A Conceptually Well-Founded Characterization of Iterated Admissibility Using an "All I Know" Operator
This work addresses a foundational problem in game theory for researchers studying epistemic characterizations of rationality, offering an incremental improvement over prior characterizations.
The paper tackles the conceptual issue in iterated admissibility (IA) related to Samuelson's concern about higher-level strategies, by providing a characterization using an 'all I know' operator that addresses this problem while using lexicographic probability sequences (LPSs), and then modifies it to work with probability structures using approximate notions.
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.