# A Note on Binary Strategy-Proof Social Choice Functions

^{1}

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## Abstract

**:**

## 1. Introduction

for every voter $i\in V$, for every profile $P\in {\mathcal{W}}^{V}$, and every preference $W\in \mathcal{W}$, letting Q be the profile identical to P except for the voter i, where ${Q}_{i}=W$, one has that the alternative $\varphi \left(P\right)$ is at least as good as $\varphi \left(Q\right)$ according to ${P}_{i}$.

## 2. The Model, Notations, Definitions

- -
- $P={\left({P}_{i}\right)}_{i\in V}$, where ${P}_{i}$ is the weak preference of the voter i in the state P;
- -
- $P=({P}_{i};{P}_{-i})$; where ${P}_{-i}$ is the restriction of P to the complement of the singleton $\left\{i\right\}$.

**Definition**

**1.**

**social choice function**is a function $\varphi :{\mathcal{W}}^{V}\to A$ from the set ${\mathcal{W}}^{V}$ of all states of the society to the set A of alternatives.

**Definition**

**2.**

**strategy-proof**(SP, for brevity) if for each voter i, for each profile P and for each profile $Q=({Q}_{i};{P}_{-i})$ identical to P in all the components different from i, the alternative $\varphi \left(P\right)$ is at least as good as the alternative $\varphi \left(Q\right)$ according to the preference ${P}_{i}$.

- -
- Alternative a is better than alternative b;
- -
- Alternative b is better than alternative a;
- -
- Alternatives a and b are indifferent to each other,

**Definition**

**3.**

**monotone**if

- -
- To check the strategy-proofness (equivalently monotonicity) of a function, it is sufficient to refer to pairs of profiles “close” to each other, in the sense that there are no other profiles between them (this is the notion of covering);
- -
- The equivalence between monotonicity and strategy-proofness does not need the universal domain assumption. Indeed, the equivalence holds true over domains without “gaps”(this is the notion of convexity).

**Definition**

**4.**

**covers**t, and we write $t\u22d6s$, if $t<s$ and there is no element u in S such that $t<u<s$.

**Definition**

**5.**

**convex**if

**Proposition**

**1.**

**Theorem**

**1.**

- (i)
- ϕ is SP;
- (ii)
- if $Q\u22d6P$ then $\varphi \left(Q\right)\le \varphi \left(P\right)$;
- (iii)
- ϕ is monotone.

**Proof.**

**Remark**

**1.**

## 3. A Recursive Construction of ${\mathsf{\Phi}}_{n}$

**Theorem**

**2.**

**Proof.**

- 0
- Preliminarly, the elements of ${\mathsf{\Phi}}_{n}$ (i.e., all the SP binary social choice functions defined on ${\{a,\sim ,b\}}^{n}$) and all the profiles are enumerated. This allows to display the elements of ${\mathsf{\Phi}}_{n}$ in a matrix $M\left(n\right)$ whose entry ${m}_{u,v}$ stands for the value of the function (numbered) u in the profile (numbered) v.

- 1
- For every element ${m}_{h,j}$ in the selected row, we highlight the elements ${m}_{\xb7,j}$ of the column j that differ from ${m}_{h,j}$. This identifies the set of rows $\{u:{m}_{u,j}\ne {m}_{h,j}\}.$
- 2
- The highlighting is realized in a way that distinguishes the case ${m}_{u,j}=a$ from the case ${m}_{u,j}=b$. To fix the ideas, we can realize this by coloring red ${m}_{u,j}$ if it is a, and blue if it is b.
- 3
- The set ${\mathsf{\Phi}}_{n}(h,a)$ (respectively, ${\mathsf{\Phi}}_{n}(h,b)$) consists of all functions occupying rows not cointaining any blue (respectively, red) component.

**Example.**We show how the above steps work in the cases $n=1,2$, i.e., with reference to formula $\left(3\right)$ in the cases $n=1,2$.

## 4. Graphic Representation of 3-Voters SP Social Choice Functions

**Definition**

**6.**

**super order-closed**if it contains the majorants of its elements. Formally: C is super order-closed if

**sub order-closed**if

**Lemma**

**1.**

**Proof.**

**we assume:**

**Remark**

**2.**

**Proposition**

**2.**

- (i)
- ϕ is SP and $\left(4\right)$ hold true.
- (ii)
- ϕ takes value a on ${\{a,\sim \}}^{V}$, $\varphi \left({b}_{V}\right)=b$, and its restriction to ${\mathcal{P}}_{b}$ is SP.

**Example. The diagram of ${\mathcal{P}}_{b}$ for $\left|V\right|=3$.**

**Visual examples of SP social choice functions**for |V| = 3.

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 5. On the Number of Binary SP Social Choice Functions with Indifference

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Remark**

**3.**

- (i)
- $X\in {\mathcal{F}}_{M},\phantom{\rule{0.166667em}{0ex}}Y\subseteq X\phantom{\rule{0.166667em}{0ex}}\Rightarrow \phantom{\rule{0.166667em}{0ex}}Y\in {\mathcal{F}}_{M}$, and
- (ii)
- $X\in {\mathcal{F}}_{M},\phantom{\rule{0.166667em}{0ex}}N\subseteq M\phantom{\rule{0.166667em}{0ex}}\Rightarrow X\cap N\in {\mathcal{F}}_{N}$.

**Proposition**

**5.**

**Remark**

**4.**

|V| | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Ded(|V|) | 3 | 6 | 20 | 168 | 7581 | 7828354 | 2414682040998 | 56130437228687557907788 |

|Φ_{|V|}| | 4 | 20 | 980 | 17792748 | - | - | - | - |

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**The subcase $\varphi \left(aba\right)=a$ and $\varphi \left(aab\right)=\varphi \left(baa\right)=b$.

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**MDPI and ACS Style**

Basile, A.; De Simone, A.; Tarantino, C.
A Note on Binary Strategy-Proof Social Choice Functions. *Games* **2022**, *13*, 78.
https://doi.org/10.3390/g13060078

**AMA Style**

Basile A, De Simone A, Tarantino C.
A Note on Binary Strategy-Proof Social Choice Functions. *Games*. 2022; 13(6):78.
https://doi.org/10.3390/g13060078

**Chicago/Turabian Style**

Basile, Achille, Anna De Simone, and Ciro Tarantino.
2022. "A Note on Binary Strategy-Proof Social Choice Functions" *Games* 13, no. 6: 78.
https://doi.org/10.3390/g13060078