The Value of Recall in Extensive-Form Games

TL;DR

This paper defines and analyzes the value of recall (VoR) for imperfect-recall extensive-form games, formalizing VoR as $VoR^{SC}(\Gamma)=\frac{u_1(\textsf{SC}(\mathscr{R}_1(\Gamma)))}{u_1(\textsf{SC}(\Gamma))}$ and examining it across solution concepts. It shows VoR can be unbounded in general, but provides tight bounds when separating the effects of absentmindedness and chance nodes, and it connects VoR to the smoothness framework to handle broader strategic reasoning. The authors identify pathologies where perfect recall can hurt all players, and propose refinements (e.g., EDT-Nash, CDT-Nash) to avoid such issues, particularly in single-player and multi-player settings. They prove VoR is NP-hard to compute or approximate in general and study partial recall refinements, establishing hardness results via reductions (e.g., from X3C). Finally, the work links VoR to related concepts like the price of anarchy, miscoordination, and abstraction, outlining practical avenues for guiding abstractions and future research in memory-augmented strategic reasoning.

Abstract

Imperfect-recall games, in which players may forget previously acquired information, have found many practical applications, ranging from game abstractions to team games and testing AI agents. In this paper, we quantify the utility gain by endowing a player with perfect recall, which we call the value of recall (VoR). While VoR can be unbounded in general, we parameterize it in terms of various game properties, namely the structure of chance nodes and the degree of absentmindedness (the number of successive times a player enters the same information set). Further, we identify several pathologies that arise with VoR, and show how to circumvent them. We also study the complexity of computing VoR, and how to optimally apportion partial recall. Finally, we connect VoR to other previously studied concepts in game theory, including the price of anarchy. We use that connection in conjunction with the celebrated smoothness framework to characterize VoR in a broad class of games.

Figures (6)

• Figure 1: A game with imperfect recall. Giving Bobble ($\color{p1color}\blacksquare$) perfect recall hurts both players. Terminals show utilities for Bobble and Alice ($\color{p2color}\CIRCLE$). Infosets are joined by dotted lines.
• Figure 2: (Left) An imperfect-recall game $\Gamma$. Boxes indicate chance nodes. (Middle) $\mathscr{R}_1(\Gamma)$, the perfect recall refinement of $\Gamma$ with respect to ${\color{p1color}\blacktriangle}\xspace$. (Right) $\Gamma$ with perfect information.
• Figure 3: Perfect recall can lead to worse CDT/EDT eq.
• Figure 4: (Left) \ref{['ex:1am_tight']}, $n=4$. (Right) Ex. \ref{['ex:1chance_tight']}, $n=2$
• Figure 5: Partial recall gives a worse EDT-Nash equilibrium: In (a), the only EDT-Nash equilibrium is playing ${\color{p1color}\textbf{L}}$; in (b), playing ${\color{p1color}\textbf{R}}$ in both infosets is also an EDT-Nash equilibrium.

Theorems & Definitions (77)

• Definition 1
• Definition 2: (Im)perfect recall
• Definition 3
• Definition 4
• Definition 5
• Lemma 6: Oesterheld22:Can
• Remark 7
• Definition 8: Game refinements/coarsenings
• Definition 9: Perfect recall refinements
• Proposition 10