Binary strategy-proof social choice functions with indifference

Abstract

We prove a representation formula that gives a new characterization of the coalitionally strategy-proof binary social choice functions. We give this characterization in the case of social choice functions selecting one of two alternatives (i.e., binary social choice). The domain of the functions we consider consists of profiles of preferences over a society of arbitrary cardinality and indifference is admitted. Strategy proofness is meant to be coalitional: No group of agents has incentives to form a coalition that can manipulate the social choice for their own advantage with false reporting.

Appendix

1.1 Proof of Theorem 4.2 for an infinite V

In V is not finite, the process described for proving Theorem 4.2 may not stop in finitely many steps.

Suppose that for an ordinal $\beta$ we have produced a double collection $\langle \varvec{{{\mathcal {I}}}},\varvec{{{\mathcal {F}}}}\rangle$, with $\varvec{{{\mathcal {I}}}}=(I_\alpha )_{\alpha <\beta }$, $\varvec{{{\mathcal {F}}}}=({{\mathcal {F}}}_\alpha )_{\alpha <\beta }$, and the corresponding sets $({{\mathcal {P}}}_\alpha )_{\alpha <\beta }$ such that

$$\begin{aligned} \{\sim _V\}\cup \bigcup _{\alpha <\beta } {{\mathcal {P}}}_\alpha \subseteq \{\phi =\psi _{\langle {{\mathcal {I}}},{{\mathcal {F}}}\rangle ,\, x}\}. \end{aligned}$$

One of the following is true.

• 1. For every profile $P\notin \{\sim _V\}\cup \bigcup _{\alpha <\beta } {{\mathcal {P}}}_\alpha$, the value $\phi (P)$ is x.

• 2. There is a profile $Q\notin \{\sim _V\}\cup \bigcup _{\alpha <\beta } {{\mathcal {P}}}_\alpha$ such that $\phi (Q)\ne x.$

In case 1, we stop since we have obtained $\phi =\psi _{\langle \varvec{{{\mathcal {I}}}},\varvec{{{\mathcal {F}}}}\rangle ,\, x}$.

In case 2, let Q be a profile such that $Q\ne \sim _V$,  $Q\notin {{\mathcal {P}}}_\alpha \,\, \forall \alpha <\beta ,$ and $\phi (Q)\ne x.$

Let $I = I(Q)$ be the set of voters that are indifferent under Q. The set I is a proper coalition distinct from all previous $I_\alpha$. Indeed, since $Q\ne \sim _V$, the set I is a proper subset of V. The implication    $I=\varnothing (\Leftrightarrow Q$ is strict) $\Rightarrow Q\in \cup _{\alpha <\beta }\, {{\mathcal {P}}}_\alpha$    tells us that I is nonempty and if $I=I_\alpha$ for some $\alpha$, then Q is a strict profile over $I^c_\alpha$ hence $Q\in {{\mathcal {P}}}_\alpha$.

Consider the restriction of $\phi$

$$\begin{aligned} \phi _{I^c}(P_{I^c}):=\phi (\sim _I, P_{I^c}). \end{aligned}$$

It is CSP and takes two values (x for $P_{I^c} =\sim _{I^c} ,\,\, \phi (Q)$ for $P_{I^c}=Q_{I^c}$). Apply Proposition 4.1 to $\phi _{I^c}$ and let ${{\mathcal {F}}}$ be a SSC family on $I^c$ such that

$$\begin{aligned}&D(a, P_{I^c})\in {{\mathcal {F}}}\Rightarrow \phi _{I^c}(P_{I^c})= a, \\&D(b, P_{I^c})\in {{\mathcal {F}}}^{\,\circ} \Rightarrow \phi _{I^c}(P_{I^c})= b. \end{aligned}$$

The new double collection $\langle \varvec{{{\mathcal {I}}}}^*,\varvec{{{\mathcal {F}}}}^*\rangle$, consisting of $I_\alpha$ for $\alpha <\beta$, $I_\beta :=I$, ${{\mathcal {F}}}_\alpha$ for $\alpha <\beta$,    ${{\mathcal {F}}}_\beta :={{\mathcal {F}}}$, has the property

$$\begin{aligned} \{\sim _V\}\cup \bigcup _{\alpha \le \beta } {{\mathcal {P}}}_\alpha \subseteq \{\phi =\psi _{\langle {{\mathcal {I}}}^*,{{\mathcal {F}}}^*\rangle ,\, x}\}; \end{aligned}$$

therefore, by transfinite induction we can conclude that a well-ordered set $\Lambda$ exists and a double collection $\langle \varvec{{{\mathcal {I}}}},\varvec{{{\mathcal {F}}}}\rangle$ indexed on $\Lambda$ such that $\phi =\psi _{\langle \varvec{{{\mathcal {I}}}},\varvec{{{\mathcal {F}}}}\rangle ,\, x}$. $\square$

1.2 On the equality $M_x=\phi _{{ \varvec{{{\mathcal {F}}}}},\, x}$

Let V be a set of cardinality n. For $k=1, \dots , n$, let us denote by ${{\mathcal {G}}}_k$ the SSC family consisting of the subsets of V with cardinality at least k. We know that ${{\mathcal {G}}}_k^{\,\circ} ={{\mathcal {G}}}_{n-k+1}$.

We show the existence of a collection $\varvec{{{\mathcal {F}}}}$ such that the SCF $M_x$ coincide with the SCF $\phi _{{ \varvec{{{\mathcal {F}}}}},\, x}$. The collection $\varvec{{{\mathcal {F}}}}$ consists of exactly n SSC families ${{\mathcal {F}}}_0, {{\mathcal {F}}}_1, \dots , {{\mathcal {F}}}_{n-1}$ on V. If we compare this result with the structure of the SCF $\phi _2$ of Example 2.4 (this function can be defined for a finite V as well), we see that the structure we obtain for $M_x$ precisely gives ${{\mathcal {F}}}_i$’s the role physical agents have in $\phi _2$ in dictating the social choice corresponding to a given profile.

Proposition 4.3

In order to have the equality $M_b=\phi _{{ \varvec{{{\mathcal {F}}}}},\, b}$ one can take the collection $\varvec{{{\mathcal {F}}}}=({{\mathcal {F}}}_i)_{0\le i\le n-1}$ as follows.

$$\begin{aligned} \varvec{{{\mathcal {F}}}}= \left\{ \begin{array} {ll} {{\mathcal {F}}}_{2i}={{\mathcal {G}}}_{p+1+i},{{\mathcal {F}}}_{2i+1}={{\mathcal {G}}}_{p-i}, \text{ with } {0\le i\le p-1}, &{}\quad \text{ if } \; n=2p \\ {{\mathcal {F}}}_0={{\mathcal {G}}}_{p+1}, {{\mathcal {F}}}_{2i+1}={{\mathcal {G}}}_{p+2+i}, {{\mathcal {F}}}_{2i+2}={{\mathcal {G}}}_{p-i}, \text{ with } {0\le i\le p-1}, &{}\quad \text{ if } \; n=2p+1.\\ \end{array} \right. \end{aligned}$$

On the other hand, for having the equality $M_a=\phi _{{ \varvec{{{\mathcal {F}}}}},\, a}$ one can take the collection $\varvec{{{\mathcal {F}}}}=({{\mathcal {F}}}_i)_{0\le i\le n-1}$ as follows.

$$\begin{aligned} \varvec{{{\mathcal {F}}}}= \left\{ \begin{array} {ll} {{\mathcal {F}}}_{2i}={{\mathcal {G}}}_{p-i},{{\mathcal {F}}}_{2i+1}={{\mathcal {G}}}_{p+1+i}, \text{ with } {0\le i\le p-1}, &{}\quad \text{ if } \; n=2p \\ {{\mathcal {F}}}_0={{\mathcal {G}}}_{p+1}, {{\mathcal {F}}}_{2i+1}={{\mathcal {G}}}_{p-i}, {{\mathcal {F}}}_{2i+2}={{\mathcal {G}}}_{p+2+i}, \text{ with } {0\le i\le p-1}, &{}\quad \text{ if } \; n=2p+1.\\ \end{array} \right. \end{aligned}$$

Proof

We show that $M_b=\phi _{{ \varvec{{{\mathcal {F}}}}},\, b}$. For this, let us suppose that n is odd: $n=2p+1$ (being the case $n=2p$ similar). Hence, the sequence of SSC families forming $\varvec{{{\mathcal {F}}}}$ is:

${{\mathcal {F}}}_0={{\mathcal {G}}}_{p+1}, {{\mathcal {F}}}_1={{\mathcal {G}}}_{p+2}, {{\mathcal {F}}}_2={{\mathcal {G}}}_p, {{\mathcal {F}}}_3={{\mathcal {G}}}_{p+3}, {{\mathcal {F}}}_4={{\mathcal {G}}}_{p-1}, \ldots , {{\mathcal {F}}}_{2i+1}={{\mathcal {G}}}_{p+2+i}, {{\mathcal {F}}}_{2i+2}={{\mathcal {G}}}_{p-i}, \dots , {{\mathcal {F}}}_{n-2}={{\mathcal {G}}}_{2p+1}={{\mathcal {G}}}_n, {{\mathcal {F}}}_{n-1}={{\mathcal {G}}}_1.$

Notice that ${{\mathcal {F}}}_0$ is self-dual and, pairwise, the families ${{\mathcal {F}}}_{2i+1}, {{\mathcal {F}}}_{2i+2}$ are dual to each other.

Hence, given a profile P, its index $\lambda (P)$ is the first element of $\{0,1,2, \dots , n-1\}$ for which the corresponding statement in the list below (where $i=0,\dots , p-1$) is true:

• 0. either $|D(a, P)|\ge p+1$ or $|D(b, P)|\ge p+1$

• 1. either $|D(a, P)|\ge p+2$ or $|D(b, P)|\ge p$

• 2. either $|D(a, P)|\ge p$ or $|D(b, P)|\ge p+2$

• ...

• $2i+1$. either $|D(a, P)|\ge p+2+i$ or $|D(b, P)|\ge p-i$

• $2i+2$. either $|D(a, P)|\ge p-i$ or $|D(b, P)|\ge p+2+i$

• ...

• $n-2$. either $|D(a, P)|\ge n$ or $|D(b, P)|\ge 1$

• $n-1$. either $|D(a, P)|\ge 1$ or $|D(b, P)|\ge n$

if one such element exists, otherwise $\lambda (P)=\infty$.

We remind that, by definition,

$$\begin{aligned} \phi _{\varvec{{{\mathcal {F}}}}, \, b}(P) = \left\{ \begin{array} {ll} a, &{}\quad \text{ if } \; D(a,P)\in {{{\mathcal {F}}}}_{\lambda (P)}\\ b, &{}\quad \text{ if } \; D(b,P)\in {{{\mathcal {F}}}}_{\lambda (P)}^{\,\circ} \\ b, &{}\quad \text{ if } \; \lambda (P)=\infty \,\,.\\ \end{array} \right. \end{aligned}$$

Now, checking that $\phi _{\varvec{{{\mathcal {F}}}}, \, b}(P) =M_b(P)$ for every P is straightforward. $\square$