Box 18, Item 1225: Draft of Rational decision theory without paradoxes

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Box 18, Item 1225: Draft of Rational decision theory without paradoxes

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Printout of draft, undated.

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One of seven papers digitised from item 1225.

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The University of Queensland's Richard Sylvan Papers UQFL291, Box 18, Item 1225

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This item was identified for digitisation at the request of The University of Queensland's 2020 Fryer Library Fellow, Dr. N.A.J. Taylor.

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For all enquiries about this work, please contact the Fryer Library, The University of Queensland Library.

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[6] leaves. 4.2 MB.

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Manuscript

Text

RATIONAL DECISION THEORY WITHOUT PARADOXES

The uniform strategy applied to remove paradoxes in several other arenas - implication,
confirmation, conditionality, explanation, logic itself — will be adapted to defuse all the usual

paradoxes vexing supposedly rational decision theory. That strategy comprises, essentially,
changing the underlying logic (which is the source of central rationality principles of logical—

type) to a modest relevant logic. It yields two important components: removal of irrelevance,
particularly in derivatives of the logical theory, for instance in this case probability and
preference theory; and reduction of excessive strength, not merely in the logical theory and its
derivatives, but especially in this case in maximization principles.
Decision theory hitherto has been, virtually without exception, classical: it has

unquestioningly assumed classical principles of consistency and maximality. Well it might be

claimed that these principles - which derive from the very classical Great Chain of Being -

serve, with but few further impeccable principles, to deliver classical logic, its derivatives such
as standard probability theory, and much else. The universality of these formative principles is

however a myth, one of the pervasive myths of modem rationality (see MR).

1. Common (i.e. equivalent) consequences arguments, and Allais' paradox.

Common consequences arguments depend upon the substitution of common, i.e.
equivalent, consequences. It is evident that what substitutions are permitted will depend not
merely upon admissible logical substitution principles, but above all upon the underlying notion
of equivalence.
A basic substitution principle takes the form
A<-*B,
F(A) /
F(B)

(S<->).

It yields, as derived cases, such further inference principles as

A<->B

f(A) = n

/

f(B) = n

and

A~B
/
G(A) = G(B),
given requisite vocabulary. But critical features of the resulting logics and theories depend
upon what the co-implication relation, <-», permits; its strength.
Over dinner at a Paris cafe one evening during an international colloquium on risky

choice, Allais asked Savage a pair of now very famous questions. First Allais asked Savage to

choose between prospects A and B:

A

complete certainty of a good outcome

$lm

B

0.10 probability of a very good outcome

$5m

0.89 probability of a good outcome
0.01 probability of a bad outcome

$lm

$0.

Savage preferred A to B. Allais next asked him to choose between prospects C and D:

C

0.89 probability of bad outcome

$0

D

2
0.11 probability of good outcome

$lm

0.90 probability of bad outcome

$0

0.10 probability of very good outcome

$5m.

Savage preferred D to C. Subsequent researchers quickly discovered that most of their
seemingly rational subjects responded initially to Allais' pair of questions as Savage had.1
The problem with these responses is that they violate classical expected utility theory, and

are accordingly classically irrational, that theory affording necessary conditions for
“rationality”. Therein too lies the Allais' paradox, that seemingly rational responses by
seemingly rational subjects are ruled irrational by the theory. For conjoint preferences of A

over B and D over C contradict the deliverance of the dominant theory, which specifies what is
rational here, according to which C over D is equivalent to A over B, and so the conjoint

preferences are inconsistent. The situation certainly supplies a paradox in the standard sense,

that preanalytic claims (that so and so definitely does not imply, does not confirm, such and

such) are contradicted by results of the dominant theory covering those claims.
The (too) brief argument to inconsistency of the seemingly rational preference rankings

under the dominant theory runs as follows:- Where U symbolizes the relevant utility function
(to which probabilities are immaterial in the present context), the preference for A over B can be
represented as
(1)

U($lm)

>

0.10U(%5m) + 0.89U($lm) + 0.01U($0),

Now it is assumed that an equivalent ranking - sometimes said to be the same ranking - results
by either adding common consequences to each side of the inequality (introduction or

combination rule) or by deleting common consequences from either side of the inequality

(elimination or deletion rule). These procedures are after all distinguishing features of dominant
expected utility theory.2 Let us then add the common consequences of a 0.89 probability of
receiving $0, and delete the common consequences of a 0.89 probability of receiving $lm.

That is, putting these operations together, let us add to each side of (1) the combination

1

The presentation follows Pope's adjustment of Allais' original account (in his 79 p.354). For
detail of subsequent research on initial responses, see also Pope 89. (Note however that Pope
tends to conflate acting rationally with acting consistently.)

2

The names given to these features of the expected utility procedure - with the different names
linked with their various axiomatic derivations - include Neumann and Morgenstern's rule for
equivalent classes, presented in their 47, p.24 and Malinvaud 52, Samuleson's independence
axiom, presented in his 53, Luce and Raiffa's substitution principle, presented in their 52,
p.26, the “combination” and “cancellation” property in Tversky and Kahneman 86 and the
joint result of “ordering” and the “reduction principle” in Fishburn 87. These critical
principles, especially their negative (reduction) forms or outputs, will come in for close
scrutiny in what follows.

(2) + 0.89U($0)
0.89U($lm).
Then the preference for A over B can be equivalently represented by the result of this
(monotonic) addition, namely,
U($lm) + 0.89 U($0) - 0.89U($lm) > 0.10U($5m) + 0.89U($lm) + 0.01U($0) +
0.89U($0) - 0.89U($lm),

whence follows by utility arithmetic,
(3)
0.11U($lm)
+ 0.89U($0m)

>

0.10U($5m) +

0.9U($0).

But (3) represents nothing but the preference of C over D. Hence, given adherence to the
dominant theory and its procedures, preference for (or choice of) A over B is equivalent to
preference for C over D, not D over C. So whoever prefers one and adheres to the procedures
is rationally obliged to prefer the other, and is rationally inconsistent in preferring its converse.

Short of postponement (Savage's immediate move, in asking for time to think, in

response to Allais), these are two main strategies:
A. Dominant options, which involve explaining away pre-analytic responses. Savage's further
moves went this this theory-saving way, through the “sure-thing” principle. From a logical

perspective, the sure-thing principle is applied in transforming cases like assessment of D vis a
vis C, which are put out of intuitive bounds, into equivalent canonical forms, which are

(intuiively) decidable. The dominant theory is sho3red up, in a typical way, by psychological
work; those who do try to assess complex non-canonical cases like C vs. D are subject to

“framing illusions”.3
A*. Subversive options, which accept the pre-analytic responses, at more or less face value,
and modify the dominant theory. There are various ways modification can take place:

B. Reject the formalisation of the argument offered as too simple, because omitting

decisive information. Such an approach is taken by Pope, who contends that an occasionally
glimpsed but generally omitted time dimension is crucial. On that approach, which certainly
does not rule out fuller formalisation, a dynamic theory is regarded as essential.
B*. Accept the style of formalisation, as not overly simplistic, but reject specific

principles used in establishing equivalence of preferences. Such an approach, though of a fully
systematic kind, is of course that course pursued here. A rival relevant static theory is
elaborated. Although a static theory is adequate, so it is claimed, to remove paradox, dynamic
implementation naturally is not excluded. While it is not essential in the present setting, in other

settings it may well be required.
We now proceed to what breaks down on the B* approach, namely subtraction principles

fail. The result is that we can have both A over B and C over D. To establish the point

3

For one fuller account of those dominant strategies, see Pope.

4
convincingly, we should supply a requisite model, [do model].

The relevant resolution sits very happily with significant empirical material which
Kahneman and Tyersky (K & T hereafter) have adduced contrasting combinative or additive
procedures with eliminative or substractive procedures,

[detail and reference].

Such

differences are strikingly exhibited not only in empirical practice (including the arithmetic

without zero of earlier human societies), but in logical theory. Two logical examples are these:
Gentzen methods, where there may be significant differences between introduction and
eliminative procedures, and, nearer to the present issue, relevant arithmetic, where negative

operations such as subtraction and division induce difficulties not encountered with

corresponding positive operatives.
K & T's positive assistance in distinguishing positive and negative diverences at work in

choice procedures is however utterly overwhelmed by their predominant (classical theory
saving) contentions. These attempt to remove all differences between ways of rectifying
classical decision theory. All are equally based on framing illusions, [work through examples,

detail this epistemological escape-chute]. Framing is a theory-solving device par excellence.

A highly favoured strategy for distancing rational decision theory from empirical
information consists in presentation of inconsistent preferences alleged to violate transitivity.

The argument runs like this: there can be no rational theory without transitivity, but actual
preferences, empirically recorded, may violate transitivity; so rational theory must discount

such empirical information. Such information is set aside, for instance as involving “framing”
distictions. There have been heroic attempts to slip around the first premiss, to design
nontransitive preference theories. No doubt something of this sort can be done; but in order to
gain a theory of some strength postulates much less pausible than transitivity are infiltrated. (In

any event, the major project of founding economic on aggregated preference theory depends
heavily, and approvently essentially on transitivity.) What should have been questioned,
however, is the second premiss, that transitivity is violated. Experimenters and other who

arrive at such a result have typically assumed consistency of subjects. Without consistency
nothing stops both the preferences empiricially found and transitivity. Such subjects have
inconsistent preferences. But, to exaggerate a little, nothing is more common than inconsistent

preferences or devises. Often, at least, preference sets are overdetermined, rather than, what

has highterto been supposed must be the case, logically underdetermined.
To allow properly for inconsistent preferences, as it is proposed to do, the underlying
logic must be paraconsistent. Otherwise, as with a classical base, the theory will trivialize

preferences. But the most satisfactory types of paraconsistent logics are relevant ones.
2. Rational decision theory and rational belief revision.

Consider the classical expansion principle - A

(A & B) V (A & ~B) and its extensions

to further parameters. - This principle is at the core of standard Bayesianism:

5
• Jeffrey's derivability axiom requires it for its generality
• causal decision theories require it for building ultimate partitions of outcomes
• standard conditional expected utility requires it, e.g. U(A) = U(A & B) + U(A &
~ B), leading to U(A / B) = U(A & B) (on which see Malinas).
The same expansion principle is critical in Lewis's triviality proof. Likewise belief revision

theory, there are negative results where similar expansion principles play a major part.
But expansion is intricated in implicational paradox, indeed it plays a crucial part in C.I.

Lewis's “independent” proof of the inevitability of positive paradox in entailment theory. That

simple argument proceeds as follows:
A-». (A & B) V (A & ~B)
(A & B) V (A & ~B)

A&(B V ~B)

A->. A & (B V ~B)
A & (B V ~B)
B V ~B
A->.B V ~B

Expansion
Distribution
by Rule Transitivity

Simplification
by Transitivity again

That is, positive paradox, and irrelevance since B may have nothing to do with A. For this
reason among other, Expansion is rejected in relevant logic (see further RLR). With its rejection
a mass of established theory, implausible theory, crumbles.

3. Conditional preference theory (Bayesian style).
Therewith much relevant relocationis required.
Relevant relocation enables the wholesale removal of much of the surrounding belt of

idealisations which shield standard theory from criticism but at the same time prevents its
practical application to real-life agents and situations. Examples of idealisation in belief and

theory revision normative theory, sharing their disabling extents are as follows: It is assumed
that

• agents are perfect (classical) reasoners

• agents have determinate (real) degrees of belief
• agents have determinate conditional degrees of beliefs for every pair of propositions
expressible in their language [ - maintenance of probabilistic coherence].

References
Allais,M., ‘The so-called Allais paradox and rational decision under uncertainty’, in M. Allais
and O. Mazen (eds), Expected Utility and the Alluis Paradox, Reidal, Dordrecht 1979, 437681.
Fishbum, P., ‘Reconsiderations in the foundations of decision under uncertainty’, Economic
Journal 97(1987) 825-841.
Lace, R., and Ruiffa, H. Games and Decisions, John Wiley, New York, 1957.
Malinvaud, E., ‘A note on von Neumann - Morgenstern's strong independence axiom’,
Econometrica 20(1952) 679.
von Neumann, J., and Morgenstern, O., Theory of Games and Economic Behaviour, Princeton
University press, New Jersey, 1947.

Pope, R., ‘The delusion of certainty in Savage's some-thing principle’ typescript Duntroon,
1989.
Routley, R., and others, Relevant Logics and Their Rivals, Ridgeview, California, 1982;
referred to as RLR.
Samuelson, P., ‘Probability, utility and the independence axiom’, Econometrica 20(1952) 670678.

Sylvan, R., ‘Modem myths concerning rationality’, ; referred to as MR.
Tversky, A., and Kahneman, D., ‘Rational choice and the framing of decisions’, Journal of
Business 59(1986) s251-s278.

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Citation

Richard Sylvan, “Box 18, Item 1225: Draft of Rational decision theory without paradoxes,” Antipodean Antinuclearism, accessed February 23, 2024, https://antipodean-antinuclearism.org/items/show/181.

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