How to Order the Alternatives, Rules, and the Rules to Choose Rules: When the Endogenous Procedural Choice Regresses

Abstract

A procedural choice problem occurs when there is no ex ante agreement on how to choose a decision rule nor an exogenous authority that is strong enough to single out a decision rule in a group. In this paper, we define the manner of procedural selection as a relation-valued procedural choice rule (PCR). Based on this definition, we then argue for some necessary conditions of a PCR. One of the main findings centers on the notion of consistency, which demands concordance between judged-better procedures and judged-better outcomes. Specifically, we found that the consistency principle and a modified version of the Pareto principle yield a simple impossibility result. We then show how the weakening of these conditions results to a degenerate PCR or the existence of a procedural veto. Finally, we show that the restriction of the preference domain to an extreme consequentialism can be seen as a positive result.

Appendix (Proofs of the Propositions)

Lemma 1.

Let K ≥ 1 be either finite or infinite. If a level-K PCR E satisfies the ‘if’ part of ILC, then for all $x \in X, k \in \left\{ {1,2, \ldots ,K} \right\}$ and f, g ∊ F ^k[x], we have fI(E ^k)g.

Proof.

We show the lemma inductively. Take arbitrary x ∊ X and $f, g \in F^{1} \left[ x \right]$. Then, by reflexivity of E ⁰, we have xE ⁰ x, or f(R ⁰)E ⁰ g(R ⁰). Therefore, by the ‘if’ part of the ILC, we have fE ¹ g. Since this argument is symmetric over f and g and does not depend on what x is, we have for all x and for all f, g ∊ F ¹[x], fI(E ¹)g.

Take any level k ∊ {1, …, K − 1}. Assume that for all f, g ∊ F ^k[x], fI(E ^k)g. Let u, v ∊ F ^k+1[x] be any x-supporting rules of level (k + 1). Then, by the completeness of E ^k+1, we have either uE ^k+1 v or vE ^k+1 u. Suppose one of these, for example uE ^k+1 v, does not hold. Then from the contraposition of ‘if’ part of the ILC, we have ¬(u(R ^k)E ^k v(R ^k)). By the completeness of E ^k, it is equivalent to v(R ^k)P(E ^k)u(R ^k). This contradicts the assumption, since u, v ∊ F ^k+1[x] implies $u\left( {R^{k} } \right), v\left( {R^{k} } \right) \in F^{k} \left[ x \right]$ and therefore the assumption demands u(R ^k)I(R ^k)v(R ^k). Therefore, we have inductively shown that fI(E ^k)g holds for all $x \in X, k \in \left\{ {1,2, \ldots ,K} \right\}$, and f, g ∊ F ^k[x]. ■

Proof of Proposition 1 [1]. 2 ≤ K < ∞: Take any x ∊ X. Consider a meta-profile R = (R ⁰, R ¹, …, R ^K−1, R ^K) such that for all i ∊ N, $F^{K} \left[ f \right]P\left( {O\left( {R_{i}^{K} } \right)} \right)F^{K} \left[ g \right]$ for some f, g ∊ F ^K−1[x]. By PWP on f and g, we have fP(E ^K−1)g. This contradicts Lemma 1, which demands that fI(E ^K−1)g. ■

[2] K = ∞: Take any x ∊ X and $k \in \varvec{N}$. Take any $R^{j} \in W\left( {F^{j} } \right)^{n} \left( {j = 0,1, \ldots ,k - 1} \right)$ and let f, g ∊ F ^k[x]. Consider a meta-profile such that for all i ∊ N and for all l ∊ {k + 1, k + 2, …}, $uP\left( {R_{i}^{l} } \right)v$ for all $u \in F^{l} \left[ f \right], v \in F^{l} \left[ g \right]$. Note that u, v ∊ F ^l[x]. At this point the PWP condition demands fP(E ^k)g while the Lemma 1 demands $fI\left( {E^{k} } \right)g$. Contradiction. ■

Proof of Proposition 2 [1]. The ‘if’ part is trivial. We show the ‘only if’ part. Suppose PCR E satisfies ILC and AF. Take any meta-profile R ∊ D, level k ∊ {1, …, K} and procedures f, g ∊ F ^k. There are two possibilities concerning the similarity of $f$ and g as a function. (1) There exists a level-k − 1 preference profile $\tilde{R}{^{k - 1}} \in W\left( {F^{k - 1} }\right)$ such that $f\left( \tilde{R}{^{k - 1}} \right) = g\left({\tilde{R}{^{k - 1}}} \right)$. Consider a meta-profile $\tilde{R} = \left( {R^{0} ,R^{1} , \ldots ,\tilde{R}{^{k - 1}} ,R^{k} , \ldots ,R^{K} } \right)$. Then, by Lemma 1, we have $fI\left( {\tilde{E}{^{k}}} \right)g$. On the other hand, we have $E^{k} |_{{\left\{ {f,g} \right\}}} = \tilde{E}{^{k}}|_{{\left\{ {f,g} \right\}}}$. Therefore, we have fI(E ^k)g. (2) Otherwise, we consider SCF h over F ^k−1 such that h(R ^k−1) = f(R ^k−1) and h(R ^′k−1) = g(R ^′k−1) for all R ^′k−1 ∊ W(F ^k−1) − {R ^k−1},. Since F ^k is the set of all possible SCFs over F ^k−1, such a SCF h is in F ^k. By applying (1) we have fI(E ^k)h and gI(E ^k)h. Thus, we have fI(E ^k)g.

Finally we must show that the PCR E is also indifferent for any alternatives x, y ∊ X. However, it is easy from the ‘only if’ part of the ILC and the above fact that fI(E ¹)g for any f, g ∊ F ¹. ■

Lemma 2.

(Arrow [ 16 ]). If a SWF $f: W\left( A \right)^{n} \to W\left( A \right)$ satisfies WP and IIA, then there exists a dictator, where:

$$ {\text{WP: }}\forall S = \left( {S_{1} , \ldots ,S_{n} } \right) \in W\left( A \right)^{n} ,\forall a,b\,\epsilon\, A,\left[ {aP\left( {S_{i} } \right)b\,\forall i\,\epsilon\, N} \right] \to aP\left( {f\left( S \right)} \right)b $$

$$ {\text{IIA}}:\,\forall S,S^{\prime} \epsilon\,W\left( A\right)^{n},\forall a, b,\epsilon\,A,S_{i} |_{{\{ a,b\} }} | =S^{\prime}_{i} |_{{\{a,b\} }} \to f\left( S \right)|_{{\{ a,b\} }} =f\left( {S^{\prime}}\right)|_{{\{ a,b\}}} $$

A dictator is an individual i ∊ N such that for all S ∊ W(A) and for all a, b ∊ A, aP(S _i )b implies aP(f(S))b.

Proof of Proposition 2 [2]. Let E be a PCR that satisfies the ‘only if’ part of ILC, PWP, AF, and PIIA. Fix (R ⁰, R ¹, …, R ^K−1) ∊ W(X) × W(F ¹) × … × W(F ^K−1) and let A be a set such that ${\text{A}}\text{ := }\left\{ {F^{K} \left[ f \right]|f \in F^{K - 1} } \right\}$. By AF, we have a function G such that for all R ^K, E ^K−1(R ⁰, …, R ^K) = G(R ^K). Moreover, by PIIA, there exists a function G ^′:W(A)ⁿ → W(F ^K−1) such that $G\left( {R^{K} } \right) = G^{'} \left( {O\left( {R_{1}^{K} } \right),O\left( {R_{2}^{K} } \right), \ldots ,O\left( {R_{n}^{K} } \right)} \right)$ for all R ^K ∊ W(F ^K). Let us consider another function $\mu :W\left( {F^{K - 1} } \right) \to W\left( A \right)$ such that for all $\tilde{R}{^{{{\text{K}} - 1}}} \in W\left({F^{K - 1} }\right)$ and f, g ∊ F ^K−1, $f\tilde{R}{^{K - 1}} g$ if and only if $F^{K} \left[f \right]\mu \left({\tilde{R}{^{K - 1}}} \right)F^{K} \left[g \right]$. Construct a composite function $\nu \text{ := }\mu \bigcirc G^{'} :W\left( A \right)^{n} \to W\left( A \right)$. This is a SWF for the set A, and it is easy to see that our PWP and PIIA condition demands the WP and IIA for SWF ν. Therefore, by Lemma 2 we have a dictator j ∊ N (of SWF ν) such that for all S ∊ W(A) and for all F ^K[f], F ^K[g] ∊ A, if $F^{K} \left[ f \right]P\left( {O\left( {R_{j}^{K} } \right)} \right)F^{K} \left[ g \right]$, then fP(ν(S))g. By the way we have constructed μ, we have fP(E ^K−1)g. Since this argument does not depend on the value of R ⁰, R ¹, …, R ^K−1 or what f and g are, we can conclude that the set of axioms yield a vetoer over any pair in F ^K−1.

We must only show the level under K − 1. Take any level l ∊ {0, 1, …, K − 2} and any alternatives/procedures x, y ∊ F ^l. Assume that $F^{K} \left[ x \right]P\left( {O\left( {R_{j}^{k} } \right)} \right)F^{K} \left[ y \right]\,$. Take f ^′ ∊ F ^K−1[x] and g ∊ F ^K−1[y] such that $F^{K} \left[ {f^{'} } \right] \in G\left( {O\left( {R_{j}^{K} } \right), B_{x} } \right)$ and $F^{K} \left[ {g^{'} } \right] \in G\left( {O\left( {R_{j}^{K} } \right), B_{y} } \right)$, where B _x : = {F ^K[h]|f ∊ F ^K−1[x]} and B _y : = {F ^K[h]|f ∊ F ^K−1[y]}. Since $O\left( {R_{j}^{K} } \right)$ is a weak ordering over $2^{{F^{K} }}$, $G\left( {O\left( {R_{j}^{K} } \right),B_{w} } \right)\left( {w = x,y} \right)$ are non-empty and we can take such f′ and g′. Now, the definition of the operator O( ) and the assumption of F ^K[x]P(O(R))F ^K[y] together yield F ^K[f′]P(O(R))F ^K[g′]. From the above paragraph we get f′P(E ^K−1)g′. Finally, iterating the ‘only if’ part of ILC we get xE ^k y.■

Proof of Proposition 3 [1]. The counterexample showed in the proof of Proposition 1 also applies under D _C . ■

[2] Let us consider a SWF S:W(X)ⁿ → W(X) which satisfies the Pareto principle: for all preference profile of level 0 R ⁰ ∊ W(X), $\left[ {xP\left( {R_{i}^{0} } \right)y\;for \;all\; i \in N} \right]$ implies xP(S(R ⁰))y. Now we define PCR E _S such that (1) for all x, y ∊ X, xE ⁰ y if and only if xS(R ⁰)y and (2) for all k ∊ {1, 2, …, K} and f, g ∊ F ^k, fE ^k+1 g if and only if f(R ^k)E ^k g(R ^k). We will show that this E _S is actually a PCR and satisfies the ILC and PWP. The completeness of each E ^k _S (k = 0, 1, …, K) is obvious. To show they are transitive, suppose E ^k _S  ∊ W(F ^k). Take any procedures f, g, h ∊ F ^k+1 and assume fE ^k+1 g and gE ^k+1 h. By (2) we have f(R ^k)E ^k g(R ^k) and g(R ^k)E ^k h(R ^k). This implies f(R ^k)E ^k h(R ^k) by the transitivity of E ^k. By (2) once again we get fE ^k+1 h. Since E ⁰ ≡ S(R ⁰) is transitive, we have inductively that E ^k ∊ W(F ^k) for all k ∊ {0, 1, …, K}. Now we show that E _S satisfies the ILC and PWP, but the former is obvious because of (2). So we show PWP. Take any k ∊ {0, 1, …, K − 1} and f, g ∊ F ^k. Suppose $F^{l} \left[ f \right]P\left( {O\left( {R_{i}^{l} } \right)} \right)F^{l} \left[ g \right]$ for all l ∊ {k + 1, …, K}. Iterating the condition of extremely consequentialist, we have for all $l \in \left\{ {{\text{k}} + 1, \ldots ,K} \right\}$. Iterating the condition of extremely consequentialist, we have for all $i \in N$ $f\left({R^{k - 1} } \right)P(R_{i}^{k - 1} )g(R^{k - 1} ),f\left({R^{k - 1}} \right)(R^{k - 2})P(R_{i}^{k - 2})g(R^{k - 1})(R^{k - 2}), \ldots,xP(R_{i}^{0})y,$ where $f \in F^{k} [x]$ and ${\text{g}} \in {\text{F}}^{\text{k}} \left[ {\text{y}} \right].$ The Pareto prinicple of ${\text{E}}^{ 0} \equiv {\text{S}}\left( {{\text{R}}^{0} } \right)$ implies ${\text{xP}}\left( {{\text{E}}^{ 0} } \right){\text{y}}$. Iteration of the contraposition of the ‘only if’ part of the ILC gives $fP(E^{k} )g$.