Are Counterfactual Decisions Relevant for Dynamically
Consistent Updating under Nonexpected Utility?*
不可预期效用下的动态一致性更新与反事实的决定有关吗？
Peter P. Wakker
CentER, Tilburg University, The Netherlands
中心、昆士兰大学，荷兰
December, 1997
1997年十月
ABSTRACT. This paper proposes a new updating method that preserves dynamic consistency in nonexpected utility. Given nonseparability of disjoint 解体events, preferences conditional on an observed 观测的；观察的event also depend on counterfactual outcomes, i.e., outcomes
that would have resulted outside of the conditioning 调节；整修；条件作用event; this point has been wel
l-understood in the literature. This paper argues that, as a consequence, also counterfactual
decisions are relevant. A new "strategic" method for updating then follows.
摘要。这篇论文提出了一种保护不可预期效用下的一致性新的更新方法。考虑到解体事件的不可分割性，对所观察事件的偏好条件，取决于反事实的结果还是会引起外调节事件的外部。这一点在文学里有很好的理解。这篇论文认为，作为一种结果也是反事实决策相关的，一种新的如下战略手段用来更新
KEYWORDS: dynamic consistency, resolute choice, updating.
Journal of Economic Literature Classification Number D81.
Please send editorial communications to:
Peter Wakker
CentER, Tilburg University
P.O. Box 90153, Tilburg 5000 LE, The Netherlands
* This paper benefitted from discussions with Rakesh Sarin and with participants in a conference on un
certainty in economics, Dept. of Economics, Johns Hopkins University, Baltimore, November 1993, and from discussions with Jean-Y ves Jaffray and attendents at a seminar on risk, uncertainty, and decision,
LAFORIA, University of Paris I and VI, France, December 1996.2
关键词：动态一致性、果断抉择，更新。
《经济文献分类号码D81。
请寄给编辑通讯:
彼得韦克尔
中心、昆士兰大学
邮箱90153号c昆士兰5000，荷兰
本文也从和关于经济学上不确定性的会议上的讨论，经济学部门，美国约翰霍普金斯大学、巴尔的摩,1993年11月，以及和Jaffray Jean-Yves和参加一个关于危险性、不确定性、和决定的研讨会的出席者的讨论。
LAFORIA，大学巴黎Ⅰ和VI，法国,1996.2
1. INTRODUCTION
editorial文章This paper proposes a new updating method for nonexpected utility, called strategic
updating. Strategic updating follows the dynamic decision principles of resolute choice advocated by McClennen (1988, 1990) and Machina (1989, 1991), in particular it preserves dynamic consistency. It deviates, however, from the method of resolute updating generally adopted in the literature and called committed updating in this paper. Under committed updating, a fixed choice is assumed at counterfactual decision nodes, so that there is no
more counterfactual decision to be taken. Under strategic updating, not only counterfactual outcomes, but also counterfactual decisions remain relevant.
Let me emphasize that strategic updating does not induce new preference behavior,
different from resolute choice. The main result of this paper, Theorem 5.1, shows that resolute choice maximizes the strategically updated functional at every decision node of
every tree. That is, strategic updating is fully in line with resolute choice. The claim of this paper is therefore that the natural way for updating resolute choice is strategic, and not Committed.
1。介绍
本文提出了一种应对不可预期的效用的更新方法，称为战略更新。战略更新的动态决策原则，遵循McClennen倡导(1988、1990、)和净化(1989—1991)，果断的选择它尤其保护动态的一致性。然而它背离从果断的更新方法一般在文学中使用在本文中称为承诺更新。在承诺更新下，一个固定的选择被假定为反事实决策的准则，因此不存在采取更多的反事实的决策。在战略更新下不仅反事实结果而且反事实的决定仍然是相关的。
让我强调战略更新不会引发新的偏好行为，不同于果断抉择。本论文的主要成果，定理5.1，显示果断选择最大化战略更新功能在每一个决策节点的每棵树。即战略更新完全符合果断的选择。这篇论文的声明因此是更新果断选择是一种自然的方式战略的反攻不是忠心承诺。
If strategic updating is considered undesirable, I hope the reader will not hold that against
the analysis of this paper, but will instead question the premise of the analysis, being
resolute choice. In that case, this paper can be interpreted as a negative result for resolute choice. I ho
pe that this paper, even if interpreted as a criticism of resolute choice, nevertheless contributes to our understanding of dynamic consistency in nonexpected utility.
The outline of the paper is as follows.
如果战略更新被认为是不良的，我希望读者不会要反对本文的分析，而是质疑果断决策分析的的前提存在在这种情况下，论文可以解释为阴性结果为果断选择。我希望这篇论文即使编译成一个批评果断抉择，不过有助于我们了解在不可预期效用下的动态的一致性。
在这篇文章的提纲如下。
Section 2 summarizes the difficulties in applying nonexpected utility to dynamic choice and updating.
第二部分总结了针对动态选择和更新在不可预期效用下应用的困难。
Section 3 defines resolute choice,
第三节定义了果断抉择,
committed updating, and strategic updating. In the next section, Example 4.1 repeates principles of revealed preference, in particular "menu independence" (every choice option
has an intrinsic value, independent of competing options). That principle underlies the
concept of utility, and has been generally accepted in preference theories such as consumer demand theory. Example 4.2 is a special case of Example 4.1. It applies the revealed3 preference principles to updating under resolute choice.
承诺更新、战略更新。在下一节里范例4.1
显示性偏好的原则，特别是“菜单”独立”(每一个选择的选择
有一个内在价值，独立于选项)。这个原则强调效用的概念，已被普遍接受的偏好理论如消费者偏好需求理论。范例4.2是范例4.1的一个特殊例子。它将已经揭示的三个优先原则应用到果更新果断决策上。
Menu independence then leads to trategic updating, and not committed updating. Theorem 5.1 states the result formally, for general decision trees. Section 6 presents some examples where committed updating does result. In each case the result is due to confounding f. 混杂；混淆actors, not representative of risk ttitude, such as extraneous 外来的；没有关联的；来自体外的commitments or hidden nodes. Section 7 presents applications of strategic updating. Section 8, finally, rephrases the argument of this paper in terms of inseparability of events, and concludes. 菜单独立导致策略更新而不是承诺更新。
定理5.1正式地阐述一般决策树的结果。第六节提出了一些承诺决策的确引起结果的例子。在每种情况下由于混杂致病因素，不代表的风险
态度，如不相干的事情，或者隐藏的节点结果都是不一样的。
第七节提出了应用战略更新。
第八节，最后，重述本文的针对不可分割的事件的观点并得出结论。
2. NONEXPECTED UTILITY IN DYNAMIC DECISIONS
Today's preference is the update of yesterday's preference (Machina, 1989, p. 1652).
Hence, decision models should be able to model updated preference. Indeed, updating is a central topic of debate in the modern nonexpected utility theories. New impulses have come from game theory, where the consistency requirement for equilibrium hinges crucially on
the method of updating after an opponent's move (Dow & Werlang, 1994; Eichberger & Kelsey, 1994; Haller, 1995; Hendon, Jacobsen, Sloth, & Tranaes, 1995; Klibanoff, 1995;
Lo, 1995; Ghirardato & Le Breton, 1996; Mukerji & Shin, 1997).
Updating is relatively simple in expected utility where, because of separability of disjoint events, the technique of dynamic optimization can be used (Bellman, 1954; Streufert,
1990).
Tractability is guaranteed because one can forget about counterfactual events from
the past (consequentialism) and implementability is guaranteed because one adheres to prior plans (dynamic consistency). One of the most serious challenges to nonexpected utility was
put forward by Hammond (1988) (see also Karni & Safra, 1989, and Sarin, 1992).
2。在动态决策下的不可预期效用
今天的偏好是昨天的偏好的更新(净化,1989,p)。
因此，决策模型应该能够模型更新的偏好。事实上，更新是一个
辩论的主题在现代不可预期效用理论下。新冲动来自博弈论即一致性严格要求平衡轴在对手行动后的更新的方法(道琼斯指数和Werlang,1994分;Eichberger &凯茜,1994分;Haller,1995分;Hendon,Jacobsen、
懒惰、和Tranaes,1995分;Klibanoff,1995分；看哪,1995分;Ghirardato和勒Breton,1996分;Mukerji和胫骨,1997)。
在期望效用里更新是相对简单的，因为解体的分割性事件，动态优化可用的技术。(行李员,1954分;Streufert,1990)。
因为人们会忘记从前反事实的事件，说以温顺是保证的，执行是保证的因为一个人持优先计划(动态的一致性)。对于不可预期效用一个最严峻的挑战是哈蒙德提出(1988)(请参见Safra
卡尼和、1989、1992、、、)。
The most common response is to preserve dynamic consistency and reduction of compound lotteries, and abandon consequentialism (Machina,1989, 1991; McClennen, 1988, 1990). The resulting approach is called resolute choice and is the subject of this paper.
Strotz (1956) suggested, in a context without uncertainty, that such an approach cannot be implemented unless a precommitment device would be available.
Machina and McClennen give arguments for implementability without such a device. The critical implication of resolute choice is that the value of a real strategy depends on counterfactual events. Suc
h dependency, while obvious in game theory (Harsanyi & Selten, 1988; Shin, 1991; Asheim, 1997), is debated in individual decision making.
最常见的反应是去保护动态一致性和复合的推测并且，这个方法叫做果断决策是本论文的主题。声称，在一个没有不确定性的环境下，这种方法不能被执行，除非一个前提承诺策略可以使用。
A drawback of resolute choice is its intractability. One's current decision depends on all
the events that might have happened in the past but didn't. Machina (1989) suggests that in
a complete analysis that is indeed the case, but counterfactual events from long ago may be ignored if their impact on a distant future has faded away.
果断决策的一个弊端是本身的棘手性。一个人当前的决定取决于所有以前可能发生但是没有发生的事件。提出在一个整体分析中事实上是一个事件。但是很早以前反事实事件可能被忽略如果他们的影响对遥远的未来渐行消失的时候。
We will follow Machina and others by not entering lifetime decision trees, but using simple decision trees in the examples and illustrations.
我们将会在例子和证明中跟着和其他人不走进人生决策树而是使用简单决策树。
Let me emphasize that this paper assumes resolute choice throughout, and
argues for normative strategic updating under that assumption. A detailed discussion of the
pros and cons of resolute choice is outside the scope of this paper.
我强调一下这篇论文假设果断决策是并且认为在此种假设下规范的策略更新详细的对此赞成和反对的探讨是本篇论文讨论之外的。
If one uses Choquet-expected utility (Schmeidler, 1989; Gilboa, 1987) to derive
decisions from nonadditive measures, then methods for updating nonadditive beliefs can generate methods for updating preferences.
如果使用
V arious update methods for nonadditive beliefs
have been proposed (Dempster, 1967; Denneberg, 1994; Gilboa & Schmeidler, 1993;
Haller, 1995; Jaffray, 1992, 1994; Lehrer, 1996; Mukerji, 1996; Shafer, 1976). For some
of these, unfortunately, Choquet expected utility would not be closed under updating (Ghirardato, 1997). As pointed out by Gilboa & Schmeidler (1993), Dempster-Shafer updating can be related to Choquet expected utility by assuming superior counterfactual5 outcomes, and Bayesian updating by assuming inferior counterfactual outcomes. For the purpose of this paper, it suffices to note that both updating methods can be related to fixed counterfactual decisions.
3. COMMITTED AND STRA TEGIC UPDA TING
Let V be the nonexpected utility functional that represents prior preference over probability distributions over outcomes. Resolute choice adopts the following prior optimization
method for solving decision trees.
(1) List all the strategies available in the decision tree.
(2) For each strategy, calculate the generated probability distribution over outcomes.
(3) Choose the best available probability distribution.
(4) Follow the belonging strategy throughout the decision tree.
Thus, the decision trees can be solved by prior "normal form" optimization over strategies.
Step 2 is based on reduction of compound lotteries, step 3 on the basic rationality principles
of revealed preference, and step 4 on dynamic consistency.
We assume henceforth that an event E has been observed, leading to a decision node that
we call node 1. Paths from node 1 onwards are denoted by X, Y, Z, R, and are called real.
The complementary event Ec is now known not to be true. Paths from the resulting decision node 2 onwards are denoted by A, B, C and are called counterfactual1. Thus, paths through
the tree can be denoted by XA, RC, etc.; they are identified with probability distributions
over outcomes.
1Counterfactual decisions are sometimes called forgone decisions in the literature.6
VE denotes the derived functional (still to be explained) that represents the decision
maker's updated preferences given E. Under resolute choice, VE should be in complete agreement with prior preference. The common method of resolute updating, routinely
followed in all papers that I am aware of, is committed updating. It assumes a fixed counterfactual strategy A, and evaluates real options R by V(RA) (Eichberger & Grant,
1997; Eichberger & Kelsey, 1996; Gilboa & Schmeidler, 1993; Jaffray, 1994, Figure 1;
Lo, 1995, 1996; Machina & Schmeidler, 1992).
In general, however, several counterfactual strategies will be available. A crucial
question for committed updating then is, of course, which counterfactual strategy A should
be chosen in the committed updating functional V(RA). The only paper that, to the best of
my knowledge, considers cases with several such counterfactual decisions, is Machina (1989; see also its twin Machina, 1991). In all examples in that paper, however, the choice between counterfactual decisions is trivially governed by stochastic dominance and hence
there is no real issue of counterfactual decisions (in Machina's Figure 12, which does have nontrivial counterfactual decisions, the decision maker is misinformed and does not know
about those decisions). Parts of the text in Machina (1989), in particular Footnote 29,
suggest the following procedure, and personal communications have confirmed that it is the generally accepted procedure in the field. First, the optimal prior strategy is determined, denoted XA throughout the rest of this paper.
2 Next, the counterfactual part A of XA is
taken as the fixed counterfactual strategy to be used in updating.
Strategic updating does not assume a fixed choice outside of E. Now any real option R is evaluated by maxCV(RC), i.e., the counterfactual strategy is optimized given R. VE(.) thus
is the pointwise optimum over all functionals V(.
R).
In summary:
2To avoid formulas with suprema, I restrict attention to decision trees where the relevant
maximization
problems have solutions. Sometimes I use formulations for the case of unique solutions, leaving the general
formulations to the reader.7
• Committed updating: VE(.) = V(.A). (3.1)
• Strategic updating: VE(.) = maxCV(.C) over all counterfactual C. (3.2)
4. A CHOICE-BASED INTERPRETA TION OF UPDA TED PREFERENCE AND AN EXAMPLE
This section discusses the pros and cons of the updating methods in the context of an example. The next section gives formal statements of the claims made here. In the first example, which does not yet consider uncertainty or dynamic choice, general foundations of revealed preference are repeated. The second example is a special case of the first, with uncertainty and resolute choice involved. The principles of the first example then naturally
lead to strategic updating. Other studies of uncertainty that invoke principles of revealed preference are Hammond (1976, 1988) and Green & Oswald (1991).
EXAMPLE 4.1 [Revealed Preference]. Consider a choice set {X,Y,Z} containing three options, where options can be anything such as commodity bundles, houses, welfare allocations, lotteries, income profiles, etc. An economist observes the choice of a rational consumer. If the consumer is only willing to choose X, then we infer X Y and X Z. The preference between Y and Z then cannot be inferred from this choice situation (Kreps,
1988). To find out about the preference between Y and Z, the economist discards option X
and presents3 the choice set {Y,Z} to the consumer. If the consumer now is only willing to choose Z, then we conclude that Z Y. In this manner, choices correspond with preferences
and vice versa.
3possibly only as a hypothetical thought experiment (Kreps, 1988)8
In the observations of the consumer's choices in the example, "ceteris paribus"
assumptions must be imposed. The removal of X from the choice set should not affect
other "relevant" aspects in the choice situation. Thus, we formulate the principle of menu independence. Choice should be based on an intrinsic value of objects. That intrinsic value should not be affected by the alternative choice options that are available, neither by their presence or absence nor by their nature. In later discussions a crucial point will indeed be
that:
The value V(Z) should be independent of X. (4.1)
The term menu-independence was proposed by Sen (1997). Other terms are monadic
value (Burks, 1977, p. 277), absence of attraction effect (Huber, Payne, & Puto, 1982),
#principle of individuation by justifiers (Broome, 1991),# and context-independence
(Tversky & Simonson, 1993; see also Tversky, 1969). The condition is a special case of framing-invariance (Kreps, 1990, p. 28) and underlies revealed preference axioms such as independence of irrelevant alternatives (Nash, 1950; Arrow, 1959) and other conditions (Samuelson, 1938; V ille, 1946; Houthakker, 1950; Sen, 1971; Tian, 1993). Classical examples have been advanced in which menu-independence is not reasonable (Luce &
Raiffa, 1957, Section 13.3; Kreps, 1990, p. 28; Sen, 1997). In such cases, however, there
is no clear meaning for preference (or nonexpected utility functionals), and standard optimization is not possible. Hence, we follow the traditions of normative analyses and

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