# A Conceptually Well-Founded Characterization of Iterated Admissibility Using an "All I Know" Operator

## TL;DR

This paper provides a new epistemic characterization of iterated admissibility using an "all I know" operator within LPSs, addressing Samuelson's concern about strategy consideration at higher levels.

## Contribution

It introduces a characterization of IA with an "all I know" operator that accounts for Samuelson's concern, using LPSs and approximate belief notions.

## Key findings

- Characterization of IA using "all I know" operator with LPSs.
- Modification of the characterization with approximate belief to address Samuelson's concern.
- Bridging the gap between probability structures and epistemic conditions for IA.

## Abstract

Brandenburger, Friedenberg, and Keisler provide an epistemic characterization of iterated admissibility (IA), also known as iterated deletion of weakly dominated strategies, where uncertainty is represented using LPSs (lexicographic probability sequences). Their characterization holds in a rich structure called a complete structure, where all types are possible. In earlier work, we gave a characterization of iterated admissibility using an "all I know" operator, that captures the intuition that "all the agent knows" is that agents satisfy the appropriate rationality assumptions. That characterization did not need complete structures and used probability structures, not LPSs. However, that characterization did not deal with Samuelson's conceptual concern regarding IA, namely, that at higher levels, players do not consider possible strategies that were used to justify their choice of strategy at lower levels. In this paper, we give a characterization of IA using the all I know operator that does deal with Samuelson's concern. However, it uses LPSs. We then show how to modify the characterization using notions of "approximate belief" and "approximately all I know" so as to deal with Samuelson's concern while still working with probability structures.