Expected multi-utility representations of preferences over lotteries

2022/10/10 by Leonetti, Paolo
#Classical Analysis and ODEs (math.CA) #FOS: Economics and business #FOS: Mathematics #Functional Analysis (math.FA) #Theoretical Economics (econ.TH)

paper · doi:10.48550/arxiv.2210.04739

Abstract

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→ R such that p\succsim q \Longleftrightarrow E_p[u] ≥ E_q[u] for all u ∈ 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~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.

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