# The Indirect-Utility Criterion for Ranking Opportunity Sets over Time

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Notation and Preliminaries

**Definition**

**1.**

**Definition**

**2.**

## 3. The Indirect-Utility Criterion over Time

**Definition**

**3.**

- either $\mathrm{max}\left({A}_{i}\right){I}_{i}\mathrm{max}\left({B}_{i}\right)$ for all $i=1,\dots ,n,$ or
- there exists $j\in N=\{1,\dots ,n\}$ such that $\mathrm{max}\left({A}_{i}\right){I}_{i}\mathrm{max}\left({B}_{i}\right)$ for all $i=1,\dots ,j-1$ and $\mathrm{max}\left({A}_{j}\right){P}_{j}\mathrm{max}\left({B}_{j}\right).$

**Axiom**

**1**(A1)

**.**

**Axiom**

**2**(A2)

**.**

**Proposition**

**1.**

**Proof.**

**Axiom**

**3**(A3)

**.**

**Theorem**

**1.**

**Proof.**

- There exists $i\in N$ such that ${a}_{j}{I}_{j}{a}_{j}^{\prime}$ for all $j=1,\dots ,i-1$ and ${a}_{i}{P}_{i}{a}_{i}^{\prime}.$An application of the definition of ${R}_{i}$ produces $(\left\{{a}_{i}\right\},{A}_{-i})\succ (\left\{{a}_{i}^{\prime}\right\},{A}_{-i})$ for all ${A}_{j}\subseteq {X}_{j}$, $j\in N\setminus \left\{i\right\}$, and in particular:$$(\left\{{a}_{1}\right\},\dots ,\left\{{a}_{n}\right\})\succ (\left\{{a}_{1}\right\},\dots ,\left\{{a}_{i}^{\prime}\right\},\dots ,\left\{{a}_{n}\right\}).$$Axiom A3 leads to:$$(\left\{{a}_{1}\right\},\dots ,\left\{{a}_{n}\right\})\succ (\left\{{a}_{1}\right\},\dots ,\left\{{a}_{i}^{\prime}\right\},{A}_{i+1}^{\prime},\dots ,{A}_{n}^{\prime})$$Taking ${A}_{j}^{\prime}=\{{a}_{j}^{\prime}\}$, $\forall j=i+1,\dots ,n$, we conclude:$$(\left\{{a}_{1}\right\},\dots ,\left\{{a}_{n}\right\})\succ (\left\{{a}_{1}\right\},\dots ,\left\{{a}_{i-1}\right\},\left\{{a}_{i}^{\prime}\right\},\dots ,\left\{{a}_{n}^{\prime}\right\}).$$$$(\left\{{a}_{1}\right\},\dots ,\left\{{a}_{i-1}\right\},\left\{{a}_{i}^{\prime}\right\},\dots ,\left\{{a}_{n}^{\prime}\right\})\sim (\left\{{a}_{1}^{\prime}\right\},\dots ,\left\{{a}_{n}^{\prime}\right\})$$$$(\left\{{a}_{1}\right\},\dots ,\left\{{a}_{n}\right\})\succ (\left\{{a}_{1}^{\prime}\right\},\dots ,\left\{{a}_{n}^{\prime}\right\}).$$
- For all $i=1,\dots ,n$, ${a}_{i}{I}_{i}{a}_{i}^{\prime}$.In this case, we have $(\left\{{a}_{1}\right\},\dots \left\{{a}_{n}\right\})\sim (\left\{{a}_{1}^{\prime}\right\},\dots ,\left\{{a}_{n}^{\prime}\right\})$, thus, we conclude.

**Example**

**1.**

**Example**

**2.**

- $\left(i\right)$
- $A\succ B\iff \exists j\in \{1,\dots ,n\}\phantom{\rule{4.pt}{0ex}}such\phantom{\rule{4.pt}{0ex}}that\phantom{\rule{4.pt}{0ex}}\mathrm{max}\left(\right)open="("\; close=")">{\cup}_{k=1}^{i}{A}_{k},\forall i=1,\dots ,j-1,\phantom{\rule{4.pt}{0ex}}and\phantom{\rule{4.pt}{0ex}}\mathrm{max}({A}_{1}\cup \dots \cup {A}_{j})P\mathrm{max}({B}_{1}\cup \dots \cup {B}_{j})$
- $\left(ii\right)$
- $A\sim B\iff \mathrm{max}({A}_{1}\cup \dots \cup {A}_{i})I\mathrm{max}({B}_{1}\cup \dots {B}_{i})\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.pt}{0ex}}for\phantom{\rule{4.pt}{0ex}}all\phantom{\rule{4.pt}{0ex}}i=1,\dots ,n.$

**Example**

**3.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

García-Sanz, M.D.; Alcantud, J.C.R.
The Indirect-Utility Criterion for Ranking Opportunity Sets over Time. *Symmetry* **2019**, *11*, 241.
https://doi.org/10.3390/sym11020241

**AMA Style**

García-Sanz MD, Alcantud JCR.
The Indirect-Utility Criterion for Ranking Opportunity Sets over Time. *Symmetry*. 2019; 11(2):241.
https://doi.org/10.3390/sym11020241

**Chicago/Turabian Style**

García-Sanz, María Dolores, and José Carlos Rodríguez Alcantud.
2019. "The Indirect-Utility Criterion for Ranking Opportunity Sets over Time" *Symmetry* 11, no. 2: 241.
https://doi.org/10.3390/sym11020241