Arrow’s decisive coalitions

Abstract

In his classic monograph, Social Choice and Individual Values, Arrow introduced the notion of a decisive coalition of voters as part of his mathematical framework for social choice theory. The subsequent literature on Arrow’s Impossibility Theorem has shown the importance for social choice theory of reasoning about coalitions of voters with different grades of decisiveness. The goal of this paper is a fine-grained analysis of reasoning about decisive coalitions, formalizing how the concept of a decisive coalition gives rise to a social choice theoretic language and logic all of its own. We show that given Arrow’s axioms of the Independence of Irrelevant Alternatives and Universal Domain, rationality postulates for social preference correspond to strong axioms about decisive coalitions. We demonstrate this correspondence with results of a kind familiar in economics—representation theorems—as well as results of a kind coming from mathematical logic—completeness theorems. We present a complete logic for reasoning about decisive coalitions, along with formal proofs of Arrow’s and Wilson’s theorems. In addition, we prove the correctness of an algorithm for calculating, given any social rationality postulate of a certain form in the language of binary preference, the corresponding axiom in the language of decisive coalitions. These results suggest for social choice theory new perspectives and tools from logic.

Appendix

In this Appendix, we sketch the proofs of the completeness theorems stated in Theorems 4.8.2, 4.8.4, and 7.4.2. A similar approach can be used to prove completeness theorems for logics that include inverse decisiveness as in Sect. 7.4.

For a logic $\mathbf {L}$, a set $\Sigma$ of formulas is $\mathbf {L}$-inconsistent if and only if there are $\sigma _1,\dots ,\sigma _n\in \Sigma$ such that $\vdash _\mathbf {L} (\sigma _1\wedge \dots \wedge \sigma _n)\rightarrow \bot$, where $\bot$ is any formula of the form $\varphi \wedge \lnot \varphi$. Otherwise $\Sigma$ is $\mathbf {L}$-consistent. We say that a formula $\varphi$ is $\mathbf {L}$-consistent if and only if $\{\varphi \}$ is $\mathbf {L}$-consistent. Finally, a set $\Sigma$ of formulas is maximally$\mathbf {L}$-consistent if and only if for every set $\Delta$ of formulas, $\Sigma \subsetneq \Delta$ implies that $\Delta$ is $\mathbf {L}$-inconsistent.

To say that a logic $\mathbf {L}$ is complete with respect to a class K of CCRs is to say that for any formula $\psi$, if $\psi$ is true of all CCRs from K, then $\psi$ is a theorem of $\mathbf {L}$. Equivalently, if $\psi$ is not a theorem of $\mathbf {L}$, then there is a CCR from K of which $\psi$ is not true. Since by propositional logic, we have $\not \vdash _\mathbf {L} \psi$ if and only if $\not \vdash _\mathbf {L} \lnot \psi \rightarrow \bot$, the formula $\psi$ not being a theorem of $\mathbf {L}$ is equivalent to $\lnot \psi$ being $\mathbf {L}$-consistent. Moreover, there being a CCR from K of which $\psi$ is not true is equivalent to there being a CCR from K of which $\lnot \psi$ is true. Thus, to prove that $\mathbf {L}$ is complete with respect to K, it suffices to show that for any formula $\varphi$, if $\varphi$ is $\mathbf {L}$-consistent, then there is a CCR from K of which $\varphi$ is true.

We will give a fairly detailed proof of Theorem 7.4.2, where $\mathbf {L}$ is $\overline{\widehat{\mathbf {T}}}$ (resp. $\overline{\widehat{\mathbf {W}}}$) and K is the class of CCRs satisfying LD, IIA, and TR (resp. FR). The proof where $\mathbf {L}$ is $\widehat{\mathbf {T}}$ or $\overline{\mathbf {T}}$ (resp. $\widehat{\mathbf {W}}$ or $\overline{\mathbf {W}}$) and K is the class of CCRs satisfying UD, IIA, and TR (resp. FR) is quite a bit simpler, as we will explain at the end.

In what follows by ‘consistent’ we mean ‘$\overline{\widehat{\mathbf {T}}}$-consistent’ or ‘$\overline{\widehat{\mathbf {W}}}$-consistent’, and by ‘$\vdash$’ we mean ‘$\vdash _{\overline{\widehat{\mathbf {T}}}}$’ or ‘$\vdash _{\overline{\widehat{\mathbf {W}}}}$’, depending on which case the reader has in mind. The proof is the same except at one point that we will flag. Indeed, the proof would be essentially the same for still other logics based on other rationality postulates (recall Sect. 8).

Suppose $\varphi$ is consistent. Let $T_1$ be the set of all terms generated by the coalition labels in $\mathsf {Coal}(\varphi )=\{a\in \mathsf {Coal}\mid a\hbox { appears in }\varphi \}$ using −, $\sqcap$, and $\sqcup$ as in Definition 4.1. Define an equivalence relation E on $T_1$ by: sEt if and only if $s\equiv t$ is a valid equation of Boolean algebra. Let $T_1/E$ be the quotient of $T_1$ by E. It follows that $T_1/E$ is a Boolean algebra in which the complement of [t] is $[-t]$, and the meet of [s] and [t] is $[s\sqcap t]$. Since $T_1$ is the set of terms generated by $\mathsf {Coal}(\varphi )$, it follows that the Boolean algebra $T_1/E$ is generated by the set of equivalence classes of elements of $\mathsf {Coal}(\varphi )$. Since the set of such equivalence classes is finite, $T_1/E$ is finite by the well-known fact that any finitely generated Boolean algebra is finite (see, e.g., Givant and Halmos 2009, p. 82). For each $x,y\in \mathsf {Alt}$ and $[t]\in T_1/E$, pick a coalition label $c_{x,y,[t]}\in \mathsf {Coal}\setminus \mathsf {Coal}(\varphi )$ such that if $\langle x,y,[t]\rangle \ne \langle x',y',[t']\rangle$, then $c_{x,y,[t]}\ne c_{x',y',[t']}$. Since $\mathsf {Alt}$ and $T_1/E$ are finite, only finitely many coalitions labels $c_{x,y,[t]}$ are needed. Crucially, $c_{x,y,[t]}$ does not appear in $\varphi$ or t. Now define

$$\begin{aligned} \Theta = \{(t\sqsubseteq c_{x,y,[t]})\wedge (\lnot \overline{D}_{x>y}(t)\rightarrow \lnot \widehat{D}_{x>y}(c_{x,y,[t]}))\mid t\in T_1,x,y\in \mathsf {Alt}\}. \end{aligned}$$

We claim that the set $\{\varphi \}\cup \Theta$ is consistent. If not, then there are $\theta _1,\dots ,\theta _{n+1}\in \Theta$ ($n\ge 0$) such that $\{\varphi ,\theta _1,\dots ,\theta _{n+1}\}$ is inconsistent, but no proper subset is inconsistent. Thus, $\vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _{n+1})\rightarrow \bot$ and hence $\vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n)\rightarrow \lnot \theta _{n+1}$ by propositional logic. The formula $\theta _{n+1}$ is $(t\sqsubseteq c_{x,y,[t]})\wedge (\lnot \overline{D}_{x>y}(t)\rightarrow \lnot \widehat{D}_{x>y}(c_{x,y,[t]}))$ for some $t\in T_1, x,y\in \mathsf {Alt}$. Thus, by propositional logic we have:

$$\begin{aligned}&\quad \,\,\,\, \vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n)\rightarrow \lnot ((t\sqsubseteq c_{x,y,[t]})\wedge (\lnot \overline{D}_{x>y}(t)\rightarrow \lnot \widehat{D}_{x>y}(c_{x,y,[t]}))\\&\Rightarrow \,\, \vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n)\rightarrow ((t\sqsubseteq c_{x,y,[t]})\rightarrow \lnot (\lnot \overline{D}_{x>y}(t)\rightarrow \lnot \widehat{D}_{x>y}(c_{x,y,[t]})) \\&\Rightarrow \,\, \vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n)\rightarrow ((t\sqsubseteq c_{x,y,[t]})\rightarrow (\lnot \overline{D}_{x>y}(t)\wedge \widehat{D}_{x>y}(c_{x,y,[t]})). \end{aligned}$$

Not only does $c_{x,y,[t]}$ not appear in $\varphi$ or t, but also $c_{x,y,[t]}$ does not appear in $\theta _1,\dots ,\theta _n$. Toward a contradiction, suppose that $c_{x,y,[t]}$ does appear in some $\theta _i$, so $\theta _i$ is the formula $(s\sqsubseteq c_{x,y,[t]})\wedge (\lnot \overline{D}_{x>y}(s)\rightarrow \lnot \widehat{D}_{x>y}(c_{x,y,[t]}))$ for some term $s\in [t]$. As an axiom of our logic, we have $s\equiv t \rightarrow (\theta _i [s/s]\leftrightarrow \theta _i[t/s])$, which is the same as $s\equiv t\rightarrow (\theta _i\leftrightarrow \theta _{n+1})$. Since $s\in [t]$, $s\equiv t$ is also an axiom of our logic, which with the previous step implies that $\theta _i\leftrightarrow \theta _{n+1}$ is a theorem of our logic. But then from the inconsistency of $\{\theta _1,\dots ,\theta _{n+1}\}$, its proper subset $\{\theta _1,\dots ,\theta _{n+1}\}\setminus \{\theta _i\}$ is also inconsistent, contradicting our assumption that no proper subset is inconsistent. Thus, we conclude that $c_{x,y,[t]}$ does not appear in $\theta _1,\dots ,\theta _n$.

On the one hand, since $\vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n)\rightarrow ((t\sqsubseteq c_{x,y,[t]})\rightarrow \widehat{D}_{x>y}(c_{x,y,[t]}))$, and $c_{x,y,[t]}$ does not appear in $\varphi \wedge \theta _1\wedge \dots \wedge \theta _n$ or t, it follows by the $\overline{D}$-generalization rule (Sect. 7.2) that

$$\begin{aligned} \vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n) \rightarrow \overline{D}_{x>y}(t). \end{aligned}$$

(1)

On the other hand, since

$$\begin{aligned}&\quad \,\,\,\, \vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n)\rightarrow ((t\sqsubseteq c_{x,y,[t]})\rightarrow \lnot \overline{D}_{x>y}(t))\nonumber \\&\Rightarrow \,\, \vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n \wedge \overline{D}_{x>y}(t))\rightarrow \lnot (t\sqsubseteq c_{x,y,[t]}), \end{aligned}$$

(2)

and $c_{x,y,[t]}$ does not appear in $\varphi \wedge \theta _1\wedge \dots \wedge \theta _n\wedge \overline{D}_{x>y}(t)$, we can apply the substitution rule to replace $c_{x,y,[t]}$ in (2) by 1 to obtain $\vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n \wedge \overline{D}_{x>y}(t))\rightarrow \lnot (t\sqsubseteq 1)$. But then since $t\sqsubseteq 1$ is a theorem of our logic, by propositional logic we obtain $\vdash \lnot (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n \wedge \overline{D}_{x>y}(t))$ and then $\vdash (\varphi \wedge \theta _1\wedge \dots \wedge \theta _n)\rightarrow \lnot \overline{D}_{x>y}(t)$. It follows, given (1), that $\{\varphi ,\theta _1,\dots ,\theta _n\}$ is inconsistent, contradicting the fact that no proper subset of $\{\varphi ,\theta _1,\dots ,\theta _{n+1}\}$ is inconsistent. Thus, we conclude that $\{\varphi \}\cup \Theta$ is consistent.

We extend the consistent set $\{\varphi \}\cup \Theta$ to a maximally consistent set $\Gamma$ using a standard construction. Fixing an enumeration $\varphi _0,\varphi _1,\dots$ of the formulas of our language, we define:

$$\begin{aligned} \Gamma _0= \{\varphi \}\cup \Theta \qquad \Gamma _{n+1}= \left\{ \begin{array}{ll} \Gamma _n\cup \{\varphi _n\}&{}\quad \hbox { if this is consistent} \\ \Gamma _n\cup \{\lnot \varphi _n\}&{}\quad \hbox { otherwise} \end{array}\right. \qquad \Gamma = \underset{n\in \mathbb {N}}{\bigcup }\Gamma _n. \end{aligned}$$

It is then a standard exercise to verify that $\Gamma$ is a maximally consistent set.

For the next step of the proof, let $T_2$ be the set of all terms (including 0 and 1) generated by the coalition labels from the finite set

$$\begin{aligned} \mathsf {Coal}(\varphi )\cup \{c_{x,y,[t]}\mid x,y\in \mathsf {Alt},[t]\in T_1/E\}. \end{aligned}$$

Define an equivalence relation $E_\Gamma$ on $T_2$ by: $sE_\Gamma t$ if and only if $s\equiv t\in \Gamma$. Then since $\Gamma$, as a maximally consistent set, contains all valid equations of Boolean algebra, it follows that the quotient $T_2/E_\Gamma$ is a Boolean algebra in which the complement of [t] is $[-t]$, and the meet of [s] and [t] is $[s\sqcap t]$. Let $\le$ be the order of this Boolean algebra, so $[s]\le [t]$ iff $[s]\sqcap [t]=[s]$. By the same argument used above to show that $T_1/E$ is finite, we have that $T_2/E_\Gamma$ is finite.

Since $T_2/E_\Gamma$ is a finite Boolean algebra (with at least two elements, since $\lnot (0\equiv 1)$ is an axiom), it is isomorphic to the powerset of a (nonempty) set V. Let f be the isomorphism sending elements of $T_2/E_\Gamma$ to subsets of V. We take the set X of alternatives to be the set $\mathsf {Alt}$ of alternative labels. We then define $\widehat{D}:X^2\rightarrow \wp (\wp (V))$ by:

$$\begin{aligned} f([t])\in \widehat{D}(x,y)\hbox { if and only if }\widehat{D}_{x>y}(t)\in \Gamma . \end{aligned}$$

(3)

This is well defined, i.e., the choice of representative from [t] does not matter, because if $\widehat{D}_{x>y}(s)\in \Gamma$ and $s\equiv t\in \Gamma$, then it is easy to see using the maximal consistency of $\Gamma$ and the principles of our logic that $\widehat{D}_{x>y}(s)\leftrightarrow \widehat{D}_{x>y}(t)\in \Gamma$ and hence $\widehat{D}_{x>y}(s)\in \Gamma$ if and only if $\widehat{D}_{x>y}(t)\in \Gamma$. With this definition of D, the fact that $\Gamma$ contains the axioms of $\overline{\widehat{\mathbf {T}}}$ implies that D satisfies the properties of Theorem 7.3 for TR (resp. the fact that $\Gamma$ contains the axioms of $\overline{\widehat{\mathbf {W}}}$ implies that D satisfies the properties of Theorem 7.3 for FR).

Next we define $\overline{D}:X^2\rightarrow \wp (\wp (V))$ by $f([t])\in \overline{D}(x,y)$ if and only if $\overline{D}_{x>y}(t)\in \Gamma$. Again this is well-defined, for the same reasons as before. The fact that $\Gamma$ contains the axiom $(\overline{D}_{x>y}(a)\wedge (a\sqsubseteq b))\rightarrow \widehat{D}_{x>y}(b)$ implies that $\langle \widehat{D},\overline{D}\rangle$ satisfies the left-to-right direction of property 6 of Theorem 7.3: if $A\in \overline{D}(x,y)$, then for all $B\supseteq A$, we have $B\in \widehat{D}(x,y)$. Proving the converse implication was the reason for our introduction of the set $\Theta$ of formulas. Suppose $f([t])\not \in \overline{D}(x,y)$, so $\overline{D}_{x>y}(t)\not \in \Gamma$ and hence $\lnot \overline{D}_{x>y}(t)\in \Gamma$ by the maximal consistency of $\Gamma$. Then given that $(t\sqsubseteq c_{x,y,[t]})\wedge (\lnot \overline{D}_{x>y}(t)\rightarrow \lnot \widehat{D}_{x>y}(c_{x,y,[t]}))\in \Gamma$, from $\lnot \overline{D}_{x>y}(t)\in \Gamma$ we have $\lnot \widehat{D}_{x>y}(c_{x,y,[t]})\in \Gamma$ by the maximal consistency of $\Gamma$, whence $\widehat{D}_{x>y}(c_{x,y,[t]})\not \in \Gamma$ by the consistency of $\Gamma$. Thus, $f([c_{x,y,[t]}])\not \in \widehat{D}(x,y)$. Since $t\sqsubseteq c_{x,y,[t]}\in \Gamma$, we also have $[t]\le [c_{x,y,[t]}]$ and hence $f([t])\subseteq f([c_{x,y,[t]}])$, which completes the proof of property 6.

Since $\langle \widehat{D},\overline{D}\rangle$ satisfies the properties of Theorem 7.3 for TR (resp. FR), it is representable by a CCR satisfying LD, IIA, and TR (resp. FR). Let $\alpha$ be the coalition labeling that sends each coalition label a to the image $f([a])\subseteq V$ of the equivalence class of [a] in $T_2/E_\Gamma$. Since the set of terms was defined inductively, one can prove by induction that the extended labeling $\dot{\alpha }$ from Definition 4.2 satisfies

$$\begin{aligned} \dot{\alpha }(t)=f([t]). \end{aligned}$$

(4)

Let $\beta$ be the alternative labeling that sends each alternative label x to itself, recalling from above that we take the set X of alternatives to be $\mathsf {Alt}$.

We claim that for any formula $\psi$, we have:

$$\begin{aligned} f\models _{\alpha ,\beta } \psi \hbox { if and only if }\psi \in \Gamma . \end{aligned}$$

(5)

The set of formulas was defined inductively, so we prove the claim by induction. For formulas of the form $t\equiv s$, we have the following equivalences:

$$\begin{aligned} f\models _{\alpha ,\beta } t\equiv s\Leftrightarrow & {} \dot{\alpha }(t)=\dot{\alpha }(s) \\\Leftrightarrow & {} f([t])=f([s])\hbox { by } (4) \\\Leftrightarrow & {} [t]=[s]\hbox { since } f \hbox { is injective} \\\Leftrightarrow & {} t\equiv s\in \Gamma \hbox { by definition of the equivalence relation }E_\Gamma . \end{aligned}$$

For formulas of the form $\widehat{D}_{x>y}(t)$, we have:

$$\begin{aligned} f\models _{\alpha ,\beta } \widehat{D}_{x>y}(t)\Leftrightarrow & {} \dot{\alpha }(t)\in \widehat{D}_f(\beta (x),\beta (y))\\\Leftrightarrow & {} f([t])\in \widehat{D}_f(\beta (x),\beta (y))\hbox { by } (4)\\\Leftrightarrow & {} f([t])\in \widehat{D}_f(x,y)\hbox { since } \beta \hbox { is the identity map}\\\Leftrightarrow & {} f([t])\in \widehat{D}(x,y)\hbox { since } \widehat{D} \hbox { is represented by } f \\\Leftrightarrow & {} \widehat{D}_{x>y}(t)\in \Gamma \hbox { by } (3). \end{aligned}$$

The proof for formulas of the form $\overline{D}_{x>y}(t)$ is analogous. For formulas of the form $\lnot \varphi$:

$$\begin{aligned} f\models _{\alpha ,\beta } \lnot \varphi\Leftrightarrow & {} f\not \models _{\alpha ,\beta } \varphi \hbox { by Definition } 4.4 \\\Leftrightarrow & {} \varphi \not \in \Gamma \hbox { by the inductive hypothesis}\\\Leftrightarrow & {} \lnot \varphi \in \Gamma \hbox { since } \Gamma \hbox { is maximally consistent}. \end{aligned}$$

The proofs for formulas of the form $\psi \wedge \chi$, $\psi \vee \chi$, etc., also use the maximal consistency of $\Gamma$, which ensures that $\psi \wedge \chi \in \Gamma$ if and only if $\psi ,\chi \in \Gamma$; $\psi \vee \chi \in \Gamma$ if and only if $\psi \in \Gamma$ or $\chi \in \Gamma$; etc. Thus, (5) holds. Then since our initial consistent formula $\varphi$ belongs to $\Gamma$, we have $f\models _{\alpha ,\beta } \varphi$. This completes the proof that every $\overline{\widehat{\mathbf {T}}}$-consistent (resp. every $\overline{\widehat{\mathbf {W}}}$-consistent) formula is true of some CCR satisfying LD, IIA, and TR (resp. FR).

The proof of Theorem or 4.8.2 or 4.8.4 is even simpler. In this case, without the interaction between $\widehat{D}$ and $\overline{D}$ to worry about, we can skip the step involving the new coalition labels $c_{x,y,[t]}$ and the set $\Theta$ of new formulas. Simply extend $\varphi$ to a maximally consistent set $\Gamma$ and continue with the rest of the proof as above. At the point where the representation result from Theorem 7.3 (for decisiveness regimes) was used, use the representation result from Theorem 3.5 (for almost decisiveness) or Theorem 3.6 (for decisiveness) instead.