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Arrow's Theorem via linear programming

Now for the linear programming proof of Arrow's Impossibility Theorem. We use the notation of previous posts. This argument is more or less lifted from Vohra et al. apart from the statement and proof of Proposition 1 below and some explanatory remarks. We take \(\Omega =\Sigma_\Gamma.\) Since all strict linear orderings of \(\Gamma\) occur, for any \(x, y, z\in\Gamma,\) we have the decisiveness inequalities \[ \begin{aligned} d_S(x, y)\le d_S (x, z)\\ d_S(z, x)\le d_S(y, x) \end{aligned} \] Recall that these inequalities followed from the existence of preference relations \(p, q\in\Omega\) such that \(x p y p z\) and \(z q x q y.\) Vohra et al. use a similar argument starting with the decisiveness inequalities, using \(\Omega=\Sigma_\Gamma\) to obtain the fact that for \(x,y,z,u\in\Gamma,\) \[d_S(x,y)\le d_S(y,z)\le d_S(z,u).\] The point is to replace expressions \(d_S(x,y)\) occur with expressions \(d_S\) in which variables are suppressed. There is another combinatorial fac…

Decisiveness VI

There is a tradition of imprecision in this subject.1 First, some notation from previous posts.

Suppose that \(f:\Omega^n\rightarrow \Sigma_\Gamma\) is an Arrovian social welfare function. Define the map \(\Delta:\Gamma^2\times\Omega^n\rightarrow 2^E\) by \[ \Delta(x, y)(P)=\lbrace j \in E: x p_j y\rbrace. \] Definition. The set \(S\subseteq E\) is weakly decisive for \(x\) over \(y\) with respect to the Arrovian social welfare function \(f\) if \[ \begin{aligned} (\forall P\in\Omega^n)(S=\Delta(x,y)(P) \Rightarrow x f(P) y) \end{aligned} \] Definition. The set \(S\subseteq E\) is decisive for \(x\) over \(y\) with respect to the ASWF \(f\) if \[ \begin{aligned} (\exists P\in\Omega^n)(S=\Delta(x,y)(P)\land x f(P) y) \end{aligned} \] The following statement is implicit in Vohra et al.2,3
Proposition. Suppose that \((x,y)\in\Gamma^2\) is a nontrivial pair (in \(\Omega\)). Let \(S\subseteq E\) be nonempty. If \(S\) is decisive for \(x\) over \(y\), then \(S\) is weakly decisive for \(x\) over \(y\).
Proof. Suppose there exists \(Q\in\Omega^n\) with \(S=\Delta (x, y)(Q) \land x f (Q) y\). Now suppose that \(S=\Delta (x, y)(P)\). Then \(\forall j\in E, x q_j y \Leftrightarrow x p_j y\). Since \(x f (Q) y\), by independence of irrelevant alternatives, \(x f (P) y\). QED.
For nontrivial \((x, y)\in\Gamma^2\) we can unambiguously define "\(S\) is decisive for \(x\) over \(y\)" by the expression "\(S=\Delta (x, y)(P)\land x f (P) y\)." The proposition shows that this condition is independent of the choice of the preference profile \(P\in\Omega^n\) such that \(S=\Delta(x,y)(P)\).

The linear program associated to an Arrovian social welfare function

We are given a collection \(\Gamma\) of at least three alternatives, denoted \(x,y,z\), a set \(\Omega\) of admissible preference relations, where \(\Omega\subseteq\Sigma_\Gamma\) is the set of preference relations, and an Arrovian soclal welfare function \(f:\Omega^n\rightarrow\Sigma_\Gamma\). From \(f\) we define a family \(\prod\nolimits_f\) of \(01\)-valued variables. \[ \begin{aligned} \prod\nolimits_f := \left\lbrace d_S(x,y)\right\rbrace_{S\subseteq E,(x, y)\in\Gamma^2} \end{aligned} \] satisfying the conditions A, B and C below.

A. If \((x,y)\in\Gamma^2\) is a nontrivial pair, then \[ d_S(x,y) = \begin{cases} 1,&\textrm{if}\,\forall P\in\Omega^n, S=\Delta(x,y)(P)\Rightarrow x f(P) y;\\ 0,&\textrm{otherwise.} \end{cases} \] B. If \((x,y)\in\Gamma^2\) is a trivial pair, then \[ d_S(x,y) = \begin{cases} 1,& \textrm{if}\,S\not=\emptyset;\\ 0,&\textrm{otherwise.} \end{cases} \] C. If \((x,x)\in\Delta_\Gamma\), then for each \(S\subseteq E\), \(d_S(x,x)=0.\) These variables satisfy the following constraints.

0. Irreflexive. \(\forall (x,x)\in\Delta_\Gamma,S\subseteq E, d_S(x,x)=0.\)
1. Unanimity. \(\forall (x,y)\in\Gamma^2\setminus\Delta_\Gamma, d_E(x,y)=1\)
2. Independence of Irrelevant Alternatives. \(\forall (x,y)\in\Gamma^2\setminus\Delta_\Gamma\), \[ d_S(x,y)+ d_{E\setminus S}(y,x)=1. \]
3. Transitivity. \(\forall (x,y),(y,z),(z,x)\in\Gamma^2\setminus\Delta_\Gamma,\) \[ d_{\Delta(x,y)(P)}(x,y) +d_{\Delta(y,z)(P)}(y, z)+d_{\Delta(z, x)(P)}(z, x)\le 2. \]

Now suppose that \(\prod\) is a collection defined by the satisfying B and C above, and in addition the constraints 1--4. Does there exist an Arrovian social welfare function \(f:\Omega^n\rightarrow\Sigma_\Gamma\) such that \(\prod = \prod\nolimits_f\)? How would we go about defining such an \(f\)? What we can do is define a map \[ \hat{f}:\Omega^n\rightarrow 2^{\Gamma^2\setminus\Delta_\Gamma} \] and show that this induces a map \(f:\Omega^n\rightarrow\Sigma_\Gamma\) by restricting the codomain to the set \(\Sigma_\Gamma\) of finite total orders on \(\Gamma\). Let \(P\in\Omega^n, (x,y)\in\Gamma^2\) and set \(S=\Delta(x,y)(P)\). Define \[ \begin{aligned} x \hat{f}(P) y \Leftrightarrow d_S(x,y)=1. \end{aligned} \] The above formula defines a relation \(\hat{f}(P)\subseteq \Gamma^2\). The relation \(\hat{f}(P)\) is irreflexive by constraint 1; thus \(\hat{f}(P)\subseteq \Gamma^2\setminus\Delta_\Gamma\). Note that by independence of irrelevant alternatives (constraint 3), for any pair \((x,y)\in\Gamma^2\setminus\Delta_\Gamma\), either \(x \hat{f}(P) y\) or \(y \hat{f}(P) x\), but not both. Hence the relation \(\hat{f}(P)\) is connected and asymmetric. The question now is whether \(\hat{f}(P)\) is transitive. Suppose otherwise; then as the relation \(\hat{f}(P)\) is connected, there exists a condorcet cycle \[ x \hat{f}(P)y, y\hat{f}(p)z, z\hat{f}(P) x. \] Hence there exist sets \(R,S,T\subseteq E\) such that \[ d_R(x,y) = d_S(y,z) = d_T(z,x) = 1, \] where by definition, \(R=\Delta(x,y)(P)\), \(S = \Delta(y,z)(P)\) and \(T=\Delta(z,x)(P)\). But that implies \[ d_{\Delta(x,y)(P)}(x,y) +d_{\Delta(y,z)(P)}(y, z)+d_{\Delta(z, x)(P)}(z, x)=3 \] contradicting constraint 4, transitivity. Hence the relation \(\hat{f}(P)\) is a strict total order relation. This defines the required map \(f:\Omega^n\rightarrow \Sigma_\Gamma\). It is straightforward to check that \(f\) satisfies unanimity and independence of independent alternatives.

Oh all right, I'll check. First observe that for \(x,y\in\Gamma\), \(x \hat{f}(P) y \Leftrightarrow x f(P) y\). For unanimity, suppose that \(P\in\Omega^n\) and that \((\forall j\in E)(x p_j y)\). Then \(E=\Delta(x,y)(P)\), and by constraint 2 and the defining relation above, \(x f(P) y\). For independence of irrelevant alternatives, let \(P,Q\in\Omega^n\), and suppose that for each \(j\in E, x p_j y \Leftrightarrow x q_j y\). We may assume that \((x,y)\) is a nontrivial pair. Let \(S = \Delta(x,y)(P) = \Delta(x,y)(Q)\). By constraint 2, \(d_S(x,y)=1\) or \(d_{E\setminus S}(x,y) = 1\), but not both. If \(d_S(x,y)=1\), then by the above defining relation, \(x f(P) y \land x f(Q)y\). If \(d_{E\setminus S}(x,y) = 1\), then as \(E\setminus S= \Delta(y,x)(P) = \Delta(y,x)(Q)\), we have \(y f(P) x \land y f(Q) x\). Thus \[ (\Delta(x,y)(P) =\Delta(x,y)(Q))\Rightarrow (x f(P) y\Leftrightarrow x f(Q) y). \] We conclude, by definition of \(f\) and the remarks following, that \(\prod = \prod\nolimits_f\).

Citations

1. Wiedijk, Freek (2009). "Formalizing Arrow's Theorem." Sadhana 34 (1): 193--220.
2. Jay Sethuraman, Teo Chung Piaw, and Rakesh V. Vohra. Integer Programming and Arrovian Social Welfare Functions. Mathematics of Operations Research Vol. 28, No. 2, May 2003, pp. 309–326.
3. Vohra, Rakesh V. "2.1 The Integer Program." Mechanism Design: A Linear Programming Approach. Cambridge: Cambridge UP, 2011. 9-10. Print

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