Timeability of Extensive-Form Games

TL;DR

This paper investigates the limitations of implementing extensive-form games with a temporal component, revealing that some games cannot be exactly timed without information leakage, and provides conditions for exact and approximate timing.

Contribution

It introduces the concept of 'timeability' of extensive-form games, offering necessary and sufficient conditions for exact timing and methods for approximate timing.

Findings

Some games are not exactly timeable due to information leakage.

Approximate timing of games is always possible but may require enormous time.

The required time for approximation can grow as a power tower, making some games practically untimeable.

Abstract

Extensive-form games constitute the standard representation scheme for games with a temporal component. But do all extensive-form games correspond to protocols that we can implement in the real world? We often rule out games with imperfect recall, which prescribe that an agent forget something that she knew before. In this paper, we show that even some games with perfect recall can be problematic to implement. Specifically, we show that if the agents have a sense of time passing (say, access to a clock), then some extensive-form games can no longer be implemented; no matter how we attempt to time the game, some information will leak to the agents that they are not supposed to have. We say such a game is not exactly timeable. We provide easy-to-check necessary and sufficient conditions for a game to be exactly timeable. Most of the technical depth of the paper concerns how to approximately time games, which we show can always be done, though it may require large amounts of time. Specifically, we show that for some games the time required to approximately implement the game grows as a power tower of height proportional to the number of players and with a parameter that measures the precision of the approximation at the top of the power tower. In practice, that makes the games untimeable. Besides the conceptual contribution to game theory, we believe our methodology can have applications to preventing information leakage in security protocols.