Expected Utility Theory with Probability Grids and ...

Expected Utility Theory with Probability Grids and Incomparabilities∗ Mamoru Kaneko†, January 20, 2018

Abstract We reformulate expected utility theory, from the viewpoint of bounded rationality, by introducing probability grids and a cognitive bound; that is, we restrict permissible probabilities only to decimal (binary) fractions of finite depths up to a given cognitive bound. We separate the measurement of utility from a pure alternative from its extension to lotteries involving more risks. Our theory is constructive, from the viewpoint of the decision maker, taking the form of mathematical induction with the measurement as the induction base and the extension as the induction step. When a cognitive bound is small, the preference relation has relatively many incomparabilities, but, as the cognitive bound becomes less restrictive, there are less incomparabilities. When there is no bound, our theory determines a complete preference relation over the set of exactly measured pure alternatives, which is considered classical expected utility theory. We give a complete characterization of incomparabilties, and a representation theorem in terms of a two-dimensional vector-valued utility function. We exemplify the theory with one experimental result reported by Kahneman-Tversky. JEL Classification Numbers: C72, C79, C91 Key words: Expected Utility, Measurement of Utility, Probability Grids, Cognitive Bound, Incomparabilities

1

Introduction

Although expected utility (EU) theory plays an important role in economics and game theory, it has been criticized for various reasons. We reconsider EU theory primarily from the viewpoint of bounded rationality due to Simon [23], [24]. We restrict permissible probabilities in EU theory to decimal (`-ary, in general) fractions up to a given cognitive bound ρ; if ρ is a natural number k }. This restriction k, the set of permissible probabilities is given as Πρ = Πk = { 100k , 101k , ..., 10 10k enables our theory to be constructive and allows us to study preference incomparabilities. When there is no cognitive bound, i.e., ρ = ∞, our theory, restricted to the set of exactly measured pure alternatives, gives complete preferences and is a fragment of classical EU theory. We first disentangle the new components of our theory. ∗ The author thanks J. J. Kline, P. Wakker, M. Lewandowski, L. Tang, A. Dominiak, S. Shiba, M. Cohen, and Y. Rebille for helpful comments on earlier versions of this paper. The author is supported by Grant-in-Aids for Scientific Research No. 26245026, Ministry of Education, Science and Culture. † Waseda University, Tokyo, Japan ([email protected])

1

Although Simon’s [23] original concept of “bounded rationality” meant a relaxation of simple utility maximization, it can be interpreted in many ways such as bounded logical inference or bounded perception ability (cf., Rubinstein [19]). In EU theory, two types of mathematical components are involved; object-components taken by a decision maker and mata-components used by the outside analyst. The former ones should be primarily targeted in EU theory, and the latter ones such as highly complex rational as well as irrational probabilities are added for analytic convenience. This addition leads to Simon’s [24] critique of EU theory as a description of “super rationality.” A distinction between object-mathematics and meta-mathematics is made in mathematical logic to analyze an extant mathematical theory itself by means of meta-mathematics (cf., Mendelson [16], p.30). This is analogously and more seriously applied to EU theory than in mathematical logic since preference comparisons in EU theory are supposed to be conducted by ordinary people with more restricted mathematical abilities than mathematicians. For example, probability values of the size 10t 2 (t = 0, ..., 102 ) are already quite accurate for ordinary people.1 In contrast, classical EU theory starts with a full real number theory and makes no distinction between components for a decision maker and those for an outside analyst. The separation of these mathematical components is a problem of degree. To capture this, we introduce the concepts of probability grids and cognitive bound ρ to EU theory. For a finite k }. ρ = k, the set of probability grids (permissible probabilities) is given as Πk = { 100k , 101k , ..., 10 10k The decision maker thinks about his evaluation of preferences with Πk for a small k to a larger k up to a bound ρ. When there is no cognitive limitation, that is, ρ = ∞, we define Π∞ = ∪∞ k=0 Πk . We describe a process of deriving preferences by the decision maker from shallow to deeper probability grids up to a cognitive bound. This approach shares motivations for “constructive decision theory” with Shafer [21], [22] and with Blume et al. [3]. These authors study Savage’s [20] subjective utility/probability theory so as to introduce certain constructive features for decision making. This paper formulates constructive decision making in an explicit manner while restricting the focus to EU theory with probability grids and having a contribution to preference incomparabilities. Brief discussions about relations between their approaches and ours is given in Section 3. In a broad sense, our treatment of probabilities falls in the field called “imprecise probabilities/similarity” (cf., Augustin et al. [2], Rubinstein [18], Tserenjigmid [25]).2 In our approach, however, each probability in Πk itself is precise, and its discrete presence is not interpreted as representing “imprecise probabilities”. The probabilities in Πk may not be fine enough for the decision maker to make a preference measurement; then he goes to finer probabilities in Πk+1 , and may repeat the process up to his cognitive bound. Hence, “imprecision” may be caused a cognitive bound. Now, consider how to describe our constructive EU theory. Constructiveness needs starting preferences; we take a hint from von Neumann-Morgenstern [26]. The authors divided the motivating argument into the following two statements, although this separation was not reflected in their mathematical development: Step B: measurement of utilities from pure alternatives in terms of probabilities; Step I : extension of these measurements to lotteries involving more risks. 1

Recall that a significance level for statistical hypothesis testing is typically 5% or 1%. In this literature, imprecision/similarity is given in the mind of a decision maker under the assumption that imprecision/similarity is defined over all real number probabilities. 2

2

: upper benchmark Benchmark lotteries

: lower benchmark

Figure 1: Measurement by the benchmarke scale We formulate these steps as mathematical induction: Step B is the inductive base measuring utilities from pure alternatives in terms of two benchmarks y, y and permissible probability grids, and Step I is the induction step extending these measurements to lotteries with more risks.3 In Fig.1, Step B is depicted; x is measured by a rough scale, y needs a more precise scale, yet both x and y are exactly measured within a given bound ρ. However, z is not exactly measured.4 These measurements are described in terms of axioms. Step I is formulated in a constructive manner from shallower to deeper layers of probability grids. Since preferences are constructed piece by piece, completeness cannot be assumed either in Step B or Step I. As the depth of a layer deepens, more preference comparisons become possible. In this process, we need to weaken the standard “independence condition,” which acts as a bridge between layers. Consider three pure alternatives y, y, y with strict preferences y  y  y. In Step B, the decision maker looks for a probability λ so that y is indifferent to a lottery [y, λ; y] = λy ∗(1−λ)y with probability λ for y and 1 − λ for y; this indifference is denoted by y ∼ [y, λ; y]. Suppose 83 25 75 λ = 10 2 ∈ Π2 . Consider another lottery d = 102 y ∗ 102 y. Step B is not applied to this since d 83 83 includes y and y. However, because of the indifference y ∼ [y, 10 2 ; y], we substitute [y, 102 ; y] for y in d and obtain d=

25 y 102

∗

75 y 102

∼

25 83 75 [y, 10 2 ; y] ∗ 102 y 102

=

2075 y 104

∗

7925 y. 104

(1)

This is the result of the standard “independence condition”. We observe two points on (1). One is a jump from the layer of depth 2 to that of depth 4, and the other is the involvement of quite precise probabilities. The “independence condition” makes a connection between layers of different depths. If we regard depth 4 as too deep, we take a cognitive bound as ρ = 2 or 3. To have a meaningful cognitive limitation, we need to avoid the jump from depth 2 to 4. With these requirements in mind, we develop our theory in a constructive manner from shallow to deeper layers of probability grids both for Step B and Step I. Since Step B may involve probability grids of various depths, the induction base may be scattered over layers. The preference relations describing Step B are listed as %B,k in PB in Table 1.1. This part is 3

Our method is applied to the subjective probability theory due to Anscombe-Aumann [1]. This method is dual to with the measurement method in terms of certainty equivalent of a lottery (cf., Kontek-Lewandowski [14] and its references). In our method, the set of benchmark lotteries up to some cognitive bound forms a base scale while in the latter, the set of monetary payments is a base scale. 4

3

based on the decision maker’s reflection on preferences in his mind. The main part of induction is a bridge between one layer to the next, which is formulated as an axiom. In Table 1.1, %k is derived from %k−1 and %B,k . Here, we emphasize that this process is described from the viewpoint of the decision maker.

Table 1.1 P B : base relation P I : constructed relations

%B,0 ↓ %0

⊆ →

%B,1 ↓ %1

k=ρ ⊆ ... ⊆ %B,k = %B,ρ ↓ → ... → %k = %ρ

Let us describe the results presented in this paper. When the cognitive bound ρ is a finite k, the resultant preference relation is given as the last %ρ = %k and, when ρ = ∞, the resultant relation is given as %∞ = ∪∞ t=0 %k . We show that in either case, %ρ is a well-defined binary relation. In Section 4, we study the relationship of our theory to classical EU theory restricting the pure alternatives Y to those exactly measured in Step B. We show that the resultant relation %∞ has an expected utility representation, which implies that %∞ is complete. Our theory is still considered constructive. Then, our theory with ρ = ∞ is uniquely extended to the full form of classical EU theory where any real numbers in [0, 1] are permissible probabilities. We argue that this extension involves some unavoidable non-constructive step. These results may be interpreted as the criticism of “super rationality” by Simon [24]. The main part of this paper is a study of incomparabilties involved in %ρ when ρ < ∞. The set, Lρ (X), of all lotteries up to cognitive bound ρ is divided into the set Mρ of measurable lotteries and the set of non-measurable lotteries Lρ (X) − Mρ . When ρ < ∞, Mρ is a proper subset of Lρ (X). We characterize incomparabilities of a non-measurable lottery f in terms of lub and glb (least upper and greatest lower bounds) of f . These results are synthesized as Theorem 6.2 whereby %ρ is represented by the two-dimensional vector function consisting of the lub and glb of each lottery f endowed with the interval order due to Fishburn [6]. The representation theorem above may be interpreted as what von Neumann-Morgenstern [26], p.29 indicated. In fact, certain papers such as Dubra, et al. [5] developed a representation theorem in terms of utility comparisons based on all expected utility functions for the relation. In this literature, incomparabilities are given in the preference relation. In contrast, in our approach, incomparabilities are changing with a cognitive bound and disappear when there are no cognitive bounds. In Section 7, we apply the results on incomparabilities to the Allais paradox, specifically, to an experimental result from Kahneman-Tversky [11]. We show that the paradoxical results remains when the cognitive bound ρ ≥ 4. However, when ρ = 2 or 3, the resultant preference relation %ρ is compatible with their experimental result. This paper is organized as follows: Section 2 explains the concept of probability grids. Section 3 gives base preference relations and formulates the derivation process. Section 4 compares our theory with classical EU theory. Section 5 discusses the measurable and non-measurable lotteries. Section 6 studies incomparabilties involved in %ρ for ρ < ∞. In Section 7, we exemplify our theory with an experimental result in Kahneman-Tversky [11]. Section 8 concludes this paper with comments on further possible studies.

4

2

Lotteries with Probability Grids and Preferences

Let ` be an integer with ` ≥ 2. This ` is the base for describing probability grids; we take ` = 10 in the main examples in this paper. The set of probability grids Πk is defined as Πk = { `νk : ν = 0, 1, ..., `k } for any finite k ≥ 0.

(2)

Here, Π0 = {0, 1}, and Π1 = { ν` : ν = 0, ..., `} is the basic set of probability grids. Each Πk is a finite set, and let Π∞ := ∪∞ t=0 Πt , which is countable. We use the standard arithmetic rules over Π∞ ; sum and multiplication are needed; for our analysis, we use these calculation rules but, for our axiomatic system itself, they are used in a restricted manner, which we mention in adequate places. However, division is problematic and is not used.5 We also allow reduction 20 2 by eliminating common factors; for example, 10 2 is the same as 10 . Hence, Πk ⊆ Πk+1 for k = 0, 1, ... The parameter k is the precision of probabilities that the decision maker uses. We 25 define the depth of each λ ∈ Π∞ by: δ(λ) = k iff λ ∈ Πk − Πk−1 . For example, δ( 10 2 ) = 2 but 20 2 δ( 102 ) = δ( 10 ) = 1. We use the standard equality = and strict inequality > over Πk . Then, trichotomy holds: for any λ, λ0 ∈ Πk , (3) either λ > λ0 , λ = λ0 , or λ < λ0 . This is equivalent to that ≥ is complete and anti-symmetric. Now, we show that each element in Πk is obtained by taking the weighted sums of elements in Πk−1 with the equal weights. This is basic for our induction method. P Lemma 2.1 (Decomposition of probabilities). Πk = { `t=1 1` λt : λ1 , ..., λ` ∈ Πk−1 } for any k (1 ≤ k < ∞). Proof. The right-hand setPis included in Πk by (2). Consider the converse. Each λ ∈ Πk is expressed as λ = 1 or λ = kt=1 ν`tt where 0 ≤ ν t < ` for t = 1, ..., k. If λ = 1, let λ1 = ... = λ` = P νk 1 ∈ Πk−1 , and λ = `t=1 1` λt . Consider the second. Let λ1 = ... = λν 1 = 1, λν 1 +1 = ν`12 +...+ `k−1 , and λt = 0 for P t = ν 1 + 2, ..., `. This definition is applied when ν 1 = 0. These λ1 , ..., λ` belong to Πk−1 and λ = `t=1 1` λt .¥ The union Π∞ = ∪∞ k=0 Πk is a proper subset of the set of all rational numbers. For example, when ` = 10, Π∞ has no recurring decimals, which are also rationals. We note that Π∞ depends upon the base `; for example, Π1 with ` = 3 has 13 , but Π∞ with ` = 10 has no element corresponding to 13 . Nevertheless, Π∞ is dense in [0, 1], which is crucial in comparing our theory with classical EU theory in Section 4.2. Let X be a set of pure alternatives. For any k < ∞, we define Lk (X) by P Lk (X) = {f : f is a function from X to Πk with x∈X f (x) = 1}.

(4)

Since Πk ⊆ Πk+1 , it holds that Lk (X) ⊆ Lk+1 (X). We denote L∞ (X) = ∪∞ k=0 Lk (X). We define the depth of a lottery f in L∞ (X) by δ(f ) = k iff f ∈ Lk (X) − Lk−1 (X). We use the same symbol δ for the depth of a lottery and the depth of a probability. It holds that δ(f ) = k if and only if maxx∈X δ(f (x)) = k. That is, δ(f ) = k if and only if some value of f has exactly depth k. This will be relevant in Section 5.  2 Πk is a subset of { kt=k ν t · `t : −` < ν t < ` for t with k1 ≤ t ≤ k2 and k1 , k2 are integers with k1 < 0 < k2 }, 1 which is a ring, that is, it is closed with respect to the three arithmetic operations +, −, and ×. See Mendelson [15], p.95. 5

5

We denote a cognitive bound by ρ, which is a natural number or infinity ∞. If ρ = k < ∞, then Lρ (X) = Lk (X), and if ρ = ∞, then Lρ (X) = L∞ (X). The latter is interpreted as the case of no cognitive limitation. When we restrict X to a subset X 0 , we define Lk (X 0 ) = {f ∈ Lk (X) : f (x) > 0 implies x ∈ X 0 }. Hence, Lk (X 0 ) is a subset of Lk (X). We call a subset S of X a support of f ∈ Lk (X) iff f (x) > 0 implies x ∈ S. This is relevant in Section 4.2.

25 75 Example 2.1. Let X = {y, y, y} and ρ = 2. Since lottery d = 10 2 y ∗ 102 y is in L2 (X) − L1 (X), 20 80 2 8 0 the depth δ(d) = 2; but since d0 = 10 2 y ∗ 102 y = 10 y ∗ 10 y ∈ L1 (X), its depth is δ(d ) = 1.

Now, we formulate a connection from Lk−1 (X) to Lk (X). We say that f = (f1 , ..., f` ) in Lk−1 (X)` = Lk−1 (X) × · · · × Lk−1 (X) is a decomposition of f ∈ Lk (X) iff P (5) f (x) = `t=1 1` × ft (x) for all x ∈ X. P Let e = ( 1` , ..., 1` ), and f is denoted by e ∗ f or `t=1 1` ∗ ft . In other words, when f = (f1 , ..., f` ) is given, f = e ∗ f is a composite lottery and is reduced to a lottery in Lk (X) by (5). In our axiomatic system, we use reduction of a composite lottery to a lottery only in this form. The next lemma connects Lk−1 (X) to Lk (X), which facilitates our induction method described in Table 1.1. Lemma 2.2 (Decomposition of lotteries). Let 1 ≤ k < ∞. Then, Lk (X) = {e ∗ f : f ∈Lk−1 (X)` }. The right-hand side is the set of composed lotteries from Lk−1 (X), and is a subset of the lefthand side Lk (X). The converse inclusion ⊆ is essential and means that each lottery in Lk (X) is decomposed to a weighted sum of some (f1 , ..., f` ) in Lk−1 (X)` with the equal weights. A proof is given in the Appendix, which involves precise construction of a decomposition f = (f1 , ..., f` ). This lemma is crucial in our theory. 25 The lottery d = [y, 10 2 ; y] in Example 2.1 has two types of decompositions: 5 10

5 ∗ [y, 10 ; y] +

5 10

∗ y and

2 10

∗y+

1 10

5 ∗ [y, 10 ; y] +

7 10

∗ y.

(6)

5 ; y] and f6 = ... = In the first, a decomposition f = (f1 , ..., f10 ) is given as f1 = ... = f5 = [y, 10 5 f10 = y. In the second, f is given as f1 = f2 = y, f3 = [y, 10 ; y] and f4 = ... = f10 = y. We use these short-hand expressions rather than a full specification of f = (f1 , ..., f10 ). The proof of Lemma 2.2 constructs the second type of a decomposition in a general manner. In Section 3.2, we show that this multiplicity causes no problem in our derivation process.

We P assume that a decomposition of f ∈ Lk (X) takes the form of f = (f1 , ..., f` ) with f = `t=1 1` ∗ ft . Keeping ` = 10, binary decompositions are not enough for Lemma 2.2, For 3 3 4 y ∗ 10 y ∗ 10 y in the above example. This f is not expressed example, consider the lottery f = 10 by a binary combination in L0 (X) = X with weights in Π1 . Without requiring a decomposition 5 6 4 5 2 8 ( 10 y ∗ 10 y) ∗ 10 ( 10 y ∗ 10 y), be in the one-level lower layer, this could be possible such as f = 10 but the requirement for decompositions to be in one-layer shallower is essential for our theory. The expression f % g means that f is strictly preferred to g or is indifferent to g. We define the strict (preference) relation Â, indifference relation ∼, and incomparability relation 1 by f f f

 g ⇐⇒ f % g and not g % f ; ∼ g ⇐⇒ f % g and g % f ;

1 g ⇐⇒ neither f % g nor g % f. 6

(7)

The incomparability relation 1 is new and is studied in the subsequent sections. Nevertheless, all the axioms are about the relations %, Â, and ∼ . The relation 1 is defined as the residual part of %. Although ∼ and 1 are sometimes regarded as closely related (cf., Shafer [21], p.469), they are well separated in Theorem 6.2.

3

EU Theory with Probability Grids

Our theory has two parts: base preference relations h%B,k iρk=0 with four axioms, and a derivation process with two axioms to derive preference relations h%k iρk=0 . The former describes the benchmark scales and measurements of pure alternatives in terms of benchmark scales, and the latter describes derived preferences over lotteries with more risks. The derivation process generates a well-defined binary relation %ρ uniquely relative to given base relations h%B,k iρk=0 . When a finite cognitive bound ρ is given, the process stops at ρ, and its resultant relation is %ρ . When ρ = ∞, the resultant relation is %∞ = ∪∞ k=0 %k .

3.1

Base preference relations

We assume that the set of pure alternatives X contains two distinguished elements y and y, called the upper and lower benchmarks. Let k < ∞. We call a lottery f in Lk (X) a benchmark lottery of depth (at most) k iff f (y) = λ and f (y) = 1 − λ for some λ ∈ Πk , which we denote by [y, λ; y]. The benchmark scale of depth k is the set Bk (y; y) = {[y, λ; y] : λ ∈ Πk } : the dots in Fig.1 express the benchmark lotteries. We define B∞ (y; y) = ∪∞ k=0 Bk (y; y). Let 0 ≤ k < ρ + 1.6 Let %B,k be a subset of

Dk = Bk (y; y)2 ∪ {(x, g), (g, x) : x ∈ X and g ∈ Bk (y; y)}.

(8)

The first part is used to describe the preferences over the benchmark scale Bk (y; y), which are uniquely determined by Axiom B1, given below. The other part is to describe the measurement of each pure alternative x ∈ X in terms of the benchmark scale, for example, if (x, g) ∈ %B,k but (g, x) ∈ / %B,k , we have a strict preference x ÂB,k g, and if (x, g) ∈ / %B,k and (g, x) ∈ / %B,k , then x and g are incomparable. We allow %B,k to be very partial. We require all pure alternatives be between the upper and lower benchmarks y and y. Axiom B0 (Benchmarks): y ÂB,0 y and y %B,0 x %B,0 y for all x ∈ X. The next states that Bk (y; y) is used as the base scale for depth k. Axiom B1 (Benchmark scale): For λ, λ0 ∈ Πk , λ ≥ λ0 ⇐⇒ [y, λ; y] %B,k [y, λ0 ; y]. It follows from this axiom that for λ, λ0 ∈ Πk ,

λ = λ0 ⇐⇒ [y, λ; y] ∼B,k [y, λ0 ; y], and λ > λ0 ⇐⇒ [y, λ; y] ÂB,k [y, λ0 ; y].

(9)

Thus, %B,k is a complete relation over Bk (y; y) by (3). This is the scale part of %B,k . This part is precise up to the cognitive bound ρ. The scale part and measurement part of %B,k are consistent in the sense of no reversals with Axiom B1. 6

Stipulating ∞ + 1 = ∞, we express the two statements “k ≤ ρ if ρ < ∞” and “k < ρ if ρ = ∞” as “k < ρ + 1”.

7

（1）

（2）

Figure 2: Base preference relations Axiom B2 (Non-reversal): For all x ∈ X and λ, λ0 ∈ Πk , [y, λ; y] %B,k x %B,k [y, λ0 ; y] =⇒ λ ≥ λ0 .

If we assume transitivity for %B,k over Dk , B2 could be derived from B1, but we adopt B2 instead of transitivity, since this is more basic.

The last requires the preferences in layer k be preserved in layer k + 1. This is expressed by the set-theoretical inclusion ⊆ in Table 1.1.

Axiom B3 (Preservation): For all f, g ∈ Dk , f %B,k g =⇒ f %B,k+1 g.

The above axioms allow still great freedom for base preference relations h%B,k iρk=0 . To see this, consider the following examples. Example 3.1. Let X = {y, y, y}. Consider two examples for h%B,k iρk=0 satisfying Axioms B0 to B3. For k = 0, the scale part is {(y, y)} and the measurement part may be given as {(y, y), (y, y)}, meaning y ÂB,0 y and y ÂB,0 y. In Fig.2, y and y are located at the points of probability 1 and 0 at k = 0, and y is between them.

Let k ≥ 1. In Fig.2, the thin solid lines give the upper and lower bounds of y: In both (1 ) and 9 7 ; y] and [y, 10 ; y] at k = 1. The scale part is: for ν, ν 0 = 0, ..., 10, (2 ), y is strictly between [y, 10 0 ν ν 9 ; y] %B,1 [y, 10 ; y] ⇐⇒ ν ≥ ν 0 . The measurement part is given as [y, 10 ; y] ÂB,1 y ÂB,1 [y, 10 7 [y, 10 ; y]. 83 The solid lines merge at k = 2 in (1 ), meaning that y becomes indifferent to [y, 10 2 ; y]. Axiom 9 7 B3 implies that [y, 10 ; y] ÂB,2 y ÂB,2 [y, 10 ; y] remain. The lines do not merge in (2 ) and y is 9 8 ; y] and [y, 10 ; y] for k ≥ 2. There are no indifferent benchmark lotteries to y at between [y, 10 any k ≥ 0.

The decision maker reflects upon his mind to look for his preferences %B,k by a thought experiment with the probability grids Πk . From the viewpoint of “bounded rationality”, we are inclined to have the view that he may stop his search for base preferences when he is happy enough for them even though they include some imprecision. In (2 ) of Example 3.1, he stops this search at depth 2. This view is also compatible with Simon’s [23] satisficing/aspiration argument, which is briefly argued in Section 8.

8

3.2

Derivation process

Let a cognitive bound ρ ≤ ∞ be given. We consider an extension from the base relations h%B,k iρk=0 to preferences over Lk (X) for k < ρ + 1. We start with %0 , and derive %k from %k−1 and %B,k , provided that %k−1 is already defined. Now, we use the following axiom for an inductive step. Let 1 ≤ k < ρ + 1. For f = (f1 , ..., f` ) and g = (g1 , ..., g` ), we write f %k g iff ft %k gt for all t = 1, ..., `. Recall that f is a decomposition of f ∈ Lk (X) when f ∈ Lk−1 (X)` and f = e ∗ f . The connection from %k−1 to %k is given as follows: Axiom C1: Let f ∈ Lk (X), g ∈ Bk (y; y), or f ∈ Bk (y; y), g ∈ Lk (X). Suppose that f , g are decompositions of f, g with f %k−1 g. Then f %k g. In addition, if ft Âk−1 gt for some t = 1, ..., `, then f Âk g. In Table 1.1, the horizontal arrows indicate this derivations. Here, reduction of compound lotteries is used only for f = e ∗ f and g = e ∗ g. It is a very weak version of the independence axiom since it connects one layer to the next only. In Section 4, we compare Axiom C1 with the independence condition in classical EU theory. The last axiom is transitivity: Let 1 ≤ k < ρ + 1.

Axiom C2 (Transitivity): For any f, g, h ∈ Lk (X), if f %k g and g %k h, then f %k h. This suggests to take the transitive closure of preferences derived at layer k. This axiom implies strict preference versions so that if at least one preference in the premise is replaced by a strict preference, then the conclusion is also strict. Now, we have the derivation process for h%k iρk=0 . Derivation process up to a cognitive bound ρ: Step 0: For any f, g ∈ L0 (X), if f %B,0 g, then f %0 g (including strict preferences7 ). Step k (1 ≤ k < ρ + 1) :

k0: for any f, g ∈ Lk (X), if f %B,k g, then f %k g (including strict preferences).

k1: f %k g is derived by C1 (including strict preferences). k2: f %k g is derived by C2.

Smallest requirement (SM) for h%k iρk=0 : each %k is obtained by a finite number of applications of Step 0 to k. The derivation process (hereafter, abbreviated as DP) proceeds inductively from step 0, and from step k − 1 to step k. Step 0 is the inductive base, which together with SM determines %0 = %B,0 . Step k also has an inductive base k0, since %B,k may include new base preferences. These inductive bases are downward arrows in Table 1.1, and step k1 is the horizontal arrow in Table 1.1, which is the induction step to derive preferences from %k−1 . Step k2 takes the transitive closure of these derived preferences. Finally, SM requires no preferences other than those constructed by k0 to k2 be added. We have one basic problem to see that each %k is a binary relation over Lk (X); that is, it is a subset of Lk (X)2 . This means that for any f, g ∈ Lk (X), exactly one of (f, g) ∈ %k and / %k appears as a part of strict (f, g) ∈ / %k holds. In our axiom system, the negation (f, g) ∈ 7

That is, if f ÂB,0 g, then f Â0 g.

9

preferences g Âk f . Therefore, it suffices to show that for any f, g ∈ Lk (X), f %k g =⇒ not (g Âk f ).

(10)

Once this is proved, %k is a well-defined binary relation over Lk (X). A proof is given in the Appendix. Then, we can see the uniqueness of a sequence h%k iρk=0 , too. Theorem 3.1 (Well-definedness). The DP generates a unique sequence of binary relations h%k iρk=0 , provided that h%B,k iρk=0 is given.

By Theorem 3.1, we have the set-theoretical description of the generated sequence h%k iρk=0 : %0 = %B,0 ; and %k = [(%k−1 )C1 ∪ (%B,k )]tr for k (1 ≤ k < ρ + 1),

(11)

where (%k−1 )C1 is the set of preferences derived from %k−1 by C1. Then we take the transitive closure of (%k−1 )C1 ∪ (%B,k ); we define, denoted by %tr , the transitive closure of % by: f %tr g ⇐⇒ for some h0 , ..., hm , f = h0 % h1 % ... % hm = g. We are interested in the resultant preference relation; recall that for ρ < ∞, it is the last relation %ρ of h%k iρk=0 , and for ρ = ∞, %ρ = ∪∞ k=0 (%k ). These make sense by the following lemma. Lemma 3.1 (Preservation of preferences). Let 1 ≤ k < ρ + 1. For any f, g ∈ Lk−1 (X), f %k−1 g implies f %k g. Proof. Let f ∈ Lk−1 (X) and g ∈ Bk−1 (y; y). Suppose f %k−1 g. Then, f, g ∈ Lk−1 (X) ⊆ P P Lk (X). Let f1 = ... = f` = f and g1 = ... = g` = g. Then, f = `t=1 1` ∗ ft and f = `t=1 1` ∗ gt .

Hence, (f1 , ..., f` ) and (g1 , ..., g` ) are decompositions of f and g. By C1, we have f %k g. In the other case where g %k−1 f, we can prove g %k f similarly. Now, let f, g ∈ Lk−1 (X). By (11) for k, there are a finite sequence of lotteries in h0 , h1 , ..., hl in Lk−1 (X) such that f = h0 %k−1 h1 %k−1 ... %k−1 hl = g and each ht %k−1 ht+1 is derived by C1 and B0 to B3; thus, at least one of ht , ht+1 belongs to Bk−1 (y; y). By the conclusion of the first paragraph, it holds that ht %k ht+1 for t = 0, ..., l − 1. By C2, f = h0 %k hl = g.¥

Here, we can give comments on relationships of our approach to Shafer [21], [22] and Blume et al. [3]. These papers are concerned with Savage’s [20] subjective probability/utility theory. We share similar motivations with theirs. In [21], [22], when basic probability/utility schemes are given, they are extended to the expected utility form. In [3], propositional logic is adopted to describe decision making (with the use of real number probabilities) and mental states are described in a semantic manner. Either approach takes some constructive aspect, but a decision making process is not explicitly formulated in a constructive manner. The following sets of pure alternatives play an important role in subsequent studies: Yk = {x ∈ X : x ∼B,k [y, λ; y] for some λ ∈ Πk } for k (0 ≤ k < ρ + 1).

(12)

This is the set of pure alternatives exactly measured by the benchmark scale Bk (y; y), k < ρ + 1. Since Πk−1 ⊆ Πk , we have, by B0, {y, y} ⊆ Y0 ⊆ Y1 ⊆ ..., but Yk+1 − Yk may be empty. In (1 ) of Example 3.1, Y0 = Y1 = {y, y} and Yk = {y, y, y} for k ≥ 2, and in (2 ) Yk = {y, y} for k ≥ 0. We let Yρ = ∪ρk=0 Yk ; in particular, when ρ = ∞, we denote Yρ simply by Y. In Section 4, we make comparisons between our theory and classical EU theory, in which the union Y = ∪∞ k=0 Yk plays a crucial role. Lemma 3.2.(1): For each y ∈ Yk , a probability λy ∈ Πk with y ∼B,k [y, λy ; y] is unique. In 10

particular, λy = 1 and λy = 0. (2): The transitive closure %tr B,k of %B,k is complete over Yk ∪ Bk (y; y).

Proof. (1): Let y ∼B,k [y, λy ; y] ∼B,k [y, λ0y ; y]. By (9), λy = λ0y . The remaining follows from B0 and (9). (2): The assertion holds by (9) when f, g ∈ Bk (y; y). Consider f = y ∈ Yk and g ∈ Bk (y; y). Then, by (12), f = y ∼B,k h for some h = [y, λ; y] ∈ Bk (y; y). Then, by (9), h %B,k g or tr g %B,k h. Thus, f %tr B,k g or g %B,k f. The case where f, g ∈ Yk is similar.¥ By Lemma 3.2, we define the utility function uo : Yk → Πk by uo (y) = λy for all y ∈ Yk .

(13)

This uo represents the relation %tr B,k over Yk , that is, for any x, y ∈ Yk , uo (x) ≥ uo (y) ⇐⇒ x tr %B,k y. This uo (·) is extended to Yk ∪ Bk (y; y) by uo ([y, λ; y]) = λ, which fully represents %tr B,k over Yk ∪ Bk (y; y). We discuss a further extension to L∞ (Y ) in Section 4.

4

Relationship to Classical EU Theory

Before our study of the behavior of %ρ for ρ < ∞, we look at the relationship between our theory and classical EU theory (cf., Milnor [9], Fishburn [7]). Our theory with ρ = ∞ allows the expected utility hypothesis over the lotteries in L∞ (Y ). The main difference is that in our theory, permissible probabilities are from Π∞ = ∪∞ k=0 Πk , while all probabilities from [0, 1] are allowed in classical theory. Our resultant relation %∞ = ∪∞ k=0 %k is uniquely extended to a relation in the sense of classical theory, but it involves non-constructive components.

4.1

Expected utility hypothesis in case with no cognitive bounds

∞ Let h%k i∞ k=0 be the preference relations constructed by the DP from base relations h%B,k ik=0 . We restrict our attention to the set of pure alternatives Y = ∪∞ k=0 Yk , where Yk is given in (12). Consider the following extension of the function uo given by (13) to L∞ (Y ) and the induced preference relation %e ; P (14) ue (f ) = y∈Y f (y)uo (y) for any f ∈ L∞ (Y );

for any f, g ∈ L∞ (Y ), f %e g ⇐⇒ ue (f ) ≥ ue (g).

(15)

The value ue (f ) belongs to Π∞ for each f ∈ L∞ (Y ), since uo is a function from Y to Π∞ . We have the following equivalence between %∞ and %e , which is proved below.

Theorem 4.1 (1): f ∼∞ [y, ue (f ); y] for all f ∈ L∞ (Y ).

(2)(Expected utility hypothesis): For any f, g ∈ L∞ (Y ), f %∞ g ⇐⇒ f %e g.

Thus, the resultant relation %∞ is complete over L∞ (Y ). When Y contains some y with y ÂB,0 y ÂB,0 y, %k is incomplete even over Lk (Y ) for any k < ∞; on the contrary, %e,k := %e ∩Lk (Y )2 is already complete over Lk (Y ). Theorem 4.1.(2) is the expected utility hypothesis, although Axiom C1 is a very weak version of independence with a specific form of reduction of compound lotteries, that is, (5). Assume 11

the full form of reduction: for any f, g ∈ L∞ (Y ) and λ ∈ Π∞ , we define λf ∗ (1 − λ)g ∈ L∞ (Y ) by (λf ∗ (1 − λ)g)(x) = λf (x) + (1 − λ)g(x) for all x ∈ X. (16)

Then, it follows from Theorem 4.1.(2) that the preference relation %∞ satisfies the full independence condition: for any f, g ∈ L∞ (Y ) and λ ∈ Π∞ (0 < λ), ID1∞ : f Â∞ g =⇒ λf ∗ (1 − λ)h Â∞ λg ∗ (1 − λ)h;

ID2∞ : f ∼∞ g =⇒ λf ∗ (1 − λ)h ∼∞ λg ∗ (1 − λ)h.

Then, Axiom C1 implies that preferences are freely carried over from shallow layers to deeper layers. However, when ρ < ∞, the reduced compound lottery λf ∗ (1 − λ)h may go beyond ρ. The classical version of these conditions are discussed in Section 4.2. Proof of Theorem 4.1: First, we show that (2) is an immediate consequence of (1). Let f, g ∈ L∞ (Y ). By (1), for some ko , f ∼k [y, ue (f ); y] and g ∼k [y, ue (g); y] for all k ≥ ko . Since %∞ = ∪∞ k=0 %k , we can take a k ≥ ko so that f %∞ g ⇐⇒ f %k g. By B1, (15), and C2, f %k g ⇐⇒ [y, ue (f ); y] %k [y, ue (g); y] ⇐⇒ ue (f ) ≥ ue (f ). In sum, f %∞ g ⇐⇒ f %e g. Now, let us prove (1). For k < ∞, let Lk (Yk ) = {f ∈ Lk (X) : f has a support S in Yk }, and L∗k (Yk ) = {f ∈ Lk (Yk ) : for t = 0, ..., k, f (y) ∈ Πk−t if y ∈ Yt − Yt−1 },

(17)

where Y−1 = ∅. We have the following two assertions: for any k ≥ 0, if y ∈ Yk − Yk−1 and f ∈ L∗k (Yk ), then f (y) = 0 or 1;

(18)

∗ L∞ (Y ) = ∪∞ k=0 Lk (Yk ).

(19)

First, we see (18); let f ∈ L∗k (Yk ). If y ∈ Yk − Yk−1 , then f (y) ∈ Π0 , i.e., f (y) = 0 or 1. Next, let us see (19). The inclusion ⊆ is essential. Let f ∈ L∞ (Y ). This f has a finite support S, and there is a k0 such that for each y ∈ S, y ∈ Yt − Yt−1 for some t ≤ k0 and also f (y) ∈ Πk0 . Let k = 2k0 . Then, for any y ∈ S, if y ∈ Yt − Yt−1 , then f (y) ∈ Πk0 ⊆ Πk−t . Thus, f ∈ L∗k (Yk ). Now, we show by induction over k = 0, ... that f ∼k [y, ue (f ); y] for all f ∈ L∗k (Yk ).

(20)

Once this is proved, we complete the proof of (1); indeed, taking any f ∈ L∞ (Y ), by (19), we have f ∈ L∗k (Yk ) for some k < ∞, and thus, f ∼k [y, ue (f ); y] by (20). Let us prove (20). Let k = 0. For any y ∈ L∗0 (Y0 ), y ∼0 y or y ∼0 , y by (12) for k = 0 and DP.(0), i.e., y ∼0 [y, 1; y] and y ∼0 [y, 0; y]. Suppose that (20) holds for k. Take any f ∈ L∗k+1 (Yk+1 ). If f (y) = 1 for y ∈ Yk+1 − Yk , then f = y ∼k+1 [y, λy ; y] by (12), which implies f = y ∼k+1 [y, ue (f ); y] because λy = ue (y). Now, suppose that f (y) = 0 for all y ∈ Yk+1 − Yk . Then, f has a support in Yk . By Lemma 2.2, f has a decomposition f = (f1 , ..., fl ). Hence, if y ∈ Yt − Yt−1 , then fl (y) = 0 for each l = 1, ..., `, which means that each fl belongs to L∗k (Yk ). By the induction hypothesis, there is a g = (g1 , ..., g` ) such that f ∼k g and gt = [y, ue (fl ); y] for l = 1, ..., `. Applying Axiom C1, P` 1 P` 1 we have f = e ∗ f ∼k+1 e ∗ g, and this becomes f ∼k+1 t=1 ` ∗ gt = t=1 ` ∗ [y, ue (ft ); y] P` 1 = [y, t=1 ` ue (ft ); y] = [y, ue (f ); y]; the last equality follows from (15).¥ 12

4.2

Extension to classical EU theory

Our theory with ρ = ∞ and Y = ∪∞ k=0 Yk still differs from classical EU theory where all probabilities from [0, 1] are permissible. In fact, our resultant relation %∞ can be uniquely extended to a preference relation in the sense of classical EU theory with the set of pure alternatives Y . Nevertheless, the extension involves non-constructive components. We first give a summary of classical EU theory and then give the extension result. Let L[0,1] (Y ) = {f : f : Y → [0, 1] is Pa lottery with a finite support S ⊆ Y }. Here, a lottery f can take any real value in [0, 1] but y∈S f (y) = 1 for some finite subset S of Y. Since Πk ⊆ [0, 1] for all k < ∞, L∞ (Y ) = ∪∞ k=0 Lk (Y ) is a subset of L[0,1] (Y ); the lotteries taking values in [0, 1] − Π∞ are newly included in L[0,1] (Y ). The set L[0,1] (Y ) is an uncountable set, but we show that L∞ (Y ) is dense in L[0,1] (Y ). We assume the reduction of compound lotteries: for any f, g ∈ L[0,1] (Y ) and λ ∈ [0, 1], λf ∗ (1 − λ)g is defined by (16). We adopt the following axiomatic system, which is one among various equivalent systems. Let %E be a binary relation over L[0,1] (Y ); and we assume NM0 to NM2 on %E . Axiom NM0 (Complete preordering): %E is a complete and transitive relation on L[0,1] (Y ). Axiom NM1 (Intermediate value): For any f, g, h ∈ L[0,1] (Y ), if f %E g %E h, then λf ∗ (1 − λ)h ∼E g for some λ ∈ [0, 1].

Axiom NM2 (Independence): For any f, g, h ∈ L[0,1] (Y ) and λ ∈ (0, 1], ID1: f ÂE g implies λf ∗ (1 − λ)h ÂE λg ∗ (1 − λ)h;

ID2: f ∼E g implies λf ∗ (1 − λ)h ∼E λg ∗ (1 − λ)h.

The difference between ID1-ID2 and ID1∞ -ID2∞ is only the domains of lotteries and permissible probabilities. Under these axioms, there is a utility function UE : L[0,1] (Y ) → R so that (21) for any f, g ∈ L[0,1] (Y ), f %E g ⇐⇒ UE (f ) ≥ UE (g), UE (f ) = Σy∈S f (y)uE (y) for each f ∈ L[0,1] (Y ),

(22)

where S is a finite support of f and uE is the restriction of UE on Y. The converse holds; if %E is given by (21) and (22), then, NM0 to NM2 hold for %E . Now, we connect this theory to our EU theory with probability grids. Let %∞ = ∪∞ k=0 %k be . Then, we have a unique the resultant relation generated by DP from base relations h%B,k i∞ k=0 extension %E of %∞ having NM0 to NM2. A proof is given below.

Theorem 4.2 (Unique extension). There is a unique binary relation %E over L[0,1] (Y ) such that for any f, g ∈ L∞ (Y ), f %∞ g =⇒ f %E g and NM0 to NM2 hold for %E .

This result is proved by denseness of L∞ (Y ) in L[0,1] (Y ) and continuity of UE (·) with respect to point-wise convergence. Denseness is a direct consequence from the denseness of Π∞ = ν ν ∪∞ k=0 Πk in [0, 1]. Continuity of UE (·) means that for any sequence {f } in L∞ (Y ), if f (y) → f (y) ν for each y ∈ Y, then limν→∞ UE (f ) = UE (f ). The function UE given by (22) is continuous. The proofs of Lemma 4.1 and Theorem 4.2 may appear constructive following %∞ . Indeed, when f ∈ L∞ (Y ), the involved probabilities in each f are described by a finite list of natural numbers. Thus, our theory is constructive up to %∞ , but the next extension step to %E is non-constructive, since probabilities newly involved in f ∈ L[0,1] (Y ) − L∞ (Y ) may be given only

13

in a nonconstructive manner.8 Lemma 4.1. L∞ (Y ) is a dense subset of L[0,1] (Y ). Proof. L∞ (Y ) is a subset of L[0,1] (Y ). We show that L∞ (Y ) is dense in L[0,1] (Y ). Now, we take any f ∈ L[0,1] (Y ). This f has a finite support S = {y0 , y1 , ..., ym } in Y. We construct a sequence ν ν {g ν }∞ ν=ν o so that g ∈ L∞ (Y ) for ν ≥ ν 0 , and for each y ∈ Y, g (y) → f (y) as ν → ∞. For any natural number ν, let zν,t = min{π t ∈ Πν : π t ≥ f (yt )} for all t = 0, ..., m. Then, we define uν,0 , ..., uν,m by ½ zν,t P if t < m uν,t = 1 − t 0} ≤ k.

(24)

Proof. We prove (24) by induction on k ≥ 0. Let k = 0. Since Y0 = L0 (Y0 ) = M0 , it holds that δ(λf ) = δ(f (y)) = 0 for all f ∈ L0 (Y0 ) = M0 . Thus, (24) holds for k = 0. Now, suppose (24) for k, and prove (24) for k + 1. Consider f ∈ Lk+1 (Yk+1 ) − Lk (Yk ). Let y ∈ Yk+1 − Yk with f (y) > 0. Suppose that f (y) = 1, i.e., f = y and δ(f (y)) = 0. Since y ∈ Yk+1 − Yk , we have δ(λy ) = k + 1. Hence, y ∈ Mk+1 ⇐⇒ 15

δ(λy ) = k + 1. Now, we suppose f ∈ Lk+1 (Yk+1 ). Let f ∈ Mk+1 . Then, f ∼k+1 g for some g ∈ Bk+1 (y; y). Then, there are decompositions f , g of f, g such that f ∼k g. For each t = 1, ..., `, since ft ∼k gt and gt ∈ Bk (y; y), we have ft ∈ Mk . By the induction hypothesis, we have δ(λy ) + δ(ft (y)) ≤ k for all y ∈ Yk for all t. Since f = e ∗ f , we have δ(f (y)) ≤ δ(ft (y)) + 1 for all t and y ∈ X. Since δ(f (y)) = 0 for all y ∈ Yk+1 − Yk by Lemma 5.1.(1), we have δ(λy ) + δ(f (y)) ≤ k + 1 for all y ∈ Yk+1 . Conversely, let δ(λy ) + δ(f (y)) ≤ k + 1 for all y ∈ Yk+1 . Let k0 = max{δ(f (y)) : x ∈ Yk+1 }. Then, k0 ≤ k + 1. Let k0 = 0. Then, f = y for some y ∈ Yk+1 and δ(λf ) ≤ k + 1. This means that f ∈ Mk+1 . Now, suppose that k0 > 0, i.e., f ∈ Lk0 (Yk0 ) − Lk0 −1 (Yk0 −1 ). If f (y) > 0 for some y ∈ Yk0 − Yk0 −1 , then f (y) = 1 by Lemma 5.1.(1) and max{δ(f (y)) : x ∈ Yk+1 } = 0, impossible by k0 > 0. Thus, f (y) = 0 for all y ∈ Yk0 − Yk0 −1 , i.e., f ∈ Lk0 (Yk0 −1 ). Now, by Lemma 2.2, we have have a decomposition f of f. Then, ft ∈ Lk0 −1 (Yk0 −1 ) for t = 1, ..., `. Since δ(λy ) + δ(f (y)) ≤ k + 1 for all y ∈ Yk+1 and k0 ≤ k + 1, we have δ(λy ) + δ(ft (y)) ≤ k for all y ∈ Yk0 and t = 1, ..., `. Hence, by the inductive hypothesis for k, we have ft ∈ Mk for t = 1, ..., `. Thus, we have gt ∈ Bk (y, y) with ft ∼k gt for t = 1, ..., m. Thus, C1, we have f = e ∗ f ∼ e ∗ g ∈ Bk (y, y). This means that f ∈ Mk+1 .¥ The assertion (24) is read in two ways. One is to fix a lottery f ∈ Lk (Y ) but to change k. Let kf = max{δ(λy ) + δ(f (y)) : f (y) > 0}. Then, (24) is written as f ∈ Mk ⇐⇒ kf ≤ k.

(25)

25 75 83 83 25 For example, when f = 10 2 y ∗ 102 y and y ∼2 [y, 102 ; y], we have kf = δ( 102 ) + δ( 102 ) = 4. Hence, (25) implies that f ∈ Mk ⇐⇒ k ≥ 4. The other way is to fix a k and to change f . Let δ(λy ) > 0 for some y ∈ Yk . Then, no matter what is chosen for k < ∞, there is an f ∈ Lk (Yk ) such that / Mk . Nevertheless, such nonmeasurability disappears in the limit δ(λy ) + δ(f (y)) > k, i.e., f ∈ ρ = ∞.

We mention the following theorem for the indifferences ∼k over Lk (X); the indifference relation ∼k occurs only in Mk and, in particular, reflexivity holds only in Mk . Theorem 5.3. Let k < ρ + 1 ≤ ∞ and f, g ∈ Lk (X). / Mk , then f ¿k g. (1) (No indifferences outside Mk ): If f ∈ (2) (Reflexivity): f ∼k f if and only if f ∈ Mk .

Proof (1): Suppose that f ∈ / Mk and g ∈ Mk . Then, g ∼k [y, λg ; y]. If f ∼k g, then f ∈ Mk by C2, a contradiction, Hence, f ¿k g. Now, let f, g ∈ / Mk . Suppose f ∼k g. By (11), there are f = h0 , h1 , ..., hm = g ∈ Lk (X) such that hl ∼k hl+1 and at least one of hl and hl+1 belongs to / Mρ and h1 ∈ Bk (y; y) ⊆ Mρ , we have, by the first Bk (y; y) for l = 1, ..., m − 1. Since h0 = f ∈ case, h0 ¿k h1 , a contradiction. Hence, f ¿k g. (2): The if part is by the definition of Mk and B1, and the only-if part (contrapositive) follows from (1) of this theorem.¥

6

Incomparabilities and their Characterizations

Now, we study how the resultant preference relation %ρ involves incomparabilities over Lρ (X) − Mρ , which happens, in particular, for ρ < ∞. Here, we characterize such incomparabilities by the concepts lub and glb of f ∈ Lρ (X). We also unifies this characterization and the representation 16

g’ g

f

f’

Figure 3: Incomparabilities of %ρ over Mρ , utilizing the interval order introduced by Fishburn [6]. Throughout this section, we assume ρ < ∞. First, we show the following lemma. Lemma 6.1. For each k ≥ 0, y %k f %k y for any f ∈ Lk (X). Proof. We show the claim by induction over k ≥ 0. Let f ∈ L0 (X) = X. By Axiom B0 and %0 = %B,0 , we have the claim for k = 0. Suppose the induction hypothesis that y %k f %k y for any f ∈ Lk (X). Consider f ∈ Lk+1 (X). Then, by Lemma 2.2, there is a vector f = (f1 , ..., f` ) ∈ Lk (X)` such that f = e ∗ f . By the induction hypothesis, y %k ft %k y for any t = 1, ..., `. By Axiom C1, we have y = e ∗ y %k+1 f = e ∗ f %k+1 e ∗ y = y.¥ Lemma 6.1 guarantees that every f ∈ Lρ (X) has upper and lower bounds in Bρ (y; y). We define the lub λf and glb λf of each f ∈ Lρ (X) by λf λf

= min{λg : g ∈ Bρ (y; y) with g %ρ f };

(26)

= max{λg : g ∈ Bρ (y; y) with f %ρ g}.

If λf and λf coincide, then f belongs to Mρ and is exactly measured; and if they differ, then f belongs to Lρ (X) − Mρ and is approximated by them. These observations are described as follows: for any f ∈ Lρ (X), λf = λf = λf ⇐⇒ f ∈ Mρ ; and λf > λf ⇐⇒ f ∈ Lρ (X) − Mρ .

(27)

The measurements have a gap, λf > λf , if and only if f is not measurable. This gap causes incomparabilities involved in %ρ . In (26), the lub and glb are defined in terms of the weak relation %ρ , but when f ∈ Lρ (X) − Mρ , these are effectively strict relation Âρ by Theorem 5.3.(1). Here, we see only the direction ⇐= of the second. Let f ∈ Lρ (X) − Mρ . Then, [y, λf ; y] = g Âρ f Âρ h = [y, λf ; y] for some g, h ∈ Bρ (y; y). By Axiom C2, we have [y, λf ; y] Âρ [y, λf ; y]; so λf > λf by (9). Now, we consider the following mutually exclusive and exhaustive cases: C: f, g ∈ Mρ ; and IC: f ∈ Lρ (X) − Mρ or g ∈ Lρ (X) − Mρ . Theorem 5.1 states that in case C, f and g are comparable. Now, we consider case IC. 17

Comparabilities/incomparabilties hold in the following manner. Suppose that f, g ∈ Lρ (X)− Mρ . In this case, we have the gaps, λf > λf and λg > λg . When these gaps are separated, for example, λf > λf ≥ λg > λg , f could be strictly preferred to g, because some lower bound of f is better than or equal to some upper bound of g. The other case, λg > λg ≥ λf > λf , is parallel, which is the case of the 3rd and 4th lines in Fig.3. These comparisons become degenerate when f ∈ Lρ (X) − Mρ and g ∈ Mρ . In Fig.3, the first and third (and the fourth) lines show the case of comparability, and the first and second lines show incomparability. We have these facts in a general manner. Recall that f 1ρ g is defined in (7), meaning that f and g are incomparable with respect to %ρ . Theorem 6.1 (Case IC). Suppose that at least one of f, g is in Lρ (X) − Mρ . Then, f Âρ g ⇐⇒ λf ≥ λg ;

(28)

f 1ρ g ⇐⇒ λf > λg and λg > λf .

(29)

Proof. First, suppose that f ∈ / Mρ and g ∈ Mρ . Then, f ¿ρ g by Theorem 5.3.(1). Consider the equivalence of (28). If f Âρ g, then f Âρ g ∼ρ [y, λg ; y], which implies λf ≥ λg by (26) and B1. Conversely, if λf ≥ λg , then f %ρ [y, λf ; y] %ρ [y, λg ; y] ∼ρ g by (26) and B1, which implies f Âρ g by C2 and f ¿ρ g. The equivalence of (29) is similar. We can prove similarly / Mρ . these assertions in the case f ∈ Mρ and g ∈ Now, suppose that f, g ∈ / Lρ (X) − Mρ . Since f ¿ρ g by Theorem 5.3.(1), we have f 1ρ g ⇐⇒ neither f Âρ g nor g Âρ f. Hence, (29) follows from (28). Now, we prove: (i): f Âρ g ⇐⇒ f Âρ h Âρ g for some h ∈ Bρ (y; y);

(ii): f Âρ h Âρ g for some h ∈ Bρ (y; y) ⇐⇒ λf ≥ λg .

These imply (28). Let us see (ii). Suppose that f Âρ h Âρ g for some h ∈ Bρ (y; y). Since λf is the lub of f , by (26), λf ≥ λh . Similarly, λh ≥ λg . Conversely, let λf ≥ λg . Then, by (26), f Âρ [y, λf ; y] %ρ [y, λg ; y] Âρ g. Hence, we can adopt [y, λf ; y] for h. Consider (i): The direction ⇐= is by C2. Now, suppose f Âρ g. By (11), there is a finite sequence of distinct g0 , ..., gm such that f = gm %ρ ... %ρ g0 = g and at least one of adjacent gl , gl+1 belongs to Bρ (y; y) for each l = 0, ..., m − 1. Hence, g1 belongs to Bρ (y; y), since f = gm and g0 = g are in Lρ (Y ) − Mρ . Since f %ρ g1 %ρ g and g1 ∈ Bρ (y; y), we have f Âρ g1 Âρ g by Theorem 5.3.(1).¥ Theorem 6.1 provides a complete characterization of incomparabilities involved in the relation %ρ . In order to synthesize this result with Theorem 5.1.(2) for the measurable domain Mρ , we consider the vector-valued function Λ(f ) = (λf , λf ) for any f ∈ Lρ (X) with the binary relation ≥ over Πρ × Πρ given by (ξ 1 , ξ 2 ) ≥ (η1 , η 2 ) ⇐⇒ ξ 2 ≥ η 1 . (30) This relation ≥ is transitive and anti-symmetric, i.e., (ξ 1 , ξ 2 ) ≥ (η1 , η 2 ) and (η 1 , η 2 ) ≥ (ξ 1 , ξ 2 ) =⇒ (ξ 1 , ξ 2 ) = (η 1 , η 2 ) (and ξ 1 = ξ 2 ) and is reflexive only for (ξ 1 , ξ 2 ) with ξ 1 = ξ 2 . Thus, ≥ is weaker than a partial ordering. This relation is the same as Fishburn’s [6] interval order. Using the function Λ(·) and ≥, we synthesize the results for cases C and IC. Theorem 6.2 (Representation). For any f, g ∈ Lρ (X), f %ρ g ⇐⇒ Λ(f ) ≥ Λ(g).

Proof. Consider case C : f, g ∈ Mρ . Since Λ(f ) = (λf , λf ) and Λ(g) = (λg , λg ), the right-hand side of (30) is λf ≥ λg . Thus, the assertion follows from Theorem 5.1. Consider case IC that 18

at least one of f, g does not belong to Mρ . Theorem 6.1 states that f Âρ g ⇐⇒ λf ≥ λg and g Âρ f ⇐⇒ λg ≥ λf . Since f ¿ρ g by Theorem 5.3.(1), we have the assertion of the theorem.¥

The relation %ρ is transitive but is neither anti-symmetric nor reflexive, though ≥ is antisymmetric.

Incomparability 1ρ and indifference ∼ρ may sound similar: indeed, Shafer [21], p.469, discussed whether incomparability 1ρ and indifference ∼ρ could be defined together and pointed out a difficulty from the constructive point of view. Theorem 6.2 gives a clear distinction between ∼ρ and 1ρ ; by Axiom C2, ∼ρ is transitive, but 1ρ not. Indeed, in Fig.3, g 1ρ f and f 1ρ f 0 but f 0 Âρ g. There are two sources for incomparabilities; X − Y and a finite ρ. Lemma 5.1.(2) implies that when f (x) > 0 for some x ∈ X − Y, this f does not belong to Mk for any k. On the other hand, even when the support of f is included in Y, f does not belong to Mk for k < kf , where kf is given in (24). Thus, f, g ∈ L∞ (Y ) are possibly incomparable with respect to %ρ ⇐⇒ ρ < max{kf , kg }. Theorem 6.2 corresponds to von Neumann-Morgenstern’s [26], p.29, indication of a possibility of a representation of a preference relation involving incomparabilities in terms of a manydimensional vector-valued function. Our result shows a specific form of their indication. If there are multiple pairs of different benchmarks, our representation theorem may be stated by a higher-dimensional vector-valued function. Dubra et al. [5] obtained the representation result in the form that an incomplete preference relation is represented by a class of expected utility functions. Here, available probabilities are given as arbitrary real numbers in the interval [0, 1] and incompleteness is allowed. Perhaps, this literature is closely related to our consideration of classical EU theory in Section 4.2. When we restrict our attention to the set of pure alternatives to Y , we have a complete preference relation, but when we consider the entire set X, the derived preference relation and its extension to L[0,1] (X) could be incomplete. Then, it is an open problem whether Theorem 6.2 can be extended or we need a representation theorem in the form of [5]. Nevertheless, some nonconstructive elements are involved here, and the restriction ρ < ∞ is natural from the viewpoint of our motivation. In the following example, we calculate the lub λd and glb λd of lottery d = Example 3.1 and an additional case. Example 6.1 (Example 3.1 continued). Recall X = {y, y, y}, and d = consider two cases

25 100 y

25 75 100 y ∗ 100 y.

∗

75 100 y

in

Here, we

9 7 83 ;y] ÂB,1 d ÂB,1 [y, 10 ;y] and y ∼B,2 [y, 10 (A) : [y, 10 2 ;y]. 9 7 77 ; y] ÂB,1 d ÂB,1 [y, 10 ; y] and y ∼B,2 [y, 10 (B): [y, 10 2 ;y]

Case (A) is (1 ) of Fig.3.1, and (B) is additional for the case of risk lovers to be considered in Section 7. Since d ∈ L2 (X) − L1 (X), we consider ρ ≥ 2. The lub and glb are given in Table 6.1: Table 6.1: the lub and glb ρ=2 ρ=3 ρ≥4

(A) λd = 25/102 & λd = 0 λd = 225/103 & λd = 175/103 λd = λd = 2075/104 19

(B) λd = 25/102 & λd = 0 λd = 225/103 & λd = 175/103 λd = λd = 1925/104

Calculation of these results will be given in the Appendix. Consider comparability/incomparability 2 between d and benchmark lottery c = [y, 10 ; y], which appears in the example in Section 7. By (27), d is measurable with respect to Bρ (y; y) if and only ρ ≥ 4. In this case, d is comparable 2 ; y] by Theorem 5.1, i.e., d Âρ c in (A) and c Âρ d in (B). When ρ = 2, 3, there with c = [y, 10 are gaps between λd and λd . Since c is in those gaps; by Theorem 6.1, d and c are incomparable.

7

An Application to a Kahneman-Tversky Example

We apply our theory to an experimental result reported in Kahneman-Tversky [11]. It is formu25 75 y ∗ 100 y; the lub and glb are given in Table lated as Example 6.1, and the key lottery is d = 100 6.1. In this application, we have an issue of how (whether) experimental subjects answer when given lotteries are incomparable for them. First, we look at the Kahneman-Tversky example, and then we make a certain postulate to interpret choice behavior of subjects. In the Kahneman-Tversky example, 95 subjects were asked to choose one from lotteries a and b, and one from c and d. In the first problem, 20% chose a, and 80% chose b. In the second, 65% chose c; and the remaining chose d. 80 vs. b = 3000 with probability 1; (80%) a = [4000, 10 2 ; 0]; (20%); 20 25 vs. d = [3000, 10 c = [4000, 10 2 ; 0]; (65%); 2 ; 0];

(35%).

The case of modal choices, denoted by b ∧ c, contradicts classical EU theory. Indeed, these choices are expressed in terms of expected utilities as: 0.80u(4000) + 0.20u(0) < u(3000)

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0.20u(4000) + 0.80u(0) > 0.25u(3000) + 0.75u(0). Normalizing u(·) with u(0) = 0, and multiplying 4 to the second inequality, we have the opposite inequality of the first, a contradiction. The other case contradicting classical EU theory is a ∧ d. EU theory itself predicts the outcomes a ∧ c and b ∧ d, depending upon the value u(3000). This is a variant of many experiments reported.9 In [11], no more information is mentioned other than the percentages. Consider three possible distributions of the answers in terms of percentages over the four cases, described in Table 7.1: the first, second, or third entry in each cell is the percentage derived by assuming 65%, 52%, or 45% for b ∧ c. The first 65% is the maximum percentage for b ∧ c, which implies 0% for a ∧ c, and these determine the 20% for a ∧ d and 15% for b ∧ d. The second entries are based on the assumption that the choices of b and c are stochastically independent, for example, 52 = (0.80 × 0.65) × 100 for b ∧ c. In the third entries, 45% is the minimum possibility for b ∧ c. We interpret this table as meaning that each cell was observed at a significant level.

a : 20% b : 80%

Table 7.1 c : 65% a ∧ c : EU: 0 //13//20 b ∧ c : paradox: 65//52//45

9

d : 35% a ∧ d : paradox: 20//7// 0 b ∧ d : EU: 15//28//35

This type of an anomaly is called the “common ratio effect” and has been extensively studied both theoretically and experimentally; typically, the independence axiom is weakened while keeping the probability space as a continuum (cf., Prelec [17] and its references).

20

We find one conflict in that every subject chose one lottery in each choice problem but our theory states that c and d are incomparable under some parameter values. The issue is how a subject behaves for the choice problem when the lotteries are incomparable for him. For such a choice problem, a person would typically be forced (e.g., unconsciously following social customs) to make a choice.10 Here, we assume that even when the lotteries are incomparable to a subject, he is forced to make a decision in an arbitrary manner. Let y = 4000, y = 0, y = b = 3000, and ρ ≥ 2. Consider the following two cases: (A) : 83 77 y ∼B,2 [y, 10 2 ; y] and (B) : y ∼B,2 [y, 102 ; y]. In comparisons between lotteries a and b, the theory predicts, independent of ρ, the choice b (or a) in case (A) (or (B)). Since (A) is the case of risk-aversion, the choice b out of a and b is expected to be more frequent than a. Comparisons between lotteries c and d depend upon ρ. In case ρ ≥ 4, it follows from Table 2075 20 6.1 that in (A), d = [y, 25 10 ; y] ∼4 [y, 104 ; y] Â4 [y, 102 ; y] = c; so d is chosen, and in (B), 2 25 ; y] Â4 [y, 1925 c = [y, 10 104 ; y] ∼4 [y, 102 ; y] = d; so c is chosen. In sum, in case ρ ≥ 4, the theory predicts only the diagonal cells b ∧ d and a ∧ c depending upon cases (A) and (B), which are the same as the predictions of classical EU theory. Thus, if all subjects have their cognitive bounds ρ ≥ 4, the theory is inconsistent with the experimental result.

2 and in either (A) or (B), λd = 225 Consider case ρ = 3. Then, it holds that λc = 10 103 > λc = 175 > λd = 103 ; thus, c and d are incomparable for him. We postulate that the subject regards d as a benchmark lottery [y, θd ; y] and θd is uniformly possible in the interval in Π3 with λd > θd > λd , with the additional assumption that if θd = λc , then c and d are equally possible. The frequencies of the choice c and d are calculated: 2 10

199 103

−

175 103

+

1 103

×

1 2

=

24.5 103

for c; and

224 103

−

200 103

+

1 103

×

1 2

=

24.5 103

for d.

Thus, c and d are equally possible to be chosen. When ρ = 2, the ratio for the choices c and d becomes 19.5 : 4.5 in either case (A) or (B). This ratio suggests that b ∧ c is a dominant case. As a whole, the tendency of choice c rather than d follows from the assumption that cognitive bounds ρ = 2 and 3 are applied to more subjects than ρ ≥ 4. This assumption is compatible with our basic presumption that people have bounded rationalities. In the base preference relations given in of (2) of Example 3.1, we do not have a benchmark lottery indifferent to y = b. The results for cases ρ = 2 and ρ = 3 are still the same as above. In case ρ ≥ 4, d is not measurable, but the resulting preference relation is the same as the above case of ρ = 3.

8

Conclusions

We developed EU utility theory with probability grids and incomparabilities. The permissible probabilities are restricted to the form of `-ary fractions up to a given cognitive bound ρ. The theory is constructive in that it starts with given base preference relations h%B,k iρk=0 and 10

In Davis-Maschler [4], Sec.6, it is reported that when a number of game theorists/economists were asked about their predictions about choices in a specific example in a cooperative game theory, only Martin Shubik refused to answer a questionnaire It was his reason that the specification in terms of cooperative game is not enough to have a precise prediction.

21

proceeds with the derivation process from one layer to the next. When there is no cognitive bound, our theory gives a complete preference relation over L∞ (Y ), and it is a fragment of classical EU theory. However, our main concern was the bounded case ρ < ∞. When ρ < ∞, the resultant preference relation %ρ over Lρ (X) is uniquely constructed but not complete as long as X contains pure alternatives strictly between the benchmarks y and y. We provided the measurable domain Mρ ; the resultant %ρ is complete over Mρ , while it involves incomparabilities outside Mρ . In Section 6, we gave a complete characterization of incomparabilities and also the representation theorem on %ρ in terms of the two-dimensional vector-valued function, utilizing Fishburn’s [6] interval order. This is interpreted as corresponding to the indication of von Neumann-Morgenstern [26], p.29. In Section 7, we applied the incomparability results to the Allais paradox, specifically, to the experimental example due Kahneman-Tversky [11]. We showed that the prediction of our theory is compatible with their experimental result, where incomparabilties involved for ρ = 2 and ρ = 3 are crucial in interpreting their results in our theory. The main part of our theory is about deductive reasoning for decision making. Bounded rationality involved here is that deduction is bounded with probability grids and a cognitive bound. The derivation of preferences in h%k iρk=0 is formulated as a mathematical induction from base preferences h%B,k iρk=0 . We may ask where is the source for h%B,k iρk=0 . Here, it is inductive, based on experiences, but not deductive. In this sense, this part is related to inductive game theory (cf., Kaneko-Matsui [12]). In particular, Kline et.al [13] provided a conceptual framework for an inductive derivation, including some deductive part, of a subjective view from one’s accumulated experiences. Our theory may be viewed from their framework; h%B,k iρk=0 may be derived from the decision maker’s experiences. This part is closely related to the case-based decision theory by Gilboa-Schmeidler [8], and the frequentist interpretation of probability (cf., Hu [10]). This derivation may be expressed in terms of Simon’s [23] satisficing/aspiration. For a pure alternative x, the decision maker evaluates the propositions for λ ∈ Πk (k ≥ 0) “x %B,k [y, λ; y]” and “[y, λ; y] %B,k x”.

(32)

and puts his satisficing degree to each. This satisficing degree comes from his experiences, possibly, frequencies of perceived sensations. He compares this degree with his aspiration level, which is, more or less, the same as the concept of a cognitive bound. If the degree is larger than or equal to the aspiration level, he treats it as his base preference. In particular, indifference x ∼B,k [y, λ; y] is important to him in that it gives the exact measurement, though the reason might be “giving up” further thoughts. If the degree does not exceed the aspiration level, he may go to a deeper level and make such evaluations again. Or without going the next level, he may accept base preferences. In Example 3.1, the precise measurement is reached at depth 2 in case (1), and in case (2), he gives up precise evaluations after depth 2. A full study of these is an open problem. Now, we mention a few possible extensions of (the deductive part of) our theory. In the present paper, we assume that the benchmarks y and y are given. The choice of the lower y could be natural, for example, the status quo. The choice of y may be more temporary in nature. In general, benchmarks y and y are not really fixed; there are different benchmarks than the given ones. We consider two possible choices of the benchmarks. One possibility is a vertical extension: we take another pair of benchmarks y and y such as 22

y %B,0 y ÂB,0 y %B,0 y. The new set of pure alternatives is given as X(y; y). Then, the set X

may be included in X(y; y). This may be straightforward when there is no cognitive bound ρ, but with a cognitive bound ρ < ∞, the relation between the original system and the new system is not simple. In the case of measurement of temperatures, the grids for the Celsius system do not exactly correspond to those in the Fahrenheit system. We may need multiple bases ` for probability units (grids), and may have multiple preference systems even for similar target problems. Another possibility is a horizontal extension. For example, y is the present status quo for a student who faces a choice problem between the alternative y of going to work for a large company and the alternative y of going to graduate school. The student may not be able to make a comparison between y and y, while he can make a comparison between detailed choices after the choice of y or y. This involves incomparabilities different from those in this paper. These possible extensions are open problems of importance.

Appendix Proof of Lemma 2.2. Let k ≥ 1. We show that if f ∈ Lk (X), then f = e ∗ f for some f ∈ Lk−1 (X)` . We can assume that δ(f ) = k; indeed, if δ(f ) ≤ k − 1, then we take f = (f, ..., f ) ∈ Lk−1 (X)` and f = e ∗ f . By δ(f ) = k ≥ 1, for each x ∈ X, f (x) is expressed as Pk vm (x) m=1 `m , where 0 ≤ vm (x) < ` for all m ≤ k. P P We “partition” the total sum x∈X km=1 vm`m(x) = 1 into ` portions so that each has the sum 1` . However, this may not be directly possible; for example, if ` = 10 and v1 (x) = 2, 2 1 then v1`(x) = 10 exceeds 1` = 10 . To avoid this difficulty, we again “partition” vm`m(x) into the 1 P sum vm (x) × `1m = `1m + ... + `1m . That is, x∈X v1`(x) is represented as the natural numbers 1 P P vm (x) 1 ..., τ 1 . In general, is 1, ..., τ 1 (τ 1 := x∈X v1 (x)) with weight `1 for each i = 1,P x∈X `m represented by a set natural numbers Dm of its cardinality x∈X vm (x), and each element in P P Dm is associated with weight `1m . In this manner, we represent the total sum x∈X km=1 vm`m(x) by the ` sets of natural numbers I1 , ..., I` with some weights so that the sum of the weights of each I1 , ..., I` is into 1` . Using these two partitions, we construct a decomposition f = (f1 , ..., f` ) of f. P P Formally, let τ 0 = 0 and τ m = m x∈X vt (x) for m = 1, ..., k. Then, let I = {1, ..., τ k }, t=1 and Dm = {τ m−1 + 1, ..., τ m } for m = 1, ..., k. We associate the weight w(i) to each i ∈ I by w(i) =

1 `m

if i ∈ Dm .

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Each vm`m(x) is represented by the vm (x) number of elements in Dm associated with weights `1m . P P P P P P P Then, i∈Dm w(i) = x∈X vm`m(x) , and km=1 i∈Dm w(i) = km=1 x∈X vm`m(x) = x∈X f (x) = 1. P Now, we define the function W over I by: W (j) = i≤j w(i) for any j ∈ I; that is, it is P P the sum of w(i)’s up to number j. Then, W (τ k ) = km=1 i∈Dm w(i) = 1. It holds that for any j ∈ I and t = 1, ..., `, t−1 t t (34) ` ≤ W (j) < ` =⇒ W (j + 1) ≤ ` .

Indeed, if j ∈ Dm , then W (j) is expressed as `sm for some positive integer s < `m , and W (j +1) = 1 t 0 W (j) + `m 0 for some m ≥ m. Thus, we have W (j + 1) ≤ ` . Using this fact and W (τ k ) = 1, we 23

find another partition I1 , ..., I` of I so that It = {σ t−1 , ..., σ t } and σ 0 = 0 < σ 1 < ... < σ ` = τ k such that P 1 (35) i∈It w(i) = ` for t = 1, ..., `.

These I1 , ..., I` are nonempty, while some of D1 , ..., Dk may be empty. Each It consists of successive elements in I. If D1 6= ∅, then I1 , ..., Iτ 1 are singleton sets, but the others may include some of D1 , ..., Dk . For each m =P1, ..., k, we label each element i in Dm = {τ m−1 + 1, ..., τ m } by ϕ(i) ∈ X. Since Dm represents x∈X vm (x), we can find a function ϕ : Dm → X so that it partitions Dm with vm (x) = |{i ∈ Dm : ϕ(i) = x}| for each x ∈ X. We define lotteries f1 , ..., f` ∈ Lk−1 (X) by: for t = 1, ..., `, ft (x) =

k |{i ∈ I ∩ D : ϕ(i) = x}| P t m for each x ∈ Y. m−1 ` m=1

(36)

If |{i ∈ It ∩ D1 : ϕ(i) = x}| = 1, then, by (35), It consists of a unique element i with w(i) = 1` , in which case |{i ∈ It ∩ Dm : ϕ(i) = x}| = 0 for m > 1. The other case is |{i ∈ It ∩ D1 : ϕ(i) = x}| = 0. Hence, the summation in (36) has at most length k − 1. Then, since |{i ∈ It ∩ Dm : ϕ(i) = x}| ν for some ≤ |{i ∈ Dm : ϕ(i) = x}| = vm (x) < ` for all m ≤ k, the value ft (x) is expressed as `k−1 k−1 ν < ` . Also, we have, by (35), P

x∈Y

ft (x) =

k |I ∩ D | k |{i ∈ I ∩ D : ϕ(i) = x}| P P P P t m t m = =`× w(i) = 1. m−1 m−1 ` ` m=1 x∈X m=1 i∈It

Thus, ft ∈ Lk−1 (X) for all t = 1, ..., `. Finally,

¥

P`

1 t=1 `

× ft (x) is calculated as

` P k P k v (x) k |{i ∈ I ∩ D : ϕ(i) = x}| ` |{i ∈ I ∩ D : ϕ(i) = x}| P P P t m t m m = = = f (y). m m m ` ` t=1 m=1 m=1 t=1 m=1 `

Proof of Theorem 3.1. We define v : X → R as follows: for any x ∈ X, v(x) = inf{λ ∈ Π∞ : [y, λ; y] %B,k x for some k < ∞}.

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Because of Axiom B0, this is well-defined. Also, v(y) = 1 and v(y) = 0 by Axiom B1. It holds that for any x ∈ X and λ ∈ Π∞ , [y, λ; y] %B,∞ x =⇒ λ ≥ v(x); and x %B,∞ [y, λ; y] =⇒ v(x) ≥ λ.

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Let us see the first. Let λ ∈ Πk and [y, λ; y] %B,k x. By (37), λ ≥ v(x). Let x %B,∞ [y, λ; y]. Next, let x %B,k [y, λ; y]. Then, there is a sequence {λν } in Π∞ with [y, λν ; y] %B,∞ x for all ν ≥ 0 and λν → v(x) as ν → ∞. Since λν ≥ λ for all ν ≥ 0 by Axiom B2, we have v(x) ≥ λ. We define the eu-function ve and the eu-preference relation %v over over L∞ (X) by P ve (f ) = x∈X f (x)v(x) for any f ∈ L∞ (X); for any f, g ∈ L∞ (X), f %v g ⇐⇒ ve (f ) ≥ ve (g).

24

(39) (40)

Since each f has a finite support, the sum in (39) is well-defined; so is %v . This implies that %v is a complete and transitive binary relation over L∞ (X). Since v(y) = 1 and v(y) = 0, we have ve ([y, λ; y]) = λ. Thus, it follows from (38) that [y, λ; y] %B,∞ x =⇒ [y, λ; y] %v x; and x %B,∞ [y, λ; y] =⇒ x ≥ [y, λ; y].

(41)

We denote the restriction of %v to Lk (X) by %v,k (= %v ∩Lk (X)2 ). Each %v,k is a complete and transitive binary relation over Lk (X). The key step is that for any k (0 ≤ k < ρ + 1) and any f, g ∈ Lk (X), f %k g =⇒ f %v,k g; and f Âk g =⇒ f Âv,k g.

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Using this, we prove that f %k g and not (f %k g) do not happen simultaneously. Suppose, on the contrary, that f %k g and not (f %k g) happen. The latter, not (f %k g), happens as a part of g Âk f. However, by (42), we have f %v,k g and g Âv,k f, which is impossible because of (40). Hence, either f %k g or not (f %k g). This implies that %k is a binary relation over Lk (X). To show (42), first, we note the following: ve (e ∗ f ) =

` X t=1

1 ` ve (ft )

for any f ∈ Lk (X)` and k ≥ 0.

(43)

P` 1 P P This follows from (39) that ve (e ∗ f ) = x∈X (e ∗ f )(x)v(x) = y∈Y t=1 ` ft (x)v(x) = P` 1 P P` P` 1 P x∈X ft (x)v(x) = x∈X and t=1 ` t=1 ` ve (ft ). Here, interchangeability of t=1 follows from the fact that f1 , ..., f` have finite supports.

The last step is to show (42) along the inductive construction of %k , k = 0, ...(< ρ + 1). For k = 0, it follows from Axiom B0 and (41) that y %v,0 x %v,0 y for any x ∈ X. Now, consider step k. Specifically, consider k0. Then, f %k g is f %B,k g. By (41), we have f %v,k g.

Consider k1; f %k g is derived by Axiom C1 with ft %k−1 gt for t = 1, ..., ` for the decompositions f = (f1 , ..., f` ) of f and g = (g1 , ..., g` ) of g. Here, the induction hypothesis is that (41) holds for each ft %k−1 gt , i.e., ft %v,k−1 gt for t = 1, ..., `. By (43), ve (f ) = ve (e ∗ f ) ≥ ve (e ∗ g) = ve (g), i.e., f %v,k g. The case of f Âk g is similar. Consider k2; f %k g is derived by Axiom C2 with f %k h and h %k g. Here, the induction hypothesis is that (41) holds for f %k h and h %k g, i.e., f %v,k h and h %v,k g. By transitivity of %v,k , we have f %v,k g.¥ Calculations of the glb and lub in Table 6.1: Here, we calculate the results in Table 6.1 83 for case (B) : y ∼B [y, 10 2 ; y].

83 25 75 Case ρ = 2 : C1 is not applied to substituting [y, 10 2 ; y] for y in d = 100 y ∗ 100 y, but may be possible if we sacrifice accuracy; that is, we substitute y and y for y in d, and obtain the following 25 75 (44) 100 y ∗ 100 y Â2 d Â2 y. 25 25 The lub and glb of d are given as λd = 10 2 and λd = 0. We verify only λd = 102 : first, 5 5 5 5 10 y ∗ 10 y Â1 10 y ∗ 10 y holds by C1 since y Â0 y and y ∼0 y by B1 and Step 0 of the DP. Hence 25 75 5 5 5 5 5 5 25 75 5 5 100 y ∗ 100 y = 10 ( 10 y ∗ 10 y) ∗ 10 y Â2 10 ( 10 y ∗ 10 y) ∗ 10 y = 100 y ∗ 100 y. This is the best upper

25

9 bound B2 (y; y). It is assumed by Example 3.1.(1) that [y, 10 ; y] Â1 y. But ρ = 2 prevents an application of Axiom C1 from improving the evaluation in (44). 175 Case ρ = 3 : Consider the second line in Table 6.1, i.e., λd = 225 103 and λd = 103 . Here, we 175 7 5 verify only λd = 103 . First, y Â1 [y, 10 ; y] by Example 3.1.(1). Then, by C1, [y, 10 ; y] = 5 5 7 5 35 25 7 5 5 5 5 10 y ∗ 10 y Â2 10 [y, 10 ; y] ∗ 10 y = [y, 102 ; y]. Again, by C1, d = 102 y ∗ 102 y = 10 [y, 10 ; y] ∗ 10 y Â3 5 35 5 175 175 10 [y, 102 ; y] ∗ 10 y = [y, 103 ; y]. Hence, d Â3 [y, 103 ; y]. By watching this derivation carefully, we see that this is the best lower bound of d in B3 (y; y). Thus, λd = 175 103 . 85 Case ρ ≥ 4 : We have no constraint on substitution of [y, 10 2 ; y] for y; we have the third 2075 line in in Table 6.1, i.e., λd = λd = 104 . We verify this following our derivation process; 5 5 83 5 5 5 5 y ∗ 10 y ∼3 425 y ∗ 575 since 10 10 y by C1 and y ∼B,2 [y, 102 ; y], we have d = 10 ( 10 y ∗ 10 y) ∗ 10 y 103 ∼4 2075 y ∗ 7825 10 y. 104

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