Step B is an introspection process to find preferences hidden in the mind of the decision maker. On the other hand, Step E is a logical process to extend base preferences found in Step B; it involves a possible difficulty generated by a new type of probability depths, \(\delta (f(x))\), interacting with a finite cognitive bound \(\rho \). This requires our axiomatic system, Axiom E1 in particular, to take a certain specific form. Keeping this remark in mind, we present our axiomatic system for Step E. Throughout this section, let \(\langle \trianglerighteq _{k}\rangle _{k<\rho +1}\) be a given base preference stream satisfying Axioms B0 to B3.
Extended preference streams
We consider how \(\trianglerighteq _{k}\) is extended to \( L_{k}(X)\) for \(k<\rho +1\). We formulate this derivation by a kind of mathematical induction from the base preferences \( \trianglerighteq _{k}\) and the previously derived relation \( \succsim _{k-1};\) this process starts at layer 0 to the last layer \(\rho \) (or goes to any layer if \(\rho =\infty \) ). These are formulated by three axioms; the first axiom E0 corresponds to the start, and the other two, E1 and E2, describe the extension process. It is shown by Theorem 4.1 that our formalism involves no logical difficulty. Then, we give one additional axiom to capture the central part determined by E0 to E2.
First, Axiom E0 is to convert base preferences \(\trianglerighteq _{k}\) to \( \succsim _{k}\) for each \(k<\rho +1\), depicted as the vertical arrows in Table 1.
Axiom E0 (Extension)(i): For any \((f,g)\in D_{0}\), \( f\trianglerighteq _{0}g\) if and only if \(f\succsim _{0}g\).
(ii): For any \(k (1\le k<\rho +1)\) and \((f,g)\in D_{k}\), if \( f\trianglerighteq _{k}g\), then \(f\succsim _{k}g\).
This states that \(\langle \trianglerighteq _{k}\rangle _{k<\rho +1}\) is the ultimate source for \(\langle \succsim _{k}\rangle _{k<\rho +1}\) in Step E. For \(k=0\), the base preferences are only the direct source for \( \succsim _{0};\) thus, (i) has both directions. For \(k\ge 1\), in addition to the base preferences, there is another source from the previous \(\succsim _{k-1};\) (ii) requires only one direction. We will show that as long as the domain \(D_{k}\) is concerned, the converse of (ii) holds for our intended preference stream \(\langle \succsim _{k}\rangle _{k<\rho +1}\).
Now, we consider the connection between layers \(k-1\) and k. For \({\widehat{f}}=(f_{1},\ldots ,f_{\ell })\) and \({\widehat{g}}=(g_{1},\ldots ,g_{\ell })\), we write \( {\widehat{f}}\succsim _{k}{\widehat{g}}\) iff \(f_{t}\succsim _{k}g_{t}\) for all \( t=1,\ldots ,\ell \). Recall that a decomposition of \(f\in L_{k}(X)\) is defined by (9)\(\mathbf {.}\) We formulate a derivation of \(\succsim _{k}\) from \(\succsim _{k-1}\) as follows: let \(1\le k<\rho +1\).
Axiom E1 (Derivation from the previous layer): Let \(f\in L_{k}(X)\), \(g\in B_{k}({\overline{y}};{\underline{y}})\), and \({\widehat{f}}\), \({\widehat{g}}\) their decompositions. If \({\widehat{f}} \succsim _{k-1}{\widehat{g}}\) or \({\widehat{g}} \succsim _{k-1}{\widehat{f}}\), then \( f\succsim _{k}g\) or \(g\succsim _{k}f\), respectively.
In layer \(k-1\), each \(f_{t}\) of \({\widehat{f}}=(f_{1},\ldots ,f_{\ell })\) is compared with the corresponding benchmark lottery \(g_{t}\) of \({\widehat{g}} =(g_{1},\ldots ,g_{\ell })\). These preferences are extended to layer k. In Table 1, the horizontal arrows indicate this derivation. A lottery \(f\in L_{k}(X)\) may involve the depth \(\delta (f(x))\) of the probability value f(x) and the depths of \(\delta ({\overline{\lambda }}_{x})\), \(\delta ({\underline{\lambda }} _{x})\) of \({\overline{\lambda }}_{x}\), \({\underline{\lambda }}_{x}\) given in b2, for each pure alternative \(x\in X\) with \(f(x)>0\). In the lottery \(d=\frac{25}{10^{2}}y*\frac{75}{ 10^{2}}{\underline{y}}\) in Example 3.1, the former is \(\delta ( \frac{25}{10^{2}})=2\) and the latter is \(\delta ({\overline{\lambda }} _{x})=\delta ({\underline{\lambda }}_{x})= \delta (\frac{77}{10^{2}})=2\) in case A. On the other hand, benchmark lotteries involve only the former depths since \(\delta (\lambda _{{\overline{y}}})=\delta (\lambda _{ {\underline{y}}})=0\) by (18). In Axiom E1, extension is always made based on the benchmark scale. In fact, Lemma 4.1 does not take this constraint into account, but Theorem 4.1 does.
Preferences extended through the benchmark scale \(B_{k}({\overline{y}}; {\underline{y}})\) in E1 are further extended by transitivity, which is the next axiom. Let \(0\le k<\rho +1\).
Axiom E2 (Transitivity): For any \(f,g,h\in L_{k}(X)\), if \( f\succsim _{k}g\) and \(g\succsim _{k}h\), then \(f\succsim _{k}h\).
We interpret Axioms E1 and E2 as inference rules with Axiom E0 as the bases for \(\succsim _{k}\). This means that the decision maker constructs \(\succsim _{0},\succsim _{1},\ldots \), step by step, using these axioms. This may involve some subtlety; Axioms E0 to E2 may lead to new unintended preferences. We will show Theorem 4.1, implying that this is not the case for the constructed preference relations.
The following are strengthened versions of E0 and E1:
E0\(^{*}:\) for all \(k<\rho +1\) and \((f,g)\in D_{k}\), \( f\trianglerighteq _{k}g\) if and only if \(f\succsim _{k}g;\)
E1\(^{*}:\) E1 holds and if the premise of E1 includes strict preferences, so does the conclusion.
Condition E0\(^{*}\) states that \(\langle \succsim _{k}\rangle _{k<\rho +1} \) is a faithful extension of \(\langle \trianglerighteq _{k}\rangle _{k<\rho +1}\) as long as the pairs of lotteries in \(D_{k}\) are concerned. The other, E1\(^{*}\), is a strengthening of E1, too. Without these, some preferences would be added in the derivation process of \(\succsim _{0},\succsim _{1},\ldots \) Note that E2 (transitivity) preserves strict preferences in the same way as E1\(^{*}\).
To prove that our extended stream \(\langle \succsim _{k}\rangle _{k<\rho +1}\ \)enjoys E0\(^{*}\), E1\(^{*}\), and E2, we first show the following lemma using the EU hypothesis, which is an auxiliary step to Theorem 4.1. A by-product is the consistency of E0\(^{*}\), E1\(^{*}\), and E2. For the lemma, a base utility stream \(\langle \varvec{\upsilon }_{k}\rangle _{k<\rho +1}\) satisfying (19) in Theorem 3.1 is given.
Lemma 4.1
(Direct application of the EU hypothesis). Let \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\) be defined as follows: for all \(k<\rho +1\),
$$\begin{aligned} f\succsim _{k}^{*}g\text { if and only if }E_{f}(\varvec{\upsilon } _{k})\ge _{I}E_{g}(\varvec{\upsilon }_{k}). \end{aligned}$$
(21)
Then, \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\) satisfying Axioms E0\(^{*}\), E1\(^{*}\), and E2.
The right-hand side is given by comparisons of the expected values of the vector-valued utility function \(\varvec{\upsilon }_{k}\). The point of the lemma is not the representation of expected utility values; instead, it is the consistency of E0\(^{*}\), E1\(^{*}\), and E2, which will be used in Theorem 4.1. The consistency of Axioms E0, E1, and E2 is straightforward since E0 takes only preferences given by B0 to B3, and E1 and E2 introduce new preferences from them. On the other hand, E0\(^{*}\ \text { and E1}^{*}\) may generate strict preferences, including negations. Hence, the consistency implied by Lemma 4.1 is a basis of our development.
It will be argued in Sect. 7 that when \(\rho =\infty \), the limit preference relation \(\succsim _{\infty }^{*}\) is determined E0, E1, and E2 under some additional condition on \(L_{\infty }(X)=\cup _{k<\infty }L_{k}(X)\).
Now, we prepare a few concepts for Theorem 4.1. Let \(\langle \succsim _{k}\rangle _{k<\rho +1}\) be a stream satisfying E0 to E2. We say that \(\langle \succsim _{k}\rangle _{k<\rho +1}\) is the smallest stream iff for any \(\langle \succsim _{k}^{\prime }\rangle _{k<\rho +1}\) satisfying E0 to E2, and \(f,g\in L_{k}(X)\), \(k<\rho +1\),
$$\begin{aligned} f\succsim _{k}g\text { implies }f\succsim _{k}^{\prime }g. \end{aligned}$$
(22)
Also, the set of preferences over \(L_{k}(X)\) derived from \(\succsim _{k-1}\) by E1 is denoted by \((\succsim _{k-1})^{\text {E1}}\), and the set of transitive closure of \(F\subseteq L_{k}(X)^{2}\) is denoted by \(F^{tr}\), i.e., \((f,g)\in F^{tr}\) if and only if there is a finite sequence \(f=h_{0}\), \(h_{1},\ldots ,h_{m}=g\) such that \((h_{t},h_{t+1})\in F\) for \(t=0,\ldots ,m-1\).
Using these concepts, we construct the smallest stream \(\langle \succsim _{k}\rangle _{k<\rho +1}\), each of which is a binary relation on \(L_{k}(X)\), and show, using Lemma 4.1, that it satisfies E0\(^{*}\) and E1\(^{*}\) as well as E2.
Theorem 4.1
(Smallest extended stream). The stream \( \langle \succsim _{k}\rangle _{k<\rho +1}\) of the sets generated by the following induction:
$$\begin{aligned} \succsim _{0}= & {} (\trianglerighteq _{0})^{tr};\text { and } \nonumber \\ \succsim _{k}= & {} [(\succsim _{k-1})^{\text {E1}}\cup (\trianglerighteq _{k})]^{tr}\text { for each }k\text { }(1\le k<\rho +1) \end{aligned}$$
(23)
is the smallest stream satisfying E0 to E2. This \(\langle \succsim _{k}\rangle _{k<\rho +1}\) satisfies E0\(^{*}\) and E1\(^{*}\).
The construction of \(\langle \succsim _{k}\rangle _{k<\rho +1}\) starts with \( \succsim _{0} = (\trianglerighteq _{0})^{tr}\), which is well defined since \(\trianglerighteq _{0}\) is a binary relation in \(D_{0}\). Then, provided that \(\succsim _{k-1}\) and \(\trianglerighteq _{k}\) are already given, \(\succsim _{k}\) is defined to be \([(\succsim _{k-1})^{\text {E1}}\cup (\trianglerighteq _{k})]^{tr}\). This is a subset of \(L_{k}(X)^{2};\) thus, it is a binary relation. Thus, the stream \(\langle \succsim _{k}\rangle _{k<\rho +1}\) is unique and is the smallest among the streams satisfying E0 to E2. Furthermore, it satisfies E0\(^{*}\) and E1\(^{*}\), where Lemma 4.1 is used. This assertion guarantees that the constructed stream \(\langle \succsim _{k}\rangle _{k<\rho +1}\) is a faithful extension of \(\langle \trianglerighteq _{k}\rangle _{k<\rho +1}\).Footnote 8
The stream \(\langle \succsim _{k}\rangle _{k<\rho +1}\) constructed in Theorem 4.1 differs from the stream \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\) given in Lemma 4.1 in that the two types of depths, \(\delta ({\overline{\lambda }}_{x})\), \(\delta ({\underline{\lambda }}_{x})\) , and \(\delta (f(x))\) are taken into account in the former, but for the latter, (21) defines the expected utility, ignoring them. These depths are interactive with cognitive bound \(\rho \). When \(\rho =\infty \), \(\succsim _{\infty }=\cup _{k<\rho +1}\succsim _{k}\) and \(\succsim _{\infty }^{*}=\cup _{k<\rho +1}\succsim _{k}^{*}\) coincide under some additional restriction on \(L_{\infty }(X)\), which will be discussed in Sect. 7.1. In general, \(\langle \succsim _{k}\rangle _{k<\rho +1}\) and \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\) differ and coincide only in partial domains, which will be discussed in Sect. 6.
In the constructing process of \(\langle \succsim _{k}\rangle _{k<\rho +1}\), E0 to E2 are all extension axioms, and the resulting stream of (23) is uniquely constructed, while there are multiple preference streams satisfying E0 to E2. It would be easier for various purposes to extract the essence of (23) by formulating one axiom. It is formulated as Axiom E3, which requires that a preference \(f\succsim _{k}g\) be based on comparisons with the benchmark scale \(B_{k}({\overline{y}};{\underline{y}})\) with either \(\trianglerighteq _{k}\) or \(\succsim _{k-1}\), that is, it eliminates preferences from other possible sources. Note that h in E3 may be the same as f or g.
Axiom E3.Let \(k<\rho +1\) and \(f,g\in L_{k}(X)\) with \( f\succsim _{k}g\). Then \(f\succsim _{k}h \succsim _{k}g\) for some \(h\in B_{k}({\overline{y}};{\underline{y}})\). When \(k\ge 1\), for the pair (f, h), \(f\trianglerighteq _{k}h\) holds or f, h have decompositions \({\widehat{f}},{\widehat{h}}\) with \({\widehat{f}}\succsim _{k-1}{\widehat{h}}\); the same holds for the pair (h, g).
The stream \(\langle \succsim _{k}\rangle _{k<\rho +1}\) given by (23) is characterized by adding E3 to E0 to E2.
Theorem 4.2
(Unique determination by E0 to E3). Any extended stream satisfying E0 to E3 is the same as the preference stream \( \langle \succsim _{k}\rangle _{k<\rho +1}\) given by Theorem 4.1.
Throughout the following, the stream given by (23) is denoted by \( \langle \succsim _{k}\rangle _{k<\rho +1}\). Other streams may have some additional superscripts such as \(^{\prime }\), \(*\).
Lemma 4.2 will be used in the subsequent analyses; (1) is the horizontal arrows in Table 1; and (2) means that \( \succsim _{k}\) is bounded in \(L_{k}(X)\) by the upper and lower benchmarks \({\overline{y}}\) and \({\underline{y}}\).
Lemma 4.2
Let \(\langle \succsim _{k}\rangle _{k<\rho +1}\) satisfies E0 to E3, and \(1\le k<\rho +1\).
- (1):
-
(Preservation of preferences): For any \(f,g\in L_{k-1}(X)\), \( f\succsim _{k-1}g\) implies \(f\succsim _{k}g\).
- (2):
-
\({\overline{y}}\succsim _{k}f\succsim _{k}{\underline{y}}\) for any \(f\in L_{k}(X)\).
The EU hypothesis is included in Axiom B1 and condition b1 along the benchmark scale \(B_{k}({\overline{y}};{\underline{y}})\), and E1 is a weak form of Axiom NM2 (independence). It follows from Lemma 4.1 and Theorem 4.1 that there are possibly multiple preference streams satisfying E0 to E2, among which some satisfy the EU representation in (21) but not in general. This is caused by two types of depths included in a lottery. For example, lottery \(d=\frac{25}{10^{2}}y*\frac{75}{10^{2}}{\underline{y}}\) involves the depths of coefficient \(\frac{25}{10^{2}}\) and of evaluation \(\lambda _{y}\). This is the reason for the EU hypothesis to hold only for some partial domain, which will be explicitly studied in Sect. 6.
Proofs
Proof of Lemma 4.1
We show that \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\) given by (21) satisfies E0\(^{*}\), E1\(^{*}\), and E2. By Theorem 3.2, we can assume that \( \upsilon _{k}({\overline{y}})=1\) and \(\upsilon _{k}({\underline{y}})=0\).
Since \(E_{f}({\underline{\upsilon }}_{k})=\lambda \) if \( f=[{\overline{y}},\lambda ;{\underline{y}}]\in B_{k}({\overline{y}};{\underline{y}})\) and \(E_{x}(\overline{\upsilon }_{k})={\overline{\upsilon }}_{k}(x)\) if \(f=x\in X\). Hence, by (19) and b2, \(f\trianglerighteq _{k}x\) if and only if \(\lambda \ge {\overline{\upsilon }}_{k}(x)\) if and only if \(E_{f}({\underline{\upsilon }}_{k})\ge E_{x}(\overline{\upsilon }_{k})\). The other cases are symmetric. Thus, E0\(^{*}\) holds for any \((f,g)\in D_{k}\).
It remains to show that \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\) satisfies E1\(^{*}\) and E2. Since (21) gives the interval order over the set \(\{[E_{f}({\overline{\upsilon }} _{k}),E_{f}({\underline{\upsilon }}_{k})]:f\in L_{k}(X)\}\), E2 holds. We show E1\(^{*}\). Let \(f\in L_{k}(X),g\in B_{k}[{\overline{y}};{\underline{y}}]\) and their decompositions \({\widehat{f}}\) and \({\widehat{g}}\) with \({\widehat{f}} \succsim _{k-1}^{*}{\widehat{g}}\). By (21), \(E_{f_{t}}(\underline{ \upsilon }_{k})\ge E_{g_{t}}({\overline{\upsilon }}_{k})\) for all \( t=1,\ldots ,\ell \). Then, \(E_{f}({\underline{\upsilon }}_{k})=E_{{\widehat{e}}*{\widehat{f}}}({\underline{\upsilon }}_{k})=\sum _{t=1}^{\ell }\tfrac{1}{\ell } E_{f_{t}}({\underline{\upsilon }}_{k})\ge \sum _{t=1}^{\ell }\tfrac{1}{\ell } E_{g_{t}}({\overline{\upsilon }}_{k})=E_{{\widehat{e}}*{\widehat{g}}}( {\overline{\upsilon }}_{k})=E_{g}({\overline{\upsilon }}_{k})\). If strict preferences are included in the decompositions, the conclusion is strict; thus, we have E1\(^{*}\). \(\square \)
Proof of Theorem 4.1
This has the three assertions: ( a) \(\langle \succsim _{k}\rangle _{k<\rho +1}\) is a sequence of a binary relations satisfying Axioms E0 to E2; (b) it is the smallest in the sense of (22) among the streams \(\langle \succsim _{k}^{\prime }\rangle _{k<\rho +1}\) satisfying E0 to E2; and (c) E0\(^{*}\), E1\( ^{*}\) hold for \(\langle \succsim _{k}\rangle _{k<\rho +1}\).
(a): E2 follows directly from (23). Consider E0. (ii) follows from (23). We show that \(\succsim _{0} =(\trianglerighteq _{0})^{tr} \) satisfies that for any \((f,g)\in D_{0}\), \(f\succsim _{0}g\) implies \(f\trianglerighteq _{0}g\). Since \(\succsim _{0} =(\trianglerighteq _{0})^{tr}\), there is a sequence \(f=h_{0}\trianglerighteq _{0}\ldots \trianglerighteq _{0}h_{m}=g\). If \(h_{t}\in X-B_{0}({\overline{y}}; {\underline{y}})\), then \(h_{t-1}\in B_{0}({\overline{y}};{\underline{y}})\) and \( h_{t+1}\in B_{0}({\overline{y}};{\underline{y}})\). By B2, \(\lambda _{t-1}\ge \lambda _{t+1}\), where \(h_{t-1}=[{\overline{y}};\lambda _{t-1},{\underline{y}}]\) and \(h_{t+1}=[{\overline{y}};\lambda _{t+1},{\underline{y}}]\). If \( h_{t},h_{t+1}\in B_{0}({\overline{y}};{\underline{y}})\), then \(\lambda _{t}\ge \lambda _{t+1}\). Hence, we can shorten the sequence to \(f=h_{0} \trianglerighteq _{0}h_{m}=g\). Thus, \(f\trianglerighteq _{0}g\).
Consider E1. Suppose that \(f\in L_{k}(X)\) and \(g\in B_{k}[{\overline{y}};{\underline{y}}]\) have decompositions \({\widehat{f}},{\widehat{g}}\) with \({\widehat{f}}\succsim _{k-1}{\widehat{g}}\). By (23), we have \( f=e*{\widehat{f}}\succsim _{k}e*{\widehat{g}}=g\). This causes no difficulty, even if \(f\in B_{k}(X)\) and \(g\in B_{k}[{\overline{y}};{\underline{y}}]\). The symmetric case \({\widehat{g}}\succsim _{k-1}{\widehat{f}}\) is similar.
(b): We prove by induction on k that \(\langle \succsim _{k}\rangle _{k<\rho +1}\) satisfies (22) for any \(\langle \succsim _{k}^{\prime }\rangle _{k<\rho +1}\) satisfying E0 to E2. When \(k=0\), we have \(\succsim _{0} = (\trianglerighteq _{0})^{tr}\) by (23). Let \(f\succsim _{0}g\), i.e., \(f (\trianglerighteq _{0})^{tr}g\), which implies that there is a sequence \(f=h_{0}\trianglerighteq _{0}h_{1}\trianglerighteq _{0}\ldots \trianglerighteq _{0}h_{m}=g\). By E0.(i), we have \(f=h_{0}\succsim _{0}^{\prime }h_{1}\succsim _{0}^{\prime }\ldots \succsim _{0}^{\prime }h_{m}=g\). By E2 for \(\succsim _{0}^{\prime }\), we have \(f\succsim _{0}^{\prime }g\).
Now, we assume that (22) holds for \(k-1\). Let \( f\succsim _{k}g\). By (23), there is a sequence \(f=h_{0}\succsim _{k}\ldots \succsim _{k}h_{m}=g\) such that each \(h_{t}\succsim _{k}h_{t+1}\) is a consequence of E1 or \(h_{t}\succsim _{k}h_{t+1}\) is \(h_{t}\trianglerighteq _{k}h_{t+1}\). In the first case, there are decompositions \({\widehat{h}}_{t}, {\widehat{h}}_{t+1}\) of \(h_{t},h_{t+1}\) such that \({\widehat{h}}_{t}\succsim _{k-1}{\widehat{h}}_{t+1}\). By the induction hypothesis, we have \({\widehat{h}} _{t}\succsim _{k-1}^{\prime }{\widehat{h}}_{t+1}\). Thus, \(h_{t}\succsim _{k}^{\prime }h_{t+1}\) by E1 for \(\succsim _{k}^{\prime }\). In the second case, \(h_{t}\trianglerighteq _{k}h_{t+1}\) implies \(h_{t}\succsim _{k}^{\prime }h_{t+1}\) by E0.(ii) for \(\succsim _{k}^{\prime }\). Hence, \( f\succsim _{k}^{\prime }g\) by E2 for \(\succsim _{k}^{\prime }\).
(c): Take \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\) given by Lemma 4.1. Since \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\) satisfies E0 to E2, it holds by (b) that for all \( k<\rho +1\) and \(f,g\in L_{k}(X)\),
$$\begin{aligned} f\succsim _{k}g\text { implies }f\succsim _{k}^{*}g. \end{aligned}$$
(24)
E0\(^{*}\): Since E0\(^{*}\) holds for \(\succsim _{k}^{*}\) by Lemma 4.1, for any \((f,g)\in D_{k}\), \(f\succsim _{k}^{*}g\) implies \(f\trianglerighteq _{k}g\). Thus, if \(f\succsim _{k}g\), then \(f\succsim _{k}^{*}g\) by (24), which implies \(f\trianglerighteq _{k}g\). For the converse, \(f\trianglerighteq _{k}g\) implies \(f\succsim _{k}g\) by (23).
\(E1^{*}\): First, we prove by induction on k that for all \(k<\rho +1\) and \(f,g\in L_{k}(X)\),
$$\begin{aligned} f\succ _{k}g\text { implies }f\succ _{k}^{*}g. \end{aligned}$$
(25)
We make the induction hypothesis that (25) holds for \(k-1\). Now, let \(f,g\in L_{k}(X)\) with \(f\succ _{k}g\). By (23), there are \( h_{0}=f,h_{1},\ldots ,h_{m}=g\) in \(L_{k}(X)\ \)such that \((h_{l},h_{l+1})\in (\succsim _{k-1})^{\text {E1}}\) or \((h_{l},h_{l+1})\in (\trianglerighteq _{k}) \) for each \(l=0,\ldots ,m-1\).
Consider case \((i): (h_{l},h_{l+1})\in (\trianglerighteq _{k})\). Then by E0\(^{*}\) for \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\), it holds that \(h_{l}\succsim _{k}^{*}h_{l+1}\), and also, if the premise is strict, it follows from E0\(^{*}\) for \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\) that \(h_{l}\succ _{k}^{*}h_{l+1}\). Now, consider case \((ii): (h_{l},h_{l+1})\in (\succsim _{k-1})^{\text {E1}}\). Let \({\widehat{h}}_{l},{\widehat{h}}_{k+1}\) be decompositions of \(h_{l},h_{l+1}\) so that \({\widehat{h}}_{l}\succsim _{k-1} {\widehat{h}}_{l+1}\) with/without strict preferences for some components. Hence, by (24) and (25) (the induction hypothesis), the same holds for \(\succsim _{k-1}^{*}\). By E1\(^{*}\) for \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\), we have \(h_{l}\succsim _{k}^{*}h_{l+1} \) and \(h_{l}\succ _{k}^{*}h_{l+1}\) if strict preferences hold for some components. At least one of \(l=0,\ldots ,m-1\), we have strict preferences for \((h_{l},h_{l+1})\in (\trianglerighteq _{k})\) or \({\widehat{h}} _{l}\succsim _{k-1}{\widehat{h}}_{l+1}\), because of \(f\succ _{k}g\). This and the above assertions in (i) and (ii) imply \(f\succ _{k}^{*}g\).
Finally, we verify E1\(^{*}\) for \(\langle \succsim _{k}\rangle _{k<\rho +1}\). Let \(f\in L_{k}(X)\) and \(g\in B_{k}({\overline{y}}; {\underline{y}})\), and let their decompositions be \({\widehat{f}},{\widehat{g}}\) with \({\widehat{f}}\succsim _{k-1}{\widehat{g}}\). Suppose that one of these preferences is strict. By E1, we have \(f\succsim _{k}g\). It suffices to show not \(g\succsim _{k}f\). However, \({\widehat{f}}\succsim _{k-1}{\widehat{g}}\) implies \({\widehat{f}}\succsim _{k-1}^{*}{\widehat{g}}\) by (24), and some components of the latter hold strictly by (25). By E1\(^{*}\) for \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\), we have \(f\succ _{k}^{*}g\), implying not \(g\succsim _{k}^{*}f\). By the contrapositive of (24), we have not \(g\succsim _{k}f\). \(\square \)
Proof of Theorem 4.2
Let \(\langle \succsim _{k}^{*}\rangle _{k<\rho +1}\) be any extended stream satisfying E0 to E3. We prove by induction on \(k<\rho +1\) that for any \(f,g\in L_{k}(X)\),
$$\begin{aligned} f\succsim _{k}^{*}g\text { if and only if }f\succsim _{k}g. \end{aligned}$$
(26)
Since \(\langle \succsim _{k}\rangle _{k<\rho +1}\) is the smallest stream satisfying E0 to E2 by Theorem 4.1, the if part holds for any \(k<\rho +1\).
Consider the only-if part. Let \(k=0\). Let \( f,g\in L_{0}(X)\) with \(f\succsim _{0}^{*}g\). Then, by E3 for \(\succsim _{0}^{*}\), we have an \(h\in B_{0}({\overline{y}};{\underline{y}})\) with \( f\succsim _{0}^{*}h\succsim _{0}^{*}g\). Thus, \(f\trianglerighteq _{0}h\trianglerighteq _{0}g\) by E0 for \(\succsim _{0}^{*}\). Hence, \(f (\trianglerighteq _{0})^{tr}g\), i.e., \(f\succsim _{0}g\) by (23). Now, we make the induction hypothesis that the only-if part holds for \( k-1. \) Let \(f\succsim _{k}^{*}g\). Then, by E3, we have \( h_{0}:=f\succsim _{k}^{*}h_{1}\succsim _{k}^{*}h_{2}:=g\) for some \( h_{1}\in B_{k}({\overline{y}};{\underline{y}})\). Let \(h_{0}=x\in X\). If \( h_{0}\trianglerighteq _{k}h_{1}\), then \(h_{0}\succsim _{k}h_{1}\) by E0\( ^{*}\) for \(\succsim _{k} \). Suppose \({\widehat{h}}_{0}\succsim _{k-1}^{*}{\widehat{h}}_{1}\) for some decompositions \({\widehat{h}}_{0}\), \( {\widehat{h}}_{1}\) of \(h_{0}\), \(h_{1}\). By the induction hypothesis, we have \( {\widehat{h}}_{0}\succsim _{k-1}{\widehat{h}}_{1}\). Thus, by E1 for \(\succsim _{k}\), we have \(h_{0}\succsim _{k}h_{1}\). In the above two cases, we have \( h_{0}\succsim _{k}h_{1}\). By the same argument, we have \(h_{1}\succsim _{k}h_{2}\). Thus, by E2 for \(\succsim _{k}\), we have \(f=h_{0}\succsim _{k}h_{2}=g\), i.e., \(f\succsim _{k}g\). \(\square \)
Proof of Lemma 4.2
(1): Let \(f\in L_{k-1}(X)\) and \(g\in B_{k-1}({\overline{y}};{\underline{y}})\). Suppose \(f\succsim _{k-1}g\). Then, \(f,g\in L_{k-1}(X)\subseteq L_{k}(X)\). Let \(f_{1}=\cdots =f_{\ell }=f\) and \(g_{1}=\cdots =g_{\ell }=g\). Then, \(f=\sum \nolimits _{t=1}^{\ell }\frac{1}{\ell } *f_{t}\) and \(g=\sum \nolimits _{t=1}^{\ell }\frac{1}{\ell }*g_{t}\). By E1, we have \(f\succsim _{k}g\). The case \(g\succsim _{k-1}f\) is similar.
Let \(f,g\in L_{k-1}(X)\) with \(f\succsim _{k-1}g\). Then, by E3 for k, \(f\succsim _{k-1}h\succsim _{k-1}g\) for some \(h\in B_{k-1}({\overline{y}};{\underline{y}})\). It follows from the conclusion of the above paragraph that \(f\succsim _{k}h\succsim _{k}g\). By E2, we have \( f\succsim _{k}g\).
(2): Let \(f\in L_{0}(X)=X\). By B0 and \(\succsim _{0} = \trianglerighteq _{0}\), we have the assertion for \(k=0\). Suppose the induction hypothesis that \({\overline{y}}\succsim _{k-1}f\succsim _{k-1} {\underline{y}}\) for any \(f\in L_{k-1}(X)\). Consider \(f\in L_{k}(X)\). Then, by Lemma 2.1, there is a vector \({\widehat{f}}\in L_{k-1}(X)^{\ell }\) such that \(f={\widehat{e}}*{\widehat{f}}\mathbf {.}\) By the induction hypothesis, \({\overline{y}}\succsim _{k-1}f_{t}\succsim _{k-1}{\underline{y}}\) for any \(t\le \ell \). By E1, \({\overline{y}}={\widehat{e}}*{\overline{y}} \succsim _{k} f={\widehat{e}}*{\widehat{f}} \succsim _{k} {\widehat{e}} *{\underline{y}}={\underline{y}}\). \(\square \)