评论"预期不确定效用"

IF 6.6 1区 经济学 Q1 ECONOMICS
Econometrica Pub Date : 2024-01-30 DOI:10.3982/ECTA21843
Simon Grant, Sh. L. Liu, Jingni Yang
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An event is deemed ideal by the DM if both it and its complement together satisfy a version of <span>Savage</span> (<span>1954</span>)'s sure thing principle.</p><p>Unfortunately, GP's characterization fails on two accounts, as their axioms neither ensure</p><p>In this note, we show that strengthening one of GP's axioms, along with a slight modification of their continuity axiom, provides a characterization of EUU maximization. But first, we present in Section 2 an example of an EUU functional involving a state-dependent interval utility and show that the preferences generated by this example, despite satisfying all of GP's axioms, cannot be represented by an EUU function of the form in (2).</p><p>Let the state space be endowed with the Lebesgue measure <i>μ</i>. Let denote the set of measurable events with respect to <i>μ</i>. Following GP, is the (interval) envelope of an act <i>f</i>, with (respectively, ) denoting the lower (respectively, upper) envelope.</p><p>We show that ≿ satisfies GP's Axioms A1–A6 which we list here for the convenience of the reader. To state them, we employ the following notation: for any pair of acts <i>f</i> and <i>g</i> and any event , <i>fCg</i> denotes the act that agrees with <i>f</i> on <i>C</i> and with <i>g</i> on the complement of <i>C</i>. We also require the following definitions.</p><p>An event <i>E</i> is <i>ideal</i> if implies for all acts <i>f</i>, <i>g</i>, <i>h</i>, and . An event <i>A</i> is <i>null</i> if for all acts <i>f</i>, <i>g</i>, and <i>h</i>. An event <i>D</i> is <i>diffuse</i> if for every non-null ideal event <i>E</i>. Let (respectively, , ) be the set of all ideal (respectively, null, diffuse) events. Let denote the set of <i>ideal simple</i> acts.<sup>1</sup></p><p>As in GP, we say an event <i>E</i> is <i>left</i> (respectively, <i>right</i>) ideal if implies (respectively, implies ). Let and be the collection of left and right ideal sets, respectively. GP's Lemma B0 establishes .</p><p>In line with GP's use of notation, events <i>E</i>, , , et cetera, denote ideal events while events <i>D</i>, , denote diffuse events. 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An event is deemed ideal by the DM if both it and its complement together satisfy a version of <span>Savage</span> (<span>1954</span>)'s sure thing principle.</p><p>Unfortunately, GP's characterization fails on two accounts, as their axioms neither ensure</p><p>In this note, we show that strengthening one of GP's axioms, along with a slight modification of their continuity axiom, provides a characterization of EUU maximization. But first, we present in Section 2 an example of an EUU functional involving a state-dependent interval utility and show that the preferences generated by this example, despite satisfying all of GP's axioms, cannot be represented by an EUU function of the form in (2).</p><p>Let the state space be endowed with the Lebesgue measure <i>μ</i>. Let denote the set of measurable events with respect to <i>μ</i>. 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引用次数: 0

摘要

Gul 和 Pesendorfer(2014 年)(以下简称 GP)在一篇充满许多重要成果和见解的创新论文中,提出了一个新颖的不确定性下的选择模型。他们考虑了一个纯主观的不确定性环境,在这个环境中,选择的对象是行为,对于每一种自然状态,这些行为都会从一组最终奖品中提供一个货币奖 x,奖品为 。在 GP 的模型中,决策者(以下简称 DM)有一个先验值 μ,该先验值定义于 ,这是一个理想事件的 σ 代数。GP 将任何理想事件 E(在 )解释为 DM 可以通过赋予其概率来精确量化该事件不确定性的事件。不幸的是,GP 的表征在两个方面都失败了,因为他们的公理既不能确保EUU 最大化,也不能确保EUU 最大化。但首先,我们将在第 2 节中举例说明一个与状态相关的区间效用的 EUU 函数,并证明这个例子所产生的偏好尽管满足 GP 的所有公理,却不能用(2)式的 EUU 函数来表示。让 μ 表示可度量事件的集合。按照 GP 的说法,是行为 f 的(区间)包络,(分别为, )表示下包络(分别为, 上包络)。我们证明≿ 满足 GP 的公理 A1-A6,为了方便读者,我们在此列出这些公理。为了说明这些公理,我们使用了以下符号:对于任何一对行为 f 和 g 以及任何事件,fCg 表示在 C 上与 f 一致的行为,以及在 C 的补码上与 g 一致的行为。如果对所有行为 f、g 和 h 来说,事件 A 都是空的,那么事件 D 就是扩散的。让 表示理想简单行为的集合。1 如在 GP 中,如果意味着 (分别意味着 ),我们说事件 E 是左(分别是右)理想的。让 和 分别是左理想集和右理想集的集合。根据 GP 的符号用法,事件 E, , , 等表示理想事件,而事件 D, , 表示扩散事件。下面是 GP 的六条公理(A1-A6)。为了验证≿ 是否满足上述六条公理,我们利用了这样一个事实:当且仅当一个事件是可度量的(即是 )的一个元素时,它才被≿ 视为理想事件。 回到公理,我们可以看到每条公理都得到了如下验证:由于由(3)产生的偏好关系≿满足 GP 的公理 1-6,因此根据 GP 的定理 1,它应该允许一个具有先验 μ 的 EUU 表示2。我们保留了 GP 的四条公理,并建议加强公理 A3 和修改公理 A6(i),而保留原来的公理 A6(ii)不变。公理 A3 的加强确保了扩散 "赌注 "的条件确定性等价物的恒定性,从而排除了上一节的(反)例子。对公理A6(i)的修改使我们能够确定理想事件集确实是一个σ代数。GP的公理A6(i)意味着阿罗单调连续性的弱化版本,它适用于理想行为和理想事件。我们的新公理 A6*(i) 是适用于所有行为和理想事件的单调连续性公理。A6*(i) 意味着理想事件的可数可加性,简化了定理 1 的证明。下一个性质确保了扩散行为之间的条件确定性等价性。为简单起见,A7 可与 A3 合并如下:3
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A Comment on: “Expected Uncertain Utility”

In an innovative paper, replete with many important results and insights, Gul and Pesendorfer (2014) (hereafter, GP) proposed a novel model for choice under uncertainty. They considered a setting of purely subjective uncertainty in which the objects of choice are acts that, for each state of nature , deliver a monetary prize x from a set of final prizes , with . We denote the set of acts by , and the decision-maker's preference relation defined over by a weak order ≿.

In GP's model, the decision-maker (hereafter, DM) has a prior μ defined over , a σ-algebra of what they referred to as ideal events. GP interpreted any ideal event E (in ) as one for which the DM can precisely quantify that event's uncertainty by assigning it the probability . An event is deemed ideal by the DM if both it and its complement together satisfy a version of Savage (1954)'s sure thing principle.

Unfortunately, GP's characterization fails on two accounts, as their axioms neither ensure

In this note, we show that strengthening one of GP's axioms, along with a slight modification of their continuity axiom, provides a characterization of EUU maximization. But first, we present in Section 2 an example of an EUU functional involving a state-dependent interval utility and show that the preferences generated by this example, despite satisfying all of GP's axioms, cannot be represented by an EUU function of the form in (2).

Let the state space be endowed with the Lebesgue measure μ. Let denote the set of measurable events with respect to μ. Following GP, is the (interval) envelope of an act f, with (respectively, ) denoting the lower (respectively, upper) envelope.

We show that ≿ satisfies GP's Axioms A1–A6 which we list here for the convenience of the reader. To state them, we employ the following notation: for any pair of acts f and g and any event , fCg denotes the act that agrees with f on C and with g on the complement of C. We also require the following definitions.

An event E is ideal if implies for all acts f, g, h, and . An event A is null if for all acts f, g, and h. An event D is diffuse if for every non-null ideal event E. Let (respectively, , ) be the set of all ideal (respectively, null, diffuse) events. Let denote the set of ideal simple acts.1

As in GP, we say an event E is left (respectively, right) ideal if implies (respectively, implies ). Let and be the collection of left and right ideal sets, respectively. GP's Lemma B0 establishes .

In line with GP's use of notation, events E, , , et cetera, denote ideal events while events D, , denote diffuse events. The following are GP's six Axioms (A1–A6).

To verify ≿ satisfies the above six axioms, we utilize the fact that an event is deemed ideal by ≿ if and only if it is measurable (i.e., an element of ).

Returning to the axioms, we see each is verified as follows:

Since the preference relation ≿ generated by (3) satisfies GP's Axioms 1–6, it follows from GP's Theorem 1 that it should admit an EUU representation with prior μ.2

We retain four of GP's axioms and propose strengthening Axiom A3 and modifying Axiom A6(i) while leaving the original Axiom A6(ii) unchanged. The strengthening of Axiom A3 ensures the constancy of conditional certainty equivalents of diffuse “bets” which rules out the (counter-)example from the previous section. The modification of Axiom A6(i) enables us to establish that the set of ideal events is indeed a σ-algebra.

GP's Axiom A6(i) implies a weaker version of Arrow's monotone continuity that applies to ideal acts and ideal events. Our new A6*(i) is the monotone continuity axiom applied to all acts and ideal events.

It is straightforward to show that A6*(i) implies the countable additivity of ideal events and simplifies the proof of Theorem 1. The next property ensures the conditional certainty equivalence between diffuse acts. Its role is similar to that of P3 in Savage's axiomatization of subjective expected utility.

For simplicity, A7 can be combined with A3 into the following:3

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来源期刊
Econometrica
Econometrica 社会科学-数学跨学科应用
CiteScore
11.00
自引率
3.30%
发文量
75
审稿时长
6-12 weeks
期刊介绍: Econometrica publishes original articles in all branches of economics - theoretical and empirical, abstract and applied, providing wide-ranging coverage across the subject area. It promotes studies that aim at the unification of the theoretical-quantitative and the empirical-quantitative approach to economic problems and that are penetrated by constructive and rigorous thinking. It explores a unique range of topics each year - from the frontier of theoretical developments in many new and important areas, to research on current and applied economic problems, to methodologically innovative, theoretical and applied studies in econometrics. Econometrica maintains a long tradition that submitted articles are refereed carefully and that detailed and thoughtful referee reports are provided to the author as an aid to scientific research, thus ensuring the high calibre of papers found in Econometrica. An international board of editors, together with the referees it has selected, has succeeded in substantially reducing editorial turnaround time, thereby encouraging submissions of the highest quality. We strongly encourage recent Ph. D. graduates to submit their work to Econometrica. Our policy is to take into account the fact that recent graduates are less experienced in the process of writing and submitting papers.
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