# Decomposition and Arrow-Like Aggregation of Fuzzy Preferences

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Preorders on a Set

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

**Proof.**

#### 2.2. The Arrovian Model in Social Choice

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Theorem**

**2**

**Theorem**

**3.**

- (i)
- Unanimity: For every $x,y\in X$ and every profile $\mathcal{P}=({\succsim}_{1},\dots ,{\succsim}_{n})\in \mathcal{A}$ such that $x{\succ}_{i}y$ holds for every $1\le i\le n$, it holds that $xF\left(\mathcal{P}\right)y$,
- (ii)
- Independence of irrelevant alternatives (Fishburn’s version): For any $x,y\in X$ and $\mathcal{P}=({\succsim}_{1},\dots ,{\succsim}_{n});{\mathcal{P}}^{\prime}=({\succsim}_{1}^{\prime},\dots ,{\succsim}_{n}^{\prime})\in \mathcal{A}$ if the restrictions to $\{x,y\}$ of ${\succ}_{i}$ and ${\succ}_{i}^{\prime}$ agree for all $1\le i\le n$, then $xF\left(\mathcal{P}\right)y=xF\left({\mathcal{P}}^{\prime}\right)y$.
- (iii)
- Non-dictatorship: There is no $k\in \{1,\dots n\}$ such that for every $x,y\in X$ and $\mathcal{P}=({\succsim}_{1},\dots ,{\succsim}_{n})\in \mathcal{A}$ it holds that $x{\succ}_{k}y\Rightarrow xF\left(\mathcal{P}\right)y$.

#### 2.3. Fuzzy Sets

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

**Proposition**

**1.**

**Proof.**

## 3. Decomposition of Fuzzy Binary Relations

**Definition**

**13.**

**Definition**

**14.**

- (i)
- Boundary conditions: For any $t\in [0,1]$, it holds that $0{\cup}_{f}t=t{\cup}_{f}0=t$.
- (ii)
- Monotonicity: For all $a,b,c,d\in [0,1]$ with $a\le c$ and $b\le d$ it holds true that $a{\cup}_{f}b\le c{\cup}_{f}d$.

**Remark**

**1.**

**Definition**

**15.**

**Proposition**

**2.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

- (i)
- $\mathcal{P}(x,y)\le \mathcal{R}(x,y)$ for all $x,y\in U$,
- (ii)
- $\mathcal{P}(x,y)=0\Rightarrow \mathcal{R}(x,y)=\mathcal{I}(x,y)$, for any $x,y\in U$,
- (iii)
- $\left[\mathcal{I}\right(x,y)\le \mathcal{I}(z,w\left)\right]\wedge \left[\mathcal{P}\right(x,y)\le \mathcal{P}(z,w\left)\right]\Rightarrow \left[\mathcal{R}\right(x,y)\le \mathcal{R}(z,w\left)\right]$ holds true for every $x,y,z,w\in U$,
- (iv)
- $\mathcal{P}(x,y)>0\iff \mathcal{R}(x,y)>\mathcal{R}(y,x)$ for every $x,y\in U$.

**Proof.**

**Definition**

**16.**

## 4. Fuzzy Preferences

**Example**

**3.**

**Definition**

**17.**

- (i)
- reflexive if for all $x\in U$ it holds that $\mathcal{R}(x,x)=1$,
- (ii)
- transitive if for every $x,y,z\in U$ it holds that $\left[\mathcal{R}\right(x,y)\ge \mathcal{R}(y,x\left)\right]\wedge \left[\mathcal{R}\right(y,z)\ge \mathcal{R}(z,y\left)\right]\Rightarrow \left[\mathcal{R}\right(x,z)\ge \mathcal{R}(z,x\left)\right]$,
- (iii)
- connected with respect to ${\cup}_{f}$ if for every pair $x,y\in U$ it holds that $\mathcal{R}(x,y){\cup}_{f}\mathcal{R}(y,x)=1$,
- (iv)
- complete if for every $x,y\in U$ it holds that $max\left\{\mathcal{R}\right(x,y),\mathcal{R}(y,x\left)\right\}=1$,
- (v)
- connected if $\mathcal{R}(x,y)+\mathcal{R}(y,x)\ge 1$ holds true for all $x,y\in U$.

**Remark**

**2.**

**Definition**

**18.**

- (i)
- $\mathcal{R}$ is reflexive, transitive and ${\cup}_{f}$-connected,
- (ii)
- $\mathcal{P}$ is asymmetric, $\mathcal{I}$ is symmetric and $\mathcal{R}$ decomposes as $\{\mathcal{P},\mathcal{I}\}$ with respect to ${\cup}_{f}$,
- (iii)
- $\{\mathcal{P},\mathcal{I}\}$ is an admissible decomposition of $\mathcal{R}$.

**Remark**

**3.**

## 5. Arrow-Like Aggregation of Fuzzy Preferences

**Definition**

**19.**

#### 5.1. The Fuzzy Arrovian Model

**Definition**

**20.**

- (i)
- Independence of irrelevant alternatives if for any two profiles $\left({\mathsf{\Lambda}}_{i}\right)$ and $\left({\mathsf{\Lambda}}_{i}^{\prime}\right)$ that belong to ${\mathcal{A}}^{n}$ and $x,y\in U$ we have that if ${\mathsf{\Lambda}}_{i\rceil \{x,y\}}\approx {\mathsf{\Lambda}}_{i\rceil \{x,y\}}^{\prime}$ for any $i\in N$, then $f{\left(\left({\mathsf{\Lambda}}_{i}\right)\right)}_{\rceil \{x,y\}}\approx f{\left(\left({\mathsf{\Lambda}}_{i}^{\prime}\right)\right)}_{\rceil \{x,y\}}$. (Here ${\mathsf{\Lambda}}_{i\rceil \{x,y\}}$ denotes the restriction of ${\mathsf{\Lambda}}_{i}$ to the subset $\{x,y\}$ of alternatives. Given two profiles $\left({\mathsf{\Lambda}}_{i}\right)=({\mathcal{R}}_{i},{\mathcal{P}}_{i},{\mathcal{I}}_{i})$ and $\left({\mathsf{\Lambda}}_{i}^{\prime}\right)=({\mathcal{R}}_{i}^{\prime},{\mathcal{P}}_{i}^{\prime},{\mathcal{I}}_{i}^{\prime})$, the notation ${\mathsf{\Lambda}}_{i}\approx {\mathsf{\Lambda}}_{i}^{\prime}$ means that ${\mathcal{R}}_{i}\approx {\mathcal{R}}_{i}^{\prime}$ as well as ${\mathcal{P}}_{i}\approx {\mathcal{P}}_{i}^{\prime}$ and ${\mathcal{I}}_{i}\approx {\mathcal{I}}_{i}^{\prime}$, in the sense of Definition 12).
- (ii)
- Pareto if for every profile $\left({\mathsf{\Lambda}}_{i}\right)\in {\mathcal{A}}^{n}$ and any $x,y\in U$ it holds that if ${\mathcal{P}}_{i}(x,y)>0$ for any $i\in N$, then ${\mathcal{P}}_{f}\left(\left({\mathsf{\Lambda}}_{i}\right)\right)(x,y)>0$.
- (iii)
- Dictatorship if there exists $k\in {N}_{n}$, called dictator, such that for every $\left({\mathsf{\Lambda}}_{i}\right)$ and $x,y\in U$ we have that if ${\mathcal{P}}_{k}(x,y)>0$ then ${\mathcal{P}}_{f}\left(\left({\mathsf{\Lambda}}_{i}\right)\right)(x,y)>0$.

#### 5.2. Pseudo-Fuzzy Preferences

**Proposition**

**5.**

**Proof.**

**Remark**

**4.**

- (i)
- if $1=\mathcal{R}(x,y)$ whereas $\mathcal{R}(y,x)=0$, we denote it by $x{\succ}_{\mathsf{\Lambda}}^{+}y$,
- (ii)
- if $1=\mathcal{R}(x,y)>\mathcal{R}(y,x)>0$, we denote it by $x{\succ}_{\mathsf{\Lambda}}^{0}y$,
- (iii)
- if $1>\mathcal{R}(x,y)>\mathcal{R}(y,x)>0$, we denote it by $x{\succ}_{\mathsf{\Lambda}}^{-}y$,
- (iv)
- if $1=\mathcal{R}(x,y)=\mathcal{R}(y,x)$, we denote it by $x{\sim}_{\mathsf{\Lambda}}^{+}y$,
- (v)
- if $1>\mathcal{R}(x,y)=\mathcal{R}(y,x)$, we denote it by $x{\sim}_{\mathsf{\Lambda}}^{0}y$.

**Definition**

**21.**

**Definition**

**22.**

**Proposition**

**6.**

**Proof.**

**Remark**

**5.**

**Definition**

**23.**

**Theorem**

**4.**

**Proof.**

**Definition**

**24.**

**Remark**

**6.**

#### 5.3. Towards a Fuzzy Arrovian Impossibility Theorem

**Definition**

**25.**

- (i)
- the property of independence of irrelevant alternatives if given any pair of profiles $\left({p}_{i}\right),\left({p}_{i}^{\prime}\right)\in {E}^{n}$ and $x,y\in U$ such that ${p}_{i\rceil \{x,y\}}={p}_{i\rceil \{x,y\}}^{\prime}$ holds true for every $i\in N=\{1,\dots ,n\}$, then $h{\left(\left({p}_{i}\right)\right)}_{\rceil \{x,y\}}=h{\left(\left({p}_{i}^{\prime}\right)\right)}_{\rceil \{x,y\}}$,
- (ii)
- the property of unanimity if given any $\left({p}_{i}\right)\in {E}^{n}$ whose associated asymmetric parts are $\left({\succ}_{i}\right)$, and $x,y\in U$ such that $x{\succ}_{i}y$ holds true for every $i\in N=\{1,\dots ,n\}$, then $x\succ y$ also holds, (with ≻ being the asymmetric part of $h\left(\left({p}_{i}\right)\right))$,
- (iii)
- the property of dictatorship if there exist $k\in N=\{1,\dots ,n\}$ such that for every $\left({p}_{i}\right)\in {E}^{n}$ and $x,y\in U$, it holds true that if ${\succ}_{k}$ stands for the asymmetric part of ${p}_{k}$, then $x{\succ}_{k}$ implies that $x\succ y$ (with ≻ denoting the asymmetric part of $h\left(\left({p}_{i}\right)\right)$).

**Definition**

**26.**

**Proposition**

**7.**

**Proof.**

**Definition**

**27.**

**Definition**

**28.**

**Definition**

**29.**

**Definition**

**30.**

**Proposition**

**8.**

**Proof.**

**Proposition**

**9.**

**Proof.**

**Remark**

**7.**

**Proposition**

**10.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Arrow, K.J. A difficulty in the concept of social welfare. J. Political Econ.
**1950**, 58, 328–346. [Google Scholar] [CrossRef] - Arrow, K.J. Social Choice and Individual Values; Wiley: New York, NY, USA, 1951. [Google Scholar]
- Arrow, K.J. Social Choice and Individual Values, 2nd ed.; Wiley: New York, NY, USA, 1963; (In this second edition some flaw arising in the original proofs of Arrow’s impossibility problem given in [1,2] were mended and corrected. The mistakes in the former proofs had been pointed out by Julian Blau in [4]). [Google Scholar]
- Blau, J.H. The existence of social welfare functions. Econometrica
**1957**, 25, 302–313. [Google Scholar] [CrossRef] - Kelly, J.S. Arrow Impossibility Theorems; Academic Press: New York, NY, USA, 1978. [Google Scholar]
- Kelly, J.S. Social Choice Theory. An Introduction; Springer: Berlin, Germany, 1988. [Google Scholar]
- Dutta, B. Fuzzy preferences and social choice. Math. Soc. Sci.
**1987**, 13, 215–229. [Google Scholar] [CrossRef] - Gibilisco, M.B.; Gowen, A.M.; Albert, K.E.; Mordeson, J.N.; Wierman, M.J.; Clark, T.D. Fuzzy Social Choice Theory. In Studies in Fuzziness and Soft Computing; Springer International Publishing: Cham, Switzerland, 2014; Volume 315. [Google Scholar]
- Mordeson, J.N.; Gibilisco, M.B.; Clark, T.D. Independence of irrelevant alternatives and fuzzy Arrow’s theorem. New Math. Nat. Comput.
**2012**, 8, 219–237. [Google Scholar] [CrossRef] - Montero, F.J. The impact of fuzziness in social choice paradoxes. Soft Comput.
**2008**, 12, 177–182. [Google Scholar] [CrossRef] - Barrett, C.R.; Pattanaik, P.K.; Salles, M. On the structure of fuzzy social welfare functions. Fuzzy Sets Syst.
**1986**, 19, 1–10. [Google Scholar] [CrossRef] - Cholewa, W. Aggregation of fuzzy opinions: An axiomatic approach. Fuzzy Sets Syst.
**1985**, 17, 249–258. [Google Scholar] [CrossRef] - Fung, L.W.; Fu, K.S. An Axiomatic Approach to Rational Decision Making in a Fuzzy Environment; Zadeh, L.A., Fu, K.S., Tanaka, K., Shimura, M., Eds.; Fuzzy Sets and Their Applications to Cognitive and Decission Processes; Academic Press: New York, NY, USA, 1975; pp. 227–256. [Google Scholar]
- Montero, F.J. A note on Fung-Fu’s theorem. Fuzzy Sets Syst.
**1985**, 17, 259–269. [Google Scholar] [CrossRef] - Montero, F.J. Aggregation of fuzzy opinions in a non-homogeneous group. Fuzzy Sets Syst.
**1988**, 25, 15–20. [Google Scholar] - Billot, A. Aggregation of preferences: The fuzzy case. Theory Decis.
**1991**, 30, 51–93. [Google Scholar] [CrossRef] - Billot, A. Economic Theory of Fuzzy Equilibria: An Axiomatic Analysis; Lecture Notes in Economics and Mathematical Systems; Springer: Berlin, Germany, 1992. [Google Scholar]
- Dasgupta, M.; Deb, R. An Impossibility Theorem with Fuzzy Preferences; de Swart, H., Ed.; Logic, Game Theory and Social Choice; Tilburg University Press: Tilburg, The Netherlands, 1999; pp. 482–490. [Google Scholar]
- Duddy, C.; Piggins, A. On some oligarchy results when social preference is fuzzy. Soc. Choice Welf.
**2018**, 51, 717–735. [Google Scholar] [CrossRef] - Fono, L.A.; Andjiga, N.G. Fuzzy strict preference and social choice. Fuzzy Sets Syst.
**2005**, 155, 372–389. [Google Scholar] [CrossRef] - Fono, L.A.; Donfack-Kommogne, V.; Andjiga, N.G. Fuzzy Arrow-type results without the pareto principle based on fuzzy pre-orders. Fuzzy Sets Syst.
**2009**, 160, 2658–2672. [Google Scholar] [CrossRef] - Richardson, G. The structure of fuzzy preferences: Social choice implications. Soc. Choice Welf.
**1998**, 15, 359–369. [Google Scholar] [CrossRef] - Fishburn, P.C. Arrow’s impossibility theorem: Concise proof and infinite voters. J. Econom. Theory
**1970**, 2, 103–106. [Google Scholar] [CrossRef] - Ekeland, I. Éléments d’ Économie Mathématique; Hermann: Paris, France, 1979. [Google Scholar]
- Banerjee, A. Fuzzy preferences and Arrow-type problems in social choice. Soc. Choice Welf.
**1994**, 11, 121–130. [Google Scholar] [CrossRef] - Ovchinnikov, S. Structure of fuzzy binary relations. Fuzzy Sets Syst.
**1981**, 6, 169–195. [Google Scholar] [CrossRef] - Díaz, S.; De Baets, B.; Montes, S. Transitivity and negative transitivity in the fuzzy setting. Adv. Intell. Soft Comput.
**2011**, 107, 91–100. [Google Scholar] - Ovchivnnikov, S. Numerical representation of fuzzy transitive relations. Fuzzy Sets Syst.
**2000**, 126, 225–232. [Google Scholar] [CrossRef] - Agud, L.; Catalán, R.G.; Díaz, S.; Induráin, E.; Montes, S. Numerical representability of fuzzy total preorders. Int. J. Comput. Intell. Syst.
**2012**, 5, 996–1009. [Google Scholar] [CrossRef] [Green Version] - Montes, I.; Díaz, S.; Montes, S. On complete fuzzy preorders and their characterizations. Soft Comput.
**2010**, 15, 1999–2011. [Google Scholar] [CrossRef] - Fono, L.A.; Andjiga, N.G. Utility function of fuzzy preferences on a countable set under max-star transitivity. Soc. Choice Welf.
**2007**, 28, 667–683. [Google Scholar] [CrossRef] - Duddy, C.; Perote-Peña, J.; Piggins, A. Arrow’s theorem and max-star transitivity. Soc. Choice Welf.
**2011**, 36, 25–34. [Google Scholar] [CrossRef] [Green Version]

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Raventós-Pujol, A.; Campión, M.J.; Induráin, E.
Decomposition and Arrow-Like Aggregation of Fuzzy Preferences. *Mathematics* **2020**, *8*, 436.
https://doi.org/10.3390/math8030436

**AMA Style**

Raventós-Pujol A, Campión MJ, Induráin E.
Decomposition and Arrow-Like Aggregation of Fuzzy Preferences. *Mathematics*. 2020; 8(3):436.
https://doi.org/10.3390/math8030436

**Chicago/Turabian Style**

Raventós-Pujol, Armajac, María J. Campión, and Esteban Induráin.
2020. "Decomposition and Arrow-Like Aggregation of Fuzzy Preferences" *Mathematics* 8, no. 3: 436.
https://doi.org/10.3390/math8030436