Expected multi-utility representations of preferences over lotteries
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913194021224448 |
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| author | Leonetti, Paolo |
| author_facet | Leonetti, Paolo |
| contents | Let $\succsim$ be a binary relation on the set of simple lotteries over a countable outcome set $Z$. We provide necessary and sufficient conditions on $\succsim$ to guarantee the existence of a set $U$ of von Neumann--Morgenstern utility functions $u: Z\to \mathbf{R}$ such that $$ p\succsim q \,\,\,\Longleftrightarrow\,\,\, \mathbf{E}_p[u] \ge \mathbf{E}_q[u] \,\text{ for all }u \in U $$ for all simple lotteries $p,q$. In such case, the set $U$ is essentially unique. Then, we show that the analogue characterization does not hold if $Z$ is uncountable. This provides an answer to an open question posed by Dubra, Maccheroni, and Ok in [J. Econom. Theory~\textbf{115} (2004), no.~1, 118--133]. Lastly, we show that different continuity requirements on $\succsim$ allow for certain restrictions on the possible choices of the set $U$ of utility functions (e.g., all utility functions are bounded), providing a wide family of expected multi-utility representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_04739 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Expected multi-utility representations of preferences over lotteries Leonetti, Paolo Classical Analysis and ODEs Theoretical Economics Functional Analysis Let $\succsim$ be a binary relation on the set of simple lotteries over a countable outcome set $Z$. We provide necessary and sufficient conditions on $\succsim$ to guarantee the existence of a set $U$ of von Neumann--Morgenstern utility functions $u: Z\to \mathbf{R}$ such that $$ p\succsim q \,\,\,\Longleftrightarrow\,\,\, \mathbf{E}_p[u] \ge \mathbf{E}_q[u] \,\text{ for all }u \in U $$ for all simple lotteries $p,q$. In such case, the set $U$ is essentially unique. Then, we show that the analogue characterization does not hold if $Z$ is uncountable. This provides an answer to an open question posed by Dubra, Maccheroni, and Ok in [J. Econom. Theory~\textbf{115} (2004), no.~1, 118--133]. Lastly, we show that different continuity requirements on $\succsim$ allow for certain restrictions on the possible choices of the set $U$ of utility functions (e.g., all utility functions are bounded), providing a wide family of expected multi-utility representations. |
| title | Expected multi-utility representations of preferences over lotteries |
| topic | Classical Analysis and ODEs Theoretical Economics Functional Analysis |
| url | https://arxiv.org/abs/2210.04739 |