Abstract
It is a familiar argument that advocates accommodating the paradoxes of decision theory by abandoning the “independence” postulate. After all, if we grant that choice reveals preference, the anomalous choice patterns of the Allais and Ellsberg problems (reviewed in section “Review of the Allais and Ellsberg “Paradoxes”) violate postulate P2 (“sure thing”) of Savage’s (The foundations of statistics. Wiley, New York, 1954) system.
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Notes
- 1.
By assuming (*), we fix it that M/ ∼ (no longer assumed to be a mixture set) has a countable dense subset in the < −order on M/ ∼, e.g., the rational-valued sure-dollar equivalents. Then our first two postulates ensure a real-valued utility u on M with the property that L 1 < L 2 if and only if u(L 1) < u(L 2). The point is that, without “independence,” the usual Archimedean axiom is neither necessary nor sufficient for a real valued utility. See Fishburn (1970, Section 3.1) for details, or Debreu (1959, Chapter 4), who discusses conditions for u to be continuous. Debreu uses a “continuity” postulate in place of (2) that, in our setting, requires that if the sequence [L i ] converges (in distribution) to the lottery L i , and L i < L k, then all but finitely many of the L i < L k . If we extend ≲ to general distributions over R , Debreu’s continuity postulate entails countably additive probability. See Seidenfeld and Schervish (1983) for some discussion of the decision-theoretic features of finitely additive probability.
- 2.
Recall, lottery L 2 (first order) stochastically dominates lottery L 1 if L 2 can be obtained from L 1 by shifting probability mass from less to more desirable payoffs. More precisely, L 2 stochastically dominates L 1 if, as a function of increasingly preferred rewards, the cumulative probability distribution for L 2 is everywhere less than (or equal to) the cumulative probability distribution for L 1. Of course, whenever L 2 stochastically dominates L 1, there is a scheme for payoffs, in accord with the two probability measures, where L 2 weakly dominates L 1.
- 3.
To define the generalized mixture-set M′, it suffices to define the operation of convex-combination of two (generalized) lotteries. This is done exactly as in Anscombe and Aumann’s (1963) treatment of “horse lotteries.” Horse lotteries, the generalized postulates (l)–(3) for horse lotteries, and, with the addition of two minor assumptions (precluding a preference-interaction between payoffs and states), the subjective expected-utility theory that results, are discussed by Fishburn (1970, Chapter 13) and briefly in section Sequential coherence of Levi’s decision theory here.
- 4.
These preferences are in conflict with Savage’s (1954) “sure-thing” postulate P2. P2 is inconsistent with the following two preferences:
-
(i)
L right < L mix.
-
(ii)
L left ∼ L right, given the coin lands heads up.
Consider the four-event partition generated by whether the coin lands heads (H) or tails (T), and whether the $10 is in the left (L) or right (R) pocket. Then, by (i), the first row (below) is preferred to the second. Savage’s theory uses “called-off” acts to capture conditional preference. Thus, by (ii), the agent is indifferent between the third and fourth rows.
HL
HR
TL
TR
L mix
$10
$0
$0
$10
L right
$0
$10
$0
$10
L left | H
$10
$0
$0
$0
L right | H
$0
$10
$0
$0
-
(i)
- 5.
Let u be a utility on payoffs and assume u is positive. Denote by E u (L 1) the expected utility of lottery L 1 under utility function u. Denote by \( {L_1}^{-1} \) the lottery that has payoffs with (multiplicative) inverse utility to L 1. Samuelson’s (1950) “Ysidro” ranking, ≲γ, on lotteries is given by the function
$$ Y\left({L}_1\right)\kern0.5em =\kern0.5em {\left[{E}_u\left({L}_1\right)/{E}_u\left({L_1}^{-1}\right)\right]}^5. $$Not only does ≲γ satisfy the ordering, Archimedean, and mixture dominance postulates while failing independence but in addition ≲γ respects stochastic dominance!
- 6.
The result is elementary and has been noted by many, including Kahneman and Tversky (1979, p. 283–284). Suppose π is not linear so that π(p + q) > π(p) + π(q). Then by letting the value υ(r 2) approach the value υ(r 1), the agent is required (strictly) to prefer L 1: P 1(r 1) = (1 – [p + q]), P 1(r2) = p + q, and P 1(r 1) = 0 – over L 2: P 2(r 1) = P 1(r 1), P 2(r 2). = p, P 2(r 3) = q, even though L 2 stochastically dominates L 1. The argument for the other case is similar: π(p + q) < π(p) + π(q).
- 7.
Hammond’s (1976, Section 3.3) felicitous phrase is that the agent uses “sophisticated” versus “myopic” choice.
- 8.
McClennen (1986, 1988, forthcoming) sketches a program of “resolute” choice to govern sequential decisions when independence fails. I am not very sure how resolute choice works. Part of my ignorance stems from my inability to find a satisfactory answer to several questions.
As I understand McClennen’s notion of resolute choice, the agent’s preferences for basic lotteries change across nodes in a sequential decision tree. (Then, the premises of the argument in Section 4 do not obtain.) In terms of the problem depicted in Fig. 4, at node A the agent resolves that he will choose L 1 at (a) of node B, and by so resolving increases its value at (a) of node B above the $5.50 alternative.
There are several difficulties I find with this proposal. I suspect that the new value of L 1 at (a) will be fixed at $6.00, and likewise for L 2 at (b). (The details of resolute choice are lacking on this point, but this suspicion is based on the observation that a minor variation in the sequential incoherence argument applies unless these two lotteries change their value from node A to node B as indicated. Just modify the construction so that the rejected cash alternative at B is $6.00 – δ.) Then the assessed value of $6.00 for L 3 (a mixture of the lotteries L 1 and L 2, now valued at $6.00 each) is in accord with postulate (2). Such resolutions mandate that changes in preferences agree, sequentially, with the independence postulate. In terms of sequential decisions, is it not the case that resolute choice requires changes in values to agree with the independence postulate?
A second problem with resolute choice directs attention at the reasonableness of these mandatory changes in values. For example, consider the Ellsberg-styled choice problem described in Section 3.2. Cast in a sequential form, under this interpretation of resolute choice, if L mix is most preferred, then the agent is required to increase the value for the option “take the contents of the right pocket,” given that the coin lands heads up, over the value it has prior to the coin flip.
But the coin flip is irrelevant to a judgment of where the money is. However uncertain the agent is prior to the coin flip, is he not just as uncertain afterwards? Concern with uncertainty in the location of the money is the alleged justification for a failure of independence when comparing the three terminal options: L left, L right, L mix, and declaring L mix (strictly) better than the other two. What justifies the preference shift, given the outcome of the coin flip, when L right becomes equivalued with an even-odds lottery over $10 and $0 despite the same state of uncertainty about the location of the money before and after the coin flip?
Third, by what standards can the agent reassess the merits of a resolution made at an earlier time? In other words, how is the agent to determine whether or not to ignore an earlier judgment: the judgment to commit himself to a resolute future choice and thereby to change his future values. Once the future has arrived, why not instead frame the decision with the current choice node as the initial node without altering basic values? Unless this issue is addressed, the question of how to ratify a resolution is left unanswered, and the problem remains of how to make sense of the earlier resolution once the moment of choice is at hand.
- 9.
By contrast, Raiffa’s (1968, pp. 83–85) classic objection to the failure of independence in the Allais paradox depends upon a reduction of extensive to normal forms. Also, in his interesting discussion, Hammond (1984) requires the equivalence of extensive and normal forms through his postulate of “consequentialism” in decision trees. These authors defend a strict expected-utility theory, in which the equivalence obtains. Likewise, LaValle and Wapman (1986) argue for the independence postulate with the aid of the assumption that extensive and normal forms are equivalent.
The analysis offered here does not presume this equivalence, nor does avoidance of sequential incoherence entail this equivalence, since, e.g., it is not satisfied in Levi’s theory either – though his theory avoids such incoherence. Hence, for the purpose of separating decision theories without independence from those without ordering, it is critical to avoid equating extensive and normal forms of decision problems.
- 10.
One may propose that, by force of will, an agent can introduce new terminal options at an initial choice node, corresponding to the “normal” form version of a sequential decision given in “extensive” form. Thus, for the problem depicted in Fig. 20.3, the assumption is that the agent can create the terminal option L 3 at node A by opting for plan 1 at A and then choosing L 1 at (a) and L 2 at (b).
Whatever the merit of this proposal, it does not apply to the sequential decisions discussed here, since, by stipulation, the agent cannot avoid reconsideration at nodes B. There may be some problems in which agents can create new terminal options at will, but that is not a luxury we freely enjoy. Sometimes we have desirable terminal options and sometimes we can only plan. (See Levi’s [1980, Chapter 17] interesting account of “using data as input” for more on this subject.)
- 11.
- 12.
Levi (1980, Section 5.6) offers a novel rule, here called rule’, for determining expectation-inequalities when the partition of states, π, is finite but when events may have subjective probability 0. The motive for this emendation is to extend the applicability of “called-off” bets (Shimony 1955) to include a definition of conditional probability given an event of (unconditional) probability 0. Also, it extends the range of cases where a weak-dominance relation determines a strict preference.
Given a probability/utility pair (P, \( \mathcal{U} \)), maximizing †-expected utility (with rule’) includes a weak-order that satisfies the independence axiom, though, †-expectations may fail to admit a real-valued representation, i.e., the “Archimedean” axiom is not then valid. Under rule†, given a pair (P, \( \mathcal{U} \)), †-expectations are represented by a lexicographic ordering of a vector-valued quantity.
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Acknowledgments
I have benefitted from discussions with each of the following about the problems addressed in this essay: W. Harper. J. Kadane, M. Machina, P. Maher, E. F. McClennen, M. Schervish; and I am especially grateful to I. Levi.
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Editors’ Note
Subjective expected-utility theory provides simple and powerful guidance concerning how to make rational decisions in circumstances involving risk. Yet actual decision making often fails, as has been well known for decades, to conform to the theory’s recommendations. If subjective expected-utility theory represents the ideal of rational behavior, these failures may simply show that people often behave irrationally. Yet if the gap between ideal and actual behavior is too wide, or if behavior that on the best analysis we can make is rational but not consistent with subjective expected-utility theory, then we may come to doubt some of the axioms of the theory. Two main lines of revision have been suggested: either weakening the “ordering” axiom that requires preferences to be complete or surrendering the so-called independence principle. Although the issues are highly abstract and somewhat technical, the stakes are high; subjective expected-utility theory is critical to contemporary economic thought concerning rational conduct in public as well as private affairs.
In the preceding article, “Decision Theory without ‘Independence’ or without ‘Ordering’: What Is the Difference?” Teddy Seidenfeld argued for the sacrifice of ordering rather than independence by attempting to show that abandoning the latter leads to a kind of sequential incoherence in decision making that will not result from one specific proposal (Isaac Levi’s) for abandoning ordering. In their comments in this section, Edward McClennen, who supports surrendering the independence postulate rather than ordering, and Peter Hammond, who argues against any weakening of subjective expected-utility theory, discuss Seidenfeld’s argument from their quite different theoretical perspectives.
Economics and Philosophy, 4, 1988, 292–297. Printed in the United States of America.
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Seidenfeld, T. (2016). Decision Theory Without “Independence” or Without “Ordering”. In: Arló-Costa, H., Hendricks, V., van Benthem, J. (eds) Readings in Formal Epistemology. Springer Graduate Texts in Philosophy, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-20451-2_20
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