Future-blindness and the product topology

TL;DR

The paper addresses infinite-horizon evaluation under extreme tail-insensitivity by defining $N$-blindness and eventual blindness. It develops a topological framework on $l^ fty$, showing that the finest topology making all continuous preferences eventual blind is the product topology $ au^p$, while the finest $N$-blind topology $ au^b_N$ is generated by the seminorms $p_k(x)=|x_k|$ for $k=0, olinebreak olinebreak olinebreak N$ with dual $c_{00}^N$. It then establishes that eventual blindness corresponds to product-topology continuity, and its dual is $c_{00}$, which has implications for price representations in infinite dimensions. The results extend the literature on myopia and the Mackey topology by providing a behavioral interpretation of product-topology continuity and clarifying how tail-insensitive preferences interact with equilibrium existence in $l^ abla_ ext{infty}$-type spaces.

Abstract

We study future-blind preferences, which are preferences that heavily discount the future, within the space of infinite consumption streams. We give two definitions: $N$-blindness, where agents ignore periods beyond a fixed date $N$, and eventual blindness, where all but finitely many dates are neglected. Using a topological approach, we show that the finest topology ensuring eventual blindness coincides with the product topology. This provides a behavioral foundation for continuity in the product topology, which was considered for studying equilibrium existence in infinite-dimensional spaces. Finally, we characterize the dual spaces under these topologies.

Theorems & Definitions (42)

• Definition 3.1
• Proposition 3.1
• proof
• Definition 3.2: brown1981myopic
• Definition 3.3: brown1981myopic
• Definition 3.4
• Proposition 3.2
• proof
• Example 1
• Definition 3.5