An element in the choice
algebra represents the opportunity to choose from among the events in the option set.
Considerable work has been done (primarily by economists) in exploring the properties
of choices represented by such menus. When a collection of menus is defined, a
function which maps each menu to its chosen subset is called a "choice function." The
question typically addressed by economists has been, "When does a choice function
define a binary preference relation?" In this paper, I will be working in the other
direction. We will assume that a preference relation exists, and use that relation to
define a choice function.
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The binary preference relation on the algebra of propositions ℰ is denoted by the
symbol “≽” 3. Naturally, every subset of an option set is also a member of the choice
algebra, so the choice functions have both a domain and range within the choice algebra
ℭ. We can define a choice function ƒ from a preference relation by:
[1]
ƒ: ℭ → ℭ : ƒ(C) = {c ∈ C | x ∈ C ⇒ c ≽ x }
In other words, given an option set C from the choice algebra ℭ, the chosen
subset of C includes any elements of C which are preferable to all other element of C.
Such elements are called “maximal” or “undominated”. Here are some useful
observations:
1.
If the chosen subset ƒ(C) has more than one member, then each
member of ƒ(C) is indifferent to all the others4.
2.
If C is finite, then ƒ(C) will never be empty. At least one element of C
will be maximal.
3.
If C is a singleton, then it sole member is itself maximal, and ƒ(C)=C.
4.
If C is infinite, then C may contain a sequence of increasingly
preferable members, and not contain a limit of that sequence. In such
a case, no members are maximal and ƒ(C) will be the empty set.
While the cases in (4) must be handled separately, we will see that they do not
pose a difficulty.
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Open Choices
When an option set does not contain any undominated element, I will the
decision problem "open”. In an open decision there are no choosable elements, and the
chosen subset is empty. Even if an agent made a (perhaps arbitrary) selection of a suboptimal element, there is always a better option for each potential choice.
Another twist, which does not work, is to choose all the elements whose value is
above a certain threshold. But such a chosen subset would contain elements which are
not indifferent. Consider the proposition which is the disjunction of such a subset.
Then we can choose one of the dispreferred disjuncts, and form a new proposition
which lacks just that dispreferred disjunct. Now the reduced proposition will be
preferred to the original, because of the averaging property5. Thus we can construct a
sequence of increasingly preferable chosen subsets, and again to select one rather than
another is arbitrary.
Although an open decision has no "solution" in the ideal sense 6 the open
decisions may be assigned an unambiguous value. Consider first the case of an option
set which is infinite but has an upper bound. An example using real numbers would be:
O1 = { 1, 1.5, 1.75, 1.875, ... }
The option set O1 has no maximal element, but at the same time, no element of
O1 is as desirable as a proposition with value equal to two. Here are two observations
regarding the set O1:
1. For any p in E st. p < 2, there is some o in O1 s.t. o > p
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2. For and p in E st. p >= 2, there is no o in O1 st. o > p
To have the opportunity to choose from among the elements of O1 must be
preferable to simply getting any element of E with is strictly less desirable than two. But
simply getting a proposition of desirability equal to two is preferable to any possible
outcome of the decision problem posed by O1.
Next, consider an option set containing an infinite sequence of elements with
consistently increasing value. An example using real numbers would be:
O2 = { 1, 3, 5, 7, ... }
Again, the option set O2 has no maximal element. However, in this case there is
no upper bound to the value of elements in O2. The relevant property of O2 is:
3. For any p in E, there is some o' in O2 st. o' > p
That is to say, the opportunity to choose an element from O2 must be preferable
to simply getting any proposition in E. Open decisions which are unbounded become
maximal elements in the choice algebra. All these unbounded decision problems are in
a single equivalence class, and are indifferent to each other.
Preference for Choice Events
Now we will formally extend the preference relation defined over the event
algebra ℰ, to the comparison of option sets in ℭ. Take an arbitrary option set C, and
consider the events in its chosen subset, ƒ(C). Since the elements of this subset are
members of the complete Boolean algebra ℰ, the disjunction of all the elements in ƒ(C) is
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also an element of ℰ. If the chosen subset of C is not empty, we will give the choice
event C a position in the preference relation by setting it equal in preference to the
disjunction of its chosen subset:
[2]
If ƒ(C) ≠ ∅, then C ≈ sup( ƒ(C) ).
In this expression, “sup()” refers to the Boolean operation of the event algebra ℰ.
The sup() or “supremum” corresponds to the potentially infinite disjunction of elements
from the chosen subset ƒ(C)7.
Now let us consider cases when ƒ(C ) is the empty set. These cases occur only
when C does not contain any maximal element. This may occur in two ways. Either
there is no upper bound for the preference of elements in C, or, there is an upper bound
but not within C itself. If there is no upper bound for the preference of elements in C,
then we will define the preference of C by:
[3]
If ∄e ∈ ℰ ∍ ∀c ∈ C, e ≽ c, then for any D ∈ ℭ, C ≽ D
That is to say, an unbounded open decision problem is weakly preferred to any
other choice. The unbounded choices are maximal, when the preference relation is
extended to the choice algebra.
The last case to consider is the case of a bounded option set which does not
contain its limiting value. If C has any upper bound e ∈ ℰ, then by the continuity
assumption8 for preference in ℰ, there is a unique equivalence class defined by the least
upper bound LUB(C) = [e0] such that e0 is preferable to every element in C, and no other
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element is lesser than e0 while still being an upper bound of C. Now we may define C as
strongly preferable to each of its elements, but dis-preferred to its least upper bound:
[4]
If LUB(C) = [e0], and no e ∈ [e0] is also in C, then {e0} ≻ C and ∀c ∈ C,
C ≻ {c}
Taking the transitive closure completes the extension of the preference relation
to CA. From these definitions some immediate consequences are:
THEOREM I: Preference in ℭ is transitive, asymmetric, and connected.
THEOREM 2: Preference in ℭ is everywhere continuous but nowhere dense.
THEOREM 3: Preference in ℭ is homomorphic to the order relation among rightterminated open and closed half-lines of the real numbers, in terms of their right
endpoints.
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Option Sets and Matrix Representation
Now let's compare the option set formalism to the decision matrix scheme which
is more familiar in the philosophical literature on decision theory. We will see that the
two approaches are equivalent. There are many versions of matrix representation under
discussion in recent books and articles, but here I will consider only the system of James
Joyce's Foundations of Causal Decision Theory9. For Joyce, a decision problem is a fourtuple <Ω,A,S,O>, where Ω is a Boolean sigma-algebra, Ai and Sj are partitions of the
act-space and state-space, and Oij is the array of outcomes which obtain when act Ai is
selected in state of nature Sj
The matrix description translates directly into a option set. The elements of the
option set are the "rows" of the decision matrix. More precisely, suppose that I and J are
index sets, respectively, for the act partition Ai and the state partition Sj. The Cartesian
product I x J is then the index set for the outcomes Oij. Now for any four-tuple
<Ω,A,S,O> we define a option set by
C = { Ci, i ∈ I | Ci = Union(j ∈ J)( Ai & Sj & Oij ) }
Joyce considers only cases where Ω is a sigma-algebra and the index sets are
countable. However, in this paper we are discussing the case envisioned by Jeffrey, in
which the Boolean algebra of propositions is complete. In the complete case, unions of
any arbitrary cardinality are allowed.
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Constructing a matrix from an option set can be equally trivial, provided we
don’t expect too much structure. In general, a detailed partition of states of nature will
not be available for an option set, nor will the set of options form a partition of available
acts, as Joyce’s system requires.
In the special cases where the elements of the option set are mutually disjoint,
Joyce’s partition requirement will be satisfied. Then a single state of nature, the
necessary proposition “T”, will suffice as sole member of the State-partition Si. The
same proposition will also serve as every member of the outcome array Oij. Thus, the
option set generates a one-column matrix-form decision problem. The formal structure
is given by:
D = < ℰ, C, {T}, {T} >
For the more general case of choice sets which have non-disjoint elements, the
Ω for a Joyce-type matrix cannot be taken directly from the proposition algebra ℰ.
Instead, we must assume a family of “ethically neutral,” mutually incompatible
propositions ℬ, equal in cardinality to ℰ and both probabilistically and causally
independent of the propositions in ℰ. Intuitively, the propositions in the “partition set”
ℬ might represent verbal performances such as, “the agent chooses e1” or “the agent
chooses e2”. It is convenient to take the propositions of ℬ as indexed by the
propositions in ℰ. Now, for any option set C = {ci}, the Boolean algebra for the matrixform representation is generated by the set:
M = { m ∈ ℬ ∪ ℰ | mi = bci & ci }
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That is to say, the propositions needed to frame the decision problem as a
matrix are formed by taking the conjunction of each member of the option set C, with
the appropriately indexed member of the independent partition set ℬ. Then we again
have a single-column matrix to represent the decision problem, where Ω is the Boolean
algebra generated by M:
D = < Ω, M, {T}, {T} >
Conclusion
We have seen that decision problems may be equally well represented by the
simple menus of economic theory, as by the more elaborate matrix representations of
philosophical decision theory. We have also seen that decision problems may have a
value, even when they lack a solution.
1
For a classic exposition, see R. Duncan Luce and Howard Raiffa, Games and Decisions
(New York: Wiley) 1957.
2
The formalism of decision sets and choice functions goes back at least to 1938. cf.
Paul Samuelson, “A Note on the Pure Theory of Consumer’s Behavior, “ Economica, N.S
vol. 5 (1938). For a survey see: Amartya Sen, “Choice Functions and Revealed
Preference,” Review of Economic Studies (1970)
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3
The preference relation is assumed to satisfy the hypotheses of Bolker's existence and
uniqueness theorems. Cf. R. C. Jeffrey, The Logic of Decision, (Chicago: University of
Chicago Press) 1983.
4
Because preference is an averaging relation, each member of the chosen subset is also
indifferent to any disjunction of other members. Intuitively, we might say that any
lottery among some of these indifferent elements is indifferent to any other.
5
Cf. R. C. Jeffrey, op. cit., p. 146
6
There is an extensive literature (originating in the work of Herbert Simon) on
“satisficing” – the selection of sub-optimal elements from decision problems. I will not
discuss those concepts in this paper.
7
Cf. R. C. Jeffrey, op. cit., p. 148
8
Ibid.
9
James M. Joyce, The Foundations of Causal Decision Theory (Cambridge: Cambridge
University Press), 1999