Global IIA, Sen’s Impossibility Theorem and State-Salient Decision Rules

1. Introduction

The motivation for the framework of this paper i.e., decision making with state (or criteria) dependent strict rankings of alternatives, can be found in Lahiri (2019). Our mathematical framework is the same as that of mathematical voting theory originating in the work of Pattanaik (1970), although our interpretation (as in Lahiri (2019)) is different.

In this paper we provide a simple proof of an axiomatic characterizations of state-salient decision rules on the domain of all strict preference profiles, which in the context of the mathematically identical framework of voting theory, due to Pattanaik (1970) and then Denicolo (1985), is about a voting rule being dictatorial if it satisfies the Pareto property and Global Independence of Irrelevant Alternatives. Our proof is different from the one available in Denicolo (1985) as well as the one available in Lahiri (2003), the latter being applicable in the same context as the one that this paper is concerned with. Our proof here uses Sen’s Impossibility Theorem, that is available along with a proof in chapter 8 of Kelly (1088), and a significant extract from the proof of theorem 1 in Lahiri (2023). The extract, proves a result that is stated as lemma 1 in this paper.

In the rest of the paper, we will refer to a “state of nature” as a “son” “states of nature” as “sons”.

An alternative is said to be Pareto-dominated if it is ranked worse than another alternative in all strictly ranked sons. A winner selection rule is said to satisfy Pareto property if at no preference profile in its domain, a Pareto-dominated alternative is a winner.

Given a winner selection rule and a pair of distinct alternatives, a son is said to be decisive for the pair, if at every preference profile in the domain of the winner selection rule, the alternative that is ranked worse among the pair in the son, is not chosen.

A winner selection rule is said to be mildly state-salient if there two distinct sons such that each is decisive for a pair of distinct alternatives, where the two pairs of distinct alternatives may or may not be disjoint.

Sen’s Impossibility Theorem (see chapter 8 of Kelly (1988)) says that there does not exist any winner selection rule on a domain satisfying “the strict domain condition” (i.e., containing all strict preference profiles) that is mildly state salient and satisfies the Pareto property.

A winner selection rule is said to satisfy Global Independence of Irrelevant Alternatives (as in Denicolo (1985)), if given two distinct alternatives and two preference profiles with the preference relation between the two alternatives being the same in all sons for both preference profiles, then an implication of one of the two alternatives being a winner and the other not being a winner at one preference profile is that the latter is not a winner at the other preference profile.

Our lemma 1 says that if any winner selection rule on the set of all strict preference profiles, satisfies Pareto property and Global Independence of Irrelevant Alternatives, then given any alternative there exists a son which is decisive for all pairs of distinct alternatives, where no pair contains the given alternative.

An immediate consequence of these Sen’s Impossibility Theorem and Lemma 1, is that any winner selection rule on the domain of all strict preference profiles satisfies Pareto property and Global Independence of Irrelevant Alternatives if and only if it is a state-salient decision rule.

Our perception is that the approach adopted here is superior from a pedagogical point of view to the self-contained proof of theorem 1 available in Lahiri (2023) for the following reasons:

(a) Sen’s Impossibility Theorem is a “must read” classic which has a very simple and elegant proof. Further, the proof does not use any result that is additionally required to prove our main result here.

(b) Steps 1 and 2 of the proof of theorem 1 in Lahiri (2023), which is what comprises the proof of lemma 1, is sufficiently demanding (taxing) for an average reader to ask for a breather after that.

2. Framework, Definitions and Assumptions

Suppose X is a non-empty finite set of alternatives containing at least three alternatives and N = {1,…,n} is a set of ‘n’ equiprobable sons for some positive integer n ≥ 2. Let us recall here a nomenclature we introduced in the introductory sentence: “son” is an abbreviation of “state of nature” and hence, we refer to “states of nature” as “sons.”

A son j corresponds to the elementary event {j}.

A strict preference relation on X, is a horizontal list of the alternatives in X such that no two alternatives in the horizontal list are the same.

The interpretation of a strict preference relation is that if x and y are two distinct alternatives, then x is strictly preferred to y at the strict preference relation if and only if x is to the left of y.

For example if X = {x₁,…,x_m} then for any one-to-one function π from {1,…,m} onto itself, x_π(1) x_π(2)… x_π(m) means x_π(i) is “strictly preferred” to x_π(j) if and only if i < j (i.e., x_π(i) is to the left of x_π(j)).

Sometimes it useful to include the dumb preference relation, i.e., the preference relation where all alternatives are equivalent to each other.

Given a preference relation R, strict or dumb R and an alternative x∈X, let rk(x,R) = 1+ cardinality{y∈X\{x}| y is “strictly preferred” to X}. rk(x,R) is said to be the rank of x at R.

If y is strictly preferred to x at a preference relation, then we also say that y is ranked better than x at the preference relation or x is ranked worse than y at the preference relation

Thus, if R is the dumb preference relation, then rk(x,R) = 1 for all x∈X. Further, if X = {x,y,z,…} has more than three alternatives then the strict preference relation R = xyz… says that rk(x,R) = 1, rk(y,R) = 2, rk(z,R) = 3 and for w∈X\{x,y,z}, x, y and z are all strictly preferred to w at R with rk(w,R) > 3.

A preference profile on X is an ordered n-tuple of preference relations (R₁,…, R_n) which are either strict or dumb, such that for each i∈N, the i^th coordinate of the n-tuple represents the preference relation in the i^th son.

It is customary to represent a preference profile by R_N.

R_N = ((R₁, n₁),…, (R_K, n_K)) denotes the preference profile, where for a positive integer K, n_k for k∈{1,…,K} is a positive integer satisfying $\sum _{\mathrm{k}=1}^{\mathrm{K}}{\mathrm{n}}_{\mathrm{k}}$ = n and there is a partition {E₁,…, Ek_,…, E_K} of N such that for each k∈{1,…,K}, E_k is an event and R_k is the preference relation, at all sons in E_k. In this case for each k∈{1,…,K}, the event E_k occurs with probability ${\displaystyle \frac{n_k}{n}}$. When n_k = 1 for all k∈{1,…,K}, then clearly K = n, with E_k being a singleton for all k∈N and then instead of writing ((R₁, 1),…, (R_n, 1)) we express the preference relation as (R₁,…, R_n).

Another possible representation is R_N = ((R₁, E₁),…, (R_K, E_K)), where for a positive integer K, {E₁,…, E_k,…, E_K} is a partition of N, where for each k∈{1,…,K}, E_k is an event and R_k is the preference relation, at all sons in E_k. In this case for each k∈{1,…,K}, the event E_k occurs with probability ${\displaystyle \frac{n_k}{n}}$, n_k being the cardinality of E_k. If E_k is a singleton for all k∈{1,…,K}, then clearly K = n, and we express the preference relation as (R₁,…, R_n).

Given a preference profile R_N, a son in which the preference relation is “not dumb” is said to be a strictly-ranked son at R_N.

A strict preference profile on X is a preference profile such that all sons are strictly ranked at the preference profile.

Let ${\mathcal{L}}^{\mathrm{N}}$ denote the set of all strict preference profiles.

A non-trivial preference profile on X is a preference profile which has at least one strictly-ranked son at the preference profile.

Note 1: In what follows we restrict our attention to non-trivial preference profiles.

A domain denoted by $\mathcal{D}$ is a non-empty subset of the set of all non-trivial preference profiles.

A domain $\mathcal{D}$ is said to satisfy the strict domain condition if it contains all strict preference profiles.

A winner selection rule on a domain $\mathcal{D}$ is a function f from $\mathcal{D}$ to Ψ(X), where Ψ(X) is the set of all non-empty subsets of X.

The set assigned by a winner selection rule to a preference profile in its domain is said to be the winner set at the preference profile (assigned by the winner selection rule). A member of a winner set at a preference profile is said to be a winner (at the preference profile).

Given a winner selection rule and two-distinct alternatives x,y, a son i is said to be decisive between x and y if at every preference profile R_N in its domain where i is a strictly-ranked son: [xR_iy implies y is not a winner] & [yR_ix implies x is not a winner].

A son is said to be locally decisive for a winner selection rule if there is pair of distinct alternatives between which it is decisive.

A winner selection rule is said to satisfy Pareto property if for all x,y∈X with x≠y and R_N in its domain: [xR_iy for all i∈N] implies [y is not a winner at R_N].

Since we have defined a domain as a non-empty subset of the set of all non-trivial preference profiles, the statement [xR_iy for all i∈N] in the definition of Pareto property implies that for at least one i∈N, which is a “strictly ranked son” so the R_i is a strict preference profile and hence x is strictly preferred to y at R_i.

Note 2: It is not possible for a winner selection rule on a domain satisfying the strict domain condition to satisfy the Pareto property and have two distinct sons i and j that are both decisive between the same pair of distinct alternatives. For if at a strict preference profile that ranks x and y in the first two ranks at all sons, with x first and y in i and y first and x second in j, then the winner set at such a profile would have to be the empty set, contrary to the definition of a winner selection rule. However, it is entirely possible that the two distinct pairs of alternatives are not disjoint.

The following definition can be found in Lahiri (2019).

A winner selection rule said to be a state-salient decision rule (S-SDR), if there is a son that is decisive between every pair of distinct alternatives for the winner selection rule.

It follows immediately from Note 2 and the definition of an S-SDR that if an S-SDR satisfies Pareto property on a domain that satisfies the strict domain condition, then a son that is decisive must be unique.

The content of the following assumption is available in Denicolo (1985).

A winner selection rule f is said to satisfy Global Independence of Irrelevant Alternatives (GIIA) if for all R_N, ${\mathrm{R}}_{\mathrm{N}}^{\mathrm{\text{'}}}$ in its domain and x,y∈X with x ≠ y: [R_i|{x,y} = ${\mathrm{R}}_{\mathrm{i}}^{\mathrm{\text{'}}}$|{x,y} for all i∈N, x∈f(R_N), y∉f(R_N)] implies [y is not a winner at ${\mathrm{R}}_{\mathrm{N}}^{\mathrm{\text{'}}}$].

The following definition is what Chapter 8 of Kelly (1988) begins with.

A winner selection rule f is said to be mildly state salient if there are at least “two distinct sons” that are locally decisive.

Note 3: If X = {x,y,z,w}, son 1 is decisive between x and y, son 2 is decisive between z and w and R_N is any preference profile satisfying xR₁y and zR₂w, where both 1 and 2 are strictly ranked sons, then the winner set at R_N must be a non-empty subset of {x,z}. If x = z, so that X = {x,y,w}, then the winner set at R_N is {x}.

3. Sen’s Impossibility Theorem

Recall that a winner selection rule is mildly state salient, if there are at least two distinct sons that are locally decisive.

The following theorem along with a proof can be found in Chapter 8 of Kelly (1988).

Sen’s Impossibility Theorem: There does not exist any winner selection rule on a domain satisfying the strict domain condition, that is mildly state salient and satisfies the Pareto Property.

For the sake of completeness, the proof in Kelly (1988) is (almost as it is) reproduced in an appendix of this paper.

An immediate consequence of Sen’s Impossibility theorem is the following.

Corollary of Sen’s Impossibility Theorem: If for a winner selection rule that satisfies Pareto Property on a domain satisfying the strict domain condition, there exists a son that is decisive between a pair of distinct alternatives, then there does not exist any other son that is also decisive for one or more pairs of distinct alternatives.

4. Axiomatic Characterization of State-Salient Winner Selection Rules

What follows till the end of the proof of lemma 1, is an extract from the proof of theorem 1 in Lahiri (2023).

For x∈X, let ${\mathfrak{D}}^{\mathrm{N}}$(x) = {R_N∈${\mathcal{L}}^{\mathrm{N}}$| for all i∈N, rk(x,R_i)∈{1,m}}.

Lemma 1: For a winner selection rule on ${\mathcal{L}}^{\mathrm{N}}$ satisfying Pareto property and GIIA, for each x∈X, there exists j(x)∈N such j(x) is decisive for all pairs of distinct alternatives in X\{x}.

Proof: Suppose f is a winner selection rule on ${\mathcal{L}}^{\mathrm{N}}$ satisfying Pareto property and GIIA.

Step 1: Let x∈X.

Since by Pareto property, rk(x,R_i) = 1 for all i∈N implies f(R_N) = {x}, let j(x) = min{j∈N| there exists R_N∈${\mathfrak{D}}^{\mathrm{N}}$(x) satisfying{i|rk(x,R_i) = 1} = {i∈N|i ≤ j} and f(R_N) = {x}}.

By GIIA, if for R_N∈${\mathcal{L}}^{\mathrm{N}}$ and w∈X\{x}, we have {i∈N|xR_iw} = {i∈N| i ≤ j(x)}, then, w∉f(R_N).

Let y,z∈X\{x} with y≠z and R_N∈${\mathcal{L}}^{\mathrm{N}}$ such that {i|xR_iz} = {i∈N| i ≤ j(x)}, zR_iy for all i∈N and xR_jy for all i ≤ j(x), the relationship between x and y being otherwise arbitrary. Thus, xR_izR_iy for all i ≤ j(x) and zR_ix for all i > j(x) if j(x) < n. Further, suppose rk(x,R_i), rk(y,R_i), rk(z,R_i) ∈{1,2,3} for all i ∈N.

By Pareto property, f(R_N) ⊂{x,y,z} and by Pareto property (once again) y∉f(R_N).

By GIIA, z∉f(R_N).

Thus f(R_N) = {x}.

By GIIA, for all R_N∈${\mathcal{L}}^{\mathrm{N}}$ and y∈X\{x}, xR_jy for all j ≤ j(x) implies y∉f(R_N).

Thus, for all x∈X, there exists j(x)∈N such that for all R_N∈${\mathcal{L}}^{\mathrm{N}}$ and y∈X\{x}, xR_jy for all j ≤ j(x) implies y∉f(R_N).

Step 2: Let x∈X and y,z∈X\{x} with y ≠ z and let R_N∈${\mathcal{L}}^{\mathrm{N}}$ be such that xR_iz, xR_iy, relationship between z and y arbitrary for all i < j(x) (if j(x) > 1), R_j(x) = zxy…, zR_ix, for all i > j(x), relationship between x, y and y,z arbitrary for i > j(x). Further, suppose rk(x,R_i), rk(y,R_i), rk(z,R_i) ∈{1,2,3} for all i∈N.

By Pareto property, f(R_N) ⊂{x,y,z}.

By the definition of j(x), and GIIA combined with xR_iz for all i < j(x), zR_ix for all i ≥ j(x), we get that f(R_N) ≠ {x}.

Since, xR_iy for all i ≤ j(x) by the concluding statement of Step 1, we get that y∉f(R_N).

Thus, z∈f(R_N).

By GIIA, for all y,z∈X\{x} with y ≠ z: [R_N∈${\mathcal{L}}^{\mathrm{N}}$ and zR_j(x)y] implies [y∉f(R_N)].

Thus, for each x∈X, there exists j(x)∈N such for all y,z∈X\{x} with y ≠ z: [${\mathrm{R}}_{\mathrm{N}}^{"}$∈${\mathcal{L}}^{\mathrm{N}}$ and z${\mathrm{R}}_{\mathrm{j}\left(\mathrm{x}\right)}^{"}$y] implies [y∉f(${\mathrm{R}}_{\mathrm{N}}^{"}$)]. Q.E.D.

An immediate consequence of Sen’s Impossibility Theorem and Lemma 1 is the following result which is available with a different proof in Lahiri (2023).

Theorem 1: A CF on ${\mathcal{L}}^{\mathrm{N}}$ satisfies Pareto property and GIIA if and only if it is an S-SDR.

Proof: Let f be a winner selection rule on ${\mathcal{L}}^{\mathrm{N}}$.

If f is an S-SDR, then it is easy to see that it satisfies Pareto property and GIIA.

Hence suppose f satisfies Pareto property and GIIA.

By Lemma 1, for each x∈X, there exists j(x)∈N such j(x) is decisive for all pairs of distinct alternatives in X\{x}.

Let x, y, z be any three distinct alternatives in X.

Thus, j(x) is decisive for all pairs of distinct alternatives in X\{x}, j(y) is decisive for all pairs of distinct alternatives in X\{y} and j(z) is decisive for all pairs of distinct alternatives in X\{z}.

By Corollary of Sen’s Impossibility Theorem, j(x) = j(y) = j(z) = j (say), and j is decisive between any pair of distinct alternatives.

Thus, f is an S-SDR. Q.E.D.