Level-k Thinking in the Extensive Form

TL;DR

This paper extends level-$k$ thinking to games in extensive form by introducing strong level-$k$ thinking, which updates beliefs about opponents' lower-level reasoning at reached information sets and adheres to a strong rationalizability principle. It then systematically compares strong level-$k$ thinking to level-$k$ thinking in normal form, strong rationalizability, strong $oldsymbol{ riangle}$-rationalizability, iterated admissibility, backward rationalizability, backward level-$k$ thinking, and backward induction, highlighting where forward induction adds predictive power and where results may diverge. Through theoretical propositions and illustrative examples (e.g., Reny, HMS; BoS with outside options), it shows that strong level-$k$ can yield outcome refinements and different predictions depending on initial beliefs and utilities, often producing non-robust results to small payoff changes. The authors also reanalyze prior BoS experiments to gauge the empirical bite of forward induction under uniform first-level beliefs, finding that certain experimental patterns align with strong level-$k$ predictions for some players and with other concepts for others. Overall, the work provides a dynamic, information-set-aware extension of level-$k$ thinking that enriches the comparative toolbox for analyzing bounded rationality in extensive-form games and guides future experimental tests of forward-induction phenomena.

Abstract

Level-k thinking has been widely applied as a solution concept for games in normal form in behavioral and experimental game theory. We consider level-k thinking in games in extensive form. Player's may learn about levels of opponents' thinking during the play of the game because some information sets may be inconsistent with certain levels. In particular, for any information set reached, a level-k player attaches the maximum level-l thinking for l < k to her opponents consistent with the information set. We compare our notion of strong level-k thinking with other solution concepts such as level-k thinking in the associated normal form, strong rationalizability, Delta-rationalizability, iterated admissibility, backward rationalizability, backward level-k thinking, and backward induction. We use strong level-k thinking to reanalyze data from some prior experiments in the literature.

Figures (10)

• Figure 1: Reny (1992) Game
• Figure 2: HMS Game
• Figure 3: Battle-of-the-sexes with an outside option I (BoS I)
• Figure 4: Battle-of-the-sexes with an outside option II
• Figure 5: Battle-of-the-sexes with an outside option III

Theorems & Definitions (21)

• Definition 1: Level-$k$ Thinking in the Normal Form
• Definition 2: Rationalizability
• Proposition 1
• Proposition 2
• Example 1
• Definition 3: Strong Level-$k$ Thinking
• Proposition 3
• Example 2
• Example 3
• Proposition 4